On the mechanics of crystalline solids (original) (raw)
Related papers
Effect of Hydrostatic Pressure on the Elastic Behavior of Cubic Crystals
physica status solidi (b), 1966
Finite deformation theory is used to calculate the effective second order elastic constants (cik′) of a cubic crystal subjected to very high pressures. We obtain cik′ = cik + A η + B, where A and B depend on the second order and higher order elastic constants and η depends on the pressure.
Crystal plasticity for dynamic loading at high pressures and strains
Arxiv preprint arXiv: …, 2008
A crystal plasticity theory was developed for use in simulations of dynamic loading at high pressures and strain rates. At pressures of the order of the bulk modulus, compressions o(100%) may be induced. At strain rates o(10 9)/s or higher, elastic strains may reach o(10%), which may change the orientation of the slip systems significantly with respect to the stress field. Elastic strain rather than stress was used in defining the local state, providing a more direct connection with electronic structure predictions and consistency with the treatment of compression in initial value problems in continuum dynamics. Plastic flow was treated through explicit slip systems, with flow on each system taken to occur by thermally-activated random jumps biased by the resolved stress. Compared with simple Arrhenius rates, the biased random jumps caused significant differences in plastic strain rate as a function of temperature and pressure, and provided a seamless transition to the ultimate theoretical strength of the material. The behavior of the theory was investigated for matter with approximate properties for Ta, demonstrating the importance of the high pressure, high strain rate contributions.
Nonlinear elasticity of prestressed single crystals at high pressure and various elastic moduli
Physical Review B, 2021
A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect of finite rotations is fully taken into account and analyzed. Different types of the bulk and shear moduli under different constraints are defined and connected to the effective properties of polycrystalline aggregates. Expressions for elastic energy and stress-strain relationships for small distortions with respect to pre-stressed configuration are derived in detail. Under hydrostatic initial load, general consistency conditions for elastic moduli and compliances are derived that follow from the existence of the generalized tensorial equation of state under hydrostatic loading obtained from single or polycrystal. It is shown that B moduli can be found from the expression for the Gibbs energy. However, higher order elastic constants defined from the Gibbs energy do not have any meaning since they do not directly participate in any of known equations, like stress-strain relationships and wave propagation equation. Deviatoric projection of B can also be found from the expression for the elastic energy for isochoric small strain increments and the missing components of B can be found from the consistency conditions. Numerous inconsistencies and errors in known works are analyzed.
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
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.
Plastic deformation of minerals at high pressure
Mineral behaviour at extreme conditions, 2005
Mechanical properties of real materials are controlled by crystal defects such as point defects, dislocations, stacking faults and grain boundaries. Taken individually, these defects can be described at the fundamental level through their atomic and electronic structures, which can be found by solving the Schrödinger equation. First-principles calculations and molecular dynamics are used to address such problems. At the scale of a grain, the mechanical properties are often the result of the collective behaviour of these defects in response to the loading conditions. Newly developed three-dimensional dislocation dyna-EMU Notes in Mineralogy, Vol. 7 (2005), Chapter 16, 389-415 the model, at the expense of predictive accuracy. There are many atomic-scale methods available to study point and planar defects in mantle minerals (for examples of studies of defects in forsterite see but here we concentrate on the study of dislocations. Most applications of these techniques for dislocation modelling have been concerned with simple metallic systems and semiconductors. Minerals usually represent a more complicated case because of their large unit cells, low symmetries and complex crystal chemistries. This complexity makes the atomistic approach even more relevant, as such fundamental issues as plastic shear anisotropy (which is responsible for crystal preferred orientations), dislocation mobilities and Peierls stresses need to be addressed at this scale.
Thermodynamically based constitutive equations for single crystals
It i,s shown that a gener&Iized Schmid law for d,uctile single crystals consistent with the entropy inequali,ty should be formulated in terms of the Eshetby stress tensor. For actiae slip systems the cansti,tutive equations in rate form are deriued. The conditions of strain localization is obtained.
Nonlinear elasticity of pre-stressed single crystals: resolving an old mess
arXiv (Cornell University), 2021
A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined, and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect of finite rotations is fully taken into account and analyzed. Different types of the bulk and shear moduli under different constraints are defined and connected to the effective properties of polycrystalline aggregates. Expressions for elastic energy and stress-strain relationships for small distortions with respect to a pre-stressed configuration are derived in detail. Under initial hydrostatic load, general consistency conditions for elastic moduli and compliances are derived that follow from the existence of the generalized tensorial equation of state under hydrostatic loading obtained from single crystal or polycrystal. It is shown that B moduli can be found from the expression for the Gibbs energy. However, higher-order elastic moduli defined from the Gibbs energy do not have any meaning since they do not directly participate in any known equations, like stress-strain relationships and wave propagation equation. The deviatoric projection of B can also be found from the expression for the elastic energy for isochoric small strain increments, and the missing components of B can be found from the consistency conditions. Numerous inconsistencies and errors in the known works are analyzed.
On the selection of active slip systems in crystal plasticity
International Journal of Plasticity, 2005
The capabilities of existing rate-independent and rate-dependent constitutive models to select the active slip systems at the corners of non-smooth theories play a crucial role in predicting localisation phenomena. Even though the description of crystal plasticity within the context of modern continuum mechanics goes back to the early 1960s, there is no universally accepted solution as to how to identify a unique set of active slip systems. Furthermore, some recently proposed integration schemes have neither been compared with other methods nor tested under complex multiaxial stress conditions thus rendering a direct assessment difficult. In this work, the predictive capabilities of existing crystal plasticity and visco-plasticity formulations and algorithms when subjected to complex multiaxial loading paths are investigated, and their relative accuracies established. In order to compare consistently the performance of different models, a generic thermodynamics-based crystallographic framework, which incorporates current formulations as special cases, is proposed. Several two-dimensional boundary value problems for elasto-plastic and elasto-viscoplastic FCC crystals are selected as benchmark cases. The effects of multiaxial loading paths, latent hardening, and dissipated energy on the selection of active slip systems at sharp yield surface corners are investigated. The differences in the predicted behaviour were found to be associated with both the particular form of the single crystal formulations and the algorithms used in their numerical implementations. Experimental data (E.P. Busso).