Multiscale modeling of intergranular fracture in aluminum: constitutive relation for interface debonding

Intergranular fracture is a dominant mode of failure in ultrafine grained materials. In the present study, the atomistic mechanisms of grain-boundary debonding during intergranular fracture in aluminum are modeled using a coupled molecular dynamics—finite element simulation. Using a statistical mechanics approach, a cohesive-zone law in the form of a traction–displacement constitutive relationship, characterizing the load transfer across the plane of a growing edge crack, is extracted from atomistic simulations and then recast in a form suitable for inclusion within a continuum finite element model. The cohesive-zone law derived by the presented technique is free of finite size effects and is statistically representative for describing the interfacial debonding of a grain boundary (GB) interface examined at atomic length scales. By incorporating the cohesive-zone law in cohesive-zone finite elements, the debonding of a GB interface can be simulated in a coupled continuum–atomistic model, in which a crack starts in the continuum environment, smoothly penetrates the continuum–atomistic interface, and continues its propagation in the atomistic environment. This study is a step toward relating atomistically derived decohesion laws to macroscopic predictions of fracture and constructing multiscale models for nanocrystalline and ultrafine grained materials.     

A continuum shape sensitivity method for fracture analysis of isotropic functionally graded materials

This paper presents a new continuum shape sensitivity method for calculating mixed-mode stress-intensity factors for a stationary crack in two-dimensional, linear- elastic, isotropic FGMs with arbitrary geometry. The method involves the material derivative concept taken from continuum mechanics, the mutual potential energy release rate, and direct differentiation. Since the governing variational equation is differentiated prior to discretization, resulting sensitivity equations are independent of approximate numerical techniques, such as the finite element method, boundary element method, mesh-free method, or others. The discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires multiplication of displacement vectors and stiffness sensitivity matrices. By judiciously selecting the velocity field, the method only requires displacement response in a subdomain close to the crack tip, thus making the method computationally efficient. Seven finite-element based numerical examples, which comprise mode-I and mixed-mode deformations and/or single or multiple interacting cracks, are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method. Comparisons have been made between stress-intensity factors predicted by the proposed method and available reference solutions in the literature, generated either analytically or numerically using various other fracture integrals or analyses. Excellent agreement is obtained between the results of the proposed method and previously obtained solutions. Therefore, shape sensitivity analysis provides an attractive alternative to fracture analysis of cracks in homogeneous and non-homogeneous materials.     

Modeling of fracture behavior in polymer composites using concurrent multi-scale coupling approach

ABSTRACT

Embedded statistical coupling method was originally developed to provide computational efficiency, to decrease coupling complexities, and to avoid the need to discretize the continuum model to atomic scale resolution in concurrent multi-scale modeling. An embedded statistical coupling method scheme is relatively easy to implement within a conventional finite element method code and has been tested in standard solid lattice structures. However, this method encounters difficulties when being implemented for amorphous materials like polymers, due to the fact that they lack specific ordered lattice structure and atoms may not be covalently bonded with each other, which are the requirements of common coupling schemes. Therefore, a new approach needs to be developed to resolve this problem. In this article, details of a modified embedded statistical coupling method approach for atomistic-continuum coupling developed to perform simulations of macroscale crack growth in polymers is presented. The presence of the continuum domain surrounding the molecular dynamics region allows for the application of far field loading, and prevents stress wave reflections from the external boundary impinging back on the crack tip. In our approach, a material point method, which is a meshless particle-in-cell method based on an arbitrary Euler-Lagrange scheme and has been proven to have good performance in large deformation problems, is used to model the continuum domain. It is concurrently coupled with molecular dynamics, a widely used method in atomistic simulations, using a so-called handshake region. Anchor points, the equilibrium positions of the constrained particles, which are designed to transmit displacements and forces between nanoscale and macroscale model, are defined in the handshake region. A concurrently coupled material point method-molecular dynamics simulation of crack propagation inside a polymer is performed to verify this new coupling approach, thereby providing a better understanding of the fracture mechanisms at the nanoscale to predict the macro-scale fracture toughness of a polymer system. Results are presented for concurrently coupled simulation of crack initiation and crack propagation in a di-functional cross-linked thermoset polymer, EPON 862.     

Direct estimation of generalized stress intensity factors using a three‐scale concurrent multigrid X‐FEM

A concurrent multigrid method is devised for the direct estimation of stress intensity factors and higher‐order coefficients of the elastic crack tip asymptotic field. The proposed method bridges three characteristic length scales that can be present in fracture mechanics: the structure, the crack and the singularity at the crack tip. For each of them, a relevant model is proposed. First, a truncated analytical reduced‐order model based on Williams' expansion is used to describe the singularity at the tip. Then, it is coupled with a standard extended finite element (FE) method model which is known to be suitable for the scale of the crack. A multigrid solver finally bridges the scale of the crack to that of the structure for which a standard FE model is often accurate enough. Dedicated coupling algorithms are presented and the effects of their parameters are discussed. The efficiency and accuracy of this new approach are exemplified using three benchmarks. Copyright © 2010 John Wiley & Sons, Ltd.     

J-integral analysis of cord-rubber serpentine belt using neural-network-based material modelling

A known factor that limits the performance of automotive front‐end accessory serpentine belt drive is cracking of the elastomer located in the rib tip. In this paper, fracture experiments were conducted using single‐edge notched tension (SENT) specimens to study the fracture behaviour of a belt rib compound. A finite‐element modelling method utilizing singular elements for crack in rubber solid was proposed and implemented in both plane‐stress and 3D solid models using ABAQUS. A newly developed neural‐network‐based model was used to represent a nonlinear elastic belt rib rubber compound. The crack finite‐element model, along with the neural‐network‐based material model, was verified with analytical and experimental results. A global–local finite‐element procedure was developed to evaluate the J‐integral for mode‐I through‐the‐thickness crack in V‐ribbed belt rib. Effects of pre‐crack length, pulley pre‐load and backside pulley displacement were investigated.     

Three-dimensional analysis of interface cracks in rubber materials

Three-dimensional and plane stress finite element analyses were carried out to investigate the stress fields and fracture parameters of interface cracks in rubber materials. The tearing energy computations for any arbitrarily shaped 3D crack front in dissimilar materials were obtained using the virtual crack extension method. The finite element results obtained are validated against existing alternate solutions and experimental data for cracks in homogeneous as well as bimaterial cases. The effects of different rubber material models, tearing energy distributions, crack extension angles, and three-dimensional regions at the crack tip for interface cracks are presented and discussed. It is shown that nonlinear materials have larger three-dimensional effects near the interface crack front, and that this effect increases as the material mismatch increases.     

Extended finite element method for three‐dimensional crack modelling

An extended finite element method (X‐FEM) for three‐dimensional crack modelling is described. A discontinuous function and the two‐dimensional asymptotic crack‐tip displacement fields are added to the finite element approximation to account for the crack using the notion of partition of unity. This enables the domain to be modelled by finite elements with no explicit meshing of the crack surfaces. Computational geometry issues associated with the representation of the crack and the enrichment of the finite element approximation are discussed. Stress intensity factors (SIFs) for planar three‐dimensional cracks are presented, which are found to be in good agreement with benchmark solutions. Copyright © 2000 John Wiley & Sons, Ltd.     

Simulations of dynamic crack propagation in brittle materials using nodal cohesive forces and continuum damage mechanics in the distinct element code LDEC

Experimental data indicates that the limiting crack speed in brittle materials is less than the Rayleigh wave speed. One reason for this is that dynamic instabilities produce surface roughness and microcracks that branch from the main crack. These processes increase dissipation near the crack tip over a range of crack speeds. When the scale of observation (or mesh resolution) becomes much larger than the typical sizes of these features, effective-medium theories are required to predict the coarse-grained fracture dynamics. Two approaches to modeling these phenomena are described and used in numerical simulations. The first approach is based on cohesive elements that utilize a rate-dependent weakening law for the nodal cohesive forces. The second approach uses a continuum damage model which has a weakening effect that lowers the effective Rayleigh wave speed in the material surrounding the crack tip. Simulations in this paper show that while both models are capable of increasing the energy dissipated during fracture when the mesh size is larger than the process zone size, only the continuum damage model is able to limit the crack speed over a range of applied loads. Numerical simulations of straight-running cracks demonstrate good agreement between the theoretical predictions of the combined models and experimental data on dynamic crack propagation in brittle materials. Simulations that model crack branching are also presented.     

