Nonlocal elastodynamics and fracture

Fracture can be viewed as a collective interaction across large and small length scales. With the application of enough stress or strain to a brittle material, atomistic scale bonds will break, leading to fracture of the macroscopic specimen. From a modeling perspective fracture should appear as an emergent phenomena generated by an underlying field theory eliminating the need for a supplemental kinetic relation describing crack growth. The displacement field inside the body for points $x$ at time $t$ is written $u(x,t)$. The perydynamic approach (Silling 2000), allows for this and is described by the nonlocal balance of linear momentum of the form \begin{equation}\label{eqn-uf} \begin{aligned} \rho{{u}_{tt}}(x,t) = \int_{{\mathcal H}_{\epsilon}({x})} {\mathbf{f}}(y,x)\;dy + \mathbf{b}(x,t) \end{aligned} \end{equation} where $\mathcal{H}_{\epsilon}({x})$ is a neighborhood of $x$, $\rho$ is the density, $\mathbf{b}$ is the body force density field, and $\mathbf{f}$ is a material-dependent constitutive law that represents the force density that a point $y$ inside the neighborhood exerts on $x$ as a result of the deformation field. The radius $\epsilon$ of the neighborhood is referred to as the horizon. Here all points satisfy the same field equation (1). The displacement fields and fracture evolution predicted by the nonlocal model should agree with the dynamic fracture of specimens when the length scale of non-locality is sufficiently small. In this respect numerical simulations are compelling, see for example (Bobaru and Zhang 2015), (Silling and Askari 2005), and (Trask et al. 2019).

The displacement for the nonlocal theory is examined in the limit of vanishing non-locality. This is done for a class of peridynamic models with nonlocal forces derived from double well potentials see, (Lipton, 2016). The term double well describes the force potential between two points. One of the wells is degenerate and appears at infinity while the other is at zero strain. For small strains the nonlocal force is linearly elastic but for larger strains the force begins to soften and then approaches zero after reaching a critical strain. This type of nonlocal model is called a cohesive model.

We theoretically investigate the limit of the displacements for the cohesive model as the length scale $\epsilon$ of nonlocal interaction goes to zero. All information on this limit is obtained from what is known from the nonlocal model for $\epsilon>0$. In this paper the single edge notch specimen is considered as given in Fig. 1 and the target theory governing the evolution of displacement fields is identified when $\epsilon=0$.

Fig. 1 Precracked symmetric specimen pulled from top and bottom.

One of the hallmarks of peridynamic simulations is localization of defect sets with horizon as $\epsilon\rightarrow 0$. Theoretically localization of the jump set of the displacement is established as $\epsilon\rightarrow 0$ in (Lipton, 2014), (Lipton, 2016) where the limiting displacement is shown to be an $SBD^2(D)$ valued function for almost all times $t\in [0,T]$. The nonlocal cohesive model converges to a dynamic model having bounded Griffith fracture energy associated with brittle fracture and elastic displacement fields satisfying the elastic wave equation (Lipton, 2014), (Lipton, 2016) away from the fractures. This can be seen for arbitrarily shaped specimens with smooth boundary in two and three dimensions. However the explicit traction law relating the crack boundary to the elastic field lies out side the scope of that analysis.

This project (Lipton and Jha 2021) builds on earlier work and provides a global description of the limit dynamics describing elastic fields surrounding a crack for the single edge notch pulled apart by traction forces on its top and bottom edges. The objective of this paper is to show that the elastic fields seen in the nonlocal model are consistent with those in the local model in the limit of vanishing horizon. The analysis given here shows that it is possible to recover the boundary value problem for the linear elastic displacement given by Linear Elastic Fracture Mechanics inside a cracking body as the limit of a nonlocal fracture model. To illustrate this a family of initial value problems given in the nonlocal formulation is prescribed. The family is parameterized by horizon size $\epsilon$. The crack motion for $\epsilon>0$ is prescribed by the solutions of the nonlocal initial value problem. It is shown that up to subsequences that as $\epsilon\rightarrow 0$ the displacements associated with the solution of the nonlocal model converge in mean square uniformly in time to the limit displacement $u^0(x,t)$. It is proved in (Lipton and Jha 2021) that the limit displacement field $u^0(x,t)$ satisfies:

• Prescribed inhomogeneous traction boundary conditions.
• Balance of linear momentum as described by the linear elastic wave equation off the crack.
• Zero traction on the sides of the evolving crack.
• The set on which the elastic displacement jumps is a subset of the crack set.
• The limiting crack motion is determined by the sequence of nonlocal problems for $\epsilon>0$ and is obtained in the $\epsilon=0$ limit.