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Comparison of experimental and computed tensile creep curves for a single crystal of orientation 0 0 1. The experimental creep curve is obtained for constant nominal stress creep loading and the computed creep curves are shown for both constant nominal stress (CNS) and constant true stress (CTS) creep loading. The computed curves are for a fully dense material.
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Experimental observations on tensile specimens in Srivastava et al (2012 in preparation) indicated that the growth of initially present processing induced voids in a nickel-based single crystal superalloy played a significant role in limiting creep life. Also, creep tests on single crystal superalloy specimens typically show greater creep strain ra...
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Context 1
... N = 5 and the parameters used in equation (22) are β ss = 0.998 and t 0 = 1.35 × 10 4 s. Figure 2 shows the experimental tensile creep curve of l/ l 0 versus time, where l 0 is the initial length of the specimen gauge section and l is the change in length of the gauge section with the loading applied in the 0 0 1 direction. For comparison purposes two computed curves for a fully dense material using the parameter values given above are also plotted: one with constant nominal stress and one with constant true stress. ...
Context 2
... corresponds to E e = 0.061 for χ = 3 and L = −1 and to E e = 0.058 for χ = 0.33 and L = −1. The steady-state effective creep strain rate, ˙ E ss , is essentially independent of the value of the Lode parameter and is almost same for χ 0.75 as for the fully dense material in figure 2 which is ˙ E ss = 0.235 × 10 −7 s −1 . For greater values of the stress triaxiality χ there is an effect of χ on ˙ E ss with ˙ E ss increasing to ˙ E ss = 0.438 × 10 −7 s −1 for χ = 3. ...
Context 3
... particular, the responses for L = −1 and L = 1 differ significantly. The steady-state effective strain rate for a fully dense material with constant N 1 in figure 2 is ˙ E ss = 0.355×10 −7 s −1 . There is a small effect of porosity (with f 0 = 0.01) on the steady-state creep rate for χ = 0.33; ˙ E ss = 0.395 × 10 −7 s −1 for χ = 0.33 and L = −1. ...
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... For example, for a porous material, with matrix exhibiting tension-compression asymmetry, the F.E. results obtained using cubic unit-cells showed the importance of the matrix tension-compression asymmetry on the rate of void growth and confirm the predictions of the dilatational model of Cazacu-Stewart [7]. Moreover, anisotropy and creep in porous crystals were investigated with cubic unit-cells in [4] and the results were further explained using an analytical model for porous crystals in [5]. The F.E. results provide insights on the effects of stress triaxiality and Lode parameter on void evolution for ductile single crystals in the dislocation creep. ...
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... Theoretical results will be validated performing unit cell analysis. This versatile numerical methodology was first introduced by Koplik and Needleman [24] and Brocks et al. [25], and allows to easily study the effect of various micromechanical ductile parameters such as second phase particles and void population [26][27][28][29], void shape or cell shape [8], material anisotropy [30], the effects of complex stress states [31][32][33][34] or matrix hardening rate behavior [35][36][37]. Numerical simulations were carried out using the commercial finite element code ABAQUS/Standard [38]. ...
In this paper, the classical Gurson model for ductile porous media is extended for strain-rate-dependent materials. Based on micromechanical considerations, approximate closed-form macroscopic behavior equations are derived to describe the viscous response of a ductile metallic material. To this end, the analysis of the expansion of a long cylindrical void in an ideally plastic solid introduced by McClintock (J Appl Mech 35:363, 1968) is revisited. The classical Gurson yield locus has been modified to explicitly take into account the strain rate sensitivity parameter for strain rate power-law solids. Two macroscopic approaches are proposed in this work. Both models use the first term of a Taylor series expansion to approximate integrals to polynomial functions. The first proposed closed-form approach is analytically more tractable than the second one. The second approach is more accurate. In order to compare the proposed approximate Gurson-type macroscopic functions with the behavior of the original Gurson yield locus, numerical finite element analyses for cylindrical cells have been conducted for a wide range of porosities, triaxialities, and strain rate sensitivity parameters. The results presented evidence that, for large values of the rate sensitivity parameters, the proposed extended Gurson-type models have the important quality to better predict the behavior of rate sensitive materials than the classical one. They also provide simpler and accurate alternatives to more traditional viscoplastic models.
... However, it will be shown later in 'Void coalescence' section that model does incorporate the effect of the presence of void and its growth on the plastic slip activity due to a combined effect of local stress concentrations, triaxialities and shape change as reported in literature (see, e.g. Asim et al., 2017;Mbiakop et al., 2015;Srivastava and Needleman, 2012). ...
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... with the c M [i] given by (53), and the stress variablesτ ...
... In addition, we can see that f /f 0 grows faster for larger values of X σ in the range X σ > 0, while f /f 0 decreases faster for smaller values of X σ in the range X σ < 0, as expected. Furthermore, the ISO results for f /f 0 are in good qualitative agreement with the corresponding FEM results for X σ > 0. In this context, it should be noted that, in addition to the FEM results of Srivastava and Needleman (2015) (depicted with solid circles), the corresponding FEM results of Srivastava and Needleman (2012) are also shown in Fig. 5(a) (with cross symbols) for X σ = 3 and 1/3 (X σ = 2/3 was not considered by Srivastava and Needleman (2012)). It can be seen that, while for X σ = 1/3 the two sets of FEM results are rather similar, for X σ = 3 they are somewhat different. ...
... In particular, for X σ = 3, the porosities for both the ISO and FEM increase rapidly with strain (see Fig. 5(a)) and, hence, the effect of the void distribution becomes progressively more significant. As pointed out by Srivastava and Needleman (2012), the increase of w at larger strains in the FEM simulation is induced by the necking of the inter-void ligament in the x 1 and x 2 directions (simultaneously), leading to rapid increases of the void radii in these directions. However, there is no well-defined inter-void ligaments for random microstructures and such "local" effects can not be captured by the ISO homogenization model. ...
In part I of this work (Song and Ponte Castañeda, 2017a), a new homogenization-based constitutive model was developed for the finite-strain, macroscopic response of porous viscoplastic single crystals. In this second part, the new model is first used to investigate the instantaneous response and the evolution of the microstructure for porous FCC single crystals for a wide range of loading conditions. The loading orientation, Lode angle and stress triaxiality are found to have significant effects on the evolution of porosity and average void shape, which play crucial roles in determining the overall hardening/softening behavior of porous single crystals. The predictions of the model are found to be in fairly good agreement with numerical simulations available from the literature for all loadings considered, especially for low triaxiality conditions. The model is then used to investigate the strong effect of crystal anisotropy on the instantaneous response and the evolution of the microstructure for porous HCP single crystals. For uniaxial tension and compression, the overall hardening/softening behavior of porous HCP crystals is found to be controlled mostly by the evolution of void shape, and not so much by the evolution of porosity. In particular, porous HCP crystals exhibit overall hardening behavior with increasing porosity, while they exhibit overall softening behavior with decreasing porosity. This interesting behavior is consistent with corresponding results for porous FCC crystals, but is found to be more significant for porous HCP crystals with large anisotropy, such as porous ice, where the non-basal slip systems are much harder than the basal systems.
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... The model has been successfully used by many groups in the literature to simulate damage due to dilatational-void-growth. There are many studies available in the literature that have modeled discrete micro-void-growth and coalescence using finite element and other multiscale modeling techniques [14,27,29]. As mentioned earlier, most of the abovementioned studies deal with hydrostatic loading and the loading rate were assumed to be quasi-static. ...
Fragmentation occurs in structures when subjected tolarge-strain-rate loading that commonly occurs in ballistic andasteroid impacts. In ductile materials, fragmentation is preceded bylocalized plastic deformation forming multiple adiabaticshear-bands. The shear-band-spacing thus formed controls the finalfragment-size distribution. The observed fragment-size distributionis different from the theoretically predicted values. The abovediscrepancy can be deciphered by simulating multiple-shear-bandformation that accounts for both thermal- andshear-void-growth-softening (which is often ignored). We model theshear-void-growth-softening by using the extended Gurson-modelNahshon–Hutchinson model)) in our method of characteristics-basedfinite-difference solution methodology. We perform parametricsimulations varying the shear-void-growth-rate. Our results indicatethat the number of shear-bands increases linearly with increase inthe shear-void-growth-rate for any applied strain-rate. The averageshear-band-spacing thus predicted is reduced, signifying theimportant role of shear-void-growth-softening in the simulations ofmultiple-shear-band formation.
... The constitutive response of metallic alloys is strongly affected by voids originating in the manufacturing process, that have an important effect on the lifetime as well as deformability of materials. Indeed, the growth of initially present processing induced voids in a nickel based single crystal superalloy as well as in standard polycrystals played a significant role in limiting creep life, as recently shown by experimental observations (Srivastava et al., 2012; Kondori and Benzerga, 2014a, b) at high enough temperatures on tensile specimens. The presence of voids (or cracks) in metals is known to be one of the major causes of ductile failure, as addressed in earlier works by Mc Clintock (1968), Rice and Tracey (1969) and Gurson (1977). ...
... Such anisotropic matrix systems have known slip directions and contain usually a small volume fraction of impurities. When these material systems are subjected to external loads impurities fracture or decohere leading to the creation of voids of various shapes, which in turn evolve in size, shape and orientation during the deformation process (Srivastava and Needleman, 2012). This complex evolution of microstructure together with the evolution of the rate-dependent matrix anisotropy is critical in the prediction of the eventual failure of the specimen under monotonic and cyclic loading conditions. ...
A rate-(in)dependent constitutive model for porous single crystals with arbitrary crystal anisotropy (e.g., FCC, BCC, HCP, etc.) containing general ellipsoidal voids is developed. The proposed model, denoted as modified variational model (MVAR), is based on the nonlinear variational homogenization method, which makes use of a linear comparison porous material to estimate the response of the nonlinear porous single crystal. Periodic multi-void finite element simulations are used in order to validate the MVAR for a large number of parameters including cubic (FCC, BCC) and hexagonal (HCP) crystal anisotropy, various creep exponents (i.e., nonlinearity), several stress triaxiality ratios, general void shapes and orientations and various porosity levels. The MVAR model, which involves a priori no calibration parameters, is found to be in good agreement with the finite element results for all cases considered in the rate-dependent context. The model is then used in a predictive manner to investigate the complex response of porous single crystals in several cases with strong coupling between the anisotropy of the crystal and the (morphological) anisotropy induced by the shape and orientation of the voids. Finally, a simple way of calibrating the MVAR with just two adjustable parameters is depicted in the rate-independent context so that an excellent agreement with the FE simulation results is obtained. In this last case, this proposed model can be thought as a generalization of the Gurson model in the context of porous single crystals and general ellipsoidal void shapes and orientations.
... Beginning with the pioneering studies of Needleman (1972), Tvergaard (1981), Koplik and Needleman (1988), micromechanical finite-element (FE) analyses of unit cells have been used to provide fundamental understanding of the mechanical response of porous solids (e.g. Richelsen and Tvergaard, 1994;Zhang et al., 2001;Srivastava and Needleman, 2012;Alves et al., 2014, etc.). In all these micromechanical studies, it was assumed that the plastic flow of the matrix (voidfree material) is governed by the von Mises criterion. ...
... To detect the onset of void coalescence we use the procedure outlined in Srivastava and Needleman (2012), which is based on monitoring the evolution of the relative inter-void ligament size. Since for an isotropic material, the greatest reduction in ligament size occurs in the direction of minimum applied stress, the evolution of the relative inter-void ligament size in the x direction i.e. l r 1 ¼ l 1 =l 0 1 ¼ ðC 1 À r 1 Þ=ðC 0 1 À r 0 Þ with the macroscopic effective strain E e was examined ( Fig. 6(a)). ...