Fracture in magnetoelectroelastic materials using the extended finite element method

Static fracture analyses in two‐dimensional linear magnetoelectroelastic (MEE) solids is studied by means of the extended finite element method (X‐FEM). In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and the crack‐tip asymptotic functions to the standard finite element approximation using the framework of partition of unity. In this study, media possessing fully coupled piezoelectric, piezomagnetic and magnetoelectric effects are considered. New enrichment functions for cracks in transversely isotropic MEE materials are derived, and the computation of fracture parameters using the domain form of the contour interaction integral is presented. The convergence rates in energy for topological and geometric enrichments are studied. Excellent accuracy of the proposed formulation is demonstrated on benchmark crack problems through comparisons with both analytical solutions and numerical results obtained by the dual boundary element method. Copyright © 2011 John Wiley & Sons, Ltd.     

Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral

This paper presents a successful implementation of the virtual crack closure integral method to calculate the stress intensity factors of an interfacial crack. The present method would compute the mixed-mode stress intensity factors from the mixed-mode energy release rates of the interfacial crack, which are easily obtained from the crack opening displacements and the nodal forces at and ahead of the crack tip, in a finite element model. The simple formulae which relate the stress intensity factors to the energy release rates are given in three separate categories: an isotropic bimaterial continuum, an orthotropic bimaterial continuum, and an anisotropic bimaterial continuum. In the example of a central crack in a bimaterial block under the plane strain condition, comparisons are made with the exact solution to determine the accuracy and efficiency of the numerical method. It was found that the virtual crack closure integral method does lead to very accurate results with a relatively coarse finite element mesh. It has also been shown that for an anisotropic interfacial crack under the generalized plane strain condition, the computed stress intensity factors using the virtual crack closure method compared favorably with the results using the J integral method applied to two interacting crack tip solutions. In order for the stress intensity factors to be used as physical variables, the characteristic length for the stress intensity factors must be properly defined. A study was carried out to determine the effects of the characteristic length on the fracture criterion based the mixed-mode stress intensity factors. It was found that the fracture criterion based on the quadratic mixture of the normalized stress intensity factors is less sensitive to the changes in characteristic length than the fracture criterion based on the total energy release rate along with the phase angle.This work has been supported by ONR, with Dr. Y. Rajapakse as the program official.     

Finite element simulation of dynamic crack propagation process using an arbitrary Lagrangian Eulerian formulation

In this paper, the arbitrary Lagrangian Eulerian formulation is employed for finite element modelling of dynamic crack propagation problem. The application phase simulation of computational dynamic fracture is applied to model by which the crack propagation history and variation of crack velocity are predicted using the material dynamic fracture toughness. The dynamic solution of problem is accomplished using the implicit time integration method. The convective terms due to mesh‐material motion are taken into account via the convection equation. A robust and efficient mesh motion technique, that its equations need not to be solved at every time step, is employed in Eulerian phase. The mesh connectivity is preserved during the analysis. So, the successive remeshing of model is eliminated. When the dynamic fracture criterion is satisfied for crack growth, the presented algorithm allows the crack to advance by splitting the material particle at the crack tip. The dynamic energy release rate is calculated at each time step to determine dynamic stress intensity factor. The predicted results are compared with those obtained through the experimental study and remeshing technique cited in the literature. The proposed computational algorithm leads to an accurate and efficient simulation of dynamic crack propagation process.     

Modelling multiple cohesive crack propagation using a finite element–scaled boundary finite element coupled method

This paper presents an extension of the recently-developed finite element–scaled boundary finite element (FEM–SBFEM) coupled method to model multiple crack propagation in concrete. The concrete bulk and fracture process zones are modelled using SBFEM and nonlinear cohesive interface finite elements (CIEs), respectively. The CIEs are automatically inserted into the SBFEM mesh as the cracks propagate. The algorithm previously devised for single crack propagation is augmented to model problems with multiple cracks and to allow cracks to initiate in an un-cracked SBFEM mesh. It also addresses crack propagation from one subdomain into another, as a result of partitioning a coarse SBFEM mesh, required for some mixed–mode problems. Each crack in the SBFEM mesh propagates when the sign of the Mode-I stress intensity factor at the crack tip turns positive from negative. Its propagation angle is determined using linear elastic fracture mechanics criteria. Three concrete beams involving multiple crack propagation are modelled. The predicted crack propagation patterns and load–displacement curves are in good agreement with data reported in literature.     

