text
stringlengths 0
1.19M
|
|---|
= 0.2. The boundary conditions are such that the structure is clamped at both ends; recall Eq. , Df means a generic ceramic/metallic or graded property (i.e. Young’s modulus, density, etc.) whereas subscripts m and c identify the metallic and ceramic phases, respectively. The shear elasticity modulus can be calculated () with common linear elastic formulas, i.e. Gf |
= 0.5Ef/(1 + |
ν).After the substitution of one of the aforementioned possibilities (i.e. P1 or P2) into Eq. and then rearranging terms, it is possible to arrive to the dimensionless eigenvalue equation:where, KA and MA are the matrices of stiffness and mass coefficients for an arch of a given subtended angle. The natural frequency coefficients are defined according ω¯=LωρMax/EMax, where ω is the circular frequency (rad/seg), whereas L is the length of the subtended arc, ρMax and EMax are the maximum density and Young’s Modulus of the FGM structure. shows the comparison of natural frequencies calculated with the present 1D model and with a 3D finite element approach, for the case of material distribution with power K |
= 1. The first four natural frequency coefficients are plotted with respect to the parameter αL |
= |
L/R, which is the subtended angle between ends. As it is clearly seen, both approaches are in good agreement in the whole range of subtended angles. It can be also said that frequencies calculated with the curved beam model have percentage difference less than 2% with respect to the 3D FEM approximation. Besides, the 1D formulation employed less than 30 times of computational resources and time employed by the 3D FEM approach.In this section studies about free and forced vibrations of thick rings are performed. The ring has hollow rectangular cross-section with the layered ceramic/metalic configuration shown in . The basic material properties are indicated in . The measures of the cross-section are such that h |
= 0.1 m, b |
= |
h/2 and t |
= 6 mm (each layer has 2 mm). The Young’s modulus, defined according to the modified rule of mixture (MROM), is given in Eq. whereas the density and the Poisson’s coefficient, given in Eq. , are defined according the conventional rule of mixtures (ROM). See the work of E=VmEmqσ+Ec+(1-Vm)qσ+EmEcVmqσ+Ec+(1-Vm)qσ+Em,Vm and Vc |
= 1 − |
Vm are the volume fractions of metallic and ceramic phases, respectively. In Eq. , q is the ratio of stress to strain transfer between the two phases (). For Al/SiC phases, this ratio has been experimentally determined (The kinematics of a thick shear deformable ring, according to Eq. , can be represented, for each value of i, by one of the modal shape possibilities P1 or P2 described in Eq. . After the substitution of one of the aforementioned possibilities into Eq. and then rearranging terms, it is possible to arrive to the dimensionless eigenvalue equation:where, KR and MR are the matrices of stiffness and mass coefficients; ω¯∗=RωρMax/EMax is the dimensionless frequency coefficient, whereas ω is the circular frequency (rad/seg), R is the radius of the ring, ρMax and EMax are the maximum density and Young’s Modulus of the layered cross-section.It should be recalled that with the use of P1 or P2 it is possible to reach to the same frequencies for i |
> 0. If the wave number is i |
= 0 two different cases appear depending on the selected displacement shape, i.e. P1 or P2. The breathing mode can be obtained with the possibility P1 and with i |
= 0, however with the alternative P2 is not possible to obtain the aforementioned breathing mode. In the case of shear deformable rings it is possible to obtain a mode of pure shear deformation (with possibility P2 and i |
= 0, not present with P1), which is not present in straight or curved (open) shear deformable beams. show typical mode shapes calculated with alternatives given in Eq. . Although, the breathing modes in rings are higher modes, depending on the geometric properties they normally appear among the third to the tenth axial-bending natural frequencies. On the other hand pure shear modes are out of the range of usual set employed to perform further calculations in the case of thin rings but not in the case of thick rings. As it was said, if i |
= 1 in both possibilities of Eq. the rigid modes (i.e. implying null frequency in the first eigenvalue) are obtained; clearly observed in one can see the first six dimensionless frequency coefficients of layered rings compared to the case of completely homogeneous hollow rectangular cross-section (i.e. all three layers having the material properties of the outer layer). In it is possible to see the variation with respect to the geometry ratio h/RG of the first five natural frequency coefficients. It has to be mentioned that a given number of frequency may change its mode shape depending on the material and geometric parameters involved. Thus the first frequency maintain the same mode shape corresponding to the first eigenvalue of the case shown in (c). Remember that the same value is obtained with anyone of the alternatives present in Eq. . However, for the following frequencies there are changes in the vibration patters depending on the value of geometric ratio h/RG. For example the second frequency coefficient in (c) between h/RG |
∈ [0.01, 0.70], the mode of (a) between h/RG |
∈ [0.70, 1.00] and a higher mode for h/RG |
> 1.00. Other responses can be evidenced in the following frequencies, although with awkward shapes. This characteristics can be clearly identified in , where the variations with respect to h/RG of the frequencies associated with a given mode shape are illustrated. Notice in this figure the points intersecting doted curves and solid curves. In these points changes arise in the vibrational pattern of rings as a consequence of a change in the parametric ratio. Besides, note that as the rings are thicker the Pure shear modes appear among the first 10–20 modes normally involved in transient and forced vibrations. the sketch of a ring with arbitrary external radial loads is shown. The external loads, applied in the circumferential neutral axis, have to be self-equilibrated. Thus, if the generic external load can be expressed as:If:Ω1=Ω3=Ω13⇒q3=q1β1β3Ω2=Ω4=Ω24⇒q4=q2β2β4., Ωj, j |
= 1, … , 4 are the driving circular frequencies.For the first example under the presence of external dynamic load, a ring with h/RG |
= 0.5 and the cross-section shape of is selected. The radial loads are q1 |
= |
q2 |
= 1000 N/m and distributed in angular spans of β1 |
= |
π/16, β2 |
= |
π/12, β3 |
= |
π/4 and β4 |
= |
π/3. The driving frequencies are Ω13 |
= 50 rad/seg and Ω24 |
= 30 rad/seg. The initial displacements and velocities are assumed null, consequently Yij(0) = 0 and Y˙ij(0)=0 (i |
= 0, 1, 2, … ; |
j |
= 1, 2). two different convergence studies are presented. Thus, shows the values of radial displacements at different angular locations with respect to the number of modes involved in the calculation. shows the time history of the radial displacement (at α |
= 0) employing the first two, three and twenty pairs of modes. The values calculated with the approach of twenty pairs of modes are the ones employed for comparison purposes, because beyond the use of ten pairs of modes the percentage errors fall below 0.01%. Thus, values calculated with more that 20 pairs of modes can be accepted as a very good approximation. Note that when three modes are employed the vibratory behavior, in qualitative terms, is the same of those obtained with more modes. shows the vibrational pattern of a thick ring with the material, geometric and loading features of the previous example. The radial displacement is depicted at three different angular locations. Notice that due to the non-uniform loading condition, the vibratory pattern at different angular locations has more oscillatory character however within the same period. shows the radial displacement at α |
= |
π/3 for three different ratios of horizontal and vertical driving circular frequencies. These last two figures were obtained performing calculations with 20 pairs of modes in order to achieve accurate results.In this paper studies devoted to characterize the vibration patterns of shear deformable thick metallic/ceramic rings have been addressed through the introduction of a one dimensional model. This model contains as especial particular cases some models of curved beam that consider shear deformation. The material properties have been considered in generalized form taking into account that the model contains as particular cases thin/thick rings/curved beam with homogeneous materials as well as orthotropic materials. The tangential stresses, needed to calculate the shear stiffness and the shear coefficient, have been consistently deduced. Besides, a superposition scheme has been introduced for multilayered metallic/ceramic cross-sections. This superposition scheme leads to a substantial reduction in computational effort and cost in comparison to other approaches, and on the other hand, offers the possibility to generalize the consistent calculation of tangential stresses in multilayered ceramic/metallic cross-sections; which is a mandatory task for 1D models based allowing for shear deformability. The present 1D model has been compared with a 3D finite element approach showing a good matching and also a substantial reduction in the calculation procedure. Parametric studies have been carried out in order to characterize the free vibration patterns and the transient vibrations in thin and thick rings. These studies show the importance to employ a sufficiently large number of modes in the calculation of the forced vibratory behavior of thick rings.The present appendix is aimed to identify the distribution of tangential stresses in the cross-section of curved beams and/or rings, in order to consistently calculate the corresponding shear coefficient. The particular case of a hollow rectangular cross-section of constant thickness t is developed in the following lines.For this particular case, φ1(yG, |
z) = 1 is adopted. According to the symmetry with respect to the plane π2 (see ), the right side is divided into 5 sub-domains. In each sub-domain j |
= 1, … , 5 the following equilibrium equations (see Eqs. while φ1(yG, |
z) = 1 then U |
= −V RG. Besides the Eq. , in the contours of the five sub-domains 16 reciprocity conditions (symmetry, boundary and continuity conditions) have to be verified. For example:τ4rα-h/2,z=0,…,τ1zαyG,b/2=0,…,τ1rαh/2-t,z=τ3rαh/2-t,z,…,τ4zαyG,b/2-t=τ5zαyG,b/2-t,…,τ2zαyG,0=τ5zαyG,0=0.The following expressions of the tangential stresses are adopted:τjrα=1r2Cj+rG1j+r22VQ,j=1,…,5,τjzα=1r2G0j+G1j+RGzVQ,j=1,…,5.The previous tangential stresses identically satisfy the equilibrium Eq. τ3zα=0,G02=G05=0,G01=-G04=bp(h-t)4t,G11=-RG+2G01b,G14=-RG+2G04b,G12=G111+b(h-t)2t(ri+q3),G15=G141-b(h-t)2t(re+q4),C1=ri4t2q3t+b(h-t),C4=re4t2q4t-b(h-t),C2=q3ri2,C5=q4re2,C3=q4q32-b(h-t)4., i.e. the definition of V, and calculating α¯0 and α¯2 according to Eq. and taking into account that A=Rα¯0,J=Rα¯2,R=Ω/α¯0 and Ω |
= 2t(b |
+ |
h |
− 2t), it is possible to arrive to:Thus the distribution of the tangential stresses along the hollow rectangular cross-section of a thick ring is completely characterized with In the case of hollow rectangular multilayered cross-sections (see ) with constant thickness t |
= |
t1 |
+ |
t2 |
+⋯+ |
tN (each ti, i |
= 1, … , |
N constant), it is possible to incorporate the calculation scheme of in a superposition method in order to consistently obtain the tangential stresses as well as the shear coefficient.The superposition scheme assumes the following hypotheses:The cross-section is divided in N layers Γj, each one with constant thickness tj, j |
= 1, … , |
N.The material properties are constant in the whole layer Γjj |
= 1, … , |
N, but not necessarily the same in all layers.According to the assumption (B.2) the centroid of individual layers are all coincident.The methodology can be employed when the following condition is verified:For this case, it is verified U |
= −RGV. However, notice that each layer can have different material properties, clearly implying that φ1(yG, |
z) is variable by steps. Thus in concordance with the second assumption, it is possible to write:where φj1(yG, |
z), is the elastic material property of the j-th layer, φ01(yG, |
z) is the reference elastic material property, and κj is a fraction of the reference value.Defining N sub-domains Δj in the following form:That is, each sub-domain Δj, j |
= 1, … , |
N, contains the previous domain with and increasing thickness such that ΔN contains the whole cross-section, and obviously this domain has thickness t. Moreover, for sub-domain Δj, j |
= 1, … , |
N, the tangential stresses are related to a fraction Qj of the shear force Q, such that:Then, the tangential stresses in each sub-domain Δj, j |
= 1, … , |
N, can be calculated by means of the following superposition formula:τrα(y,z)=∑i=nN(τrα)iτzα(y,z)=∑i=nN(τzα)i,(y,z)∈Δn,n=1,2,…,N,where under the summation symbol, each (τrα)i and (τzα)i means the tangential stresses calculated applying the method of in the N sub-domains under the conditions Q |
= |
Qj, homogeneous section, and t=∑i=1jti,j=1,…,N.Notice that tangential stresses given in Eq. have to verify the following equilibrium equation for the whole non-homogeneous section, which is the same Eq. for the corresponding sub domains Δj, j |
= 1, … , |
N:Then a system in terms of the Qj, j |
= 1, … , |
N can be defined, whose solution lead to the following:Dj=κj-κj+1∏i=1NViVj,j=1,…,N-1,DN=κN∏i=1NViVN,whereas Vj, j |
= 1, … , |
N are parameters calculated for each sub-domain with method of . The parameters Vj and Dj, j |
= 1, … , |
N should verify the following condition:It should be soundly remarked that the use of this superposition methodology is quite important from a practical viewpoint, since the employment of a conventional and consistent calculation of the tangential stresses should verify equilibrium equations, boundary, symmetry and continuity conditions in N(2N |
+ 3) sub-domains. For example in the example shown in , one should affect all those conditions in 27 domains. As it can be inferred, the use of this superposition scheme allows the sparing of a huge amount of computational resources, especially time. Moreover, this superposition scheme can be employed, conceptually speaking, for other layered hollow cross-sections. However under the requisite of being symmetric with respect to the plane π2 in Ultra-high-speed processing of nanomaterial-reinforced woven carbon fiber/polyamide 6 composites using reactive thermoplastic resin transfer moldingHigh-speed molding of continuous carbon fiber-reinforced composites was carried out by thermoplastic resin transfer molding using a very-low-viscosity thermoplastic monomer. After dispersing 0.1 wt.% of various nanomaterials in ε-caprolactam, a monomer of anionic polymerized polyamide 6 (A-PA6), the processing temperature and catalyst-activator contents were optimized based on the polymerization induction time required for resin injection in the mold. Additionally, the dispersion stability, solidification time, monomer conversion, crystallinity and mechanical properties of the composites were evaluated. Higher dispersion stability of the nanomaterials in A-PA6 resulted in a significantly shortened process cycle time – on the order of seconds –and higher crystallinity in the composite. In addition, carbon fibers treated with atmospheric plasma were used to increase the rate of resin impregnation into fibers. These factors not only increased composite fabrication productivity but also enhanced its mechanical properties.In recent years, legislation to regulate automobile fuel consumption and emissions have been strengthened in developed countries. Therefore, lightweight fiber-reinforced composites have attracted much interest from the automobile industry. In particular, efforts are underway to apply long fiber-reinforced thermoplastic to seatback frames and battery storage for hybrid vehicles []. However, further improvement of the physical properties of these composites is necessary to meet the future demand for lighter weighing composites, and thus, continuous fiber-reinforced thermoplastics is necessary.Despite the fact that thermoplastic composites have many advantageous properties such as high impact resistance [], they cannot be produced effectively along with the continuous fibers because of the difficulty associated with impregnating high-viscosity molten resin into continuous fiber textile. Reaction injection molding is a widely practiced processing method for fiber-reinforced thermoset composites; however, a similar concept for thermoplastics, known as thermoplastic resin transfer molding (T-RTM), has only recently been developed []. Anionic ring-opening polymerization of ε-caprolactam (CPL) occurs much faster than the hydrolytic reaction []; therefore, it is most suitable for the T-RTM process. The CPL, a precursor to anionic polymerized polyamide 6 (A-PA6), is injected in the low-viscosity monomer state into a mold charged with a fiber textile preform. Upon injection, the monomer reacts immediately with the pre-mixed catalyst and activator, leading to simultaneous polymerization and crystallization below the polymer melting and crystallization point. The viscosity of the resin used in T-RTM is 100 times lower than that required for the reaction processing of thermoset resins, and in particular, 160,000 times lower than that needed for the melting process of engineering thermoplastics []. Therefore, the T-RTM process not only has few product shape limitations, but is also very advantageous for resin impregnation into fibers and fast production.For accelerating the reaction rate, it is typical to increase the catalyst-activator contents, or increase the polymerization temperature. An inert environment also increases the reaction rate, because oxygen and moisture in air deactivate the anions and active sites, as previously reported []. The increasing polymerization temperature increases the activation energy required for polymerization, inducing rapid reaction, but it decreases the polymer crystallinity and mechanical properties []. However, low polymerization temperature not only decreases the polymerization rate but also increases the crystallization rate, which induces reactive groups to be trapped inside the growing crystals before they are polymerized, thus causing low monomer conversion [An improvement in resin impregnation through the surface treatment of fibers can also reduce the manufacturing process time. Instead of conventional surface treatment methods, which have a long treatment time and a limited sample size, atmospheric plasma treatment has recently shown many advantages such as continuous in-line processing capability and no vacuum chamber. Also, it can increase the polar functional groups [] and roughness on the fiber surface, thereby making the fiber surface hydrophilic []. These factors result in an increased resin impregnation rate.Nanomaterials (NMs) and fine particles have been applied as additives in various research fields, and they have especially used in polymer research field owing to their excellent properties as reinforcing agents []. In particular, NMs act as nucleating agents to promote the nucleation of crystallites within the polymer [], thereby increasing the crystallization temperature and melting temperature []. These factors lead to the stabilization of the crystal structure [], and increase in crystallinity as well as enhancement of the mechanical properties []. Furthermore, they increase the crystallization rate, which can improve the productivity and decreasing product shrinkage and distortion [Fast polymerization and low viscosity are necessary for the T-RTM. Therefore, plain woven carbon fiber (WCF)-reinforced composites were developed using various NMs and plasma treatment to reduce the process cycle time without deteriorating the crystallinity and mechanical properties. The fabrication process parameters of NM/A-PA6/WCF composites were optimized with respect to polymerization induction time – which is the initial slow stage of a chemical reaction, after which the reaction accelerates and therefore governs the overall processing time – and dispersion stability, crystallinity and mechanical properties were investigated.Polyacrylonitrile-based plain WCFs (T-300, Toray Corporation, Japan) were used in this study. ε-Caprolactam (C2204, Sigma-Aldrich, USA), which has a low moisture content (<100 ppm) and a low melting point (68 °C), was used as a precursor of A-PA6. Because sodium-based catalysts provide very short polymerization induction times as compared to magnesium-bromide-salt-based catalysts, which delay the onset of polymerization by several tens of minutes, a sodium-based catalyst (Addonyl® Kat_NL, Rhein Chemie, Germany) was used. Addonyl® 8120, an activator containing the blocked 1,6-hexamethylene diisocyanate, was used along with the catalyst. Various as-received NMs, including multiwalled carbon nanotube (MWCNT, CM-250, Hanwha Chemical, Korea), reduced graphene oxide (rGO, V30-100), graphene oxide (GO, V30, Standard Graphene, Korea), nanoclay (682632; surface-modified by aminopropyltriethoxy silane and octadecylamine, Sigma-Aldrich, USA), and exfoliated graphite nanoplatelet (xGnP, C750, XG Sciences, USA), were used as additives.Anionic ring-opening polymerization is sensitive to moisture [], and thus, every experiment was performed in a moisture-free environment of a glove box. First, the catalyst and activator were put separately into two closed tanks A and B. Then, the CPL and NMs were put, half each, into the two containers. Next, the containers were placed on a heating device and dispersed for 20 min above the melting point of CPL (69 °C) using a magnetic stirrer and a homogenizer. When the temperature reached 10 °C lower than the mold temperature, which was set equal to the polymerization temperature, the mixtures in tanks A (CPL/NM/catalyst) and B (CPL/NM/activator) were mixed for 3 s, and the resulting mixture was injected into the mold cavity containing 9 plies of WCF at a vacuum pressure of 75 kPa. The surface treatment of the WCF was processed using an atmospheric plasma system (API, Korea) with a nozzle height of 11 mm and x-axis nozzle speed of 20 m/min before placing the WCF in the mold. Schematics of the T-RTM and plasma treatment processes are presented in The cavity size of the mold was 300 × 120 × 2.5 mm (length × width × thickness), and the mold had two resin inlets and outlets. In this cavity, 9-ply WCF was laminated, and the minimum necessary time for producing composites was determined. The process requires at least 3–5 s for mixing, 5–10 s for resin infusion, and 5 s of spare time. Thus, it was confirmed that an approximate total 20–25 s was required as a minimum induction time. In other words, if the reaction accelerates before 20 s and the resin viscosity exceeds 1 Pa s, the solidification is completed during the impregnation of the resin into fibers, such that a proper specimen cannot be obtained. However, if the reaction proceeds beyond 25 s, it does not fulfil the purpose of productivity improvement. For these reasons, the polymerization temperature, catalyst and activator content with induction time of 20–25 s were optimized for each specimen.The NM-dispersed CPL was dissolved in diethyl ether (99%, E0697, SAMCHUN Chemical, Korea), and then a Turbiscan (Turbiscan Lab Expert, Formulaction Inc., USA), was used to rapidly evaluate the dispersion stability by an optical measurement method using multiple light scattering []. It can analyze both particle migration phenomena such as sedimentation and creaming, and particle size variation phenomena such as flocculation and coalescence. Hence, a qualitative evaluation is possible. In this study, each specimen was measured at intervals of 5 min at 50 °C for 1 h, and then the dispersion stability of the samples was compared by quantitative evaluation of the Turbiscan stability index (TSI).The degree of monomer conversion (Mc) of the A-PA6 reinforced with NMs was determined by two methods, extraction and evaporation techniques []. The residual CPL after polymerization was extracted from the synthesized A-PA6 using soxhlet extractor with methanol and deionized water for 10 h each. After drying in a vacuum oven at 90 °C overnight, the monomer conversion was calculated by comparing sample weight before and after extraction. The other technique for measuring the monomer conversion was to weigh the sample before and after heating in the furnace up to 230 °C. Because the boiling point of CPL is below the Tm of PA6, the residual CPL is evaporated when temperature reaches 220 °C. Hence, the A-PA6 after heating was expected to be free of CPL. Therefore, the final degree of conversion for both techniques was determined by the following equation.Mc=SampleweightafterremovingresidualmonomerSampleweightbeforeremovingresidualmonomer×100%The crystallization temperature, nucleation temperature, melting temperature, and percent crystallinity were measured using differential scanning calorimetry (DSC, TA Instruments Q800, USA). The DSC analysis was performed under a nitrogen atmosphere, over a wide temperature range from −10 to 250 °C with a heating rate of 10 °C/min. The percent crystallinity (Xc) of the A-PA6 was determined using the endothermic peak of the DSC thermogram by the following equation.This is the ratio of the melting heat enthalpy of the measured sample (ΔHm, experiment) and 100% crystalline PA6 (ΔHo, literature = 230 J/g) [For measuring the composition change of the plasma-treated WCF surface, X-ray photoelectron spectroscopy (XPS, K-Alpha, Thermo Fisher Scientific, USA) was used in this study. High-resolution spectra such as O1s, C1s, and N1s were observed. To analyze the polar functional groups in detail, the C1s scan was deconvoluted into five Gaussian peaks using a fitting software. Atmospheric plasma treatment typically makes the fiber surface hydrophilic, and hence a contact angle detector (Phoenix 300, SEO, Korea) was employed to evaluate the hydrophilicity of the WCF surface after plasma treatment.To determine the total production cycle time, the required time for the manufacturing process of the NM/A-PA6/WCF composite was measured by dividing it into four parts: mixing time, resin infusion time, induction time, and solidification time. The sum of these times was taken as the production cycle time, and the demolding time was excluded from the measurement.The in-plane shear strength, tensile, and flexural properties were determined using a universal testing machine (UTM, 5982, Instron, USA), and both the specimen information and test method were obtained from ASTM standards. A crosshead speed of 0.9 mm/min was constantly applied to pull the tensile test samples, which had a 0–90° fiber weaving direction. The testing condition of the in-plane shear strength was same as the tensile test, but the specimens had a different weaving direction (±45° laminate). The shear strength was determined using the following equation.where Pm is the maximum in-plane load, τ12m is the in-plane shear strength, and A is the cross-sectional area of specimen. In the flexural test, the crosshead displacement rate of 1 mm/min was applied to push the center of the flexural specimens consistently, and the flexural modulus and strength were evaluated.The polymerization temperature, catalyst-activator contents with 20–25 s polymerization induction time, were optimized for each specimen, and the results are shown in -(a). Several studies on anionic polymerization have shown that the reaction generally takes place between 130 and 160 °C, and higher temperature leads to faster reaction and lower crystallinity. Hence, in this study, the preparation of neat A-PA6 was carried out in the range of 130–150 °C by varying the concentration of the catalyst and activator and by referring to the optimized values obtained from the literature []. As can be seen from the results, the induction time for neat A-PA6 decreased with increasing temperature. At 130 °C, the reaction was initiated only with a high amount of catalyst (5–6.3 wt.%) and activator (3–3.77 wt.%), and no reaction occurred with other concentrations. Moreover, the reaction did not occur under the two lowest contents of catalyst and activator at 135 °C. Above 140 °C, the reaction took place at all conditions, but the deviation of the induction time was large depending on the catalyst-activator contents. At even higher temperatures, the induction time of most conditions was within 20 s, owing to the excessively rapid reaction. The measurement results of the crystallinity, induction time, and solidification time at 3.7 wt.% catalyst and 2.9 wt.% activator are shown in -(b). As a result, the higher the polymerization temperature, the higher the polymerization rate, leading to shorter polymerization induction and solidification times. However, crystallinity was greatly reduced because polymerization was completed before crystallization was sufficiently progressed. Additionally, too high a polymerization rate was unsuitable for our experimental setup or for producing large parts. Although crystallinity was high at low temperatures, the polymerization induction and solidification times increased dramatically, considerably reducing the productivity. Therefore, for the fabrication of neat A-PA6, the most appropriate conditions were a catalyst content of 3.7 wt.%, activator content of 2.9 wt.%, and processing temperature of 140 °C.In the case of specimens containing 0.1 wt.