An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures
Introduction
Peridynamics is a nonlocal continuum theory of solid mechanics based on integral equations, originally proposed to address elasticity problems involving discontinuities and long-range forces (Silling, 2000). A key feature of peridynamics is the ability to naturally describe discontinuities such as cracks (Silling and Bobaru, 2004, Silling and Askari, 2005, Askari et al., 2008, Parks et al., 2008, Silling et al., 2010, Ha and Bobaru, 2010) and phase transition boundaries (Dayal and Bhattacharya, 2006) within a single continuum framework. This is in contrast to classical theories of continua which require additional constraints across surfaces of discontinuity. Peridynamics has been used in a variety of problems in continuum mechanics including membrane and fiber models (Silling and Bobaru, 2004), phase transitions (Dayal and Bhattacharya, 2006), inter-granular fracture (Askari et al., 2008), meso-scale modeling of material response (Askari et al., 2008), and heat transfer (Bobaru and Duangpanya, 2010).
The most promising peridynamic theory appears to be the so-called state-based peridynamic formulation (Silling et al., 2007, Silling and Lehoucq, 2010), which makes it possible to incorporate very general material models. Within the state-based formulation, a distinction is made between ordinary and non-ordinary peridynamics constitutive models, where ordinary materials are defined by the restriction that bond forces always act in a direction parallel to the bond. Non-ordinary formulations allow for far more general constitutive responses (Warren et al., 2009). In particular, the constitutive correspondence formulation (Silling et al., 2007, Foster et al., 2010) is a subset of non-ordinary formulations aimed at incorporating classical nonlinear constitutive models using the nonlocal approximation to the deformation gradient tensor . For example, state-based formulations of viscoplasticity models (Foster et al., 2010) and ductile damage models for metals (Foster, 2009, Tupek et al., 2013) have recently been proposed and effectively used in simulations based on particle discretizations. In principle, general constitutive laws can be formulated based on nonlocal versions of classical nonlinear tensor strain measures directly computed from this nonlocal . However, it will be shown below that the nonlocal deformation gradient allows for modes of deformation which are physically impossible and yet undetectable by the theory irrespective of the chosen strain measure.
The issue of unphysical deformation modes leading to matter interpenetration has been observed in numerical simulations of peridynamics based on particle discretizations (Silling and Askari, 2005, Parks et al., 2008). In those references, matter interpenetration was effectively handled by the use of so-called short-range forces. From a theoretical point of view, short-range forces in peridynamics are justified in situations involving true short-range interactions (e.g. to describe contact mechanics). However, it can be argued that the addition of short range forces to avoid matter interpenetration constitutes a numerical artifact not easily justified in the theory of peridynamics, which is an essentially nonlocal theory predicated on the fundamental role of long-range interactions. In addition, it can be presumed that effective elimination of matter interpenetration in numerical simulations requires ad hoc tuning of the short-range force intensity to the specific problem at hand. It is therefore desirable to find a solution to the problem of matter interpenetration that is rooted directly within the nonlocal theory. It is this specific issue that this paper attempts to address.
Toward this end, we propose an extended constitutive correspondence formulation of peridynamics which generalizes the constitutive framework to models expressed in terms of generalized strain tensors and their work-conjugate stresses. More specifically, we introduce nonlinear measures of bond elongation which are inherently singular when the matter interpenetration constraint is violated and, thus, avoid unphysical deformations. The extended constitutive correspondence formulation is then expressed in terms of generalized nonlocal Seth–Hill strain tensors (Seth, 1964, Hill, 1968) which are based on the bond-level strain measures and which are shown to be exact in the uniform infinitesimal limit. Importantly, it is also shown that the extended constitutive correspondence framework is ordinary and supports general inelastic and anisotropic materials models.
We start by reviewing the state-based formulation of peridynamics, and the constitutive frameworks for ordinary peridynamic materials and constitutive correspondence in Section 2. In Section 3 we show by way of example the kinematic deficiency present in the original correspondence formulation. Section 4 is devoted to the new peridynamic bond-strain measures, the corresponding family of nonlocal strain tensors and their properties, including a demonstration that the new formulation fixes the violation of the matter interpenetration constraint in cases where the original version fails. In Section 5, the extended constitutive correspondence formulation is then stated and shown to be ordinary. We conclude the presentation with a brief summary in Section 6.
Section snippets
Review of state-based peridynamics
For completeness, we briefly review the state-based peridynamics formulation, primarily following the notation and approach in Silling et al. (2007) and Silling and Lehoucq (2010). State-based peridynamics is a nonlocal continuum theory which describes the dynamics of a continuum body. Assume the body occupies the region in the reference configuration at time t=0 and the region at time t.
Consider material points in the reference configuration . From the perspective of point
Limitation in the kinematics of constitutive correspondence
In this section we investigate a basic fundamental limitation of the kinematic assumption of the constitutive correspondence formulation. Specifically, we demonstrate using several examples how unphysical deformation modes may be undetectable by the nonlocal deformation gradient .
- 1.
Sub-horizon material collapse: Consider the extreme situation depicted in Fig. 2 where a small volume of material collapses to a single point . The peridynamic deformation vector-state in this case is
Peridynamic bond-strain measures
In the previous section, we concluded that in the existing constitutive correspondence framework unphysical deformations may result in a kinematically admissible . In other words, violating the kinematic constraint (2) on the bond-level does not imply that the resulting nonlocal deformation gradient violates the tensor-level kinematic constraint (1), i.e. The source of this limitation is that averages the material deformation over the horizon in such a
Constitutive models based on the nonlocal strain tensor
A straightforward approach for formulating constitutive models in terms of the proposed strain measures is based on the peridynamic correspondence concept: the classical strain energy density function as originally formulated in terms of a classical strain tensor must be evaluated using the corresponding nonlocal strain tensor , i.e. . The work-conjugate stress measure then follows as Finally, the peridynamic force vector-state
Conclusion
The ability to naturally handle field discontinuities has previously been put forward as a key advantage of the peridynamic theory over classical continuum theories (Silling et al., 2007, Silling and Lehoucq, 2010). However, without careful consideration this flexibility may go too far and result in peridynamic formulations in which highly unphysical deformation modes (i.e. matter interpenetration) are allowed. Although in previous numerical simulations these issues have been effectively
Acknowledgments
This work was supported by the U.S. Army through the Institute for Soldier Nanotechnologies under Contract DAAD-19-02-D-0002 and by the Office of Naval Research under Grant N00014-07-1-0764.
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