Peridynamic Functionally Graded and Porous Materials: Modeling Fracture and Damage

Abstract

In this chapter, we present two peridynamic models for composite materials: a locally homogenized model (FH-PD model, based on results reported in Cheng et al. (Compos Struct 133: 529–546, 2015)) and an intermediately homogenized model (IH-PD model). We use these models to simulate fracture in functionally graded materials (FGMs) and in porous elastic materials. We analyze dynamic fracture, by eccentric impact, of a functionally graded plate with monotonically varying volume fraction of reinforcements. We study the influence of material gradients, elastic waves, and of contact time and magnitude of impact loading on the crack growth from a pre-notch in terms of crack path geometry and crack propagation speed. The results from FH-PD and IH-PD models show the same cracking behavior and final crack patterns. The simulations agree very well, through full failure, with experiments. We discuss advantages offered by the peridynamic models in dynamic fracture of FGMs compared with, for example, FEM-based models. The models lead to a better understanding of how cracks propagate in FGMs and of the factors that control crack path and its velocity in these materials. The IH-PD model has important advantages when compared with the FH-PD model when applied to composite materials with phases of disparate mechanical properties. An application to fracture of porous and elastic materials (following Chen et al. (Peridynamic model for damage and fracture in porous materials, 2017)) shows the major effect local heterogeneities have on fracture behavior and the importance of intermediate homogenization as a modeling approach of crack initiation and growth.

Appendix A

Numerical Discretization

The peridynamic equations can be discretized using the finite element method, or any other method appropriate to compute solutions to integro-differential equations (or integration equations in static model). This approach, however, soon hits well-known obstacles and difficulties for problems with evolving topologies, like those in dynamic fracture and fragmentation. Instead, meshfree-type discretizations are preferred in peridynamics simulations of dynamic failure of materials. The discretization proposed in Silling and Askari (2005) uses the midpoint integration scheme (equivalent to a one-point Gaussian integration) for the domain integral. Numerical simulations are performed using the following discretized equation:

$$ \int_{H_{\boldsymbol{x}}}\boldsymbol{f}\left(\hat{\boldsymbol{x}}-\boldsymbol{x},\boldsymbol{u}\left(\hat{\boldsymbol{x}},t\right)-\boldsymbol{u}\left(\boldsymbol{x},t\right)\right)\mathrm{d}{V}_{\hat{\boldsymbol{x}}}\simeq \sum_{j\in Fam(i)}c\left({\xi}_{ij}\right){s}_{ij}{V}_{ij} $$

where Fam(i) is the family of nodes j with their area (volume in 3D) covered, fully or partially, by the horizon region of nodes i, ξ_ij is the bond length between nodes i and j, s_ij is the relative elongation for the same bond, and V_ij is the area of node j estimated to be covered by the horizon of node i.

Note that node j may not be fully contained within the horizon of node I, so a partial volume integration, first introduced in Hu et al. (2010) and also shown in Zhang and Bobaru (2015), is used here to improve the accuracy of midpoint quadrature scheme. The main advantage of this algorithm compared with one that simply checks whether a node is inside or outside the horizon region is that as the grid density increases (for a fixed horizon value), the numerical convergence (in terms of strain energy density, for example) is monotonic (see Hu et al. 2010).

Both dynamic (see section “Numerical Studies for Dynamic Crack Propagation in FGMs”) and static (see section “Quasi-Static Fracture in Brittle Porous Elastic Materials”) simulations are performed in this work. In the dynamic fracture simulations of the FGM plate (section “Numerical Studies for Dynamic Crack Propagation in FGMs”), we apply Velocity-Verlet method with a time interval of 0.05 μs. For the quasi-static fracture tests in section “Quasi-Static Fracture in Brittle Porous Elastic Materials,” the energy minimization method (see Shewchuk 1994; Zhang et al. 2016; Le and Bobaru 2017) is used, and the conjugate gradient (CG) method with secant line search is adopted to minimize the strain energy of the system. For all static simulations in this chapter, the CG method is used with a convergence tolerance defined by: $\frac{\left|{W}_i-{W}_{i-1}\right|}{W_{i-1}}<{10}^{-6}$, in which W_i and W_i − 1 are the total strain energy at current (i) and previous (i−1) CG iterations.

m-convergence and δ-convergence: For a fixed horizon, the ratio m = δ/Δx describes how accurate the numerical quadrature for the integral in Eq. 1 will be. We call this ratio “the horizon factor.” We recall that in the m-convergence we consider the horizon δ fixed and take m → ∞. The numerical PD approximation will converge to the exact nonlocal PD solution for the given δ. In the case of δ-convergence, the horizon δ → 0 while m is fixed or increases with decreasing δ. For δ-convergence and in problems with no singularities, the numerical PD solutions are expected to converge to the classical local solution.