The peridynamic model of solid mechanics is a nonlocal theory containing a length scale. It is based on direct interactions between points in a continuum separated from each other by a finite distance. The maximum interaction distance provides a length scale for the material model. This paper addresses the question of whether the peridynamic model for an elastic material reproduces the classical local model as this length scale goes to zero. We show that if the motion, constitutive model, and any nonhomogeneities are sufficiently smooth, then the peridynamic stress tensor converges in this limit to a Piola-Kirchhoff stress tensor that is a function only of the local deformation gradient tensor, as in the classical theory. This limiting Piola-Kirchhoff stress tensor field is differentiable, and its divergence represents the force density due to internal forces. The limiting, or collapsed, stress-strain model satisfies the conditions in the classical theory for angular momentum balance, isotropy, objectivity, and hyperelasticity, provided the original peridynamic constitutive model satisfies the appropriate conditions.

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Convergence of Peridynamics to Classical Elasticity Theory

This paper treats the dynamics of phase transformations in elastic bars The specific issue studied is the compatibility of the field equations and jump conditions of the one-dimensional theory of such bars with two additional constitutive requirements: a kinetic relation controlling the rate at which the phase transition takes place and a nucleation criterion for the initiation of the phase transition A special elastic material with a piecewise-linear, non-monotonic stress-strain relation is considered, and the Riemann problem for this material is analyzed For a large class of initial data, it is found that the kinetic relation and the nucleation criterion together single out a unique solution to this problem from among the infinitely many solutions that satisfy the entropy jump condition at all strain discontinuities

Kinetic relations and the propagation of phase boundaries in solids

The notion of the driving traction on a surface of strain discontinuity in a continuum undergoing a general thermomechanical process is defined and discussed. In addition, the associated constitutive notion of a kinetic relation, in which the normal velocity of propagation of the surface of discontinuity may be a given function of the driving traction and temperature, is introduced for the special case of a thermoelastic material.

On the driving traction acting on a surface of strain discontinuity in a continuum

A phenomenological model describing the mechanical behavior of polycrystalline solids undergoing stress-induced phase transformations is presented in the setting of three-dimensional (3-D) media. The model, which is developed within the framework of Generalized Standard Materials, has been built on a few basic assumptions, namely an admissibility condition for thermodynamic forces and a locking constraint for phase transformations. The macroscopic kinematic consequences of stress-induced phase transformations are described by a tensor named transformation strain. Numerical integrations of the resulting set of equations show that many features of shape memory alloys obtained by certain authors for CuAlZnMn alloys under non-proportional loadings can be qualitatively described by the present model.

Three-dimensional model for solids undergoing stress-induced phase transformations

In this paper the existence of minimizers in nonlinear elasticity is established under assumptions on the stored energy that permit the formation of new holes in the body. Such cavities have been observed in experiments on elastomers, and a mathematical theory for radially symmetric cavities has been developed by Ball. Here the full three-dimensional problem is considered and an additional, physically motivated, energy term that is proportional to the area of the boundary of the deformed body is included. The minimizers lie in a subclass of those maps in W1, p, 2<p<3, that are one-to-one almost everywhere and preserve orientation. Roughly speaking, this subclass consists of those maps in which cavities in one part of the body are not filled by material from other parts of the body. Such maps are shown to be much more regular than expected. In particular, some ideas of Sverak are used to show that each map in this subclass has a representative which is continuous outside a set of Hausdorff dimension 3 — p and that this representative also satisfies Lusin's condition (N), i.e., it maps Lebesgue null sets onto such sets. It is also shown that the distributional Jacobian of such a map is a measure which is the sum of a measure that is absolutely continuous with respect to Lebesgue measure and (at most) a countable number of Dirac measures.

An existence theory for nonlinear elasticity that allows for cavitation

In this paper, we carry out an explicit analysis of a bifurcation problem for a solid circular cylinder composed of a particularcompressible nonlinearly elastic material. This problem is concerned with the bifurcation of a solid body into a configuration involving an internal cavity. A discussion of its physical interpretation is then carried out. In particular, it is shown that this model may be used to describe the nucleation of a void from apre-existing micro-void.

A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void

The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials

The equations governing the equilibrium of a finitely deformed elastic solid are derived from the Principle of Minimum Potential Energy. The possibility of the deformation gradient and the stresses being discontinuous across certain surfaces in the body — “equilibrium shocks” — is allowed for. In addition to the equilibrium equations, natural boundary conditions and traction continuity condition, a supplementary jump condition which is to hold across the surface of discontinuity is derived. This condition is shown to imply that a stable equilibrium shock must necessarily be dissipation-free.

An admissibility condition for equilibrium shocks in finite elasticity

The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials

This investigation is concerned with the plane strain deformation of an infinite slab, containing a circular cavity, within the theory of finite elastostatics for a particular homogeneous isotropic compressible material, the so-called Blatz-Ko material. The body is subjected to uniform pressure, either internal or external. Exact closed-form solutions for the axisymmetric deformation and stress fields are obtained. In the case of internal pressure, it is found that the applied pressure may not exceed a certain maximum value pmax. At a value of pressure pe (<pmax), the governing equations lose ellipticity at the cavity wall. For greater values of pressure this solution remains smooth, though involving both elliptic and non-elliptic regions. Non-existence of axisymmetric solutions with discontinuous strain fields is established. The possibility of bifurication into a surface mode is considered and it is shown that this occurs at a value of pressure slightly smaller than pe. Such surface wrinkling leads to a periodic distribution of points of stress concentration, from which shear bands may initiate.

Rohan Abeyaratne

Papers

A bifurcation problem for a compressible nonlinearly elastic medium: growth of a micro-void

The finite deformation of internally pressurized hollow cylinders and spheres for a class of compressible elastic materials

An admissibility condition for equilibrium shocks in finite elasticity

The pressurized hollow sphere problem in finite elastostatics for a class of compressible materials

Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium