A numerical fracture analysis of indentation into thin hard films on soft substrates

In this paper, finite element simulations of spherical indentation of a thin hard film deposited on a soft substrate are carried out. The primary objective of this work is to understand the mechanics of fracture of the film due to formation of cylindrical or circumferential cracks extending inwards from the film surface. Also, the role of plastic yielding in the substrate on the above mechanics is studied. To this end, the plastic zone development in the substrate and its influence on the load versus indentation depth characteristics and the stress distribution in the film are first examined. Next, the energy release rate J associated with cylindrical cracks is computed. The variation of J with indentation depth and crack length is investigated. The results show that for cracks located near the indenter axis and at small indentation depth, J decreases over a range of crack lengths, which implies stability of crack growth. This regime vanishes as the location of the crack from the axis increases, particularly for a substrate with low yield strength. Finally, a method for combining experimental load versus indentation depth data with simulation results in order to obtain the fracture energy of the film is proposed.     

A general mixed-mode brittle fracture criterion for cracked materials

In order to predict the fracture direction and fracture loadings of cracked materials under the general mixed-mode state, this paper presents a new general mixed-mode brittle fracture criterion based on the concept of maximum potential energy release rate (MPERR). This criterion can be easily degraded to the pure-mode fracture criterion, and can also be reduced to the commonly accepted fracture criteria for the mixed-mode I/II state. In order to validate the proposed criterion, we have carried out the experiments with aluminium alloy specimens under various mixed-mode loading conditions. The experimental results agree well with the predictions of the proposed criterion.     

A numerical study on the impact resistance of composite shells using an energy based failure model

This paper presents a numerical study on the impact resistance of composite shells laminates using an energy based failure model. The damage model formulation is based on a methodology that combines stress based, continuum damage mechanics (CDM) and fracture mechanics approaches within a unified procedure by using a smeared cracking formulation. The damage model has been implemented as a user-defined material model in ABAQUS FE code within shell elements. Experimental results obtained from previous works were used to validate the damage model. Finite element models were developed in order to investigate the pressure and curvature effects on the impact response of laminated composite shells.     

Boundary effect on concrete fracture and non-constant fracture energy distribution

This paper extends the local fracture energy concept of Hu and Wittmann [29] and [30], and proposes a bilinear model for boundary or size effect on the fracture properties of cementitious materials. The bilinear function used to approximate the non-constant local fracture energy distribution along a ligament is based on the assumption of the proportionality of the local fracture energy to the fracture process zone (FPZ) height and characterises the FPZ height reduction when approaching a specimen back boundary. The bilinear function consists of a horizontal straight line of the intrinsic fracture energy G_F and a declining straight line that reduces to zero at the back boundary. It is demonstrated that using the bilinear model, the size-independent fracture energy G_F can be estimated from the fracture energy data measured on laboratory-size specimens, and the intersection of these two linear functions, defined as the transition ligament, represents the influence of the back boundary on the fracture properties. It is also demonstrated that the specimen size alone is not sufficient to characterise the size effect in the fracture properties observed on laboratory-size specimens.     

Effects of atoms on brittle fracture

This article aims to answer two related sets of questions. First: in principle, how large an effect can structure at the atomic scale have upon the fracture of two macroscopically identical samples? The answer to this question is that the effects can be very large. Perfectly sharp cracks can be pinned and stationary under loading conditions that put them far beyond the Griffith point. Crack paths need not obey the rule K _II=0. Crack speeds can vary from zero to the Rayleigh wave speed under identical loading conditions but depending upon microscopic rules. These conclusions are obtained from simple solvable models, and from techniques that make it possible to extrapolate reliably from small numerical calculations to the macroscopic limit. These techniques are described in some detail. Second: in practice, should any of these effects be visible in real laboratory samples? The answer to this second question is less clear. The qualitative phenomena exhibited by simple models are observed routinely in the fracture of brittle crystals. However, the correspondence between computations in perfect two-dimensional numerical samples at zero temperature and imperfect three-dimensional laboratory specimens at nonzero temperature is not simple. This paper reports on computations involving nonzero temperature, and irregular crack motion that indicate both strengths and weaknesses of two-dimensional microscopic modeling.     

A hybrid method to assess interface debonding by finite fracture mechanics

Adhesive connections are potentially weak locations in many kinds of engineering structures. Since adhesive joints can be regarded locally as bimaterial notches, the assessment of the hazard of crack nucleation, initiation and propagation in the vicinity of bimaterial notches and the reliability of the junctions is an important problem. An essential requirement in this context is a sufficient criterion for crack nucleation. The present contribution proposes a modified approach based on Leguillon’s hypothesis in order to provide a feasible criterion. A crack at a notch is assumed to be initiated and to grow if and only if both the released energy and the local stresses exceed critical values. Thus, simulating virtual crack growth along an interface of two dissimilar bonded materials, the integrity of the bond is revisable. The approach enables the determination of characteristic lengths for freshly nucleated cracks forming the base for any further integrity assessment. As an example, the concept is applied to the analysis of an adhesive bond of metallic and ceramic materials under severe thermal loading conditions as they occur, among other examples, in high temperature fuel cell technology. It is shown that the failure hazard of the adhesive joint can be reduced significantly by an appropriate local design.     

