Peridynamic Material Models Research Articles

Peridynamic modeling of damage and non-shock ignition behavior of confined polymer bonded explosives under impact loading

A mechanical-thermal-chemical coupled peridynamic (PD) modeling for analyzing the damage and non-shock ignition behavior of polymer bonded explosives (PBXs) subject to impact loading has been presented. The friction and heat conduction and Arrenhenius law are embedded into the PD theory, ensuring that the simulation model is capable of capturing major features of dynamic damage response of PBX. A PD material model considering plasticity and the asymmetric behaviors of PBX under tensile and compressive loadings is proposed in this study. For a ring-shaped PBX 9501 sample confined by the steel structures, the simulation results indicate that the PBX 9501 in the regions towards the impact face and support face suffers from the serious damage. Meanwhile, the corresponding average temperatures in these two regions are also higher than their counterparts in other regions due to the more intense interaction between PBX 9501 and the steel confinement structures. The ignition time decreases with the increasing impact velocity within the range of 40–120 m s−1. The internal steel shell response is analyzed according to its velocity history within a representative box. It is found that the deformation process of internal steel shell is closely related to the complicated flow behaviors of PBX 9501.

Peridynamic material model calibration based on digital image correlation experimental measurements

The non-local theory of peridynamics, based on integral equation of motion, offers a computation alternative for materials with discontinuities. In this study, the peridynamic material model parameters are calibrated using full field Digital Image Correlation measurements applied to simple tensile test experiments, circumventing thus the need for more challenging test setups/methods for the determination of the peridynamic material parameters (especially the critical stretch failure parameter). A two stage algorithm in the optimization software LS-Opt together with a MATLAB stage for performing the peridynamic simulation and an Excel stage for defining the outputs of the simulation are implemented to minimize the differences between the peridynamic model and experimental results. The material model parameters determined based on the simple test setup and the calibration procedure proposed here can then be used for modeling more challenging geometries and load cases in peridynamics.

Peridynamic Model for Single-Layer Graphene Obtained from Coarse-Grained Bond Forces

An ordinary state-based peridynamic material model is proposed for single-sheet graphene. The model is calibrated using coarse-grained molecular dynamics simulations. The coarse-graining method allows the dependence of bond force on bond length to be determined, including the horizon. The peridynamic model allows the horizon to be rescaled, providing a multiscale capability and allowing for substantial reductions in computational cost compared with molecular dynamics. The calibrated peridynamic model is compared to experimental data on the deflection and perforation of a graphene sheet by an atomic force microscope probe.

A machine-learning framework for peridynamic material models with physical constraints

As a nonlocal extension of continuum mechanics, peridynamics has been widely and effectively applied in different fields where discontinuities in the field variables arise from an initially continuous body. An important component of the constitutive model in peridynamics is the influence function which weights the contribution of all the interactions over a nonlocal region surrounding a point of interest. Recent work has shown that in solid mechanics the influence function has a strong relationship with the heterogeneity of a material’s micro-structure. However, determining an accurate influence function analytically from a given micro-structure typically requires lengthy derivations and complex mathematical models. To avoid these complexities, the goal of this paper is to develop a data-driven regression algorithm to find the optimal bond-based peridynamic model to describe the macro-scale deformation of linear elastic medium with periodic heterogeneity. We generate macro-scale deformation training data by averaging over periodic micro-structure unit cells and add a physical energy constraint representing the homogenized elastic modulus of the micro-structure to the regression algorithm. We demonstrate this scheme for examples of one- and two-dimensional linear elastodynamics and show that the energy constraint improves the accuracy of the resulting peridynamic model.

Inelastic peridynamic model for molecular crystal particles

The peridynamic theory of solid mechanics is applied to modeling the deformation and fracture of micrometer-sized particles made of organic crystalline material. A new peridynamic material model is proposed to reproduce the elastic–plastic response, creep, and fracture that are observed in experiments. The model is implemented in a three-dimensional, meshless Lagrangian simulation code. In the small deformation, elastic regime, the model agrees well with classical Hertzian contact analysis for a sphere compressed between rigid plates. Under higher load, material and geometrical nonlinearity is predicted, leading to fracture. The material parameters for the energetic material CL-20 are evaluated from nanoindentation test data on the cyclic compression and failure of micrometer-sized grains.

Propagation of a Stress Pulse in a Heterogeneous Elastic Bar

The propagation of a wave pulse due to low-speed impact on a one-dimensional, heterogeneous bar is studied. Due to the dispersive character of the medium, the pulse attenuates as it propagates. This attenuation is studied over propagation distances that are much longer than the size of the microstructure. A homogenized peridynamic material model can be calibrated to reproduce the attenuation and spreading of the wave. The calibration consists of matching the dispersion curve for the heterogeneous material near the limit of long wavelengths. It is demonstrated that the peridynamic method reproduces the attenuation of wave pulses predicted by an exact microstructural model over large propagation distances.

Deriving Peridynamic Influence Functions for One-dimensional Elastic Materials with Periodic Microstructure

The influence function in peridynamic material models has a large effect on the dynamic behavior of elastic waves and in turn can greatly effect dynamic simulations of fracture propagation and material failure. Typically, the influence functions that are used in peridynamic models are selected for their numerical properties without regard to physical considerations. In this work, we present a method of deriving the peridynamic influence function for a one-dimensional initial/boundary-value problem in a material with periodic microstructure. Starting with the linear local elastodynamic equation of motion in the microscale, we first use polynomial anzatzes to approximate microstructural displacements and then derive the homogenized nonlocal dynamic equation of motion for the macroscopic displacements, which is easily reformulated as linear peridyamic equation with a discrete influence function. The shape and localization of the discrete influence function are completely determined by microstructural mechanical properties and length scales. By comparison with a highly resolved microstructural finite element model and the standard linear peridynamic model with a linearly decaying influence function, we demonstrate that the influence function derived from microstructural considerations is more accurate in predicting time-dependent displacements and wave dynamics.

Peridynamic Modelling of Fracture in Polycrystalline Ice

ABSTRACTIn this study, a peridynamic material model for a polycrystalline ice is utilised to investigate its fracture behaviour under dynamic loading condition. First, the material model was validated by considering a single grain, double grains and polycrystalline structure under tension loading condition. Peridynamic results are compared against finite element analysis results without allowing failure. After validating the material model, dynamic analysis of a polycrystalline ice material with two pre-existing cracks under tension loading is performed by considering weak and strong grain boundaries with respect to grain interiors. Numerical results show that the effect of microstructure is significant for weak grain boundaries. On the other hand, for strong grain boundaries, the effect of microstructure is insignificant. The evaluated results have demonstrated that peridynamics can be a very good alternative numerical tool for fracture analysis of polycrystalline ice material.

On the stability of the generalized, finite deformation correspondence model of peridynamics

A class of peridynamic material models known as constitutive correspondence models provide a bridge between classical continuum mechanics and peridynamics. These models are useful because they allow well-established local constitutive theories to be used within the nonlocal framework of peridynamics. A recent finite deformation correspondence theory (Foster and Xu, 2018) was developed and reported to improve stability properties of the original correspondence model (Silling et al., 2007). This paper presents a stability analysis that indicates the reported advantages of the new theory were overestimated. Homogeneous deformations are analyzed and shown to exibit unstable material behavior at the continuum level. Additionally, the effects of a particle discretization on the stability of the model are reported. Numerical examples demonstrate the large errors induced by the unstable behavior. Stabilization strategies and practical applications of the new finite deformation model are discussed.

A generalized, ordinary, finite deformation constitutive correspondence model for peridynamics

A finite deformation constitutive framework for peridynamics that uses generalized Kirchhoff stress measures as intermediate quantities in computing the peridynamic material point interactions is proposed. It is shown to be a generalization of a previous finite deformation constitutive correspondence theory and has additional useful properties. The new model is shown to be an ordinary material model and therefore conserves angular momentum without additional restrictions. Additionally, it does not suffer from well-known “surface effects” present in some peridynamic material models, has improved material stability properties, and allows for inclusion of elasto-plastic behavior in a straightforward manner.

Modeling shockwaves and impact phenomena with Eulerian peridynamics

Most previous development of the peridynamic theory has assumed a Lagrangian formulation, in which the material model refers to an undeformed reference configuration. In the present work, an Eulerian form of material modeling is developed, in which bond forces depend only on the positions of material points in the deformed configuration. The formulation is consistent with the thermodynamic form of the peridynamic model and is derivable from a suitable expression for the free energy of a material. It is shown that the resulting formulation of peridynamic material models can be used to simulate strong shock waves and fluid response in which very large deformations make the Lagrangian form unsuitable. The Eulerian capability is demonstrated in numerical simulations of ejecta from a wavy free surface on a metal subjected to strong shock wave loading. The Eulerian and Lagrangian contributions to bond force can be combined in a single material model, allowing strength and fracture under tensile or shear loading to be modeled consistently with high compressive stresses. This capability is demonstrated in numerical simulation of bird strike against an aircraft, in which both tensile fracture and high pressure response are important.

