Adaptive coupling between damage mechanics and peridynamics: A route for objective simulation of material degradation up to complete failure

The objective (mesh-independent) simulation of evolving discontinuities, such as cracks, remains a challenge. Current techniques are highly complex or involve intractable computational costs, making simulations up to complete failure difficult. We propose a framework as a new route toward solving this problem that adaptively couples local-continuum damage mechanics with peridynamics to objectively simulate all the steps that lead to material failure: damage nucleation, crack formation and propagation. Local-continuum damage mechanics successfully describes the degradation related to dispersed microdefects before the formation of a macrocrack. However, when damage localizes, it suffers spurious mesh dependency, making the simulation of macrocracks challenging. On the other hand, the peridynamic theory is promising for the simulation of fractures, as it naturally allows discontinuities in the displacement field. Here, we present a hybrid local-continuum damage/peridynamic model. Local-continuum damage mechanics is used to describe “volume” damage before localization. Once localization is detected at a point, the remaining part of the energy is dissipated through an adaptive peridynamic model capable of the transition to a “surface” degradation, typically a crack. We believe that this framework, which actually mimics the real physical process of crack formation, is the first bridge between continuum damage theories and peridynamics. Two-dimensional numerical examples are used to illustrate that an objective simulation of material failure can be achieved by this method.     

Ductile-to-brittle transition in tensile failure of particle-reinforced metals

We present an analytical micromechanical model designed to simulate the tensile stress-strain behaviour and failure of damaging composites containing a high volume fraction of reinforcing particles. One internal damage micromechanism is considered, namely particle fracture, which is assumed to obey a Weibull distribution. Final composite tensile failure occurs when one of two possible failure criteria is reached, given by (i) the onset of tensile instability, or (ii) an “avalanche-like” propagation of particle breaks to neighbouring particles. We show that an experimentally observed transition from failure by tensile instability to abrupt failure resulting from an increase of matrix strength can be mimicked by the model because local load-sharing (i.e. load transfer from a broken particle to its immediate neighbours) is accounted for.     

The brittle fracture criterion based on the maximum tensile stress on the surface of blunt crack tip

The brittle fracture criterion is developed for a blunted crack. The curvature radius of the blunt crack tip is suggested as a characteristic length for brittle materials, and then the fracture toughness of the brittle materials can be determined from the cohesion strength and the characteristic length of the materials.     

Microcrack interaction brittle rock subjected to uniaxial tensile loads

A micromechanics-based model is proposed to describe unstable damage evolution in microcrack-weakened brittle rock material. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress–strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to be consistent with the experimental results.     

A micromechanical damage and fracture model for polymers based on fractional strain-gradient elasticity

We formulate a simple one-parameter macroscopic model of distributed damage and fracture of polymers that is amenable to a straightforward and efficient numerical implementation. We show that the macroscopic model can be rigorously derived, in the sense of optimal scaling, from a micromechanical model of chain elasticity and failure regularized by means of fractional strain-gradient elasticity. In particular, we derive optimal scaling laws that supply a link between the single parameter of the macroscopic model, namely, the critical energy-release rate of the material, and micromechanical parameters pertaining to the elasticity and strength of the polymer chains and to the strain-gradient elasticity regularization. We show how the critical energy-release rate of specific materials can be determined from test data. Finally, we demonstrate the scope and fidelity of the model by means of an example of application, namely, Taylor-impact experiments of polyurea 1000 rods.     

Damage and self-similarity in fracture

Consider applications of damage mechanics to material failure. The damage variable introduced in damage mechanics quantifies the deviation of a brittle solid from linear elasticity. An analogy between the metastable behavior of a stressed brittle solid and the metastable behavior of a superheated liquid is established. The nucleation of microcracks is analogous to the nucleation of bubbles in the superheated liquid. In this paper we have applied damage mechanics to four problems. The first is the instantaneous application of a constant stress to a brittle solid. The results are verified by applying them to studies of the rupture of chipboard and fiberglass panels. We then obtain a solution for the evolution of damage after the instantaneous application of a constant strain. It is shown that the subsequent stress relaxation can reproduce the modified Omori’s law for the temporal decay of aftershocks following an earthquake. Obtained also are the solutions for application of constant rates of stress and strain. A fundamental question is the cause of the time delay associated with damage and microcracks. It is argued that the microcracks themselves cause random fluctuations similar to the thermal fluctuations associated with phase changes.     

