Dynamic Brittle Fracture from Nonlocal Double-Well Potentials: A State-Based Model

  • Robert LiptonEmail author
  • Eyad Said
  • Prashant K. Jha
Reference work entry


We introduce a regularized model for free fracture propagation based on nonlocal potentials. We work within the small deformation setting, and the model is developed within a state-based peridynamic formulation. At each instant of the evolution, we identify the softening zone where strains lie above the strength of the material. We show that deformation discontinuities associated with flaws larger than the length scale of nonlocality δ can become unstable and grow. An explicit inequality is found that shows that the volume of the softening zone goes to zero linearly with the length scale of nonlocal interaction. This scaling is consistent with the notion that a softening zone of width proportional to δ converges to a sharp fracture set as the length scale of nonlocal interaction goes to zero. Here the softening zone is interpreted as a regularization of the crack network. Inside quiescent regions with no cracks or softening, the nonlocal operator converges to the local elastic operator at a rate proportional to the radius of nonlocal interaction. This model is designed to be calibrated to measured values of critical energy release rate, shear modulus, and bulk modulus of material samples. For this model one is not restricted to Poisson ratios of 1∕4 and can choose the potentials so that small strain behavior is specified by the isotropic elasticity tensor for any material with prescribed shear and Lamé moduli.


Free fracture model Nonlocal interactions Double-well potentials State-based peridynamics 



This material is based upon the work supported by the US Army Research Laboratory and the US Army Research Office under contract/grant number W911NF1610456.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and Center for Computation and TechnologyLouisiana State UniversityBaton RougeUSA
  2. 2.Department of MathematicsLouisiana State UniversityBaton RougeUSA

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