Douglas J. Bammann

Calculation of stress in atomistic simulation

Atomistic simulation is a useful method for studying material science phenomena. Examination of the state of a simulated material and the determination of its mechanical properties is accomplished by inspecting the stress field within the material. However, stress is inherently a continuum concept and has been proven difficult to define in a physically reasonable manner at the atomic scale. In this paper, an expression for continuum mechanical stress in atomistic systems derived by Hardy is compared with the expression for atomic stress taken from the virial theorem. Hardys stress expression is evaluated at a fixed spatial point and uses a localization function to dictate how nearby atoms contribute to the stress at that point; thereby performing a local spatial averaging. For systems subjected to deformation, finite temperature, or both, the Hardy description of stress as a function of increasing characteristic volume displays a quicker convergence to values expected from continuum theory than volume averages of the local virial stress. Results are presented on extending Hardys spatial averaging technique to include temporal averaging for finite temperature systems. Finally, the behaviour of Hardys expression near a free surface is examined, and is found to be consistent with the mechanical definition for stress.

A discussion of stress rates in finite deformation problems

Abstract It has recently been shown that the finite elastic-plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed, while the solution for isotropic hardening gives a monotonically increasing stress. This paper analyzes this response on the basis of continuum mechanical descriptions of the problem. Three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress. In light of the equations for the evolution of the yield surface, the hypoelastic solution to the simple shear problem for each of the three stress rates is presented. It is shown that the use of the Jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution. An alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress.

A model for finite-deformation plasticity

SummaryWe propose a new large deformation viscoplastic model which includes the effects of static and dynamic recovery in its strain rate response as well as the plastic spin in its rotational response. The model is directly obtained from single slip dislocation considerations with the aid of a maximization procedure and a scale invariance argument. It turns out that the evolution of the back stress and the expression for the plastic spin are coupled within the structure of the theory. The model is used for the prediction of nonstandard effects in torsion, namely the development of axial stress and strain as well as the directional softening of the shear stress. The comparisons between the present continuum model and both experiments and self-consistent polycrystalline calculations are very encouraging.

A model of crystal plasticity containing a natural length scale

A crystal plasticity model is developed which is embedded with a natural length scale. The model is developed within the framework of Coleman-Gurtin thermodynamics of internal state variables. The normal multiplicative decomposition of the deformation gradient utilized in crystal plasticity is expanded to add an extra degree of freedom. The internal state variables introduced in the free energy include the elastic strain associated with the statistically stored dislocations, and the curvature, which is derived from the continuum theory of dislocations. In this theory, the curvature is the curl of the elastic rotation associated with the polar decomposition of the elastic deformation gradient. This leads to an internal stress field that results from the presence of geometrically necessary dislocations and possesses an inherent length scale, in addition to the normal mechanical resistance that is proportional to the square root of the statistically stored dislocations. A crystal plasticity model incorporating these internal stresses is implemented into a FEM code. The results of the models prediction of the formation of misoriented cells, whose size is determined by the length scale of the model, as well as the prediction of the gradients of misorientation at the interface of the two cells is compared with bi-crystal experiments on Aluminum. During the compression of these bi-crystals, the dislocation patterns that develop encompass local lattice rotations or cells of misorientation that have a hierarchical of length scales. Therefore, the kinematics of the model are generalized to include an additional rotational degree of freedom, and additional tensor state variables are derived from the kinematics. This model has many similarities with the multipolar dislocation/disclination theory of Eringen and Claus and may be viewed as a finite deformation extension of that theory. The relationship of this model, which now has multiple length scales, to existing deformation theory type of strain gradient models is then discussed, and the length scales are related to actual physical mechanisms.

A damage model for ductile metals

Abstract A physically-based theory of damage for ductile metals is outlined. It rests upon a direct extension of the authors recently proposed viscoplastic model for finite deformations to include the effects of dislocation—void interactions as they manifest themselves in void nucleation, growth, and coalescence. Emphasis is put on illustrating the general structure of the present framework within which coupling effects of texture development, void formation, and adiabatic heating can be considered and their role to the localization of deformation and failure can be evaluated. No special attention is placed on justifying the various growth laws and simplifying assumptions pertaining to the detailed structure of the model, for example the manner that spatial gradients of the damage variable enter into the theory. Such simplifications, however, facilitate the solution of the relevant equations for a case of homogeneous triaxial state of stress permitting a qualitative comparison with experimental data obtained for Bridgeman-notch specimens.