Reprint of “Cracks in inhomogeneous materials: Comprehensive assessment using the configurational forces concept”

The concept of configurational forces is applied to demonstrate the application of the concept of configurational forces in the numerical simulation of crack growth and fracture processes. It is shown, how material property variations at an interface affect the crack driving force and how the criterion of maximum dissipation is used to evaluate the direction of crack propagation. Fatigue crack growth experiments were conducted on diffusion welded bimaterial specimens consisting of a high-strength steel and soft ARMCO iron. Two cases are considered: (1) specimens with an interface perpendicular to the initial crack orientation, and (2) specimens with an inclined interface. The numerical simulation with the concept of configurational forces show that not only variations of the elastic modulus and/or the yield stress have a tremendous influence on the crack driving force, the crack growth rate, and the curvature of the crack path, but also the thermal residual stresses that resulted from a rather small difference of the coefficient of thermal expansion.     

Prediction of energy release rates for crack growth using FEM-based energy derivative technique

This paper presents a new method, named energy derivative technique, to calculate energy release rate for a variety of crack growth scenarios. The new method is based on energy conservation principle for crack growth, and is applicable to crack development in any quasi-static condition in which dynamic energy for crack growth is negligible. The method has the advantage over existing finite element-based methods in that the former does not require an elaborate fine mesh in the vicinity of a crack tip, and is not limited to linear deformation behaviour. Several case studies are presented to demonstrate validity of the method, which are (i) growth of penny-shaped crack for linear elastic fracture behaviour, (ii) crack growth in rubber sheet under tension for nonlinear elastic fracture behaviour, (iii) delamination in end-notched flexure specimen with friction, and (iv) crack growth with plastic deformation in double-edge-notched plate under tension. Results from these case studies show excellent agreement with data available in the literature, which were determined using either analytical or other FEM-based techniques.     

A discrete constitutive model for transverse and shear damage of symmetric laminates with arbitrary stacking sequence

A damage constitutive model in conjunction with a 2-D finite element discretization is presented for predicting onset and evolution of matrix cracking and subsequent stiffness reduction of symmetric composite laminates with arbitrary stacking sequence subjected to membrane loads. The formulation uses laminae crack densities as the only state variables, with crack growth driven by both mechanical stress and residual stress due to thermal expansion. The formulation is based on fracture mechanics in terms of basic materials properties, lamina moduli, and critical strain energy release rates G_IC and G_IIC, only. No additional adjustable parameters are needed to predict the damage evolution. Spurious strain localization and mesh size dependence are intrinsically absent in this formulation. Thus, there is no need to define a characteristic length. Comparison of model results to experimental data is presented for various laminate stacking sequences. Prediction of crack initiation, evolution, and stiffness degradation compare very well to experimental data.     

An incremental energy minimization state update algorithm for 3D phenomenological internal‐variable SMA constitutive models based on isotropic flow potentials

An incremental energy minimization approach for the solution of the constitutive equations of 3D phenomenological models for shape memory alloys (SMA) is presented. A robust algorithm for the solution of the resulting nonsmooth constrained minimization problem is devised, without introducing any regularization in the dissipation or chemical terms. The proposed algorithm is based on a thorough detection of the singularities relevant to the incremental energy formulation, in conjunction with a Newton–Raphson method equipped with a Wolfe line search dealing with regular solutions. The saturation constraint on the transformation strain is treated by means of an active set strategy, thus avoiding any need for a two‐stage return‐mapping algorithm. A parametrization of the saturation constraint manifold is introduced, thus reducing the problem dimensionality, with improved computational performance. Finally, an efficient algorithm for the computation of the dissipation function in terms of Haigh–Westergaard invariants is presented, allowing for a quite general choice of deviatoric transformation functions. Numerical results confirm the robustness and consistency of the proposed state update algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

A procedure for superposing linear cohesive laws to represent multiple damage mechanisms in the fracture of composites

The relationships between a resistance curve (R-curve), the corresponding fracture process zone length, the shape of the traction-displacement softening law, and the propagation of fracture are examined in the context of the through-the-thickness fracture of composite laminates. A procedure for superposing linear cohesive laws to approximate an experimentally-determined R-curve is proposed. Simple equations are developed for determining the separation of the critical energy release rates and the strengths that define the independent contributions of each linear softening law in the superposition. The proposed procedure is demonstrated for the longitudinal fracture of a fiber-reinforced polymer-matrix composite. It is shown that the R-curve measured with a Compact Tension Specimen test cannot be predicted using a linear softening law, but can be reproduced by superposing two linear softening laws.     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

Finite fracture mechanics: A coupled stress and energy failure criterion

The aim of the present paper is to introduce a new failure criterion in the framework of Finite Fracture Mechanics. Criteria assuming that failure of quasi-brittle materials is affected by stress or energy flux acting on a finite distance in front of the crack tip are widely used inside the scientific community. Generally, this distance is assumed to be small compared to a characteristic size of the structure, i.e. to any length describing the macroscale. A key point of the present paper is to analyse what happens if the smallness assumption does not hold true. The proposed approach relies on the assumption that the finite distance is not a material constant but a structural parameter. Its value is determined by a condition of consistency of both energetic and stress approaches. The model is general. In order to check its soundness, an application to the strength prediction for three point bending tests of various relative crack depths and of different sizes is performed. It is seen that, for the un-notched specimens, the present model predicts the same trend as the Multi-Fractal Scaling Law (MFSL). Finally, a comparison with experimental data available in the literature on high strength concrete three point bending specimens is performed, showing an excellent agreement. It is remarkable to observe that the method presented herein is able to provide the fracture toughness using test data from un-notched specimens, as long as the range of specimen sizes is broad enough.     

Mesh size sensitivity of the combined FEM/DEM fracture and fragmentation algorithms

In the context of the combined finite-discrete element method a number of algorithms aimed at modelling progressive failure, fracture and fragmentation of solids under extreme loading conditions have been proposed in the last few years. The fracture patterns obtained by recently proposed algorithms are impressive. However, sensitivity of these algorithms to both mesh size and mesh orientation remains an open question. The aim of this paper is to further investigate sensitivity to mesh size of the recently proposed so called combined single and smeared crack model. Sensitivity to mesh orientation is outside scope of this paper and is discussed only qualitatively.     

A framework for fracture modelling based on the material forces concept with XFEM kinematics

A theoretical and computational framework which covers both linear and non‐linear fracture behaviour is presented. As a basis for the formulation, we use the material forces concept due to the close relation between on one hand the Eshelby energy–momentum tensor and on the other hand material defects like cracks and material inhomogeneities. By separating the discontinuous displacement from the continuous counterpart in line with the eXtended finite element method (XFEM), we are able to formulate the weak equilibrium in two coupled problems representing the total deformation. However, in contrast to standard XFEM, where the direct motion discontinuity is used to model the crack, we rather formulate an inverse motion discontinuity to model crack development. The resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion, which may be used to simulate linear as well as non‐linear fracture in one and the same formulation. In fact, the linear fracture formulation can be retrieved from the non‐linear cohesive zone formulation simply by confining the cohesive zone to the crack tip. These features are clarified in the two numerical examples which conclude the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

Simple bond energy approach for non-destructive measurements of the fracture toughness of brittle materials

A simple bond breakage model for computing the fracture surface energy and toughness of a wide variety of brittle materials is presented and correlated against values reported in the literature for the single crystalline forms of these same materials. The correlation shows that this simple model can provide an accurate estimate for both the fracture surface energy and toughness of these materials. It is further shown that this simple model can be extended to amorphous materials with reasonable accuracy by normalizing the fracture surface energy of the crystalline material by the ratio of the density of the amorphous material vs. the density of the single crystalline material. Applications to thin film low-k materials and capabilities for non-destructive measurements are also discussed.     

Evaluation of directional mesh bias in concrete fracture simulations using continuum damage models

In the present comparative study, we investigate the influence of directional mesh bias on the results of failure simulations performed with isotropic and anisotropic damage models. Several fracture tests leading to curved crack trajectories are simulated on different meshes. The isotropic damage model with a realistic biaxial strength envelope for concrete is highly sensitive to the mesh orientation, even for fine meshes. The sensitivity is reduced if the definition of the damage-driving variable (equivalent strain) is based on the modified von Mises criterion, but the corresponding biaxial strength envelope is not realistic for concrete. The anisotropic damage models used in this study capture reasonably well arbitrary crack trajectories even if the biaxial strength envelope remains close to typical experimental data. Their superior performance can be at least partially attributed to their ability to capture dilatancy under shear, which is revealed by a comparative analysis of the behavior of individual models under shear with restricted or free volume expansion.     

Numerical study of mixed-mode fracture in concrete

In the present paper, a finite element code based on the microplane model for concrete is used for the analysis of typical mixed-mode geometries: a Single-Edge-Notched beam, a Double-Edge-Notched specimen and a Dowel-Disk specimen. A local smeared fracture finite element analysis is carried out. As a regularization procedure, the crack band method is used. The principal objective of the study was to investigate whether the smeared fracture finite element code is able to predict mixed-mode fracture of concrete with no optimisation of the material model parameters. Comparison between experimental and numerical results shows that the used code predicts structural response and crack patterns realistically for all cases investigated. Moreover, it is shown that for most of the studied geometries a mixed-mode fracture mechanism dominates at crack initiation, however, with increase of the crack length mode-I fracture becomes dominant and the specimens finally failed in mode-I fracture.