Theory and numerics for finite deformation fracture modelling using strong discontinuities

A general finite element approach for the modelling of fracture is presented for the geometrically non‐linear case. The kinematical representation is based on a strong discontinuity formulation in line with the concept of partition of unity for finite elements. Thus, the deformation map is defined in terms of one continuous and one discontinuous portion, considered as mutually independent, giving rise to a weak formulation of the equilibrium consisting of two coupled equations. In addition, two different fracture criteria are considered. Firstly, a principle stress criterion in terms of the material Mandel stress in conjunction with a material cohesive zone law, relating the cohesive Mandel traction to a material displacement ‘jump’ associated with the direct discontinuity. Secondly, a criterion of Griffith type is formulated in terms of the material‐crack‐driving force (MCDF) with the crack propagation direction determined by the direction of the force, corresponding to the direction of maximum energy release. Apart from the material modelling, the numerical treatment and aspects of computational implementation of the proposed approach is also thoroughly discussed and the paper is concluded with a few numerical examples illustrating the capabilities of the proposed approach and the connection between the two fracture criteria. Copyright © 2005 John Wiley & Sons, Ltd.     

Fracture mechanics‐based progressive damage modelling of adhesively bonded fibre‐reinforced polymer joints

A quasi‐static progressive damage model for prediction of the fracture behaviour and strength of adhesively bonded fibre‐reinforced polymer joints is introduced in this paper. The model is based on the development of a mixed‐mode failure criterion as a function of a master R‐curve derived from the experimental results obtained from standard fracture mechanics joints. Consequently, the developed failure criterion is crack‐length and mode‐mixity dependent, and it takes into account the contribution of the fibre‐bridging effect. Energy release rate values for adhesively bonded double‐lap joints are obtained by using the virtual crack closure technique method in a finite element model, and the numerically obtained strain energy release rate is compared to the critical strain energy release rate given by the mixed‐mode failure criterion. The entire procedure is implemented in a numerical algorithm, which was successfully used for predicting the strength and R‐curve response of adhesively bonded double‐lap structural joints made of pultruded glass fibre‐reinforced polymers and epoxy adhesives.     

Investigation on the mechanical properties and fracture toughness of graphite

In the present study, mechanical properties and fracture toughness of graphite as a brittle material were investigated. At first, some specimens were examined in two perpendicular directions to derive Young's modulus and ultimate tensile strength. Then, graphite fracture toughness tests were conducted using some three‐point bending specimens with a sharp machined V‐notch by two different methods. The first method is based on the applied force at the moment of fracture, and the second one uses energy released during the test. Moreover, a technique was adopted to reduce differences between the two methods. It was observed that considering the effect of dehydration of the specimens, the fracture toughness was reduced by about 8%. Finally, crack growth simulation of the experiment was performed and indicated that finite element analysis predicts about 25% lower crack length values when critical energy release rate is utilized as a crack growth criterion instead of fracture toughness. In other words, the required input displacement for crack growth would be overestimated by using the critical energy release rate criterion.     

Steady-state crack growth in rubber-like solids

The fracture toughness of rubber-like materials depends on several factors. First there is the surface energy required to create new crack surface at the crack tip. Second, a significant amount of energy is dissipated through viscoelastic processes in the bulk material around the crack tip. Third, if the crack propagates very rapidly, inertia effects will come into play and contribute to the fracture toughness. In the present study, a computational framework for studying high-speed crack growth in rubber-like solids under conditions of steady-state is proposed. Effects of inertia, viscoelasticity and finite strains are included. The main purpose of the study is to study the contribution of viscoelastic dissipation to the total work of fracture required to propagate a crack in a rubber-like solid. The model was fully able to predict experimental results in terms of the local surface energy at the crack tip and the total energy release rate at different crack speeds. In addition, the predicted distributions of stress and dissipation around the propagating crack tip are presented.     