An internal variable model of viscoplasticity

Abstract A time and temperature dependent plasticity model is formulated in a Lagrangian system to describe finite deformation. The history dependence and large strain behavior are incorporated through the introduction of one tensor internal variable. Linear decomposition of the stretching tensor into elastic and plastic parts is assumed and a constitutive equation is formulated for the plastic stretching which predicts a pseudo yield point. The model assumes a temperature change based upon he occurrence of adiabatic processes. Model prediction in comparison to strain rate change tests and tension and torsion tests is discussed.

A Geometric Framework for the Kinematics of Crystals With Defects

Presented is a general theoretical framework capable of describing the finite deformation kinematics of several classes of defects prevalent in metallic crystals. Our treatment relies upon powerful tools from differential geometry, including linear connections and covariant differentiation, torsion, curvature and anholonomic spaces. A length scale dependent, three-term multiplicative decomposition of the deformation gradient is suggested, with terms representing recoverable elasticity, residual lattice deformation due to defect fields, and plastic deformation resulting from defect fluxes. Also proposed is an additional micromorphic variable representing additional degrees-of-freedom associated with rotational lattice defects (i.e. disclinations), point defects, and most generally, Somigliana dislocations. We illustrate how particular implementations of our general framework encompass notable theories from the literature and classify particular versions of the framework via geometric terminology.

A Nonlocal Phenomenological Anisotropic Finite Deformation Plasticity Model Accounting for Dislocation Defects

A phenomenological, polycrystalline version of a nonlocal crystal plasticity model is formulated. The presence of geometrically necessary dislocations (GNDs) at, or near, grain boundaries is modeled as elastic lattice curvature through a curl of the elastic part of the deformation gradient. This spatial gradient of an internal state variable introduces a length scale, turning the local form of the model, an ordinary differential equation (ODE), into a nonlocal form, a partial differential equation (PDE) requiring boundary conditions. Small lattice elastic stretching results from the presence of dislocations and from macroscopic external loading. Finite deformation results from large plastic slip and large rotations. The thermodynamics and constitutive assumptions are written in the intermediate configuration in order to place the plasticity equations in the proper configuration for finite deformation analysis.

On the kinematics of finite-deformation plasticity

SummaryA theory of finite deformation plasticity is developed which involves a multiplicative decomposition of the deformation gradient through the assumption that there exists a stress-free configuration which can be used to separate the elastic and plastic components of the response. By using the polar decomposition on the usual indeterminate elastic and plastic deformation tensors, two uniquely defined stress-free configurations can be identified. The structure of this theory is compared with that of a spatial theory involving the polar decomposition of the total deformation gradient. It is shown that for the special case of linear response between the stress and the elastic strain, the two theories are indistinguishable in terms of their stress responses.

Anholonomic Configuration Spaces and Metric Tensors in Finite Elastoplasticity

Abstract Deformation mappings are considered that correspond to the motions of lattice defects, elastic stretch and rotation of the lattice, and initial defect distributions. Intermediate (i.e., relaxed) configuration spaces associated with these deformation maps are identified and then classified from the differential-geometric point of view. A fundamental issue is the proper selection of coordinate systems and metric tensors in these configurations when such configurations are classified as anholonomic. The particular choice of a global, external Cartesian coordinate system and corresponding covariant identity tensor as a metric on an intermediate configuration space is shown to be a constitutive assumption often made regardless of the existence of geometrically necessary crystal defects associated with the anholonomicity (i.e., the non-Euclidean nature) of the space. Since the metric tensor on the anholonomic configuration emerges necessarily in the definitions of scalar products, certain transpose maps, tensorial symmetry operations, and Jacobian invariants, its selection should not be trivialized. Several alternative (i.e., non-Euclidean) representations proposed in the literature for the metric tensor on anholonomic spaces are critically examined.