% NMs, the induction time was increased, with the result that suitable induction time was not obtained in the range of 140–145 °C, except when high concentrations of catalyst and activator were used. In addition, the use of large amounts of catalyst and activator is generally inappropriate, because it causes reductions in the molecular weight, toughness, and melting temperature of A-PA6 []. Thus, the appropriate induction time (20–25 s) was achieved by increasing the temperature to 150 °C with same catalyst and activator content as neat A-PA6. In the case of GO-reinforced A-PA6, the temperature had to be increased above 155 °C to meet the condition, and therefore the catalyst content was slightly increased as an alternative.Increasing the polymerization temperature reduced the solidification time, although it deteriorated the crystallization properties. However, using the NMs in anionic polymerization was anticipated to greatly reduce the solidification time without loss of crystallinity, owing to their excellent properties as a nucleation agent []. Thus, NM-reinforced A-PA6s, prepared at 150 °C, were characterized.A combination of a backscattering and transmission sensor with a vertical scanner enables the detection physical heterogeneities over the whole sample height with a vertical resolution up to 20 μm. Therefore, nascent destabilization phenomena by variations of the particle size or local variations of the volume fraction were detected, and then dispersibility was investigated. Most NMs had a black color, with the exception of nanoclay, leading to light absorption, and thus the dispersion stability was evaluated by change in the transmission profile with respect to the backscattering profile. The variation in transmission levels was measured by distinguishing the sample bottle into three sections: top, middle, and bottom. These results are shown in in terms of the mean value of transmission intensity over time. The transmission level of all samples increases at the top section, owing to a decrease in the concentration of nanoparticles, whereas it has a low level at the bottom section owing to an increase in nanoparticle concentration as part of sediment formation. In particular, in the CNT-dispersed CPL case, a very sharp increase rate is detected before 5 min. This indicates a fast sedimentation rate. The middle section also shows a similar tendency to the top section, but a sharp slope of the CNT sample is found between 5 and 10 min, after which the transmission intensity becomes idle. This means that the sedimentation in the top section is almost complete within 5 min and the sedimentation in the middle section is almost complete within 5–10 min. Conversely, the transmission level at an initial measurement of the CNT sample in the bottom section is very high compared with other NMs, owing to the creaming phenomenon, but it plunges within 5 min by accelerated sedimentation. These phenomena are representative of the degradation of dispersion stability, indicating that the CNT has poor dispersibility. In the case of nanoclay, the final transmission levels in all sections are similar, because the particle size variation phenomena such as flocculation are more dominant than the particle migration phenomena. The flocculation is also a factor decreasing dispersion stability. The slope of the graph of the top section indicates the sedimentation rate as mentioned above, and the calculated results are presented in . The unit is transmission level (%) per hour, and the sedimentation rate of the CNTs is even faster than other NMs; in particular, it is approximately 900 times faster than the xGnPs. Hence, if the CNTs are used as an additive to the T-RTM process, a pretreatment process such as oxidation for improving dispersibility is considered to be necessary. The other sedimentation rates decrease in the order of nanoclay, rGO, GO, and xGnP. In particular, the xGnP-dispersed sample has the best dispersion stability, because it has low transmission variation in all sections and the lowest rate of sedimentation. For a more accurate and comprehensive comparison, the TSI, a quantitative analysis provided by Formulaction Inc., is indicated in . The TSI is typically useful to distinguish, compare, and evaluate the dispersion stability between different samples, and a lower TSI value indicates better dispersion stability. The results show that, as expected, the xGnPs have the best dispersion stability, whereas the CNTs have the worst dispersion stability. In the case of other NMs, it was difficult to judge which NM is more stable, but by using the TSI, it is quantitatively confirmed that dispersion stability is better in the order of GO, rGO and nanoclay.The monomer conversion of A-PA6 was increased by adding the NMs at the extraction method, but the conversion of the CNT-reinforced A-PA6 sample was lower than neat A-PA6 at the evaporation method, as shown in . According to Nylon Plastics Handbook, edited by Melvin I. Kohan [], increasing the polymerization temperature reduced the degree of crystallinity because the thermal motion of the polymer chains became too high to let them crystallize, which resulted in a decrease in the monomer conversion. In this study, although the polymerization temperature of NM-reinforced A-PA6 was 10 °C higher than the neat A-PA6, the monomer conversion of NM-reinforced samples was higher than the neat sample, owing to the nucleating role of NMs, which will be discussed in the next section. Another reason, the slow solidification (polymerization) rate of neat A-PA6, can be also considered because active groups in the monomer can be buried inside a growing crystal before they can polymerize by high crystallization rate at 140 °C []. For these reasons, most NM-reinforced A-PA6 samples had a higher conversion, but the CNT-reinforced sample had similar or lower conversion as compared with neat A-PA6. This is due to the poor dispersion stability, leading to an increase in amorphous phase by preventing the diffusion of CPL.. The NMs can stabilize the crystalline sites thereby promoting the rate of the crystallization, and increasing the degree of crystallinity. Thus, all the NM-reinforced A-PA6s have the higher degree of crystallinity than neat A-PA6. From the endothermic peaks, it is confirmed that the melting temperature is shifted to a higher temperature for samples reinforced with xGnP, nanoclay, and GO, whereas in CNT- and rGO-reinforced samples, the melting point decreases slightly as compared with the neat sample. This is due to the low dispersion stability and high volume fraction, which can make nanoparticles agglomerated together easily, leading to a reduction the interfacial area of the NMs available for nucleation. The exothermic peak shows that the addition of the NMs increases the nucleation temperature and crystallization temperatures of the A-PA6; particularly, the GO- and xGnP-reinforced A-PA6 samples have the highest nucleation and crystallization temperatures, as well as narrow and high peak shapes. It means that they have a much higher rate of crystallization as compared to other samples. Therefore, when the polymerization rate is increased, the crystallization rate is also increased due to the NMs, which allows sufficient crystallization to proceed while polymerization is completed. Consequently, although the NM-reinforced A-PA6s were polymerized at a temperature 10 °C higher than neat A-PA6, which is one of the main factors decreasing crystallinity, they had higher crystallinity. Both the degree of crystallinity and melting temperature are correlated with monomer conversion. Therefore, the GO-, nanoclay-, and xGnP-reinforced samples exhibit better crystallization properties in common with monomer conversion.The change in the composition of the WCF surface was analyzed using XPS. Over the entire range of the XPS scan (0–1400 eV), the C1s (280–295 eV), O1s (525–540 eV), N1s (394–410 eV), and Si2p (97–107 eV) peaks were present in all samples in common, and the C1s-deconvoluted peaks comprised C–C (284.5 eV), C–OH (286.1 eV), C=O (287.7 eV), and COOH (288.9 eV) peaks were investigated to confirm an effect of plasma treatment. From the results of the XPS analysis (see ), the oxygen functional groups such as hydroxyl, carbonyl, and carboxyl groups considerably increase after plasma treatment. Because these functional groups can make the fiber surface hydrophilic and react with A-PA6 and NMs to form hydrogen bonding, the increase in the functional groups through the atmospheric plasma treatment promotes the resin impregnation rate into the fibers and improves the interfacial adhesion between the fibers and resin []. Moreover, the result of the water contact angle measurement presented in also confirms that the plasma treatment positively affects the fiber surface area and the changes in surface composition, thereby reducing the contact angle., the manufacturing time was measured by dividing the process into four parts (mixing, infusion, polymerization induction, and solidification), and the results are shown in the . The mixing time was always fixed at 3 s, and the induction time was already measured during the optimization described in Section . Therefore, we will focus here on the resin infusion time and solidification time. As anticipated, the resin infusion time into the mold, where the nine plies of WCF are laminated, is the shortest for neat CPL. However, the CPL dispersed with NMs with the higher volume fraction and lower dispersion stability (e.g., CNT, rGO) at the same wt.% increases resin infusion time. Moreover, plasma-treated fibers reduce the resin infusion time slightly, because the polar functional groups on the fiber surface increase the resin impregnation rate into fibers. Due to the small space of the mold cavity, the time gap was not very noticeable, except for the CNT and rGO samples. The ratio of the resin infusion time to the total process cycle time is very low, but it is expected that the effect of reducing the resin infusion time through plasma treatment can be clearly obtained at large-sized products. As for the solidification time, which accounts for the largest portion of the total processing time, the time gap between A-PA6/WCF and NM/A-PA6/WCF composites is very large because not only the polymerization temperature of the NM/A-PA6/WCF composite is higher than the A-PA6/WCF composite, but also the NMs accelerate the reaction by increasing the crystallization rate. In particular, the GO-reinforced composite has the shortest solidification time, because the slightly higher catalyst content is used to achieve the proper induction time. The xGnP-reinforced composite with the highest dispersion stability and crystallization rate has the second shortest solidification time. In conclusion, the total process cycle time of the NM/A-PA6/plasma-treated WCF composite decreases by more than two times (235%) compared to the neat A-PA6/WCF composite, without loss of crystallinity.Twelve kinds of specimens depending on NMs and plasma treatment were tested to investigate the tensile, in-plane shear, and flexural properties. From the results of the tensile tests (see ), the CNT-reinforced specimen with poor dispersion stability shows slight decreases in elastic modulus and tensile strength compared to the neat specimen, owing to agglomerated CNT particles caused by van der Waals attractive force and entanglement at high temperature; this leads to a reduction in the interfacial bonding between the fiber surface and resin. The highest elastic modulus and ultimate tensile strength are shown in the specimens reinforced with GO and xGnP, respectively, increasing by 28.8 and 18.0%, respectively, compared to the A-PA6/WCF specimen. In addition, the test results show that the plasma treatment not only reduces resin infusion time but also increases the mechanical properties. After plasma treatment, the elastic modulus and strength increased by 4.1 and 5.4% on average, respectively.The in-plane shear strength results (see ) show that the shear strength is increased in the NM/A-PA6/WCF specimens, excluding the CNT-reinforced specimen. Moreover, in S–S curves before and after the plasma treatment (), the fracture of the composites made by surface-treated fibers occurs later than for the non-treated specimens. This is because the polar functional groups such as hydroxyl, carbonyl, and carboxyl increased on the fiber surface, leading to increased interfacial bonding properties. Hence, after plasma treatment, in-plane shear strength increased by 4.4% on average.The three-point bending test results show that the flexural properties of the NM-reinforced composite are proportional to the crystallization properties and the dispersion stability. In , the maximum flexural modulus and flexural strength are increased by 29.2 and 30.7%, respectively, for the xGnP-reinforced A-PA6/plasma-treated WCF composite as compared to the A-PA6/WCF composite. This is somewhat different from the results of the previous tensile test, in which the xGnP-reinforced specimen had no highest modulus. Before the plasma treatment, the xGnP-reinforced A-PA6/WCF composite has relatively high flexural modulus, which is consistent with the other test results, but it has the highest value after plasma treatment. This implies that the improvement in interfacial bonding between the resin and fiber by plasma treatment is even more effective for the xGnP-reinforced A-PA6/WCF composite as compared with the other NM-reinforced composites, because xGnPs shows the best dispersion stability in resin. Generally, the flexural properties of fiber-reinforced composites are more influenced by the polymer matrix and interfacial boding properties than the intrinsic fiber properties. Therefore, the average improvement rate of the flexural modulus (10.2%) and strength (5.2%) by the plasma treatment is higher than for the other test results. The GO- and xGnP-reinforced A-PA6/plasma-treated WCF composites not only showed the best mechanical properties, but also provided the shortest process cycle time. Thus, the GO and xGnP are the most suitable nano-additives for the T-RTM process using PA6.Polymerization and crystallization occurred simultaneously during the anionic ring-opening polymerization of CPL in the mold, causing a rapid reaction and yielding composites with excellent mechanical properties. Higher polymerization temperature results in a faster production rate, however, this might not lead to polymers with a high crystallinity and good mechanical properties. Thus, a compromise between productivity and good mechanical properties is inevitable. To overcome these limitations, the NMs were used as a reinforcing agent, allowing the polymerization to take place at a higher temperature by delaying the induction time. This not only dramatically reduced the solidification time but also increased the mechanical properties by preventing the decrease in crystallinity owing to the role of the NMs as a great nucleation agent. Moreover, by using the fibers treated by atmospheric plasma, the resin transfer rate in the mold was increased, which contributed to shortening the overall process cycle time. Additionally, the mechanical properties were improved by increasing the polar functional groups on the fiber surface. In conclusion, when NMs and plasma treatment are applied to the T-RTM process simultaneously, not only was the process cycle time reduced by a factor of more than two (235%), the mechanical properties were also enhanced.A microstructure based approach to model effects of surface roughness on tensile fatigueThis paper presents a novel approach to model the effects of surface roughness on fatigue of tensile specimens. Voronoi tessellation was employed to simulate material microstructure and generate rough surfaces. Continuum damage mechanics was utilized to model progressive material degradation due to fatigue loading. In order to evaluate and corroborate the results from the finite element fatigue damage model, an experimental investigation was conducted on 4130 steel specimens under fully reversed loading. SEM Imaging was performed to obtain grain morphology parameters to model microstructure accurately using Voronoi grains. Ra measured from optical surface profilometry was utilized to model roughness. Three levels of roughness were chosen for this study. Damage parameters for fatigue modeling were obtained from the experimental SN results of the smooth specimens having Ra = 0.1 μm. The same parameters were used to model fatigue of numerically generated rough domains and predict a reduction in fatigue lives with increase in Ra. The randomness originating from microstructure and surface profile predicts a scatter in fatigue lives. The fatigue failure mechanism predicted by the model is in close agreement with experimental observations. The reduction in endurance limit with increasing roughness level as predicted by the model provides better accuracy than the widely used empirical relationship for surface finish factor ka.Most failures in rotating machinery are caused by fatigue of machine components under time-varying loads In general, fatigue process consists of three stages: (i) crack initiation, (ii) crack propagation and (iii) final fracture. For high cycle fatigue, a significant portion of fatigue life and its variability is attributed to crack initiation stage Here, Se′ represents reference fatigue limit obtained from rotating bending fatigue test of mirror-polished specimen and Se represents the corresponding value of reduced fatigue limit due to surface effects where Sut is ultimate tensile strength of material. Parameters A and B are dependent on surface finish type (fine-ground, machined, hot-rolled or as-forged). Similarly, Johnson Researchers have pointed out that the widely used surface finish factor relationships provided above are too conservative in some cases and have provided improved versions for such relationships. Shareef and Hasselbusch In addition to the empirical formulations