A non-ordinary state-based peridynamic method to model solid material deformation and fracture

By T. L. Warren, Stewart A. Silling, Abe Askari, Olaf Weckner, Michael A. Epton, Jifeng Xu. 2009

388 citations recorded by the source corpus.

Abstract In this paper, we develop a new non-ordinary state-based peridynamic method to solve transient dynamic solid mechanics problems. This new peridynamic method has advantages over the previously developed bond-based and ordinary state-based peridynamic methods in that its bonds are not restricted to central forces, nor is it restricted to a Poisson’s ratio of 1/4 as with the bond-based method. First, we obtain non-local nodal deformation gradients that are used to define nodal strain tensors. The deformation gradient tensors are used with the nodal strain tensors to obtain rate of deformation tensors in the deformed configuration. The polar decomposition of the deformation gradient tensors are then used to obtain the nodal rotation tensors which are used to rotate the rate of deformation tensors and previous Cauchy stress tensors into an unrotated configuration. These are then used with conventional Cauchy stress constitutive models in the unrotated state where the unrotated Cauchy stress rate is objective. We then obtain the unrotated Cauchy nodal stress tensors and rotate them back into the deformed configuration where they are used to define the forces in the nodal connecting bonds. As a first example we quasi-statically stretch a bar, hold it, and then rotate it ninety degrees to illustrate the methods finite rotation capabilities. Next, we verify our new method by comparing small strain results from a bar fixed at one end and subjected to an initial velocity gradient with results obtained from the corresponding one-dimensional small strain analytical solution. As a last example, we show the fracture capabilities of the method using both a notched and un-notched bar.

Subfields: Civil and Structural Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/f8e0440154e55a422b1f2257ec22a29c39caf62e

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In Civil and Structural Engineering

In Mechanics of Materials

Citations in the corpus

A morphing approach to couple state-based peridynamics with classical continuum mechanics (2016, methodology): “The SPD model has been employed to study plasticity [7, 26], damage and fracture [27, 28] of materials.”
An improved peridynamic approach for quasi-static elastic deformation and brittle fracture analysis (2015, methodology): “To eliminate the limitations including the limited Poisson's ratio, a reformulation of the bond-based PD relation, called the statebased peridynamic model, was proposed by Silling and coworkers [29–31], and it has been employed to describe the mechanical and fracture behavior of more general materia”
Viscoplasticity using peridynamics (2010, neutral): “[7] have recently implemented the linear elastic and elastic–plastic linear hardening models into Emu and showed convergence to analytic solutions of bending in a peridynamic bar.”
Dual‐horizon peridynamics (2015, methodology): “With the help of the shape tensor proposed in the NOSB-PD, the fundamental tensors in continuum mechanics can be easily introduced in peridynamics, including the deformation gradient tensor or velocity gradient [38, 39].”
Meshfree Methods: Progress Made after 20 Years (2017, neutral): “Recently, plasticity, viscoplasticity, and continuum damage mechanics have been incorporated in this context (Foster et al. 2010; Tupek et al. 2013; Warren et al. 2009).”
Non-ordinary state-based peridynamic analysis of stationary crack problems (2014, methodology): “Discretization and numerical implementation In a discrete system, the nonlocal deformation gradient, FðxjÞ, at a particle j can be expressed using Riemann sums over the total number of particles, m, within the horizon, H, as [20] FðxjÞ 1⁄4 Xm”
A non-ordinary state-based peridynamics formulation for thermoplastic fracture (2016, methodology): “[22] for incorporating the elasto-plastic linear hardening constitutive model.”
An extended constitutive correspondence formulation of peridynamics based on nonlinear bond-strain measures (2014, background): “Non-ordinary formulations allow for far more general constitutive responses [12].”