NUMERICAL AND THEORETICAL STUDY OF THE PROPERTIES OF A LINEAR ELASTIC PERIDYNAMIC MATERIAL

The peridynamic theory is a generalization of classical continuum mechanics and takes into account the interaction between material points separated by a finite distance within a peridynamic horizon . The parameter corresponds to a length scale and is treated as a material property related to the microstructure of the body. This work concerns a study of the properties of a linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which considers both length and relative angle changes, and is based upon a free energy function that contains four material constants. Using convergence results of the peridynamic theory to the classical linear elasticity theory in the limit of vanishing sequences of and a correspondence argument between the proposed free energy function and the strain energy density function from the classical theory, expressions were obtained relating three peridynamic constants to the classical elasticity constants of an isotropic linearly elastic material. To evaluate the fourth peridynamic material constant, the correspondence argument is used together with the deformation field of an elastic beam subjected to pure bending. This work also concerns the validation of the proposed linearly elastic peridynamic model through numerical simulations of mechanical problems formulated in the context of both the classical linear elasticity and peridynamic theories. Simulation results will be presented at the meeting

Boundary Layer Effects in a Finite Linearly Elastic Peridynamic Bar

Abstract The peridynamic theory is an extension of the classical continuum mechanics theory. The peridynamic governing equations involve integrals of interaction forces between near particles separated by finite distances. These forces depend upon the relative displacements between material points within a body. On the other hand, the classical governing equations involve the divergence of a tensor field, which depends upon the spatial derivatives of displacements. Thus, the peridynamic governing equations are valid not only in the interior of a body, but also on its boundary, which may include a Griffith crack, and on interfaces between two bodies with different mechanical properties. Near the boundary, the solution of a peridynamic problem may be very different from the classical solution. In this work, we investigate the behavior of the displacement field of a unidimensional linearly elastic bar of length L near its ends in the context of the peridynamic theory. The bar is in equilibrium without body force, is fixed at one end, and is subjected to an imposed displacement at the other end. The bar has micromodulus C, which is related to the Young's modulus E in the classical theory and is given by different expressions found in the literature. We find that, depending on the expression of C, the displacement field may be singular near the ends, which is in contrast to the linear behavior of the displacement field observed in the classical linear elasticity. In spite of the above, we show that the peridynamic displacement field converges to its classical counterpart as a length scale, called peridynamic horizon, tends to zero

Wiggly strain localizations in peridynamic bars with non-convex potential

We discuss the static states of a peridynamic nonlinear elastic bar of finite length in a hard device, which represents a continuum description of a complex hierarchical structure with interacting long-range crosslinkers, of the type encountered in biological systems. The nonlocal character of the model requires that edge conditions are defined on a boundary layer with the same length of the horizon, affecting the solution in the bulk interior. Assuming that the constituent microscopic ligaments contain bistable units governed by a non-convex potential, we show that the development of coexisting folded-unfolded phases, either synchronized or unsynchronized, induces in the displacement field the formation of undulations at a micro-scale of the length of the horizon, associated with strain localizations triggered at the bar ends. The equilibrium paths, found numerically with a pseudo-arc-length continuation method, become unstable within a certain range of elongation, suggesting the possible occurrence of a negative-stiffness response

A STUDY OF PENALTY FORMULATIONS USED IN THE NUMERICAL APPROXIMATION OF A RADIALLY SYMMETRIC ELASTICITY PROBLEM

We consider a class of two-dimensional problems in classical linear elasticity for which material overlapping occurs in the absence of singularities. Of course, material overlapping is not physically realistic, and one possible way to prevent it uses a constrained minimization theory. In this theory, a minimization problem consists of minimizing the total potential energy of a linear elastic body subject to the constraint that the deformation field must be locally invertible. Here, we use an interior and an exterior penalty formulation of the minimization problem together with both a standard finite element method and classical nonlinear programming techniques to compute the minimizers. We compare both formulations by solving a plane problem numerically in the context of the constrained minimization theory. The problem has a closed-form solution, which is used to validate the numerical results. This solution is regular everywhere, including the boundary. In particular, we show numerical results which indicate that, for a fixed finite element mesh, the sequences of numerical solutions obtained with both the interior and the exterior penalty formulations converge to the same limit function as the penalization is enforced. This limit function yields an approximate deformation field to the plane problem that is locally invertible at all points in the domain. As the mesh is refined, this field converges to the exact solution of the plane problem.Fundacao de Amparo a Pesquisa do Estado de Sao Paulo (FAPESP)[2007/03753-9]University of Minnesota Supercomputing Institute (UMSI

Analytical formulae for electromechanical effective properties of 3-1 longitudinally porous piezoelectric materials

A unidirectional fiber composite is considered here, the fibers of which are empty cylindrical holes periodically distributed in a transversely isotropic piezoelectric matrix, The empty-fiber cross-section is circular and the periodicity is the same in two directions at an angle pi/2 or pi/3. Closed-form formulae for all electromechanical effective properties of these 3-1 longitudinally periodic porous piezoelectric materials are presented. The derivation of such expressions is based on the asymptotic homogenization method as a limit of the effective properties of two-phase transversely isotropic parallel fiber-reinforced composites when the fibers properties tend to zero. The plane effective coefficients satisfy the corresponding Schulgasser-Benveniste-Dvorak universal type of relations, A new relation among the antiplane effective constants from the solutions of two antiplane strains and potential local problems is found. This relation is valid for arbitrary shapes of the empty-fiber cross-sections. Based on such a relation, and using recent numerical results for isotropic conductive composites, the antiplane effective properties are computed for different geometrical shapes of the empty-fiber cross-section. Comparisons with other analytical and numerical theories are presented. (c) 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved