Bond-Based Peridynamics Does Not Converge to Hyperelasticity as the Horizon Goes to Zero
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Bond-Based Peridynamics Does Not Converge to Hyperelasticity as the Horizon Goes to Zero J.C. Bellido1
· J. Cueto1 · C. Mora-Corral2
Received: 9 January 2020 © Springer Nature B.V. 2020
Abstract Bond-based peridynamics is a nonlocal continuum model in Solid Mechanics in which the energy of a deformation is calculated through a double integral involving pairs of points in the reference and deformed configurations. It is known how to calculate the limit of this model when the horizon (maximum interaction distance between the particles) tends to zero, and the limit turns out to be a (local) vector variational problem defined in a Sobolev space, of the type appearing in (classical) hyperelasticity. In this paper we impose frame-indifference and isotropy in the model and find that very few hyperelastic functionals are -limits of the bond-based peridynamics model. In particular, Mooney-Rivlin materials are not recoverable through this limit procedure. Mathematics Subject Classification 35Q74 · 49J45 · 74B20 · 74G65 Keywords Bond-based peridynamics · Hiperelasticity · Gamma-convergence as the horizon goes to zero
1 Introduction Peridynamics is a nonlocal continuum model for Solid Mechanics proposed by Silling in [30]. The main difference with classical elasticity ([10]) relies on the nonlocality, reflected in the fact that points separated by a positive distance exert a force upon each other. In this model the use of gradients is avoided by computing internal forces by integration instead of differentiation, and consequently the elastic energy is the result of a double integration. A main feature is that deformations are not assumed to be smooth, in contrast with classical continuum mechanics, where they are required to be, at least, weakly differentiable as functions of a Sobolev space. This makes peridynamics a suitable framework for problems
B J.C. Bellido 1
Departamento de Matemáticas, ETSI Industriales and INEI, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain
2
Departamento de Matemáticas, Universidad Autónoma de Madrid, Facultad de Ciencias, 28049 Madrid, Spain
J.C. Bellido et al.
where discontinuities appear naturally, such as fracture, dislocation, or, in general, multiscale materials. Since the pioneering paper [30], the development of peridynamics has been really overwhelming, both from a theoretical and a numerical-practical point of view. Some references on this are [17, 18, 33, 34] and the two books [16, 19]. The original model proposed in [30] was the so-called bond-based model, in which the elastic energy is given by a double integral depending on pairs of points in the reference and deformed configurations. It is assumed the existence of a function w, named as pairwise potential function, such that the total (macroelastic) energy of any deformation u : → Rm of the deformable solid ⊂ Rn is given by ˆ ˆ w(x − x , u(x) − u(x )) dx dx , (1)
∩B(x,δ)
where δ > 0 is the horizon, a model parameter which measures the maximum interaction distance between the particles. Physica
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