Analytical and numerical analysis of frictional damage in quasi brittle materials

Frictional sliding and crack growth are two main dissipation processes in quasi brittle materials. The frictional sliding along closed cracks is the origin of macroscopic plastic deformation while the crack growth induces a material damage. The main difficulty of modeling is to consider the inherent coupling between these two processes. Various models and associated numerical algorithms have been proposed. But there are so far no analytical solutions even for simple loading paths for the validation of such algorithms. In this paper, we first present a micro-mechanical model taking into account the damage-friction coupling for a large class of quasi brittle materials. The model is formulated by combining a linear homogenization procedure with the Mori–Tanaka scheme and the irreversible thermodynamics framework. As an original contribution, a series of analytical solutions of stress–strain relations are developed for various loading paths. Based on the micro-mechanical model, two numerical integration algorithms are exploited. The first one involves a coupled friction/damage correction scheme, which is consistent with the coupling nature of the constitutive model. The second one contains a friction/damage decoupling scheme with two consecutive steps: the friction correction followed by the damage correction. With the analytical solutions as reference results, the two algorithms are assessed through a series of numerical tests. It is found that the decoupling correction scheme is efficient to guarantee a systematic numerical convergence.     

A new 2D discrete model applied to dynamic crack propagation in brittle materials

In this work, a 2D discrete model (DM) applied to the dynamic crack propagation in brittle materials is developed and implemented. The proposed model is based on a particular discretization of Navier’s equations, presenting similarities to the Born model, with the advantage that the constants appearing in it are explicitly related to the elastic properties. This model overcomes the limitations in the choice of Poisson’s ratio present in other discrete models. Three numerical examples are presented to show the capability of this method in modelling wave propagation and dynamic fracture problems. The obtained results are in agreement with experimental and numerical results reported by other researchers.     

Coupled modeling of damage growth and permeability variation in brittle rocks

This paper presents a coupled model for anisotropic damage and permeability evolution by using a micro–macro approach. The damage state is represented by a second order tensor. The evolution of damage is determined from a crack propagation criterion. The free enthalpy function of cracked material is obtained by using micromechanical considerations. It is assumed that cracks exhibit normal aperture which is coupled with the crack growth due to asperities of crack faces. By using Darcy’s law for macroscopic fluid flow and assuming laminar flow in microcracks, the overall permeability of the RVE is obtained by a volume averaging procedure taking into account crack aperture in each orientation.     

Constitutive relationship of brittle rock subjected to dynamic uniaxial tensile loads with microcrack interaction effects

A micromechanical model is proposed to describe both stable and unstable damage evolution in microcrack-weakened brittle rock material subjected to dynamic uniaxial tensile loads. The basic idea of the present model is to classify the constitution relationship of rock material subjected to dynamic uniaxial tensile loads into four stages including some of the stages of linear elasticity, pre-peak nonlinear hardening, rapid stress drop, and strain softening, and to investigate their corresponding micromechanical damage mechanisms individually. Special attention is paid to the transition from structure rearrangements on microscale to the macroscopic inelastic strain, to the transition from distribution damage to localization of damage and the transition from homogeneous deformation to localization of deformation. The influence of all microcracks with different sizes and orientations are introduced into the constitutive relation by using the statistical average method. Effects of microcrack interaction on the complete stress-strain relation as well as the localization of damage for microcrack-weakened brittle rock material are analyzed by using effective medium method. Each microcrack is assumed to be embedded in an approximate effective medium that is weakened by uniformly distributed microcracks of the statistically-averaged length depending on the actual damage state. The elastic moduli of the approximate effective medium can be determined by using the dilute distribution method. Micromechanical kinetic equations for stable and unstable growth characterizing the ‘process domains’ of active microcracks are taken into account. These ‘process domains’ together with ‘open microcrack domains’ completely determine the integration domains of ensemble averaged constitutive equations relating macro-strain and macro-stress. Theoretical predictions have shown to consistent with the experimental results.     

脆性材料动态破坏过程的数值模拟

根据连续介质力学基本原理和离散元法基本思想，针对脆性材料动态破坏过程问题，推导出基于圆盘单元的正交各向异性离散元二维计算模型，并利用此模型计算了各向异性薄板在冲击载荷下的应力波传播问题。通过将计算结果与LS－DYNA程序相比较，表明了本文提出的离散元模型的精确性。另外，对钢弹侵彻下滑凝土圆板破坏过程这个典型的发生连续介质向非连续介质转化的动力学问题进行了数值模拟和动画显示，并同侵彻实验中出现的几种破坏形式进行了分析和对比，从而证明了本计算方法是计算和模拟脆性材料出现从连续介质向非连续介质转变的动态破坏问题的有力工具。     

Cohesive fracture simulation and failure modes of FRP–concrete bonded interfaces

A work-of-fracture method using three-point bend beam (3PBB) specimen, commonly employed to determine the fracture energy of concrete, is adapted to evaluate the mode-I cohesive fracture of fiber reinforced plastic (FRP) composite–concrete adhesively bonded interfaces. In this study, a bilinear damage cohesive zone model (CZM) is used to simulate cohesive fracture of FRP–concrete bonded interfaces. The interface cohesive process damage model is proposed to simulate the adhesive–concrete interface debonding; while a tensile plastic damage model is used to account for the cohesive cracking of concrete near the bond line. The influences of the important interface parameters, such as the interface cohesive strength, concrete tensile strength, critical interface energy, and concrete fracture energy, on the interface failure modes and load-carrying capacity are discussed in detail through a numerical finite element parametric study. The results of numerical simulations indicate that there is a transition of the failure modes controlling the interface fracture process. Three failure modes in the mode-I fracture of FRP–concrete interface bond are identified: (1) complete adhesive–concrete interface debonding (a weak bond), (2) complete concrete cohesive cracking near the bond line (a strong bond), and (3) a combined failure of interface debonding and concrete cohesive cracking. With the change of interface parameters, the transition of failure modes from interface debonding to concrete cohesive cracking is captured, and such a transition cannot be revealed by using a conventional fracture mechanics-based approach, in which only an energy criterion for fracture is employed. The proposed cohesive damage models for the interface and concrete combined with the numerical finite element simulation can be used to analyze the interface fracture process, predict the load-carrying capacity and ductility, and optimize the interface design, and they can further shed new light on the interface failure modes and transition mechanism which emulate the practical application.     

Evolution of nanoscale defects to planar cracks in a brittle solid

Fracture of a solid is a highly multiscale process that associates atomic scale bond breaking with macroscopic crack propagation, and the process can be dramatically influenced by the presence of defects in materials. In a nanomaterial, defect formation energy decreases with the reduction of material size, and therefore, the role of defects in crack formation and subsequent crack growth in such materials may not be understood from the classical laws of fracture mechanism. In this study, we investigated the crack formation process of a defective (with missing atoms) nanostructured material (NaCl) using a series of molecular dynamics (MD) simulations. It was demonstrated that simple defects in the form of several missing atoms in the material could develop into a planar crack. Subsequently, MD simulations on failures of nanosized NaCl with pre-defined planar atomistic cracks of two different lengths under prescribed tensile displacement loads were performed. These failure loads were then applied on the equivalent continuum models, separately, to evaluate the associated fracture toughness values using the finite element analysis. For small cracks, the fracture toughness thus obtained is cracksize dependent and the corresponding critical energy release rate is significantly smaller than Griffith’s theoretical value. Explanation for this discrepancy between LEFM and the atomistic model was attempted.     

Fracturing damage process in dynamic split experiments of a brittle glass

In this study, the 3-dimensional discrete element method is firstly introduced to explain the fracturing damage process of the dynamic split experiment of a special brittle glass ZnS. The corresponding dynamic split experiment is also performed by using the split Hopkinson pressure bar. Then the numerical results correspond closely to those obtained by experiments, and the fracturing damage mode shows that the sample under high strain rate loading would crack along vertical diameter in the band region between two loading edges, which differs from the static damage mode. Furthermore, by comparing a group of contrast numerical tests, the numerical results prove that loading area upon the top side of samples would influence the fracture mode of dynamic split experiments, which indicates that the narrow loading plane is better.     

