A two-scale extended finite element method for modelling 3D crack growth with interfacial contact

3D fatigue crack growth problems are nowadays handled using X-FEM coupled with level set techniques. It is also well established that such an approach allows mesh-independent crack modelling and no remeshing during crack propagation. However, when contact and friction occur along the crack faces, a discretization of the internal variables linked to the interface law is necessary. The interface discretization is generally constructed from the finite elements cut by the crack. As a consequence, a mesh dependency between the bulk discretization and the interface discretization is introduced. However, the dimension of the possible non-linearities arising at the crack interface (like confined plasticity or unilateral contact with friction) may be several orders of magnitude finer than the crack size. A finer discretization is thus required to accurately capture these non-linearities. The aim of the present paper is to develop a method considering the 3D cracked structure and the crack interface as two independent global and local problems characterized by different length scales and different behaviors. Here, the interface is seen as an autonomous entity with its own discretization, variables and constitutive law. A formulation involving three-fields is used. The interface is linked to the global problem in a weak sense in order to avoid instabilities in the contact solution. Two iterative strategies are considered to solve the contact problem. Two-dimensional and three-dimensional numerical examples are presented to demonstrate the ability of the model to solve the contact at the crack interface with or without propagation at a given level of accuracy.     

An extended finite element method for modeling crack growth with frictional contact

A new technique for the finite element modeling of crack growth with frictional contact on the crack faces is presented. The eXtended Finite Element Method (X-FEM) is used to discretize the equations, allowing for the modeling of cracks whose geometry are independent of the finite element mesh. This method greatly facilitates the simulation of a growing crack, as no remeshing of the domain is required. The conditions which describe frictional contact are formulated as a non-smooth constitutive law on the interface formed by the crack faces, and the iterative scheme implemented in the LATIN method [Nonlinear Computational Structural Mechanics, Springer, New York, 1998] is applied to resolve the nonlinear boundary value problem. The essential features of the iterative strategy and the X-FEM are reviewed, and the modifications necessary to integrate the constitutive law on the interface are presented. Several benchmark problems are solved to illustrate the robustness of the method and to examine convergence. The method is then applied to simulate crack growth when there is frictional contact on the crack faces, and the results are compared to both analytical and experimental results.     

An efficient model of mixed-mode delamination in laminated composites including bridging mechanisms

A refined model is developed to analyse delamination in composite laminates accounting for bridging stresses at crack faces. The analysis adopts a first-order shear deformable layer-wise kinematics for the laminate and an interface model simulating mixed-mode fracture in the presence of a bridged delamination. A penalised interface simulates adhesion between layers and provides energy release rates through its strain energy density while a two-parameter softening interface with a limit displacement models bridging stresses. Delamination evolution analysis is performed by solving the non-linear boundary value problem resulting from a stress analysis coupled with opportune propagation conditions. Numerical examples are presented for composite laminates subjected to both pure mode and mixed-mode loading conditions and the results are compared with those obtained adopting classic delamination models. Analytical formulae for energy release rate evaluation are also proposed to carry out an investigation of the main factors governing accuracy in predicting delamination growth. The proposed approach captures important effects which are not included in classic delamination models. The accuracy of the model is assessed by comparisons with 2D finite element results obtained by using delamination interface elements. The finite element model agrees well with results obtained by using the proposed approach.     

Finite element computations of anisotropic continuum damage in reinforced concrete

A coupled damage-plasticity concept for the modeling of reinforced concrete structures is presented. Concrete is considered in the framework of mixture theory as a composite consisting of cement matrix and aggregate. Anisotropic damage in the cement matrix and plasticity at the contact interface between cement matrix and aggregate describe the nonlinear constitutive behavior of concrete. Reinforcement is taken into account with three-dimensional rebar-elements. Damage at the interface between concrete and reinforcement is described within the framework of a microplane theory. The proposed concept is verified for some reinforced concrete structures.     

Thermodynamic-based interface model for cohesive brittle materials: Application to bond slip in RC structures

The main objectives of this work concern a new formulation for an inelastic constitutive model for bond-slip phenomena between brittle cohesive materials and reinforcement through a nondimensional zero-thickness joint finite element. The bond-slip behavior assumes the small deformation and plane strains case. It is firmly placed within the framework of thermodynamics and continuum damage mechanics accounting for frictional sliding. The finite element interface model is based on a 2D-degenerated four-node quadrilateral element which is able to account for interface normal stress as well as for the nonlinear hysteretic tangential-normal coupling of bonding. One of the most important advantages of the proposed model consists in a sufficiently rich kinematics structure for contact allowing for the implementation of refine stress-strain constitutive relations.     