obtained purely from experimental data, several analytical approaches have been developed to model the effect of roughness and other surface finish factors on fatigue. Murakami Most of the existing analytical approaches model roughness either as defect, notch or pre-existing crack and calculate stress concentration factor explicitly to predict fatigue lives. The current paper proposes a novel approach to model roughness and its effect on fatigue using a microstructure based model. The model uses damage mechanics to find crack initiation/s and fatigue life of specimens, given material microstructure and roughness measure Ra. Damage mechanics introduces a local damage variable, D into material constitutive equations to account for degradation in material properties as fatigue progresses This paper describes a novel analytical approach based on damage mechanics to model the effect of surface roughness on fatigue. Fatigue experiments on 4130 steel tensile specimen were conducted under fully reversed axial loading for three different roughness levels and the results were compared against model predictions. Experimentally observed grain morphology was used to model material microstructure using Voronoi tessellations. Voronoi grains were then discretized using constant strain triangular finite elements. Surface roughness was generated by exploiting the tessellation process and using Ra level obtained from surface profilometry observations. The randomness implemented in microstructure modeling ensures that no two simulation domains are identical in terms of grain morphology or roughness profile. The fatigue damage model predicts multiple crack initiation points wherein usually one of the cracks propagates to become final fracture. The observations from experimental investigation validate this feature of model prediction. Additionally, due to the randomness incorporated in modeling, crack initiation spots and fatigue life are different for every simulation domain. To characterize the variability in fatigue lives, a minimum of 33 numerically generated domains were analyzed at every load and roughness level. The damage parameters used in simulation were obtained from experimental S-N curve of smooth specimens (Ra = 0.1 μm). The same parameters were used to predict fatigue life distributions of numerically generated rough specimens at two higher roughness levels (Ra = 0.5 μm, 1.5 μm). The fatigue life predictions were compared with experimental results. The reduction in endurance limit, given by the surface finish factor, ka is predicted by the model with better accuracy than empirical methods.The fatigue modeling procedure requires parameters pertaining to material damage, grain morphology and roughness of critical surfaces. This section describes how these parameters are obtained through experimental evaluation. Specimen geometry, testing procedure and the process followed to produce specimens at different roughness levels are also discussed.The specimens used in this investigation were machined from 1.6 mm thin sheets of AISI 4130 steel. depicts the specimen geometry. The gauge area has non-uniform cross section and has continuous radius between the ends creating maximum stress concentration in the central region. It ensures that the failure occurs in this region and hence, the microstructure modeling was restricted to the central zone, as discussed in . Additionally, this helped in tracking crack initiation by focusing a camera on the central section. All the specimens were produced by milling in such a way that the tensile axis is aligned with the sheet rolling direction.In order to obtain the grain morphology parameters for microstructure modeling, the sheet was polished to mirror finish, followed by etching with 2% Nital solution to reveal grain boundaries. (a) is an SEM image of grain morphology. It can be observed that the grains are not equiaxed in shape and are mildly elongated in the loading direction, due to cold rolling. The planimetric method (b) shows the corresponding numerically generated microstructure which mimics these experimentally observed grain morphology parameters. illustrates the surface profiles and Ra measurement for all three roughness levels. contains Ra values for all three sets, averaged over 20 observations for each set. For convenience, specimens belonging to the three roughness levels are named as Set A (Ra = 0.1 μm), Set B (Ra = 0.5 μm) and Set C (Ra = 1.5 μm) in this paper.All specimens were tested in high cycle fatigue regime using MTS Landmark servo-hydraulic axial fatigue test rig. Fatigue tests were performed with load control under fully reversed loading. Peak Value Control function was used in the machine to ensure that the specified load levels are actually achieved in each cycle during testing. All tests were conducted at 20 Hz frequency. Nominal stress amplitudes in different tests varied from 250 MPa to 350 MPa while a zero mean stress was maintained throughout all tests. This range of stress amplitude was chosen to cover the high cycle fatigue regime of 104–106 cycles. Due to fully reversed nature of loading, approximately one out of ten tests resulted in buckling of the specimen which were ignored in fatigue life data. All tests were run until failure of the specimen which was controlled by setting cut off maximum displacement of 3 mm. When a test exceeded 106 cycles, it was stopped and infinite life was assumed. Few tests used tighter displacement cut off as stopping criterion so that cracks can be observed before the specimen failed.This section presents the approach used to model microstructure and surface roughness using Voronoi tessellations. The FE modeling approach is also explained, followed by a description of the fatigue damage model.Microstructural topological features such as grain size and orientation affect fatigue life scatter observed in polycrystalline materials, including steel illustrates schematically the procedure used to construct simulation domains of tensile specimen. First, seed points confirming to specimen geometry were distributed in 2D space as shown in (a). The indicated zone is the region of interest since critical stresses, damage and cracks are confined to this zone. Seed distribution in this region was controlled to result in the desired topological properties of material microstructure, such as average grain size and topological orientation. Next, seed points within distance δ from the specimen boundary were reflected across the boundary as shown in (b). This is to ensure that the boundary is automatically generated while performing Voronoi tessellation. (c) shows the output of Voronoi tessellation performed using ‘voronoin’ function in Matlab. Finally, the cells generated outside the specimen geometry were removed and the desired domain was obtained as shown in shows different kinds of grain morphology which can be obtained by controlling initial seed distributions. (a) was obtained using uniformly distributed seed points and corresponds to equiaxed grains. (b) represents non-equiaxed grains obtained by stretching the seed point distribution in Y-direction. The extent of stretch needed is determined by the average aspect ratio of the desired grain morphology. The elongation in grain morphology of the microstructure model in current paper is not pronounced in since the average aspect ratio is close to 1.The process described in the previous section is used to generate smooth boundaries of the domain since external seed points are perfect reflections of their internal counterparts. Thus, Voronoi tessellation results in auto-generation of the reflection boundary, by its very definition. In order to generate surface roughness, the reflection step in the construction process was modified. Instead of true reflection across the boundary, a perturbed reflection was utilized to perform Voronoi tessellation. The perturbation was performed only in the region of interest indicated in (a) since fatigue failure is localized in this zone. demonstrates the process used to generate roughness on curved boundary C in the region of interest. Let rin denote the position vector of an internal seed point eligible for reflection (within distance δ fromC), measured from the center of curvature of C. The position where it intersects C is denoted by rb, which has the magnitude of radius of the curve C.The above equations outline the reflection process used to obtain external seed points which results in smooth boundary after performing Voronoi Tessellation. For perturbed reflection, Δr′ was used instead of Δr, which has its magnitude given byand has the same direction as Δr. Here, N (|Δr|, σ¯) denotes Gaussian probability distribution with mean |Δr| and standard deviation σ¯. For every rin, Δr′ was computed using Eq. and the corresponding reflected external seed point was obtained using the following equation.This leads to perturbation of external seed points and results in rough boundary C' after Voronoi tessellation. The level of roughness generated in this manner is proportional to the standard deviation σ¯ of the Gaussian distribution. It should be noted that the choice of σ¯ = 0 automatically results in smooth boundary since Eqs. In this investigation, roughness level is characterized by its arithmetic measure Ra. Therefore, in order to compare modeling and experimental results, it is crucial that the experimentally measured Ra on the surface of interest along loading direction Here, z is measured from the mean profile C and L is the length of the curve C in the region of interest. The evaluation of integral in Eq. was achieved by realizing that the region in-between curves C' and C can be divided into trapezoids and ‘dual-triangles’, as shown in can be expressed as the sum of the absolute area of these trapezoids ΣAtr, and dual-triangles ΣAdt, where Atr and Adt can be calculated asHere, b1 and b2 denote the lengths of parallel bases of a trapezoid/dual-triangle and a is the altitude. If ntr and ndt denote the number of trapezoids and dual-triangles on C' respectively, Eq. Ra=∑intr(bi,1+bi,2)ai2+∑intrbi,12+bi,22ai2(bi,1+bi,2)∑i(ntr+ndt)aiThe above formula was used to calculate Ra on C'. Since the computed Ra is proportional to standard deviation σ¯, a desired value of Ra was obtained by modifying σ¯. Average Ra computed for each set of numerically generated specimens using this method are reported in After generating the microstructure topology model, Voronoi grains were discretized into constant strain triangles to perform finite element analysis in ABAQUS. The vertices of each Voronoi cell were joined to its centroid in order to segment the cell into triangles. In order to simulate the fully reversed load controlled fatigue cycling, sinusoidal constant amplitude load with zero mean was applied on top edge of the specimen model in Y direction in the form of time-varying rectangular pressure distribution, while the bottom edge was constrained to move only in X direction. Additionally, the central node of bottom edge of the specimen was fixed to avoid rigid body motion. A full load cycle was divided into four time steps to record the stress histories and capture the entire range of stress reversal Δσ. These details of the model are presented in Fatigue damage under cyclic loading leads to gradual degradation of material properties due to development of micro-cracks and voids. Damage mechanics provides a suitable framework to model these microscopic fatigue failure mechanisms in an empirical fashion. It defines a thermodynamic state variable, D to measure material degradation due to fatigue. In general, the damage variable is incorporated in material constitutive relationships in the following form,where σij, Cijkl, Iklmn, Dklmn, ∊mn denote stress, stiffness, identity, damage and strain tensors, respectively. Under the assumptions of isotropic damage, Eq. where D is a scaler between 0 and 1. As fatigue cycles progress, the value of D increases from 0 ( corresponding to pristine state of the material) to 1 (the completely damaged state).The damage evolution law used in the current investigation is given by the following model proposed by Xiao et al Here, N is number of cycles, Δσ is the critical stress range, and σr and m are material parameters. The parameter σr is referred to as resistance stress since it indicates the ability of material to resist damage accumulation The choice of critical stress quantity Δσ depends on the type of fatigue loading which causes failure. In tensile fatigue, the normal tensile stress leads to generation and propagation of cracks. Applying continuum theory of elasticity would provide deterministic results for location and magnitude of critical stress reversal. Fatigue life calculated in this fashion would be deterministic. However, stochastic effects are observed when normal stresses are calculated with respect to inherent planes of weakness such as grain boundaries, which is characteristic of the microstructure topology. Additionally, the stochastic nature of surface profile for a given level of roughness generates unique stress concentration contours for every numerical domain generated. These two factors give rise to scatter in the numerical results of fatigue lives.The material damage parameters σr and m were determined from experimental S-N data of set A specimen shown in . Using Basquin’s law, the experimental data can be curve fitted usingwhere Δσ is the stress range. In fully reversed loading, it can be related to the stress amplitude, σa by the following relationIt should be noted that the power law fit displayed in is for σa while Basquin’s law parameters a and b were obtained from power fit of Δσ. Integrating The same parameters were used in fatigue life prediction of numerically generated rough specimens – set B and C, to test the accuracy of roughness modeling in this investigation. and damage based elastic response of the tensile specimen need to be solved simultaneously to perform fatigue analysis. To achieve computational feasibility, ‘jump-in cycles’ procedure proposed by Lemaitre lists all the parameters used for fatigue modeling of tensile specimens. depicts experimentally obtained SN results of specimen belonging to each roughness level. Using Basquin’s law given by Eq. to power fit the experimental results of set A specimen yields a = 971.9 MPa and b = −0.0442. These parameters were used to calculate the reference damage parameters using Eq. corresponding to smooth specimen viz. σr = 1117.6 MPa and m = 22.6. The damage parameters were later utilized in fatigue analysis of numerically generated rough specimens in order to predict their fatigue lives and compare with corresponding experimental data in . Fatigue tests which did not lead to failure are shown with arrows in this figure indicating infinite life. As depicted in the figure, increasing roughness levels lead to decrease in fatigue lives of specimen. Additionally, when roughness level is substantial (Set 3, Ra = 1.5 μm), the SN curve is steeper indicating that effect of roughness is more pronounced at higher fatigue lives The failure mode was analyzed by inspecting the fractured surface of one of the specimens under SEM. The three stages of fatigue namely crack initiation, crack propagation and final fracture can be observed in (a). Crack initiated at the central gage section of specimen having minimum cross sectional area and at the corner of rectangular cross section. The initiation spot and beach marks corresponding to initial phase of propagation can be clearly observed in (c) displays the later stages of crack propagation showing cracks traversing towards left direction and developed into the final fracture phase as seen in (d). These observations are consistent with the classical description of fatigue fracture surfaces of high cycle fatigue in tensile specimens having rectangular cross section In order to