Florin Bobaru

Peridynamics for multiscale materials modeling

The paper presents an overview of peridynamics, a continuum theory that employs a nonlocal model of force interaction. Specifically, the stress/strain relationship of classical elasticity is replaced by an integral operator that sums internal forces separated by a finite distance. This integral operator is not a function of the deformation gradient, allowing for a more general notion of deformation than in classical elasticity that is well aligned with the kinematic assumptions of molecular dynamics. Peridynamics effectiveness has been demonstrated in several applications, including fracture and failure of composites, nanofiber networks, and polycrystal fracture. These suggest that peridynamics is a viable multiscale material model for length scales ranging from molecular dynamics to those of classical elasticity.

A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities

We introduce a multidimensional peridynamic formulation for transient heat-transfer. The model does not contain spatial derivatives and uses instead an integral over a region around a material point. By construction, the formulation converges to the classical heat transfer equations in the limit of the horizon (the nonlocal region around a point) going to zero. The new model, however, is suitable for modeling, for example, heat flow in bodies with evolving discontinuities such as growing insulated cracks. We introduce the peridynamic heat flux which exists even at sharp corners or when the isotherms are not smooth surfaces. The peridynamic heat flux coincides with the classical one in simple cases and, in general, it converges to it in the limit of the peridynamic horizon going to zero. We solve test problems and compare results with analytical solutions of the classical model or with other numerical solutions. Convergence to the classical solutions is seen in the limit of the horizon going to zero. We then solve the problem of transient heat flow in a plate in which insulated cracks grow and intersect thus changing the heat flow patterns. We also model heat transfer in a fiber-reinforced composite and observe transient but steep thermal gradients at the interfaces between the highly conductive fibers and the low conductivity matrix. Such thermal gradients can lead to delamination cracks in composites from thermal fatigue. The formulation may be used to, for example, evaluate effective thermal conductivities in bodies with an evolving distribution of insulating or permeable, possibly intersecting, cracks of arbitrary shapes.

Shape sensitivity analysis and shape optimization in planar elasticity using the element-free Galerkin method

Abstract This paper demonstrates that the element-free Galerkin (EFG) method can be successfully used in shape design sensitivity analysis and shape optimization for problems in 2D elasticity. The continuum-based variational equations for displacement sensitivities are derived and are subsequently discretized. This approach allows one to avoid differentiating the EFG shape functions. The present formulation, that employs a penalty method for imposing the essential boundary conditions, can be easily extended to 3D and/or non-linear problems. Numerical examples are presented to show the capabilities of the current approach for calculating sensitivities. The flexibility of the EFG method, that eliminates the element connectivity requirement of the finite element method (FEM), permits solving shape optimization problems without re-meshing. The problem of shape optimization of a fillet is used to demonstrate this fact. Smoother stresses and better accuracy for points close to the boundary allow for a better EFG solution compared to published results using the FEM, the boundary element method (BEM) or the boundary contour method (BCM). Furthermore, for the EFG approach, grid-optimization appears unnecessary.

Why do cracks branch? A peridynamic investigation of dynamic brittle fracture

In this paper we review the peridynamic model for brittle fracture and use it to investigate crack branching in brittle homogeneous and isotropic materials. The peridynamic simulations offer a possible explanation for the generation of dynamic instabilities in dynamic brittle crack growth and crack branching. We focus on two systems, glass and homalite, often used in crack branching experiments. After a brief review of theoretical and computational models on crack branching, we discuss the peridynamic model for dynamic fracture in linear elastic–brittle materials. Three loading types are used to investigate the role of stress waves interactions on crack propagation and branching. We analyze the influence of sample geometry on branching. Simulation results are compared with experimental ones in terms of crack patterns, propagation speed at branching and branching angles. The peridynamic results indicate that as stress intensity around the crack tip increases, stress waves pile-up against the material directly in front of the crack tip that moves against the advancing crack; this process “deflects” the strain energy away from the symmetry line and into the crack surfaces creating damage away from the crack line. This damage “migration”, seen as roughness on the crack surface in experiments, modifies, in turn, the strain energy landscape around the crack tip and leads to preferential crack growth directions that branch from the original crack line. We argue that nonlocality of damage growth is one key feature in modeling of the crack branching phenomenon in brittle fracture. The results show that, at least to first order, no ingredients beyond linear elasticity and a capable damage model are necessary to explain/predict crack branching in brittle homogeneous and isotropic materials.

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.

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 multi-layered glass systems, further complicate this problem. In this paper we use a simplified peridynamic model of a multi-layer 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.

Selecting the kernel in a peridynamic formulation: A study for transient heat diffusion

Abstract The kernel in a peridynamic diffusion model represents the detailed interaction between points inside the nonlocal region around each material point. Several versions of the kernel function have been proposed. Although solutions associated with different kernels may all converge, under the appropriate discretization scheme, to the classical model when the horizon goes to zero, their convergence behavior varies. In this paper, we focus on the particular one-point Gauss quadrature method of spatial discretization of the peridynamic diffusion model and study the convergence properties of different kernels with respect to convergence to the classical, local, model for transient heat transfer equation in 1D, where exact representation of geometry is available. The one-point Gauss quadrature is the preferred method for discretizing peridynamic models because it leads to a meshfree model, well suited for problems with damage and fracture. We show the equivalency of two definitions for the peridynamic heat flux. We explain an apparent paradox and discuss a common pitfall in numerical approximations of nonlocal models and their convergence to local models. We also analyze the influence of two ways of imposing boundary conditions and that of the “skin effect” on the solution. We explain an interesting behavior of the peridynamic solutions for different horizon sizes, the crossing of m -convergence curves at the classical solution value that happens for one of the ways of implementing the classical boundary conditions. The results presented here provide practical guidance in selecting the appropriate peridynamic kernel that makes the one-point Gauss quadrature an “asymptotically compatible” scheme. These results are directly applicable to any diffusion-type model, including mass diffusion problems.

Peridynamic 3D models of nanofiber networks and carbon nanotube‐reinforced composites

Here we employ a reformulation of the continuum mechanics theory, the peridynamic formulation (PF) in an integral form that, at the discretized level, resembles molecular dynamics (MD). The peridynamic theory is based on a continuum formulation and can capture nucleation and propagation of defects and discontinuities without ad‐hoc assumptions or special treatments needed by classical continuum theory. We analyze nanofiber networks and CNT‐reinforced polymer composites. We treat all crossovers contacts between fibers as perfect bonds. The use of repulsive short‐range forces eliminates the need for complex contact detection algorithms. We generate the fibers as 3D curves with random orientation, with or without preferred directionality. We use an object‐oriented code written in Fortran 90/95 to define the geometrical entities. The PF can capture the deformation and complex fracture behavior in fully 3D dynamic simulations. van der Waals forces are included in these calculations. The strength of the bonds b...

Designing Optimal Volume Fractions For Functionally Graded Materials With Temperature-Dependent Material Properties

We present a numerical approach for material optimization of metal-ceramic functionally graded materials (FGMs) with temperature-dependent material properties. We solve the non-linear heterogeneous thermoelasticity equations in 2D under plane strain conditions and consider examples in which the material composition varies along the radial direction of a hollow cylinder under thermomechanical loading. A space of shape-preserving splines is used to search for the optimal volume fraction function which minimizes stresses or minimizes mass under stress constraints. The control points (design variables) that define the volume fraction spline function are independent of the grid used in the numerical solution of the thermoelastic problem. We introduce new temperature-dependent objective functions and constraints. The rule of mixture and the modified Mori-Tanaka with the fuzzy inference scheme are used to compute effective properties for the material mixtures. The different micromechanics models lead to optimal solutions that are similar qualitatively. To compute the temperature-dependent critical stresses for the mixture, we use, for lack of experimental data, the rule-of-mixture. When a scalar stress measure is minimized, we obtain optimal volume fraction functions that feature multiple graded regions alternating with non-graded layers, or even non-monotonic profiles. The dominant factor for the existence of such local minimizers is the non-linear dependence of the critical stresses of the ceramic component on temperature. These results show that, in certain cases, using power-law type functions to represent the material gradation in FGMs is too restrictive.

Granular layers on vibrating plates: Effective bending stiffness and particle-size effects

Acoustic methods of land mine detection rely on the vibrations of the top plate of the mine in response to sound. For granular soil (e.g., sand), the particle size is expected to influence the mine response. This hypothesis is studied experimentally using a plate loaded with dry sand of various sizes from hundreds of microns to a few millimeters. For low values of sand mass, the plate resonance decreases with added mass and eventually reaches a minimum without particle size dependence. After the minimum, a frequency increase is observed with additional mass that includes a particle-size effect. Analytical nondissipative continuum models for granular media capture the observed particle-size dependence qualitatively but not quantitatively. In addition, a continuum-based finite element model (FEM) of a two-layer plate is used, with the sand layer replaced by an equivalent elastic layer for evaluation of the effective properties of the layer. Given a thickness of sand layer and corresponding experimental resonance, an inverse FEM problem is solved iteratively to give the effective Youngs modulus and bending stiffness that matches the experimental frequency. It is shown that a continuum elastic model must employ a thickness-dependent elastic modulus in order to match experimental values.