A mixed interface finite element for cohesive zone models

The phenomena of crack initiation, propagation and ultimate fracture are studied here under the following assumptions: (i) the crack law is modelled by means of a cohesive zone model and (ii) the crack paths are postulated a priori. In this context, a variational formulation is proposed which relies on an augmented Lagrangian. A mixed interface finite element is introduced to discretise the crack paths, the degrees of freedom of which consist in the displacement on both crack lips and the density of cohesive forces. This enables an exact treatment of multi-valued cohesive laws (e.g. initial adhesion, contact conditions, possible rigid unloading, etc.), without penalty regularisation.A special attention is paid to the convergence with mesh-refinement, i.e. the well-posedness of the problem, on the basis of theoretical results of contact mechanics and some complementary numerical investigations. Fulfilment of the LBB condition is the key factor to gain the desired properties. Moreover, it is shown that the integration of the constitutive law admits a unique solution as soon as some condition on the augmented Lagrangian is enforced. Finally, a 3D simulation shows the applicability to practical engineer problems, including in particular the robustness of the formulation and its compatibility with classical solution algorithms (Newton method, line-search, path-following techniques).     

Multi-scale boundary element modelling of material degradation and fracture

In the present paper a multi-scale boundary element method for modelling damage is proposed. The constitutive behaviour of a polycrystalline macro-continuum is described by micromechanics simulations using averaging theorems. An integral non-local approach is employed to avoid the pathological localization of micro-damage at the macro-scale. At the micro-scale, multiple intergranular crack initiation and propagation under mixed mode failure conditions is considered. Moreover, a non-linear frictional contact analysis is employed for modelling the cohesive-frictional grain boundary interfaces. Both micro- and macro-scales are being modelled with the boundary element method. Additionally, a scheme for coupling the micro-BEM with a macro-FEM is also proposed. To demonstrate the accuracy of the proposed method, the mesh independency is investigated and comparisons with two macro-FEM models are made to validate the different modelling approaches. Finally, microstructural variability of the macro-continuum is considered to investigate possible applications to heterogeneous materials.     

Numerical study of dynamic processes in a continuous medium with a crack initiated by a near-surface disturbance by means of the grid-characteristic method

The purpose of this work is to study the problem of the near-surface disturbance propagation in a massive rock containing various heterogeneities, i.e., empty or filled cracks. Numerical solutions have been obtained for problems of wave propagation in such highly heterogeneous media, including those taking into account the plastic properties of the rock that can be manifested in the vicinity of a seismic gap or a well bore. All kinds of elastic and elastoplastic waves are analyzed resulting from the propagation of the initial disturbance and the waves arising from the reflection from the cracks and from the boundaries of the integration domain. An investigation was carried out of wave identification by means of seismograms obtained at the receiver located near the ground surface. In this study, the grid-characteristic method is employed using computational grids with triangular meshes and boundary conditions formulated at the interface between the rock and the crack, and on free surfaces in an explicit form. The proposed numerical method is extremely general and is suitable for investigations of the processes of seismic waves’ interaction with heterogeneous inclusions because it ensures the construction of the most correct computational algorithms at the boundaries of the integration domain and at the medium’s interface.     

A robust and consistent remeshing-transfer operator for ductile fracture simulations

This paper addresses the numerical simulation of quasi-static ductile fracture. The main focus is on numerical and stability aspects related to discrete crack propagation. Crack initiation and propagation are taken into account, both driven by the evolution of a discretely coupled damage variable. Discrete ductile failure is embedded in a geometrically nonlinear hyperelasto-plastic model, triggered by an appropriate criterion that has been evaluated for tensile and shear failure. A crack direction criterion is proposed, which is validated for both failure cases and which is capable of capturing the experimentally observed abrupt tensile–shear transition. In a large strain finite element context, remeshing enables to trace the crack geometry as well as to preserve an adequate element shape. Stability of the computations is an important issue during crack propagation that can be compromised by two factors, i.e. large stress redistributions during the crack opening and the transfer of variables between meshes. A numerical procedure is developed that renders crack propagation considerably more robust, independently of the mesh fineness and crack discretisation. A consistent transfer algorithm and a crack relaxation method are proposed and implemented for this purpose. Finally, illustrative simulations are compared with published experimental results to highlight the features mentioned.     

Flexural analysis of reinforced fibrous concrete members using the finite element method

A numerical procedure based on the finite element method is developed for the geometric and material nonlinear analysis of reinforced concrete members containing steel fibres and subjected to monotonic loads. The proposed procedure is capable of tracing the displacements, strains, stresses, crack propagation, and member end actions of these structures up to their ultimate load ranges. A frame element with a composite layer system is used to model the structure. An iterative scheme based on Newton-Raphson's method is employed for the nonlinear solution algorithm. The constitutive models of the nonlinear material behaviour are presented to take into account the nonlinear stress-strain relationships, cracking, crushing of concrete, debonding and pull-out of the steel fibres, and yielding of the reinforcement. The geometric nonlinearity due to the geometrical change of both the structure and its elements are also represented. The numerical solution of a number of reinforced fibrous concrete members are compared with published experimental test results and showed good agreement.     

A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro–macro homogenization modeling of gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients are attributed to assigned micro-structural representative volume element (RVE), that leads to a higher-order macroscopic Cosserat continuum modeling and enables to incorporate the micro-structural size effects. The enhanced Hill–Mandel condition for gradient-enhanced Cosserat continuum is extracted as a corollary of the presented generalized Hill’s lemma. The derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the RVE in order to ensure the satisfaction of the Hill–Mandel energy condition.With the link between the discrete particle assembly and its effective Cosserat continuum in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress–strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters.     

The simulation of 3D elastic scattering produced by thin rigid inclusions using the traction boundary element method

A mixed formulation that uses both the traction boundary element method (TBEM) and the boundary element method (BEM) is proposed to compute the three-dimensional (3D) propagation of elastic waves scattered by two-dimensional (2D) thin rigid inclusions. Although the conventional direct BEM has limitations when dealing with thin-body problems, this model overcomes that difficulty. It is formulated in the frequency domain and, taking into account the 2-1/2D configuration of the problem, can be expressed in terms of waves with varying wavenumbers in the zdirection, k_z. The elastic medium is homogeneous and unbounded and it should be noted that no restrictions are imposed on the geometry and orientation of the internal crack.     

Estimation of wave responses from subvertical macrofracture systems using a grid characteristic method

This work investigates the formation and propagation of scattered waves forming a response of crack structures on the seismologic record. The initial impulse is a flat wave front propagating into the depth of the medium. The paper considers the periodic structure of the scattered wave response from a system (cluster) of subvertical cracks. The procedures of estimation of geometrical parameters of such crack structures are based on numerical experiments. We use a grid-characteristic method to calculate the triangular grids with the formulation of boundary conditions at the interface of the medium and the cracks, as well as on the borders of the region of integration with the characteristic properties of the governing equations of the hyperbolic type. This numerical method makes it possible to build the most correct numerical algorithms on the boundaries of the integration domain and on the surfaces of the media division (interfaces) and to take into account the domain of the solution’s dependence and the physics of the problem (propagation of disturbances on the characteristic directions). This is why this method is perhaps the most suitable one for the numerical solution of dynamic problems with a distinct wave character in geologically substantially inhomogeneous media, in particular, for the considered problem of seismic waves interaction with fissured structures.     

Variational formulation and nonsmooth optimization algorithms in elastodynamic contact problems for cracked body

A mathematical model of an elastodynamic contact problem for a body with a crack with unilateral restrictions and friction on the crack faces is presented in classical and weak forms. Different variational formulations of unilateral contact problems with friction based on the principles of Hamilton–Ostrogradskii and Tupin, and boundary variational principles are considered. In particular, boundary variational functionals that are used with boundary integral equations are established. Nonsmooth optimization algorithms of Udzawa type for the solution of these unilateral contact problems with friction are developed. The convergence of the proposed algorithms is studied numerically.     

Computational service life estimation of contacting mechanical elements in regard to pitting

A computational model for determining the service life of contacting surfaces in regard to surface pitting is presented. The model considers the material fatigue process leading to pitting, i.e. the conditions required for the short fatigue crack propagation originating from the initial crack in a single material grain. In view of small crack lengths observed in surface pitting, the simulation takes into account the short crack growth theory. The stress field in the contact area and the required functional relationship between the stress intensity factor and the crack length are determined by the finite element method. An equivalent model of two contacting cylinders is used for numerical simulations of crack propagation in the contact area. On the basis of numerical results, and with consideration of some particular material parameters, the probable service life period of contacting surfaces is estimated for surface curvatures and loadings that are most commonly encountered in engineering practice.     

Boundary element analysis of cracked homogeneous or bi-material structures under thermo-mechanical cycling

We present a sub-domain boundary element procedure to evaluate the failure capacity of cracked homogeneous and bi-material media under cyclic thermo-mechanical loads. The boundary integral equations of uncoupled, time-dependent thermo-elasticity are employed to account for the time-varying nature of the thermal load. If crack closure due to thermal distortion takes place, then the displacement and traction field may affect the heat flux between the crack faces, and the thermal and mechanical parts of the problem will need to be solved repeatedly until thermo-mechanical convergence is achieved. We present results from cases of pure mode-I fracture in homogeneous materials and for interfacial fracture in bi-materials. Our study discusses the influence of crack closure on quasi-static, sub-critical crack extension. Especially in case of interfacial cracks the type of loading, the thermal resistance between the crack faces, and the coefficient of friction are also taken into account. The results suggest that the above parameters may have a severe impact on the predicted failure capacity of cracked structures and should be considered in the evaluation of fatigue life.     

A three-dimensional self-adaptive cohesive zone model for interfacial delamination

Discrete crack models with cohesive binding forces in the fracture process zone have been widely used to address failure in quasi-brittle materials and interfaces. However, the numerical concerns and limitations stemming from the application of interface cohesive zone models in a quasi-static finite element framework increase considerably as the relative size of the process zone decreases. An excessively fine mesh is required in the process zone to accurately resolve the distribution of tractions in a relatively small moving zone. With a moderate mesh size, inefficient path-following techniques have to be employed to trace the local discretization-induced snap-backs. In order to increase the applicability of cohesive zone models by reducing their numerical deficiencies, a self-adaptive finite element framework is proposed, based on a hierarchical enrichment of the standard elements. With this approach, the planar mixed-mode crack growth in a general three-dimensional continuum, discretized by a coarse mesh, can be modeled while the set of equations of the non-linear system is solved by a standard Newton–Raphson iterative procedure. This hierarchical scheme was found to be most effective in reducing the oscillatory behavior of the global response.     

Micromechanics-based FEM simulation of fiber-reinforced cementitious composite components

Fiber bridging along cracks is an important mechanism governing the fracture toughness and the pseudo-ductility of fiber-reinforced brittle materials and structures. This paper attempts to predict structural behavior of fiber-reinforced cementitious composite (FRCC) components using the finite-element procedure with micromechanics-based constitutive modeling of the stress-displacement relation along the crack. The tensile stress-displacement relation along a Mode I (opening) crack is established based on fiber pullout curves derived from a micromechanical model. A statistical model is used to account for random fiber distribution. Two-dimensional finite-element simulations of beam behavior are performed with the finite-element package ADINA. Using the discrete crack approach, strain softening truss elements are placed along the crack to simulate the fiber bridging effect. Experiments of beams under four-point bending are performed with specimens containing different fiber volume fractions (up to 1.5%). The numerical results for the load vs deformation behavior of the beams agree well with the experimental results. The FEM procedure for micromechanics-based design and analysis of FRCC components is therefore established. Simulation of component behavior to identify the most cost-effective design can, hence, be carried out.     

Modeling delamination as a strong discontinuity with the material point method

Decohesion is an important failure mode associated with layered composite materials. Here, the energy implications of material softening are explored in a thermodynamic framework with the result that the dissipated energy (fracture energy) is greater than the plastic work of the traction on the failure surface. It is also argued that if the traction and continuum constitutive equations are solved simultaneously, the resulting algorithm is as simple as that for conventional plasticity. For numerical simulations, the material point method displays the attributes of no mesh deformation so that remeshing is not necessary and the continuous tracking of material points avoids the need for remapping history variables such as decohesion. Compatibility is invoked in a weak sense with the result that no special algorithms are needed for mesh realignment along crack surfaces or for double nodes. Example solutions exhibit no sensitivity of delamination propagation with mesh orientation.     

Modeling of Viscoelastic Contacts and Evolution of Limit Surface for Robotic Contact Interface

Viscoelastic contact is a type of contact which includes, in addition to linear or nonlinear elastic response, time-dependent response due to relaxation or creep phenomena that govern the contact behavior. The characteristics of the time-dependent relaxation of such a viscoelastic contact are typically exponentially decaying functions, and exponentially growing functions for creep, respectively. Such contacts can be found in anthropomorphic robotic fingers, soft materials, viscoelastic skin with rigid core, and human fingers and feet. In this paper, the nature of viscoelastic contacts is investigated, and the evolution of their friction limit surfaces and of the pressure distributions at the contact interface are studied. Two cases commonly found in robotic grasping and manipulation are discussed. Based on the modeling formulation, it is found that the two important parameters of analysis and modeling for such contacts, i.e., the radius of contact area and the profile of pressure distribution, can be chosen using proposed coupling equations as the viscoelastic contact interface evolves with time. The new contribution of this paper includes a proposal of coupling equations between the two important parameters to describe the viscoelastic contact interface, and a study of the evolution of limit surfaces for viscoelastic contact interface due to temporal dependency, and the implication on grasp stability. It is found from the evolution of limit surfaces that when normal force is applied with typical viscoelastic contacts, grasp becomes more stable as time elapses. The modeling can be applied to the design of fingertips and the analysis of robotic grasping and manipulation involving viscoelastic fingers