numerically investigate the effect of surface roughness on fatigue life of 4130 steel tensile specimens, at least thirty-three randomly generated specimens at each level of roughness were subjected to cyclic loading and fatigue lives were calculated until crack initiation. Damage parameters for fatigue modeling were obtained from experimental SN results of smooth specimen, as described in . At each roughness level, fatigue simulations were performed at multiple stress amplitude levels to generate the SN curve. shows numerically generated SN results obtained from power law fit on these fatigue simulations for each roughness level. It can be observed that the reduction in life due to surface roughness is successfully predicted by the fatigue damage model. Specimens generated at higher roughness level have more pronounced stress concentration zones at their boundaries. Higher stress concentration in rough domains lead to faster damage accumulation and hence shorter lives. It can be noted that the model predicted shorter fatigue lives for Set 3 specimens as compared to experimental results in low cycle portion of the SN curve. This behavior is explained in The numerically generated tensile specimen were subjected to cyclic loading. Thus, damage accumulated in each finite element at a rate proportional to the normal stress reversal amplitude at the intergranular weak plane, as governed by Eq. illustrates the state of damage in a smooth specimen after the first iteration of the jump-in cycles procedure. The critical elements with highest damage accumulation can be observed at either sides of the specimen. In this figure, the most critical element appears at the right boundary of the specimen. It must be noted that damage was calculated in the central zone of the gage section only where stresses are highest and failure is expected. As the specimen undergoes more cycles of fatigue loading, one of the elements reached the critical value of damage (Dc = 1) and the corresponding weak intergranular boundary was failed giving rise to crack initiation. Due to topological randomness inherent to every specimen, no two specimens have identical location of crack initiation. Additionally, the initiation is not synchronous at either sides of the specimen. shows different stages of fatigue failure in a rough tensile specimen predicted by the model. In this case, the location of crack initiation depends on two factors; stress concentration due to roughness, and grain boundary orientation with respect to the tensile axis. (a) depicts the stress concentration zones appearing because of rough surface profile on either sides of the specimen. In the elements inside these zones, the damage causing normal stress component at the intergranular boundary is maximum when the boundary is perpendicular to loading axis. Hence, cracks initiate perpendicular to the tensile direction, consistent with experimental observations. (b) depicts this phenomenon clearly with the first crack getting initiated on the left boundary perpendicular to the loading axis. After crack initiation, the crack tip becomes a potential location for subsequent cracks due to stress concentration. Such subsequent cracks may not be aligned perpendicular to the tensile axis. They form the path of crack propagation leading to failure of the specimen. It is also possible that new cracks appear at a different location from existing crack tips due to critical orientation of grain boundaries and stress concentration due to roughness. (c) exhibits the latter case with second crack emanating from the right side of the specimen. In fact, multiple crack initiation locations are found in the specimen during fatigue cycling, an observation consistent with experiments. Eventually, one of these cracks propagate into the specimen and caused final failure. (d) depicts this crack propagation phase. shows comparison between numerically obtained and experimentally observed crack patterns. It shows that the appearance of multiple crack initiations before fatigue failure is captured accurately by the model. The sequence of events in fatigue failure of tensile specimen described here is consistent with earlier studies performed on specimen having similar geometry For a given roughness level, Voronoi tessellation was used to generate distinct microstructures and surface profiles for every numerically synthesized domain. Weibull distributions are commonly used to characterize fatigue scatter of mechanical systems depicts two-parameter Weibull fits for fatigue lives obtained from simulation for all three roughness levels of numerically generated specimen sets at stress amplitude of 275 MPa. Simulation results for a numerically generated smooth specimen set (Ra = 0 μm) are also added in this figure. The two parameters - Weibull slope and Weibull scale, characterizing these fits are listed in . Weibull slope is a measure of scatter in the data whereas Weibull scale indicates the number of cycles with 63.2% probability of failure. The Weibull slope of predicted fatigue lives decreases with increase in roughness levels since the randomness in stress concentrations is more pronounced at higher roughness levels. For perfectly smooth specimens with Ra = 0 μm, the slope is 23.7 which is much higher than the usual observed range of 1.7–4.8 for steels depicts that at high roughness level, i.e. Ra = 1.5 μm, the reduction in life is more prominent in high cycle fatigue. This observation is consistent with the fact that the effect of roughness is more detrimental at higher fatigue lives do not show such sensitivity towards the fatigue life and are nearly parallel to each other. This is due to the assumption of elastic response in high cycle fatigue zone since the damage evolution law used in current model does not take plasticity into account. However, the model essentially captures the maximum effect of roughness of fatigue lives by predicting the modification in endurance limit with good accuracy. Thus, the fatigue damage model provides a conservative estimate of effect of roughness on fatigue lives which can be used to design machine components without the need of conducting experiments at different roughness levels. In order to test the utility of this model, the predicted SN results were used to calculate the surface finish factor ka for endurance limit modification. provides the power law fit parameters of experimental as well as predicted SN curve for all three roughness levels. Using these power law equations, the applied stress range was calculated corresponding to a life of 106 cycles to obtain ka for set B and C with respect to set A. shows the comparison of ka calculated from the current model, as well as empirical charts with experimental results. The results indicate that the ka predicted by the current model is in better agreement with the experiments. The maximum error in ka calculated from simulation and experimental results is 1.1%. In order to calculate the empirical ka using empirical charts This paper presents a novel analytical approach to generate surface roughness and model its effect on high cycle fatigue of tensile specimens. The results were validated by performing fatigue experiments on 4130 steel tensile specimens having different roughness levels under fully reversed loading. Voronoi tessellation was used to simulate microstructure as well as surface roughness. The surface roughness parameters were obtained experimentally through SEM imaging and profilometry. Continuum damage mechanics was employed to model material degradation due to fatigue cycling. The model utilized damage parameters obtained from SN results of smooth specimens and numerically generated rough specimens to predict the effect of roughness on fatigue life.Experimental investigation showed that the fatigue life reduces with increasing roughness levels, as expected. Numerical results from the fatigue damage model predicted the similar trend. The fatigue failure mechanism observed in experiments was also successfully recreated in the simulation results. This included the appearance of multiple crack initiation points during the failure. Fatigue life prediction at a given load and roughness level displayed scatter due to randomness in microstructure topology and surface roughness profile of the specimens produced by Voronoi tessellations. The scatter was characterized using two-parameter Weibull distribution. The predicted scatter increased with increasing roughness levels and complied with experimental range for steels in case of Set B (Ra=0.5 μm) and Set C (Ra=1.5 μm) specimens. Finally, the model was used to predict the surface finish factor for endurance limit modification. This prediction corroborates well with experimental data and provides better accuracy than the widely used empirical charts. However, the prediction by the numerical the model did not capture the lowered sensitivity of fatigue lives to roughness level in low cycle range, as observed in experimental results.Effect of NaOH on the vitrification process of waste Ni–Cr sludgeThis study investigated the effect of NaOH on the vitrification of electroplating sludge. Ni, the major metal in the electroplating sludge, is the target for recovery in the vitrification. Sludge and encapsulation materials (dolomite, limestone, and cullet) were mixed and various amounts of NaOH were added to serve as a glass modifier and a flux. A vitrification process at 1450 °C separated the molten specimens into slag and ingot. The composition, crystalline characteristics, and leaching characteristics of samples were measured. The results indicate that the recovery of Ni is optimal with a 10% NaOH mass ratio; the recoveries of Fe, Cr, Zn, Cu, and Mn all exhibited similar trends. The results of the toxicity characteristic leaching procedure (TCLP) show that leaching characteristics of the slag meet the requirements of regulation in Taiwan. In addition, a semi-quantitative X-ray diffraction analysis revealed that the main crystalline phase of slag changed from Ca3(Si3O9) to Na4Ca4(Si6O18) with a NaOH mass ratio of over 15%, because the Ca2+ ions were replaced with Na+ ions during the vitrification process. Na4Ca4(Si6O18), a complex mineral which hinders the mobility of metals, accounts for the decrease of metal recovery.Electroplating is a widely used surface treatment procedure for enhancing the anticorrosion, smoothness, and luster of surfaces. The main procedure of electroplating involves connecting the desired material to a cathode and immersing it into an acidic solution (that usually includes metal cations such as H2SO4, HCl, and HNO3) to coat one or more thin layers of metal on the treated surface Ni and Cr have been extensively studied as electroplating materials because they have excellent corrosion resistance properties Solidification is a conventional method for the stabilization of electroplating waste Acid leaching is economically favorable but it may cause secondary pollution if the waste acid is improperly disposed of. Therefore, high-temperature treatment is also widely employed to treat metal sludge. Previous studies indicated that metals can be separated in some types of melting furnace by evaporation or phase separation in the vitrification process Si, which acts as a glass former in a glass network, is the most important additive for vitrification. In glass chemistry, Ca is the most commonly used glass modifier for improving the glassy structure. The mass ratio of CaO to SiO2, which is defined as the basicity, thus governs the vitrification process. Na is also used as a network modifier; it is believed to decrease the aggregation extent of a glass network In this study, sludge was obtained from a Ni and Cr coating workshop in Taiwan. The sludge included degreased sludge, filter sludge, and wastewater treatment sludge, which were produced from the processes of solvent recycling, electroplating, and waste treatment, respectively Sludge from a Ni and Cr electroplating workshop was selecting as the target material for metal recovery using the vitrification process. The water content and volatile component of the sludge were 9.89% and 32.0%, respectively. In addition, the Cl, S, and Si in the sludge were 1186, 9011, and 15,970 mg/kg, respectively. The sludge was dried in a 105 °C oven for 72 h to remove the water content before vitrification. The input for vitrification included a reductant additive and NaOH. The reductant additive was composed of dolomite, limestone, and cullet in a weight ratio of 1:3:4, which was found to be the optimal condition in our previous study in (see ref. , NaOH was added to the specimens in mass ratios of 1.00%, 5.00%, 10.0%, 15.0%, 20.0%, and 25.0% of sludge with additives in a graphite crucible to determine the effect of NaOH on metal recovery. The mass of input materials for testing runs is shown in . The vitrification processes were performed in a 1450 °C high-temperature-resistant furnace. For the operation protocol, the temperature was increased from room temperature to 40 °C in 5 min, to 400 °C in 30 min, to 800 °C in 60 min, to 1400 °C in 120 min, and then held for 30 min to complete the vitrification process. When the vitrification process finished, the heater was turned off for 24 h to cool the furnace to room temperature.The products from the vitrification process (i.e., slag and ingot) were separated according to their specific weight. The slag and ingot were digested with microwaves (MARS/MARS Xpress, CEM) based on the NIEA R353.00C method (equivalent to U.S. EPA SW846 Method 3050b) In order to investigate the effect of NaOH addition on the transition of crystalline phases in slag, XRD analysis was conducted using a powder diffractometer (Geigerflex 3036). The slag powder for XRD analysis was milled until the particle size was <25 μm (200 mesh sieve). Then, a 10% mass ratio of highly pure silica was homogeneously mixed with specimens as an internal standard for crystalline semi-quantitative analysis The composition of input materials is shown in . For Ni–Cr electroplating sludge, the major metal species were Ni (191,000 mg/kg), Cr (169,000 mg/kg), and Fe (50,600 mg/kg). For dolomite and limestone, the major metal species was Ca, accounting for 241,000 and 398,000 mg/kg, respectively, with other metals possibly being impurities or additives introduced during manufacturing processes. The role of cullet is to provide Si to form the glass matrix. The levels of hazardous metals were all below 1000 mg/kg.Similar to incineration, vitrification reduces the volume and mass of input materials. In addition to Ni–Cr sludge, the additives dolomite, limestone, cullet, and sodium hydroxide (NaOH) were used as input materials in the recovery trials. Slag and ingot were the main products of the vitrification process. The slag was the main mass contributor of the vitrification output (>97.9%), with the ingot accounting for less than 2.1%. The output to input mass ratios (O/I mass ratios) for 1%, 5%, 10%, 15%, 20%, and 25% NaOH to total input mass ratio trials were 73.1%, 73.4%, 71.7%, 69.9%, 68.5%, and 67.5%, respectively. The data show that the O/I mass ratios decreased with increasing NaOH; i.e., the reduction of mass was proportional to the increase in NaOH mass with good linear correlation (pearson correlation = 0.982, p |