A position-aware linear solid constitutive model for peridynamics

A position-aware linear solid (PALS) peridynamic constitutive model is proposed for isotropic elastic solids. The PALS model addresses problems that arise, in ordinary peridynamic material models such as the linear peridynamic solid (LPS), due to incomplete neighborhoods near the surface of a body. We improved model behavior in the vicinity of free surfaces through the application of two influence functions that correspond, respectively, to the volumetric and deviatoric parts of the deformation. Furthermore, the model is position-aware in that the influence functions vary over the body and reflect the proximity of each material point to free surfaces. Demonstration calculations on simple benchmark problems show a sharp reduction in error relative to the LPS model.

Origin and effect of nonlocality in a composite

A simple demonstration of nonlocality in a heterogeneous material is presented. By analysis of the microscale deformation of a two-component layered medium, it is shown that nonlocal interactions necessarily appear in a homogenized model of the system. Explicit expressions for the nonlocal forces are determined. The way these nonlocal forces appear in various nonlocal elasticity theories is derived. The length scales that emerge involve the constituent material properties as well as their geometrical dimensions. A peridynamic material model for the smoothed displacement field is derived. It is demonstrated by comparison with experimental data that the incorporation of nonlocality in modeling dramatically improves the prediction of the stress concentration in an open hole tension test on a composite plate.

Peridynamic beams: A non-ordinary, state-based model

This paper develops a new peridynamic state based model to represent the bending of an Euler–Bernoulli beam. This model is non-ordinary and derived from the concept of a rotational spring between bonds. While multiple peridynamic material models capture the behavior of solid materials, this is the first 1D state based peridynamic model to resist bending. For sufficiently homogeneous and differentiable displacements, the model is shown to be equivalent to Eringen’s nonlocal elasticity. As the peridynamic horizon approaches 0, it reduces to the classical Euler–Bernoulli beam equations. Simple test cases demonstrate the model’s performance.

A peridynamic material model for the analysis of dynamic crack propagation in orthotropic media

A new material model for the dynamic fracture analysis of anisotropic materials has been proposed within the framework of the bond-based peridynamic theory. This model enables predicting complex fracture phenomena such as spontaneous crack nucleation and crack branching, curving and arrest, a capability inherited from the bond-based peridynamic theory. An important feature of the model is that the bond properties, i.e. the stiffness constant and critical stretch, are continuous functions of bond orientation in the principal material axes. This facilitates fracture analysis of anisotropic materials with random orientations, such as polycrystalline microstructures. Elastic and fracture behaviour of the model has been verified through simulating uniaxial tension of a composite plate and fracture of a cortical bone compact tension specimen, and making quantitative comparisons to analytical and experimental data. To further demonstrate the capabilities of the proposed model, dynamic fracture of a polycrystalline microstructure (alumina ceramic) has been simulated. The influence of the grain boundary and grain interior fracture energies on the interacting and competing fracture modes of polycrystalline materials, i.e. intergranular and transgranular fracture, has been studied.

Linearized Theory of Peridynamic States

A state-based peridynamic material model describes internal forces acting on a point in terms of the collective deformation of all the material within a neighborhood of the point. In this paper, the response of a state-based peridynamic material is investigated for a small deformation superposed on a large deformation. The appropriate notion of a small deformation restricts the relative displacement between points, but it does not involve the deformation gradient (which would be undefined on a crack). The material properties that govern the linearized material response are expressed in terms of a new quantity called the modulus state. This determines the force in each bond resulting from an incremental deformation of itself or of other bonds. Conditions are derived for a linearized material model to be elastic, objective, and to satisfy balance of angular momentum. If the material is elastic, then the modulus state is obtainable from the second Frechet derivative of the strain energy density function. The equation of equilibrium with a linearized material model is a linear Fredholm integral equation of the second kind. An analogue of Poincare’s theorem is proved that applies to the infinite dimensional space of all peridynamic vector states, providing a condition similar to irrotationality in vector calculus.

Analysis of plastic peridynamic material with RBF meshless method

The peridynamic material model (PMM) is a new way to describe the material failures with discontinuities. Earlier works, presented by S. Silling, F. Bubaru and etc. introduced the Linear Elastic Fracture Mechanics of peridynamic material. In the recent work, the isotropic hardening plastic extension of PMM is presented. To solve the nonlinear integral equations of the problem the modified Newton method is used with mesh-less spatial discretization. At last some example shows the similarities and different between the results of classical and peridynamic plasticity.

Peridynamic States and Constitutive Modeling

A generalization of the original peridynamic framework for solid mechanics is proposed. This generalization permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. This extends the types of material response that can be reproduced by peridynamic theory to include an explicit dependence on such collectively determined quantities as volume change or shear angle. To accomplish this generalization, a mathematical object called a deformation state is defined, a function that maps any bond onto its image under the deformation. A similar object called a force state is defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and force state is the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provides a means to incorporate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It also allows the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory.