An approach for incorporating classical continuum damage models in state-based peridynamics

Abstract

Peridynamics has gained significant attention as an alternative formulation for problems in solid mechanics. Recent contributions have included initial attempts to include material damage and failure. In this paper, we propose an approach to incorporate classical continuum damage models in the state-based theory of peridynamics. This has the advantage of enabling the description of the damage evolution process in peridynamics according to well-established models. The approach is based on modifying the peridynamic influence function according to the state of accumulated damage. As a result, peridynamic bonds between nonlocal material points are severed in accordance with the damage law. The peridynamic damage formulation proposed is implemented for the particular case of a well established ductile damage model for metals. The model is applied to the simulation of ballistic impact of extruded corrugated aluminum panels and compared with experiments.

Introduction

Peridynamics is a non-local continuum theory of solid mechanics originally proposed by Silling to address elasticity problems involving discontinuities and long-range forces [1]. One of its main objectives is to provide a formulation which naturally supports the presence of discontinuities in the deformation field. Another advantage of the theory is that the resulting equations of motion are naturally discretized using particle-based methods [2]. This is presumed to have advantages in problems involving severe material deformations where mesh-based discretizations fail. One of the key remaining challenges in peridynamics and associated discretization methods is how to describe material failure in ways which are consistent with established models of fracture and damage.

In the original formulation of peridynamics, usually referred to as the bond-based theory, fracture is commonly incorporated by means of a critical relative displacement criterion, i.e., when the change in distance between two particles reaches a critical value $u_c$, their bond is irreversibly broken [3]. A particle-based discretization of peridynamic was proposed in [4], where it was demonstrated that the critical bond elongation $u_c$ can be related to the fracture energy $G_0$ for brittle materials. This approach has been used for modeling fracture and failure of composites, nanofiber networks and polycrystals [5], to simulate ballistic impact on brittle plates [6], to study crack nucleation in peridynamic solids [7], and to study dynamic crack propagation and crack branching [8]. The main limitation of the bond-based peridynamic theory is that it only considers pairwise interactions between particles. As is well known, a direct consequence of this assumption is that the effective Poisson’s ratio for isotropic linear materials is fixed at the value of $\nu =0.25$ [1]. An immediate repercussion of this limitation is the inadequacy of the bond-based peridynamics formulation in situations involving incompressible deformations, e.g., plasticity.

To address these issues, Silling et al. developed the so called state-based peridynamics formulation [9], which makes it possible to incorporate general constitutive models. In particular, the new formulation introduces a constitutive correspondence framework which enables the use of traditional constitutive models formulated in terms of a continuum local measure of deformation (i.e., the deformation gradient tensor, F). Recently, this approach was used to model viscoplastic deformations in metals [10], [2]. The ability to incorporate classical constitutive models also opens the path for using classical continuum damage models within the peridynamics framework.

Existing state-based peridynamic damage modeling approaches in the literature are based on permanently modifying the peridynamic influence function by instantly setting it equal to zero and severing the bond when a failure criterion is achieved [10], [11]. Within the context of ordinary state-based peridynamics, the role of the influence function has been explored in [12], and a critical bond elongation criterion has been proposed in [11] which is similar to the damage modeling approach commonly used in bond-based peridynamics [3], [4], [5], [6], [7], [8]. A severing criterion based on a maximum elastic bond energy was proposed in [10] for the constitutive correspondence formulation and calibrated to dissipate a pre-specified fracture energy when a new surface is created. These approaches appear to be successful for modeling brittle fracture. However, there are situations (e.g. in ductile fracture), in which damage evolution and failure are known to depend on quantities such as the stress triaxiality, Lode angle, and possibly other parameters characterizing the local stress state [13], [14], [15]. It would therefore be desirable to be able to incorporate classical damage models whose primary objective is the description of damage mechanisms and their evolution in a physics-based or phenomenological manner.

A more general framework for modeling damage within peridynamics has recently been proposed [16]. The theory constitutes a thermodynamically consistent extension of state-based peridynamics where accumulated damage is represented by a damage-state. However, the requirement to introduce a peridynamic damage-state makes it impossible to use existing damage models directly, and the necessary reformulation within this framework has yet to be done for general damage models. Specifically, it is not clear how the Johnson–Cook damage model adopted in the examples in this paper should be modified to fit this general framework.

The main objective of this paper is to develop a state-based peridynamics formulation where classical (local) continuum damage models can be incorporated without modification. It is found that a direct implementation of damage models within the constitutive correspondence framework leads to instabilities associated with unphysical diffusion of the damage zone. To address this issue, we employ a peridynamic bond degradation criterion based on the accumulated material damage. As damage evolves at a material point, the peridynamic influence function for bonds in the neighborhood is decreased so that in the limit of full damage its interaction with other material points vanishes. This can be viewed as an extension of previous bond-severing criteria in state-based peridynamics to more general cases where the influence function is allowed to degrade gradually and to have a general dependence on other state variables such as plastic strain, void volume fraction, temperature, etc., thereby enabling the description of the damage evolution process. To assess the method, we consider the specific case of the Johnson–Cook plasticity and damage model [17].

In Section 2, we present the state-based peridynamic damage formulation and discuss the implementation of the method for the specific damage model considered. In Section 3, we present numerical results consisting of: (1) the simulation of a Taylor impact test which is used for verification against other numerical methods; and (2) the simulation of ballistic impact of extruded aluminum sandwich panels by steel spheres. A summary and discussion with suggestions for future research is presented in Section 4.