ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

Damage progression from impact in layered glass modeled with peridynamics

Dynamic fracture in brittle materials has been difficult to model and predict. Interfaces, such as those present in multilayered glass systems, further complicate this problem. In this paper we use a simplified peridynamic model of a multilayer glass system to simulate damage evolution under impact with a high-velocity projectile. The simulation results are compared with results from recently published experiments. Many of the damage morphologies reported in the experiments are captured by the peridynamic results. Some finer details seen in experiments and not replicated by the computational model due to limitations in available computational resources that limited the spatial resolution of the model, and to the simple contact conditions between the layers instead of the polyurethane bonding used in the experiments. The peridynamic model uncovers a fascinating time-evolution of damage and the dynamic interaction between the stress waves, propagating cracks, interfaces, and bending deformations, in three-dimensions. 5 movies (.avi files) are attached below; a zipped (.zip) file of all is 5 is also attached

Peridynamic model for dynamic fracture in unidirectional fiber-reinforced composites

We propose a computational method for a homogenized peridynamics description of fiber-reinforced composites and we use it to simulate dynamic brittle fracture and damage in these materials. With this model we analyze the dynamic effects induced by different types of dynamic loading on the fracture and damage behavior of unidirectional fiber-reinforced composites. In contrast to the results expected from quasi-static loading, the simulations show that dynamic conditions can lead to co-existence of and transitions between fracture modes; matrix shattering can happen before a splitting crack propagates. We observe matrix–fiber splitting fracture, matrix cracking, and crack migration in the matrix, including crack branching in the matrix similar to what is observed in recent dynamic experiments. The new model works for arbitrary fiber orientation relative to a uniform discretization grid and also works with random discretizations. The peridynamic composite model captures significant differences in the crack propagation behavior when dynamic loadings of different intensities are applied. An interesting result is branching of a splitting crack into two matrix cracks in transversely loaded samples. These cracks branch as in an isotropic material but here they migrate over the “fiber bonds” without breaking them. This behavior has been observed in recent experiments. The strong influence that elastic waves have on the matrix damage and crack propagation paths is discussed. No special criteria for splitting mode fracture (Mode II), crack curving, or crack arrest are needed, and yet we obtain all these modes of material failure as a direct result of the peridynamic simulations. 10 supplementary video files are attached (below)

MODELING DYNAMIC FRACTURE AND DAMAGE IN A FIBER-REINFORCED COMPOSITE LAMINA WITH PERIDYNAMICS

We propose a peridynamic formulation for a unidirectional fiber-reinforced composite lamina based on homogenization and mapping between elastic and fracture parameters of the micro-scale peridynamic bonds and the macro-scale parameters of the composite. The model is then used to analyze the splitting mode (mode II) fracture in dynamic loading of a 0° lamina. Appropriate scaling factors are used in the model in order to have the elastic strain energy, for a fixed nonlocal interaction distance (the peridynamic horizon), match the classical one. No special criteria for splitting failure are required to capture this fracture mode in the lamina. Convergence studies under uniform grid refinement for a fixed horizon size (m-convergence) and under decreasing the peridynamic horizon (_-convergence) are performed. The computational results show that the splitting fracture mode obtained with peridynamics compares well with experimental observations. Moreover, in the limit of the horizon going to zero, the maximum crack propagation speed computed with peridynamics approaches the value obtained from an analytical classical formulation for the steady-state dynamic interface debonding found in the literature

Characteristics of dynamic brittle fracture captured with peridynamics

Using a bond-based peridynamic model, we are able to reproduce various characteristics of dynamic brittle fracture observed in experiments; crack branching, crack-path instability, asymmetries of crack paths, successive branching, secondary cracking at right angles from existing crack surfaces, etc. We analyze the source of asymmetry in the crack path in numerical simulations with an isotropic material and symmetric coordinates about the pre-crack line. Asymmetries in the order of terms in computing the nodal forces lead to different round-off errors for symmetric nodes about the pre-crack line. This induces the observed slight asymmetries in the branched crack paths. A dramatically enhanced crack-path instability and asymmetry of the branching pattern are obtained when we use fracture energy values that change with the local damage. The peridynamic model used here captures well the experimentally observed successive branching events and secondary cracking. Secondary cracks form as a direct consequence of wave propagation and reflection from the boundaries

The formulation and computation of the nonlocal J-integral in bond-based peridynamics

This work presents a rigorous derivation for the formulation of the J-integral in bond-based peridynamics using the crack infinitesimal virtual extension approach. We give a detailed description of an algorithm for computing this nonlocal version of the J-integral.We present convergence studies (m-convergence and δ-convergence) for two different geometries: a single edge-notch configuration and a double edge-notch sample.We compare the results with results based on the classical J-integral and obtained from FEM calculations that employ special elements near the crack tip.We identify the size of the nonlocal region for which the peridynamic J-integral value is near the classical FEM solutions.We discuss how the boundary conditions and the peridynamic “skin effect” may influence the peridynamic J-integral value.We also observe, computationally, the path-independence of the peridynamic J-integral