Interaction-based non-local damage model for failure in quasi-brittle materials

The purpose of this paper is to present a new macroscopic approach to describe the evolving non-local interactions which are produced at the mesoscale during damage and failure in quasi-brittle materials. A new-integral type non-local model is provided where the weight function is directly built from these interactions, and therefore takes into account their evolution during the material failure intrinsically.     

From yield to fracture,failure initiation captured by molecular simulation

While failure of cracked bodies with strong stress concentrations is described by an energy criterion (fracture mechanics), failure of flawless bodies with uniform stresses is captured by a criterion on stress (yielding). In-between those two cases, the problem of failure initiation from flaws that moderately concentrate stresses is debated. In this paper, we propose an investigation of the process of failure initiation at the atomic scale by means of molecular simulations. We first discuss the appropriate scaling conditions to capture initiation, since system sizes that can be simulated by molecular mechanics are strongly limited. Then, we perform a series of molecular simulations of failure of a 2D model material, which exhibits strength and toughness properties that are suitable to capture initiation with systems of reasonable sizes. Transition from fracture failure to yield failure is well characterized. Interestingly, in some specific cases, failure exceeds yield failure which is in contradiction with most initiation theories. This occurs when stress are highly concentrated while little mechanical energy is stored in the material. This observation calls for a theory of initiation which requires that both stress and energy are necessary conditions of failure. Such an approach was proposed by Leguillon (2002). We show that the predictions of this theory are consistent with the molecular simulation results.     

CDPM2: A damage-plasticity approach to modelling the failure of concrete

A constitutive model based on the combination of damage mechanics and plasticity is developed to analyse the failure of concrete structures. The aim is to obtain a model, which describes the important characteristics of the failure process of concrete subjected to multiaxial loading. This is achieved by combining an effective stress based plasticity model with a damage model based on plastic and elastic strain measures. The model response in tension, uni-, bi- and triaxial compression is compared to experimental results. The model describes well the increase in strength and displacement capacity for increasing confinement levels. Furthermore, the model is applied to the structural analyses of tensile and compressive failure.     

Multiscale modelling of damage and failure in two-dimensional metallic foams

The fracture strength of metal foams depends sensitively on the properties of the constituent material as well as the cellular architecture. A change in microscopic properties carries over to the macroscopic scale through an alteration of the mesoscopic damage and fracture mechanisms. In this paper we study these dependencies using a modelling framework that takes all these ingredients into account. We have developed a micromechanical model based on a discrete Voronoi representation of cellular metals that incorporates power-law strain hardening and damage development of the cell wall material. The influence of the relative density and material strain hardening on the cell wall damage behavior and overall fracture response is analyzed in detail. The effect of the cellular architecture is studied by varying the cell shape anisotropy and structural randomness. We also simulate the effect of post-processing heat treatments on the solid material plastic and fracture properties and how this affects the overall fracture profile and damage development. Finally, all material and architectural effects are summarized in a strength versus ductility graph, identifying trends for improved design of metallic foams.     

Multi-scale defect interactions in high-rate failure of brittle materials,Part II: Application to design of protection materials

Micromechanics based damage models, such as the model presented in Part I of this 2 part series (Tonge and Ramesh, 2015), have the potential to suggest promising directions for materials design. However, to reach their full potential these models must demonstrate that they capture the relevant physical processes. In this work, we apply the multiscale material model described in Tonge and Ramesh (2015) to ballistic impacts on the advanced ceramic boron carbide and suggest possible directions for improving the performance of boron carbide under impact conditions. We simulate both dynamic uniaxial compression and simplified ballistic loading geometries to demonstrate that the material model captures the relevant physics in these problems and to interrogate the sensitivity of the simulation results to some of the model input parameters. Under dynamic compression, we show that the simulated peak strength is sensitive to the maximum crack growth velocity and the flaw distribution, while the stress collapse portion of the test is partially influenced by the granular flow behavior of the fully damaged material. From simulations of simplified ballistic impact, we suggest that the total amount of granular flow (a possible performance metric) can be reduced by either a larger granular flow slope (more angular fragments) or a larger granular flow timescale (larger fragments). We then discuss the implications for materials design.