A strain-rate and temperature dependent constitutive model for BCC metals incorporating non-Schmid effects: Application to tantalum–tungsten alloys (original) (raw)
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High-rate deformation of single crystal tantalum: Temperature dependence and latent hardening
Latent hardening is defined as the hardening on a secondary slip system caused by slip on a primary slip system. Latent hardening is usually quantified by a latent hardening ratio (LHR) ϭ s / p , where p is the shear stress on the primary system just before unloading, and s is the yield shear stress on the secondary system. Previous latent-hardening experimental work on bcc (1,2) and on fcc metals (3-6) has predominantly been carried out at quasi-static strain rates. In most of the previous studies, the LHR was observed to range from 1.1 to 2.5; the exceptions are the works by Nowacki and Zarka , Wu et al. (8), and Mingzhang et al. (9). In both bcc and fcc materials, coplanar systems experience essentially no latent hardening (1,3,6). In most studies the yield stress of latent systems was measured by back extrapolating the flow stress to zero plastic strain. According to Wu et al. (8) and Mingzhang et al. (9), the back extrapolation method is not an accurate measure of the yield stress of a latent system, because here the initial rapid work-hardening rate is ignored. Mingzhang et al. (9) measured p as the stress at which deviation from linearity occurred, and almost always obtained a LHR less than 1. Stroh (10) calculated the stress field for a general distribution of dislocations on the primary and secondary slip system. From this he derived the LHR to be less than 1 for all combinations of the primary and latent slip systems. Similarly Zarka (11) calculated the far-field stresses created by randomly distributed segments of dislocations, and concluded that the LHR does not exceed 1.
Comparisons of crystal hardening laws in multiple slip
International Journal of Plasticity, 1985
This paper brings together and concisely reviews results from recent analytical investigations on single crystals (variously clone alone or with students) in which predictions from different theoretical hardening laws are contrasted and compared with experimental studies. Finitely deforming f.c.c, crystals in both constrained and unconstrained multiple-slip configurations are considered. Four crystal hardening laws are given prominence. Two of these belong to a class of theories in which the physical hardening moduli relating rates-of-change of critical strengths (in the 24 crystallographically equivalent slip systems) to slip-rates are taken as symmetric. These are G. I. Taylor's classic isotropic hardening rule (proposed in 1923), which is almost universally adopted in the metallurgical literature for various approximate analyses of single and poly-crystal deformation, and a 2-parameter modification of Taylor's rule that has an empirical basis in the qualitative features of experimentally determined latent hardening in single slip. The other two hardening laws featured here belong to a class of theories that were introduced in 1977 by this author. This class requires the above modu[i to be nonsymmetric and explicitly dependent upon the current stress state in such a manner that the following consequences are assured. (1) The deformation-dependent hardening of latent slip systems necessarily develops anisotropically if there is relative rotation of gross material and underlying crystal lattice. (2) The theories admit self-adjoint boundary value problems for crystalline aggregates, hence a variational formulation. (The fact that symmetric physical hardening moduli do not permit variational formulations of polycrystalline problems was shown at the 1972 Warsaw Symposium.) The two members of this class considered here are the original (and simplest possible) theory of rotation-dependent anisotropy, which was proposed by this author in 1977 and commonly has been referred to as the "simple theory," and a modification of this theory introduced in 1982 by Peirce, Asaro and Needleman that lies between Taylor's rule and the simple theory in its predictions for finitely deforming f.c.c, crystals. (In a series of five papers during 1977-79, the simple theory was shown to universally account for the experimental phenomenon of "overshooting" in single slip in both f.c.c, and b.c.c, crystals.) Theoretical results from the various hardening rules are contrasted and compared with finite strain experiments in the metallurgical literature. Both tensile-loaded crystals in 4, 6 and 8-fold symmetry orientations and compressively loaded crystals under conditions of channel die constraint are treated. A postulate of minimum plastic work introduced in 1981 plays a prominent role in the theoretical analyses, in many cases providing a unique solution to the slip system inequalities and deformation constraints (where applicable). The rather remarkable ability of the simple theory to reconcile diverse qualitative features of both constrained and unconstrained finite deformation of f.c.c, crystals is demonstrated. Finally, conditions for total loading (all systems active) in 6-fold symmetry are investigated, and certain concepts regarding the selection of active systems under prescribed straining are critically assessed. i i4
International Journal of Plasticity, 2010
Orthotropic elasto-plastic model Finite element a b s t r a c t This paper presents a comprehensive experimental and theoretical investigation of the deformation behavior of high-purity, polycrystalline a-titanium under quasi-static conditions at room temperature. The initial material in this study was a cross-rolled plate with a strong basal texture. To quantify the plastic anisotropy and the tension-compression asymmetry of this material, monotonic tensile and compressive tests were conducted, on samples cut along different directions of the plate. A new anisotropic elastic/plastic model was developed to describe the quasi-static macroscopic response of the aggregate. Key in its formulation is the use of an anisotropic yield criterion that captures strength-differential effects and an anisotropic hardening rule that accounts for texture evolution associated to twinning. A very good agreement between FE simulations using the model developed and uniaxial data was obtained.
The mechanisms governing twin-induced strain hardening of high-purity a-titanium at room temperature were incorporated into constitutive laws to describe the evolution of both twin and slip resistance due to deformation twinning. The proposed equations were incorporated in a Taylor-type crystal plasticity model to predict mechanical behavior and texture evolution for different deformation paths. Model predictions for the overall stress-strain response and texture evolution compared well with the experimental results. Specifically, the model captured the three stages of strain hardening for uniaxial-compression and plane-strain-compression testing of a-titanium. In addition, predicted texture evolution due to the reorientation of twinned area showed excellent agreement with the observations. These findings proved the necessity of incorporating twinning and its associated hardening mechanisms in realistic constitutive descriptions to account for anisotropic strain-hardening behavior and texture evolution in materials that deform by both slip and twinning.
Acta Materialia, 2011
A new constitutive plasticity model for prismatic slip in hexagonal α-titanium is developed. In the concept pure edge and screw dislocation densities evolve on the {101¯0}〈12¯10〉 slip systems. The model considers that the screw dislocation segments have a spread out core, leading to a much higher velocity of edge compared with screw dislocations. This enables the model to describe the observed transition in strain hardening from stage I to stage II in single crystals oriented for prismatic slip. Good agreement is found between the experimentally observed and simulated stress–strain behavior.
Procedia Engineering, 2014
Multi-scale modelling offers physical insights in the relationship between microstructure and properties of a material. The macroscopic anisotropic plastic flow may be accounted for by consideration of (a) the polycrystalline nature and (b) the anisotropic grain substructure. The latter contribution to anisotropy manifests itself most clearly in the event of a change in the strain path, as occurs frequently in multi-step forming processes. Under monotonic loading, both the crystallographic texture and the loading-dependent strength contribution from substructure influence the macroscopically observed strength. The presented multi-scale plasticity model for BCC polycrystals combines a crystal plasticity model featuring grain interaction with a substructure model for anisotropic hardening of the individual slip systems. Special attention is given to how plastic deformation is accommodated: either by slip of edge dislocation segments, or alternatively by dislocation loop expansion. Results of this multi-scale modelling approach are shown for a batch-annealed IF steel. Whereas both model variants are seen to capture the transient hardening after different types of strain path changes, the dislocation loop model offers more realistic predictions under a variety of monotonic loading conditions.
Physical Mesomechanics, 2010
The paper briefly considers the structure of internal variable constitutive relations. The mesoscale model required for determination of macroscale internal variables is taken to be one of the crystal plasticity (Lins model), in which critical shear stress along slip systems assumes great importance. In this work, evolution equations for critical shear stress that take into account dislocation annihilation and reactions with the formation of Lomer Cottrell barriers are proposed thus making possible description of the Bauschinger effect and additional hardening under complex loading. The potentialities of the model are demonstrated by numerical simulation of monotonic and cyclic uniaxial loading of polycrystals.
A dislocation density-based crystal plasticity constitutive model for prismatic slip in α-titanium
Acta Materialia, 2011
A new constitutive plasticity model for prismatic slip in hexagonal a-titanium is developed. In the concept pure edge and screw dislocation densities evolve on the f1 0 1 0gh1 2 1 0i slip systems. The model considers that the screw dislocation segments have a spread out core, leading to a much higher velocity of edge compared with screw dislocations. This enables the model to describe the observed transition in strain hardening from stage I to stage II in single crystals oriented for prismatic slip. Good agreement is found between the experimentally observed and simulated stress-strain behavior.