= 0.001). As shown in , the mass of slag was proportional to the input mass. A 10% NaOH mass ratio produced the optimal ingot recovery ratio for metals.. The major metals for the slag from highest to lowest content were Ca, Cr, Fe, and Ni. The content of Ca ranged from 128,000 to 230,000 mg/kg in all specimens. The levels of Cr, Fe, and Ni for all testing rounds were 10,800–16,500, 4790–8410, and 3810–10,800 mg/kg, respectively. All other metals were lower than 1000 mg/kg. According to the composition of input materials, Ca was contributed by dolomite and limestone, and the elements Cr, Fe, and Ni were from the electroplating sludge.The highest Ca content was obtained at a 5% NaOH mass ratio; the Ca content gradually decreased with increasing NaOH content due to the dilution of encapsulation materials. However, the highest levels of Ni, Fe, and Cr were obtained at 20%, 15.0%, and 25% NaOH mass ratios, respectively. In terms of metal recovery, the separation of these metals was not very good for these NaOH mass ratios. In addition, the Si concentrations in the slag for 1%, 5%, 10%, 15%, 20%, and 25% NaOH/total input material trials were 262,000, 251,000, 249,000, 243,000, 237,000, and 229,000 mg/kg, respectively.The metal composition of ingot is shown in . The major metal species in the ingot were Ni, Fe, and Cr. The level of Ni ranged from 340,000 to 637,000 mg/kg. The masses of Fe and Cr for all testing rounds were 94,600–16,1000 and 70,000–133,000 mg/kg, respectively. With such high levels of metals, the ingot can be sent to a smelter for further recovery or can be reused as an additive for steelmaking. No Pb was detected in any testing round; Al was only detected with a 10% NaOH mass ratio. The highest mass of Ca was obtained with a 25% NaOH mass ratio, indicating that the separation of ingot and slag was not complete and that some of the encapsulation material flowed into the ingot. All other metals (including Ni, Fe, Cr, Cu, Zn, and Mn) were at their highest levels with a 10% NaOH mass ratio. A 10% NaOH mass ratio is thus the optimal condition for increasing the metal content in the ingot. According to the data, the ingot should be further treated by smelters to improve the purity of Ni for sale on the metal recycling market.The products of vitrification can be separated according to specific gravity. Metals with large specific gravity settle at the bottom ingot, and those with small specific gravity float to the top slag. Al, Ca, Mn, and Pb have small specific gravity, so they stay in the slag.The mass distributions of Ni, Fe, and Cr between the slag and ingot are shown in , respectively. The Ni, Fe, and Cr mass distributed in the ingot increased with NaOH addition of 1–10%, and then decreased for mass ratios of 10–25%. With 10% and 15% NaOH mass ratios, 77.6 and 60.1% of the Ni was collected in the ingot, respectively. The distributions of Fe and Cr exhibited the same trend as that of Ni in the ingot; the highest proportions of Fe and Cr in the ingot were 30.2% and 18.9%, respectively. In other words, the metal enrichment effect of Fe and Cr was lower than that of Ni.The mass recovery ratios of Ni, Fe, and Cr in the vitrification process are shown in . The results show that Ni, Fe, and Cr obtain their highest recovery mass ratios at 10% NaOH to total input mass. The recovery mass ratios of Ni, Fe, and Cr were 90.7%, 56.6%, and 19.0%, respectively. The trend of the metal recovery mass ratio was the same as that of metal distribution in the ingot.. Among the 9 analyzed metals, only total Pb, total Cr, and total Cu meet the EPA's TCLP regulations, with standards of 5.0, 5.0, and 15.0 mg/L, respectively. The concentrations of Pb, Cr, and Cu in TCLP extracts were non-detectable (ND), 0.15–0.45 mg/L, and ND, respectively. The results indicate that all leaching metal concentrations were below the TCLP regulation standards, which is consistent with results reported by Li et al. The patterns and volume fraction of crystalline phases in the slag are shown in , respectively. There are three stages in the transition of crystalline phases with increasing NaOH content. In stage I (1–5% NaOH mass ratio), Ca3(Si3O9) (pseudowollastonite) was the main crystalline phase (15–16%), SiO2 (quartz) was the secondary phase (4–7%), and the AVF was below 70%. For stage II (∼10% NaOH mass ratio), Ca3(Si3O9) (6%) was the only crystalline phase and AVF increased to 86% in the slag. This shows that the crystalline phases diminished or disappeared and that the slag turned into an amprphous structure. For stage III (>15% NaOH mass ratio), the crystalline phase transformed into Na4Ca4(Si6O18) (combeite) and the volume fraction increased with NaOH addition. Therefore, the AVF gradually decreased from 60% to 30%.The transition of crystalline phases and the amorphous phase indicate the role of NaOH in the vitrification process. In stage I, the amount of NaOH is too low to significantly affect the formation of crystalline phases. With increasing amount of NaOH, Na+ ions started to replace other cations, altering the glassy matrix. This explains why the AVF gradually increased with increasing NaOH content in stages I and II. With further NaOH addition, a new crystalline phase, Na4Ca4(Si6O18), appeared in stage III. The phase transformation is due to the substitution of Na+ ions for Ca+2 ions and can be expressed as:Ca6(Si6O18) + 4Na+ |
→ Na4Ca4(Si6O18) + 2Ca2+In the former two stages, the Na+ ions are insufficient to replace all Ca2+ ions. The partial substitution gives the slag a glassy amorphous structure. With sufficient Na+ ions, the complete substitution rearranges the structure into an ordered one. Therefore, Na4Ca4(Si6O18) was found and its amount increased with NaOH. Combeite has a very complex structure which may retard the movement of heavy metals in the molten material. This could explain why the metal recovery ratio decreased with NaOH addition for 10–25% NaOH mass ratio.Vitrification with NaOH additive increased the major metals (Ni, Fe, and Cr) contained in the Ni and Cr electroplating sludge in the ingot, especially Ni. The optimal mass ratio of NaOH additive was 10% of the total input. The crystalline phases of slag changed from Ca3(Si3O9) to Na4Ca4(Si4O18). A high amount of the amorphous phase improved the separation efficiency of the ingot and slag. The results show that NaOH in appropriate amounts is an efficient additive for recovering Ni and Cr in the vitrification process. The TCLP concentrations of Cr, Pb, and Cu meet EPA's regulations. Therefore, vitrification is a potential treatment process for Cr and Ni in electroplating sludge.A criterion for brittle fracture in U-notched components under mixed mode loadingA failure criterion is proposed for brittle fracture in U-notched components under mixed-mode static loading. The criterion, called UMTS, is developed based on the maximum tangential stress criterion and also a criterion proposed in the past for mode I failure of rounded V-shaped notches [Gomez FJ, Elices M. A fracture criterion for blunted V-notched samples. Int J Fracture 2004;127:239–64]. Using the UMTS criterion, a set of fracture curves are derived in terms of the notch stress intensity factors. These curves can be used to predict the mixed mode fracture toughness and the crack initiation angle at the notch tip. An expression is also obtained from this criterion for predicting fracture toughness of U-notched components in pure mode II loading. It is shown that there is a good agreement between the results of UMTS criterion and the experimental data obtained by other authors from three-point bend specimens.critical notch stress intensity factor in pure mode ISeveral failure criteria have been proposed in the past for brittle fracture under monotonic mixed-mode I/II loading in components containing a sharp crack. One of the most frequently used criteria is the maximum tangential stress (MTS) criterion proposed initially by Erdogan and Sih In the present paper, a brittle fracture criterion, called UMTS, is developed based on the maximum tangential stress (MTS) criterion and also a criterion proposed for mode I failure of rounded V-shaped notches Using the UMTS criterion, a set of fracture curves are derived in terms of the notch stress intensity factors (NSIFs). These curves can be used for more convenient predictions of the mixed mode fracture toughness and the crack initiation angle at the notch tip. An expression is also obtained from this criterion for predicting the fracture toughness of U-notched components in pure mode II loading.The stress field for sharp V-shaped notches was first presented by Williams ), an expression was developed by Filippi et al. σθθσrrσrθ(I)=Kρ,IV2πr1-λ1mθθ(θ)mrr(θ)mrθ(θ)(I)+rr0μ1-λ1nθθ(θ)nrr(θ)nrθ(θ)(I)In this expression, Kρ,IV is the mode I notch stress intensity factor (NSIF) and r0 is the distance between the origin of the polar coordinate system and the notch tip. The functions mij(θ) and nij(θ) are reported in . The eigenvalues λ1 and μ1 which depend on the notch opening angle are reported in Ref. σθθσrrσrθ(II)=Kρ,IIV2πr1-λ2mθθ(θ)mrr(θ)mrθ(θ)(II)+rr0μ2-λ2(II)nθθ(θ)nrr(θ)nrθ(θ)(II)Here, Kρ,IIV is the mode II notch stress intensity factor. The eigenvalues λ2 and μ2 are detailed in the Ref. The following expression is valid according to relation between Cartesian and curvilinear coordinate systems where 2α is the notch opening angle and ρ is the notch tip radius.The expressions for notch stress intensity factors (NSIFs) are where σθθ and σrθ are the tangential and the in-plane shear stresses, respectively. The auxiliary parameters ω1 and ω2 are given in When the notch opening angle is zero, then q |
= 2 and the rounded V-notch becomes a U-notch and one can write from Eq. , an expression for the tangential stress in U-notches can be derived as:σθθ=122πrKρ,IV32+ρrcosθ2+12cos3θ2+Kρ,IIV32-ρrsinθ2+32sin3θ2The maximum tangential stress (MTS) criterion is a well known and commonly used criterion for investigating brittle failure in specimens containing a sharp crack and loaded under mixed-mode conditions. According to this criterion, fracture occurs when the tangential stress at a critical distance rc ahead of crack attains its critical value (σθθ)c. Recent studies on different engineering materials suggest that when the crack is sharp (ρ |
= 0), the parameters rc and (σθθ)c are material parameters and independent of mode mixity (see for example In this paper, the application of the MTS criterion is extended to U-notched domains and a criterion called UMTS is developed. Considering the UMTS criterion under mixed-mode I/II loading, one can use the following equation in order to obtain the fracture initiation angle:Kρ,IV-34+ρ2rcsinθ02-34sin3θ02+Kρ,IIV34-ρ2rccosθ02+94cos3θ02=0Note that the parameter r must be substituted with rc according to the requirements of the MTS criterion. In pure mode I, Kρ,IIV vanishes and Eq. gives θ0 |
= 0 i.e. fracture initiates along the notch bisector line and the fracture initiation angle at the notch tip (θ0) is zero due to the symmetry of geometry and loading. In pure mode II loading where the notch edges slide parallel to the bisector line, the Kρ,IV is zero. Therefore, Eq. is simplified to the following expression: indicates that mode II failure initiates along the angle θ∗ which depends on the critical distance rc and the notch tip radius ρ. Considering the MTS criterion, failure at the crack or notch tip occurs when the tangential stress necessarily reaches the critical value (σθθ)c.(σθθ)c=122πrcKρ,IV32+ρrccosθ02+12cos3θ02+Kρ,IIV32-ρrcsinθ02+32sin3θ02 can be used to predict brittle fracture in U-shaped notches under mixed-mode I/II loading. For a more concise presentation of Eqs. , the following symbols are defined here:Failure criteria reported in the literature for the rounded V-notches suggest that in pure mode I loading, brittle fracture occurs when the NSIF reaches a critical value KcV,ρ called fracture toughness of rounded V-notch (see Ref. , mixed-mode brittle fracture for U-notches can be written in terms of KcV,ρ as:Kρ,IVBcosθ02+12cos3θ02+Kρ,IIVCsinθ02+32sin3θ02=AKcV,ρIt must be taken into account that, the parameter KcV,ρ depends not only on the material properties but also on the notch tip radius. Thus, it is not a material constant. The value KcV,ρ can be obtained from the following expressions where Pmax is the failure load that is determined experimentally using conventional notched test specimens. Note that, Kρ,IV(P1) is the notch stress intensity factor for an arbitrary load P1. Using Eq. and obtain the crack initiation angle θ0 from:Kρ,IV-B2sinθ02-34sin3θ02+Kρ,IIVC2cosθ02+94cos3θ02=0If rc is known for a U-notched specimen, one can draw the variations of Kρ,IV versus Kρ,IIV using Eqs. , and obtain the mixed mode fracture curves for U-shaped notches.The critical distance rc for U-notches which is measured from the origin of polar coordinate system (not from the notch tip) can be calculated by squaring both sides of Eq. For known values of KcV,ρ and (σθθ)c, the values of rc are obtained from:, rc depends on ρ and KcV,ρ. Therefore, rc for U-notches is a geometry dependent parameter and is not a fixed material property. In general, Eq. has three roots: two complex roots and one real root. Since rc is a real parameter, only the real one is acceptable. A sample fracture curve for zero notch tip radius is shown in To evaluate the suitability of the UMTS criterion, its theoretical results were compared with the experimental data reported by Gomez et al. KIC=1.7±0.1MPam12E=5.05±0.04GPaν=0.40±0.01σu=128.4±0.1MPaThe U-notched three-point-bending specimens were used for the experimental program. The geometry and loading conditions are shown in . In all the specimens the notch depth was fixed (a |
= 14 mm) as were the thickness and the height (14 and 18 mm, respectively). In order to have different mixed-mode conditions, the location of the applied load P was changed in various tests, resulting in four different values of the eccentricity (b |
= 9, 18, 27, 36 mm). To study the effect of the notch tip radius ρ, six values of the root radius were chosen: ρ |
= 0, 0.2, 0.3, 0.5, 1, 2 mm. Three tests for each geometry condition and a total of 72 static tests were performed in general As mentioned earlier, for calculating the value of rc, it is necessary to determine the parameters KcV,ρ and (σθθ)c. Gomez and Elices for PMMA at −60 °C and for various notch tip radii (ρ) are presented in . Mixed mode fracture in the U-notched components can be studied using a series of fracture curves that depend on ρ and rc. The curves are similar to the fracture curve of the classical MTS criterion which has been used frequently for analyzing the cracked components. For obtaining these curves, we define a parameter called the mode mixity (Me) as:The value of Me varies from zero (for pure mode II) to one (for pure mode I). Now, by extracting Kρ,IVKρ,IIV from Eq. Me=2πtan-1C2cosθ02+94cos3θ02B2sinθ02+34sin3θ02To achieve the fracture curve for given values of notch radius ρ and critical distance rc, one can follow the steps below:Choose an arbitrary value of the parameter Me between zero and one.Solve a linear system of equations consisting of Eqs. and find the normalized NSIFs: Kρ,IV/KcV,ρ and Kρ,IIV/KcV,ρ.Repeat steps 1–5 for other values of Me.Draw the fracture curve using the series of points calculated in step 5. shows the mixed-mode (I/II) fracture curves of the U-notched PMMA components for different notch tip radii. According to UMTS criterion, the value of rc is material dependent. Thus, the curves are plotted exclusively for PMMA at −60 °C. Using this approach, one can plot similar curves for other materials. Note that the fracture curves are displayed in non-dimensional form by dividing the NSIFs by KcV,ρ, as given in . It is useful to mention that for each notch radius, Kρ,IV is divided by its own particular KcV,ρ and not by an independent and constant value of KcV,ρ. Therefore, in pure mode I fracture the parameter Kρ,IV/KcV,ρ is equal to 1 for each value of notch radius. The points located on the fracture curve show the conditions related to the onset of brittle fracture (i.e. crack initiation from the notch tip) for various loading mode mixity. If the point situates under the related fracture curve, the fracture will not occur. But, if it locates over the curve, the fracture is expected to occur. In the case ρ |
= 0, when the U-notch becomes a crack, the fracture curve turns into the curve obtained previously by Erdogan and Sih In addition to fracture load, the other important parameter in the fracture analysis of the U-notches is the fracture initiation angle θ0 at the notch tip. In order to plot the curves of fracture initiation angle for a given notch radius ρ, the following steps can be done:Choose an arbitrary value for Me between 0 and 1. and find the fracture initiation angle θ0.Repeat steps 1–4 for other values of Me. shows the variation of the crack initiation angle (θ0) versus the mode mixity (Me) for the U-notched PMMA specimens at −60 °C for various notch tip radii. The sign of the fracture initiation angles is negative, but the absolute values are shown in . Here, the experimental results reported by Gomez et al. shows that the UMTS fracture curves together with the experimental results reported in , but for the fracture initiation angle θ0. It is seen from that despite a natural scatter in the experimental results, there is a good agreement between the theoretical findings predicted by the UMTS criterion and experimental results obtained and reported by Gomez et al. the mixed mode fracture toughness of PMMA specimens increases when the notch tip radius is increased. This is because the stress concentration at the notch tip decreases for larger notch radii requiring a larger load for fracturing the notched specimen. The notch tip radius also affects the fracture initiation angle θ0, although its effects on θ0 is not as much as on the fracture curves. shows that in pure mode I loading condition (Me |
= 1), the fracture initiation angle is equal to zero for all the notches with different tip radii. In mixed mode and pure mode II loading conditions, the fracture initiation angle is decreased as the notch radius is increased. It is shown in this figure that for ρ |
= 0 (sharp crack) the fracture initiation angle is equal to −70.51 (°) which is in agreement with predictions of the conventional MTS criterion Keff takes into account the effects of both mode I and mode II components of notch deformation and can be used as an alternative option for representation of the theoretical and experimental results in mixed mode fracture. The comparison of percentage errors for UMTS and SED criteria with respect to the experimental data are presented in for various notch radii. It is seen from that in almost all cases the results of the UMTS criterion are in better agreement with the experimental data compared to the results of the SED criterion.As mentioned earlier, the aim of this paper is to propose a failure criterion for brittle U-notched components under mixed-mode I/II loading and also to investigate the applicability of the conventional MTS criterion in U-notched domains. It was shown that the UMTS criterion can be used for predicting the fracture initiation angle as well as the onset of brittle fracture in U-notched specimens. A set of fracture curves were also presented by which the fracture load could be predicted conveniently. In addition to its simplicity over the SED criterion, the UMTS criterion could provide more accurate predictions of experimental results at least for the three-point bend experiments reported in One of the advantages of the UMTS criterion is its ability to estimate the pure mode II fracture toughness of the U-shaped notches. In pure mode II loading, Kρ,IV is zero and θ0 |
= |
θ∗ can be calculated from Eq. where Kρ,IIV is the mode II fracture toughness of the U-shaped notches. Thus, one can use Eq. for predicting the mode II fracture toughness in U-notched components.It is useful to remind that previous studies show that under certain geometry and loading conditions the size and scale of bodies that contain a sharp crack can influence their fracture toughness. Since the UMTS criterion for U-notched components is an extension of the conventional MTS criterion for sharp crack problems, it is underlined that all the analyses and results presented in this paper are confined to those notched components in which the size effect is negligible and also the critical distance is almost independent of the geometry.Finally, it must be pointed out that the three-point bend specimen (TPB) is not an ideal specimen for investigating mixed mode fracture in the full range of mode mixity. This is because the TPB specimen is not able to provide pure mode II or mode II dominant loading conditions. Thus, in order to study the effects of geometry and size on the fracture behavior of U-notched components, it is very useful to evaluate the theoretical results of UMTS criterion using other notched specimens under mixed mode loading. For example, brittle fracture analysis can be of particular interest for quasi-brittle materials like concrete or coarse-grain rocks which often possess a larger critical distance in comparison with most of the brittle materials.The classical MTS criterion for cracked specimens was extended to notched domains for predicting mixed-mode (I/II) fracture of U-shaped notches.Using the UMTS criterion, a series of fracture curves were developed in terms of notch stress intensity factors (NSIFs) and for various notch tip radii. These curves make the fracture predictions in U-notched component convenient.The theoretical predictions of the UMTS criterion were compared with the results of U-notch experiments conducted on PMMA at −60 °C. A good agreement was shown to exist between the theoretical and experimental results.Compared with the experimental results, the accuracy of the UMTS criterion in predicting Keff was generally more than the SED criterion for different values of the notch radii.(1) Functions used in the stress field for rounded V-shaped notches (mode I and II) mθθmrrmrθ(I)=11+λ1+χb1(1-λ1)(1+λ1)cos(1-λ1)θ(3-λ1)cos(1-λ1)θ(1-λ1)sin(1-λ1)θ+χb1(1-λ1)cos(1+λ1)θ-cos(1+λ1)θsin(1+λ1)θnθθnrrnrθ(I)=q4(q-1)1+λ1+χb1(1-λ1)χd1(1+μ1)cos(1-μ1)θ(3-μ1)cos(1-μ1)θ(1-μ1)sin(1-μ1)θ+χc1cos(1+μ1)θ-cos(1+μ1)θsin(1+μ1)θmθθmrrmrθ(II)=11-λ2+χb2(1+λ2)(1+λ2)sin(1-λ2)θ(3-λ2)sin(1-λ2)θ(1-λ2)cos(1-λ2)θ+χb2(1+λ2)sin(1+λ2)θ-sin(1+λ2)θcos(1+λ2)θnθθnrrnrθ(II)=14(μ2-1)1-λ2+χb2(1+λ2)χd2(1+μ2)sin(1-μ2)θ(3-μ2)sin(1-μ2)θ(1-μ2)cos(1-μ2)θ+χc2-sin(1+μ2)θsin(1+μ2)θ-cos(1+μ2)θ(2) The expressions for parameters ω1 and ω2ω1=q4(q-1)χd1(1+μ1)+χc11+λ1+χb1(1-λ1)ω2=14(μ2-1)χd2(1-μ2)-χc21-λ2+χb2(1+λ2)The values of the parameters λ1, λ2, μ1, μ2, χb1,χb2,χc1,χc2,χd1,χd2 are reported in Ref. Some critical issues for a reliable molecular dynamics simulation of nano-machiningThis paper investigates some critical issues in carrying out a reliable molecular dynamics simulation of nano-machining. Using monocrystalline copper as the workpiece material, this work shows that the Morse potential with compensation scaling can describe the interaction between copper atoms at a much better computational efficiency than the EAM potential. Based on a comprehensive study using the bond distance dynamics and cumulative error theories, the present investigation concluded that to avoid erroneous results the integration time step should be between the intervals of 4–12 fs, and that the machining speed should be less than 2062 m/s. The molecular dynamics simulation of nano-milling was then implemented as an application example.instantaneous distance between atoms i and jpair-interaction energy between atoms i and jtwo-body central potential at equilibriumcontribution to the electron density at i from atom jelectron density at the site of atom i from all the other atomsWith the ever increasing needs for miniaturized ultra-precision components in micro/nano-electro-mechanical systems (MEMS/NEMS), the fabrication of nano-scale surface features has brought about significant challenges in both the understanding of the process design mechanisms and the technical realization of production First, in an MD analysis it is important to select an appropriate interaction potential that can correctly describe the deformation of a material. In the investigation on copper with the aid of MD, for example, both the Morse potential and the embedded atom method (EAM) potential have been widely used to describe the interatomic interactions. The Morse potential is simple and computationally efficient in applications, because it takes only two-body interactions into account. In metals, however, thermal energy is mainly conducted by electron movements and hence needs to be compensated, as pointed out by Shimada et al. The second issue of primary importance is a proper selection of time step. In MD simulations, the motion equations are integrated by the finite difference method in which truncation and round-off errors are inevitably incurred , both errors are determined by the integration step size (Δt), but affected in different ways. The global truncation error (gte) rises with increasing the integration step size; but the global round-off error (gre) drops due to the decline of the number of iterations. Usually, a big integration time step leads to an unstable motion of atoms or molecules due to the correspondingly big errors occurring in the integration. On the other hand, a small integration time step causes a high computational cost The third issue that is vital to a reliable MD simulation is the selection of a rational range of machining speeds, which, on one hand, should be physically meaningful, but on the other hand, should be broad enough to allow to understand deeply the strain rate effect of a material. In a nano-scratching/cutting process, Pei et al. The aim of the present study is to investigate and clarify the aforementioned critical issues for carrying out a reliable molecular dynamics analysis of nano-mechanical machining.The reliability of a molecular dynamics simulation depends on the proper selection of an inter-atomic potential. As stated previously, the Morse potential and the EAM potential have been widely used to describe the inter-atomic interactions between copper atoms. The Morse potential is an empirical pairwise energy function of the formwhere re is the equilibrium distance between atoms i and j, rij is the instantaneous distance between atoms i and j; De is the cohesive energy between the two atoms; and α is the material constant which is obtained from elastic modulus where V(rij) is the pair-interaction energy between atoms i and j; Fi is the embedding energy of atom i; and ρi¯ is the electron density at the site of atom i from all the other atoms, which is expressed aswhere ρj(rij) is the contribution to the electron density at i from atom j. Pei et al. In metals, thermal conductivity is mainly governed by the mobility of electrons. Since the expression of the Morse potential does not consider the influence of electrons, it could lead to an inaccurate description of heat conduction, especially in high speed processing. Based on the comparison of the experiments of face turning and orthogonal 3D-cutting of a copper specimen with MD simulations, Shimada et al. where Ui is the scaled velocity of atom i; Vm is the mean square velocity; Vi is the velocity of atom i before scaling; and β1 is a constant chosen to make the heat conductivity coincide with the analytical prediction in a continuum model. They then further rescaled the velocities of atoms to keep the total energy unchanged. With this combined scaling, they demonstrated that the cutting forces from an MD simulation become reasonable in comparison with their experimentally measured values.To simulate a complex machining process, such as grinding or milling, a simple and efficient potential function is preferable. As have been highlighted previously, compared with the EAM potential, the Morse potential is prominent in its computational efficiency. To use the latter, however, it is essential to examine its capability in describing the metallic bonding, and to justify whether a scaling method can compensate the deficiency of the potential. To this end, let us first examine the nano-indentation-induced deformation in single-crystal copper, using both the Morse and the EAM potentials, because the deformation caused by a nano-indentation involves a much higher density change in the deformation zone due to the large hydrostatic stress associated with, and likely a higher electron density change, in comparison with that in other nano-machining processes. Hence, if the Morse potential can reflect well the nano-indentation, it should also be appropriate for nano-machining as to be detailed in Section 2.2.To avoid the boundary effect, the control volume of the specimen for a molecular dynamics analysis must be large enough. A test showed that for a nano-indentation under a hemispherical indenter of radius 2.14 nm, a volume of 14.4 nm × 14.4 nm × 8.7 nm consisting of 160,745 atoms is sufficient. As shown in , two layers of atoms of the control volume fixed in space (red dots) were used as the boundary atoms in order to eliminate the rigid body motion of the volume. As heat will be generated during the indentation process, two layers of thermostat atoms (green dots) were arranged to surround the Newtonian atoms of copper in order to ensure a proper heat conduction outwards, except the top (0 0 1) surface. The motions of all the other atoms obey the Newton’s second law (Newtonian atoms). The simulation model was first relaxed in a canonical ensemble where the number of moles N, volume V, and temperature T were conserved. A series of nano-indentations were carried out along the [001¯] direction on the (0 0 1) surface of the single crystal copper. A rigid indenter was initially positioned 0.3 nm above the top surface of the control volume. Using an integration time step of 10 fs, the indenter penetrated into the copper specimen along the negative z-axis with speed vi. Throughout the simulation, the average temperature was kept constant at 293 K by rescaling the velocities at every time step. shows the load–displacement curves of the loading process in the elastic region with the Morse potential, the Morse potential with compensation scaling (MCS), and the EAM potential, respectively. Although the three curves are similar, that with the Morse potential shows an obviously bigger oscillation. The curves obtained with the MCS and EAM potentials match quite well, demonstrating that the compensation scaling is effective and necessary when using the Morse potential. shows the overall response of copper to nano-indentation in a complete loading-unloading cycle when using the MCS and EAM potentials. The two curves are very similar in the plastic deformation regime of the loading and unloading processes; but the MCS result shows that the material’s resistance to deformation is somewhat higher. Such a difference could be attributed to the increase in electron density as explained below. With the penetration of the indenter into the workpiece, the atomic density of the contact zone becomes higher. As a result, the electrons which were originally in constant motion around the nucleus are overlapped, causing an increase in the electron density. From the embedding functions developed for copper Johnson described that with the increase of electron density the embedding energy becomes progressively negative In a nano-machining simulation, chips are removed instantaneously. As such, congestion of atoms ahead of a nano-machining tool would be much lower compared to that in a nano-indentation. Hence the contribution of the electron density energy in nano-machining will become much smaller; and the difference between the forces calculated with either the MCS or EAM potentials will minimise.When a simulation is reliable, its computational efficiency will be a major concern. This is particularly critical in simulating a large system. For example, with the nano-indentation simulation described above, when the indentation depth reaches 2.1 Å, the case with the MCS potential costs 2.5 h; while that with the EAM potential requires more than 6 h (nearly 2.5 times greater). The MCS potential is clearly much more efficient, yet provides consistent result with that of the EAM potential. Hence, in the following investigations into the other critical issues, the MCS potential will be used.As stated in Section 1, to simulate the machining of copper, various time step values ranging from 0.5 fs to 15 fs have been used in the literature. To avoid erroneous results, three different approaches (the algorithm constraint, bond-distance dynamics and the well-established JKR theory) will be used below to determine the most appropriate time step.Since the Verlet algorithm is good at solving differential equations and offers the virtues of simplicity and stability for moderately large time steps, this algorithm has been widely used in MD. However, it is only conditionally stable, because velocities and accelerations are assumed to be constant over a given time step. To satisfy this assumption, the period of vibration should be split into many segments (e.g., 10), i.e. the time step should be smaller than a few precent of the period of vibration of an atom, Tvwhere mr is the reduced mass of the system and K is the material stiffness (also known as the force constant) given byTo obtain the material stiffness of copper, consider a system of two copper atoms. The force between the atoms can be monitored as a function of their distance, as shown in . By definition, the stiffness of copper is the gradient at r |
= |
re and it is 20.5 N/m. Hence from Eq. (7), Tv |
= 316.5 fs which gives the maximum time step that should not exceed 10% of Tv, i.e., 31.65 fs.Let us examine the turning point of molecular dynamics by monitoring the Cu–Cu bond distance when relaxing copper atoms at different time steps. A simulation model was generated by seven copper units, consisting of a perfect copper unit surrounded by other six units next to its six surfaces to avoid the error due to surface relaxation of the central unit. The model was then relaxed for 500 iterations using time steps varying from 0.5 fs to 50 fs with an interval of 0.1 fs. shows the average bond length of copper of the central unit when using different time steps, in which the zone before 30 fs is enlarged to get a clear visualization. It is clear that a stable bond-distance is obtained between 3 fs and 26 fs, indicating that the time step should fall within this range.By considering the effect of surface adhesion energy, under static and purely elastic conditions, the JKR theory was used here and the time steps varied from 1 fs to 25 fs with an interval of 1 fs. The contact area based on the JKR function, a′, is given bywhere P is the load, γ is the energy per unit contact area, R is the indenter radius and K′ is the elastic constant. Then the contact area a0 at load zero is given byIn addition, the negative critical load Pc, which denotes the maximum adhesion load (negative value), is given bywhere ac is the corresponding contact area at Pc.In the indentation simulation by MD, the contact area between the indenter and workpiece was calculated by counting the number of C–Cu atom pairs and adding those atomic contact areas where ra is the theoretical radius of a copper atom and Nc is the number of contact pairs. Then the loading force from the JKR theory is defined by Eq. (13). Hence the standard deviation, average absolute deviation and absolute average deviation of the loading forces obtained from the JKR theory and the MD simulation with different time steps can be obtained, as shown in (a). It is clear that the deviation is reasonably small for the time steps between 4 fs and 12 fs.Global errors can be examined for a further validation. In the Verlet algorithm used to solve the Newton’s equation of motion, adding the two expressions of position x from time t forward to t+ |
Δt and that backward to t |
− Δt, giving rise tox(t+Δt)=2x(t)-x(t-Δt)+d2x(t)dt2Δt2+O(Δt4)Even though no third-order derivative exists in Eq. (15), the local truncation error still varies as Δt4. Thus the global truncation error (gte) iswhere M is the number of steps given by (t/Δt).The global error is made up of gre and gte. However, when the time step is large enough, gte dominates the global error and hence the difference in JKR and MD simulation is taken as gte. From (b), it can be seen that the standard deviation, average absolute deviation and absolute average deviation of K2 drop sharply when Δt increases from 1 fs to 3 fs. After that, K2 becomes more stable. For Δt larger than 4 fs, K2 is a tiny constant, indicating that the error is not influenced by K2. This means that 4 fs is the minimum threshold to ensure a small global error.Based on the investigations using the three approaches described above, we can now conclude that the integration time step should be between 4 fs and 12 fs in order to ensure that the MD simulations are accurate and reliable; otherwise, a simulation may not provide meaningful results as shown by the following example.To demonstrate the importance of the parameter selection established above, let us take a time step of 20 fs for a nano-milling that is outside the recommended range of 4–12 fs. The control volume of 12.3 nm × 21.7 nm × 8.7 nm consists of 204,551 atoms, which is large enough to avoid the boundary effect (a). A rigid diamond blade of dimensions 0.1 nm × 1.8 nm × 2.3 nm, shown in (b), was used as the nano-milling tool. The initial position of the tool was 0.3 nm away from the left edge of the control volume and fed along the negative x-axis with speed v while rotating around its z-axis at an angular velocity ω. The nano-machining was carried out along the [1¯00] direction on the (0 0 1) surface with a feed rate of 18 m/s and a liner cutting speed of 1413 m/s. The speed ratio was 0.0127 which is less than the threshold value of 0.02 (as to be explained in Section 4 below). The chips were removed to obtain a clear visualization of the machined surface. During the milling process, the average temperature was kept constant at 293 K by rescaling the velocities at every time step. shows a snapshot of this simulation. It can be seen that the relative positions of atoms become disordered, and the Newtonian atoms start to move out of the control volume. This is clearly not an acceptable result, caused by the improper selection of the time step.As mentioned previously, the selection of the machining speed for MD analysis should on one hand be physically meaningful, and on the other hand be broad enough to allow exploring the strain rate effect of a material in a nano-machining process. In this section, we will investigate the maximum possible nano-machining speed selection in a scientifically meaningful way.In the MD simulation of a nano-machining process, a cutting tool moves into the workpiece with a cutting speed, vc. To keep the workpiece in its solid state, the kinetic energy during the machining process must not exceed the energy required for melting. According to the equipartition theorem, the average kinetic energy of a system is (3NkT/2), where N is the total number of atoms in the system, k is the Boltzmann constant and T is the absolute temperature. Hence,If vmax is the threshold speed of nano-machining above which the material would melt, thenBased on the characteristic parameters of diamond and copper given in and for copper to be in its solid state,In general, the speed used in the MD simulation of any machining process is much higher than that used in the experimental work. There are mainly two reasons for this. One is that a high speed can be easily incorporated in a computational simulation to explore the speed effect (or the strain effect) of a material subjected to nano-machining. The other is that the computational capacity of even the most powerful super-computers does not allow a real time analysis even though the accumulation and truncation errors discussed previously could be totally ignored. It is therefore of primary importance to have a general guideline on the speed selection for a nano-machining process to provide the simulation reliability. As the deformation of copper is highly strain rate dependent compared to many other materials such as aluminium The results show that within the speed range specified above, MD simulations do reflect the strain-rate effect well (), which shows that the yield stress of copper increases with the loading speed (or strain rate). The elastic region of all the curves overlaps, and the predicted Young’s modulus of 134.4 GPa is in a very good agreement with the experimentally measured value of 125 GPa.In many mechanical manufacturing processes such as turning, grinding and milling, the trajectory of the machining tool is determined by both its linear and rotary motions, which, in turn, could bring about the complexity in a proper selection of nano-machining speed. In the following, the nano-milling process is used to demonstrate the complexity that in addition to the cutting speed the ratio of the feed rate to cutting speed is also important.To clarify this issue, let us first summarise the available process parameters of macro-, micro- and nano-milling reported in the literature . It can be seen that there are some similarities between the macro and micro milling processes. First, as illustrated in (a), their linear cutting speeds (vc) are in the same range and their feed rates (v) rise with the increase of feed per tooth (h). Secondly, (b) shows that the ratios of the feed rate to cutting speed (RFCS) are in the same zone (most of them below 0.02). However, at the nano scale, both the feed rate and cutting speed are much smaller compared to those at the macro/micro scales; and the ratio of the liner cutting speed to feed rate varies with the feed per tooth Having identified the key parameters as described in the preceding sections, let us reconsider the nano-milling process outlined in Section 3.3, but with a time step of 10 fs which falls within the rational range of 4–12 fs.(a) shows the periodic force variation in the nano-milling process. Apparently, it is similar to that in a micro/macro-milling process However, considering the atomic scale anisotropy of the material and its instant contact area with the tool during milling, the forces can still be predicted by a simple formula. Based on an analytical force model originally developed for the micro-end-milling where Km is the material coefficient that varies with the direction of machining as a result of the atomic scale anisotropy, and Ac is the atomic contact area. To obtain the material coefficient Km in different directions, we consider a quarter of a nano-milling revolution that can be divided into small cutting steps with continuously varying directions. Hence, the material coefficients Km along different directions such as [1¯00], [5¯1¯0], [5¯2¯0], [2¯1¯0], [1¯1¯0], [4¯7¯0], [2¯5¯0] and [2¯7¯0] on the (001) surface can be calculated by performing separate cutting simulations along those directions. shows the variation of Km with the cutting directions which are translated into azimuthal angles.The analytical milling force during a quarter revolution can then be predicted using Eq. (20), as illustrated in . The good agreement between the analytical prediction and MD simulation shows that MD simulations performed with rationalized parameters can provide reliable results and insightful information for nano-machining analysis.This paper has clarified some critical issues for a reliable molecular dynamic simulation in nano-machining. These include the selection of an appropriate potential function, the proper range of an integration time step, and the limit of strain rate (or machining speed). Specifically, the present investigation has brought about the following conclusions:The Morse potential with compensation scaling can well describe the interaction between copper atoms in nano-machining processes. Its application can significantly improve the computational efficiency of a large system.With the aid of the bond-distance dynamics and JKR theory, it is identified that the suitable integration time step for the simulation of copper is between 4 fs and 12 fs.For a reliable nano-machining simulation of copper, the cutting speed should be below 2062 m/s.Although the investigation presented in this study has focused on nano-machining, the methodology outlined here is applicable to the selection of reliable MD simulation parameters for many other nano-manufacturing/deformation processes.Effects of TC4 foil on the microstructures and mechanical properties of accumulatively roll-bonded aluminium-based metal matrix compositesIn the present study, Ti–6Al–4V alloy (TC4) foils were initially used as raw materials to produce aluminum-based metal matrix composites via the warm accumulative roll-bonding process. A composite with homogeneously distributed TC4 fragments in Al matrix was produced after 12 ARB cycles were performed. Microstructure analyses revealed that the thickness and length of the TC4 phase and the Al grain size in the Al/TC4 composites were refined by increasing the number of ARB cycles. Compared with the pure metal fragments in Al/pure metal systems, the TC4 fragments in Al/TC4 were thick and long because of the low plasticity–high strength of TC4 phase. The tensile properties of the Al/TC4 composites were also investigated. Results showed that the tensile strength and elongation increased with the number of ARB cycles and reached the maximum values of 147 MPa and 4.2% after 12 cycles were performed, respectively. Compared with the ARB monolithic, 1060-Al, Al/TC4 composites exhibited higher strength but lower elongation after 12 ARB cycles were conducted.Aluminium-based metal matrix composites (AMMCs) have attracted considerable attention because of their light weight, high strength, high stiffness, and wear resistance Hard particles and fragments are usually used to reinforce ARB AMMCs. Lu et al. For fragment-reinforced AMMCs, pure metal foils, such as Ti Pure metal and AZ31 Mg alloys are low strength–high plastic (LS–HP) materials. The co-deformation process of Al/LS–HP systems, as well as the microstructures and mechanical properties of Al/LS–HP AMMCs, has been thoroughly investigated Al/TC4 AMMCs were prepared by rolling TC4 foils and annealing 1060 commercially pure Al sheets with initial thicknesses of 0.1 and 1 mm, respectively. The TC4 foils and 1060-Al sheets were cut parallel to the rolling direction to produce 100 mm×100 mm strips. list the chemical compositions of the Ti foils and Al sheets used in this study.The TC4 foils and Al sheets were degreased using acetone and then wire-brushed. Two strips of Al and one strip of TC4 were stacked alternately to achieve 2.1 mm thickness, and then the strips were heated in a furnace at 280 °C for 5 min. Roll-bonding was performed without lubrication, and a laboratory rolling mill with 300 mm diameter and 220 mm barrel length was used. The rolling speed and rolling reduction ratio were 0.4 m/s and 52%, respectively. The thickness of the Al/TC4/Al ‘sandwich’ was 1 mm.The sandwich samples were cut in half, and the two halves were placed together. The stacks were heated in a furnace at 280 °C for 5 min and rolled again in the reverse rolling direction using a 50% reduction ratio. Warm rolling was used to avoid edge cracking during roll bonding. This process was defined as one ARB cycle. Hausöl et al. The microstructures of the samples were observed using scanning electron microscopy (SEM) and transmission electron microscopy (TEM). The TEM samples were prepared by mechanical grinding to a thickness of 30 μm in the rolling direction–transverse direction (RD–TD) and then by thinning with a twinjet electropolishing unit in a solution of 10 vol% acetic acid and 90 vol% methanol at room temperature. Tensile tests were conducted using an Instron mechanical testing system at room temperature, and the strain rate was 4×104 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.