Determination of dynamic fracture toughness for PMMA

A dynamic FEM (finite element method) and a strain gage method are applied to analyze the dynamic fracture toughness and SIF (stress intensity factor) for PMMA (polymethyl methacrylate). The analyses are carried out for plates with an edge crack subjected to one-point bending in a plane of the plate. A simple procedure that the present author has proposed is applied to the problem of using a triangular element of assumed constant strain on finite element analysis. The numerical simulation by FEM provides values for the applied forces as measured with the strain gages. Also, a crack initiation time is measured with the strain gage mounted around the crack tip. The dynamic fracture toughness is determined by adapting the crack initiation time to the simulation curve of the dynamic SIF calculated by the FEM. In this study, the usefulness of the method to determine the dynamic fracture toughness is investigated by comparing predictions with the experimental results for dynamic stresses and SIFs.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.     

Intelligent Condition Monitoring of Aerospace Composites: Part I - Nano Reinforced Surfaces & Interfaces

This study is part of a major research program concerning intelligent condition monitoring of aerospace composites. Specifically, it evaluates the influence of nanofillers on the reinforcement of adhesively bonded layer under mode-I fracture toughness using multiscale modelling. In this novel approach, we couple coarse–grain molecular dynamics with continuum mechanics. The molecular dynamics domain and the finite element domain are overlapped in a handshaking subdomain, The implementation of coarse–grain molecular dynamics radically reduces the size of the problem. An explicit algorithm coupling the two methodologies was developed and used to determine the energy release rates of cohesive cracks in adhesively bonded composite joints with varying amount of nano-reinforcement in the adhesive layer. Both the quality of the prediction of the multiscale model and the influence of the nanofillers are evaluated and discussed.     

Finite‐deformation irreversible cohesive elements for three‐dimensional crack‐propagation analysis

We develop a three‐dimensional finite‐deformation cohesive element and a class of irreversible cohesive laws which enable the accurate and efficient tracking of dynamically growing cracks. The cohesive element governs the separation of the crack flanks in accordance with an irreversible cohesive law, eventually leading to the formation of free surfaces, and is compatible with a conventional finite element discretization of the bulk material. The versatility and predictive ability of the method is demonstrated through the simulation of a drop‐weight dynamic fracture test similar to those reported by Zehnder and Rosakis. The ability of the method to approximate the experimentally observed crack‐tip trajectory is particularly noteworthy. © 1999 John Wiley & Sons, Ltd.     

An enriched element‐failure method (REFM) for delamination analysis of composite structures

This paper develops an enriched element‐failure method for delamination analysis of composite structures. This method combines discontinuous enrichments in the extended finite element method and element‐failure concepts in the element‐failure method within the finite element framework. An improved discontinuous enrichment function is presented to effectively model the kinked discontinuities; and, based on fracture mechanics, a general near‐tip enrichment function is also derived from the asymptotic displacement fields to represent the discontinuity and local stress intensification around the crack‐tip. The delamination is treated as a crack problem that is represented by the discontinuous enrichment functions and then the enrichments are transformed to external nodal forces applied to nodes around the crack. The crack and its propagation are modeled by the ‘failed elements’ that are applied to the external nodal forces. Delamination and crack kinking problems can be solved simultaneously without remeshing the model or re‐assembling the stiffness matrix with this method. Examples are used to demonstrate the application of the proposed method to delamination analysis. The validity of the proposed method is verified and the simulation results show that both interlaminar delamination and crack kinking (intralaminar crack) occur in the cross‐ply laminated plate, which is observed in the experiment. Copyright © 2009 John Wiley & Sons, Ltd.     

Multiscale modeling of dynamic crack propagation

A multiscale approach is employed to investigate a center-cracked specimen with the purpose to redefine fracture toughness from the atomistic perspective and to simulate different modes of crack propagation. The specimen is divided into three regions: (1) far field, modeled by classical fracture mechanics, (2) near field, modeled by a multiscale field theory and analyzed by a generalized finite element method, and (3) crack tip atomic region, modeled by molecular dynamics (MD). The exact and analytical solution of the far field is utilized to specify boundary conditions at the interface between the far field and the near field. The interaction between the near field and the crack tip region is described by full-blown interatomic forces. In this work, crystals of perovskite (Barium Titanate) and rocksalt (Magnesia) have been studied. Fracture toughness is defined as a material property associated with instability of the MD simulation. Mode I, Mode II, and mixed mode fracture have been investigated and numerical results will be presented and discussed.