Formulation and implementation of strain rate-dependent fracture toughness in context of the phase-field method

The phase-field approach is a promising technique for the realistic simulation of brittle fracture processes, both in quasi-static and transient analysis. Considering fast loading, experimental evidence indicates a strong relationship between the rate of strain and the material's resistance against fracture, which can be considered by a dynamic increase factor for the strength of the material. The paper at hand presents a novel approach within the framework of phase-field models for brittle fracture. A rate-dependent fracture toughness is formulated as a function of the rate of crack driving strain components, which results in higher strength for faster loading. Beside the increased amount of energy necessary to evolve a crack at a high strain rate loading situation, the model incorporates quasi-viscous stress-type quantities that are not directly related to the formation of the crack and exist only in the phase-field transition zone between broken and sound material. The governing strong form equations for a transient simulation are derived and the relevant information for an implementation of the model into a finite element code is outlined in detail. The performance of the model is demonstrated for static and dynamic benchmark simulations and for a comparison to experimental findings.     

FEM Modeling of Crack Propagation in a Model Multiphase Alloy

In this paper, several widely applied fracture criteria were first numerically examined and the crack-tip-region Jntegral criterion was confirmed to be more applicable to predict fracture angle in an elastic-plastic multiphase material. Then, the crack propagation in an idealized an elastic-plastic finite element method. The variation dendritic two-phase AI-7%Si alloy was modeled using of crack growth driving force with crack extension was also demonstrated. It is found that the crack path is significantly influenced by the presence of α-phase near the crack tip, and the crack growth driving force varies drastically from place to place. Lastly, the simulated fracture path in the two-phase model alloy was compared with the experimentally observed fracture path.     

The wave propagation and vibrational energy flow characteristics of a plate with a part-through surface crack

In the view of structure-borne sound, vibrational wave and energy flow characteristics of infinite thin plate of finite width with a part-through surface crack are investigated. The case of an all-over part-through crack parallel to the finite side of the plate is considered. The crack is modeled as a line spring and the flexibility of the spring is deduced from the relationship between the strain energy and stress intensity factor in fracture mechanics. The responses of both intact plate and cracked plate under the excitation of harmonic force are reduced with wave propagation approach, then the input energy flow and transmitted energy flow of intact and cracked plates are calculated. The results show that the vibrational energy flow of cracked plate is highly related to the depth and location of the part-through crack. The location and the depth of the crack can be identified by the contour lines of normalized input energy flow with different driving frequencies. The research provides theoretical basis for the crack detection by measuring the vibrational energy flow in cracked plate structures.     

An engineering approach to safety assessment for non-J-controlled crack growth

The -integral has been proposed as a new fracture parameter for non-J-controlled crack growth in previous studies. In this paper, the -integral is investigated to assess the safety of structures with defects. The estimation formula for the -integral is presented to calculate the crack driving force by referring to the engineering estimation expression of the J-integral. The material resistance curves J_R−Δa and , of A533B material, are obtained by tests and nonlinear finite element analyses. The -integral failure criterion for non-J-controlled crack growth is assumed in the paper according to the crack driving force and the material resistance curves. The method of admissible stress curve, i.e. the stress-crack length curve, is developed to simplify the conventional engineering assessment procedure by following the failure criterion for non-J-controlled crack growth. A new engineering safety assessment approach is proposed to assess the ductile fracture instability of flawed structures for non-J-controlled crack growth by the method of an admissible stress curve. Two examples for the compact tension configuration and the cylinder with a surface crack under internal pressure are examined by using the new engineering assessment approach. The new engineering safety assessment approach obviously simplifies the conventional engineering assessment procedure. By comparing with the tests, the results from these two examples show that this new engineering safety assessment approach for non-J-controlled crack growth gives a reliable prediction for the ultimate load capacity of flawed structures.     

A treatment of interfacial cracks in the presence of friction

Frictional sliding on interface crack surfaces results in weak crack tip stress singularity and zero strain energy release rate. A fracture criterion based on finite extension strain energy release rate, is proposed to capture the intrinsic fracture toughness. The finite extension strain energy release rate is shown to represent the magnitude of the singular stress field. Numerical simulations of a center crack in a bimaterial infinite medium under remote shear as well as fiber pull-out and push-out in composite materials are presented to illustrate the frictional effect in both small and large scale contacts near the crack tip.     

Characterisation of elastomers fracture behavior by energetic parameters

The knowledge of fracture behavior of elastomers necessitates the comprehension of crack initiation and propagation phenomena which pose difficulties related to the deformation of elastomers. The reliability of elastomer materials is linked to their resistance to rupture. This resistance can be evaluated using the global approach of fracture mechanics. The objective of this work is to numerically analyze by finite element method the characterization of rupture behavior of these materials on the basis of energetic parameters. Consideration is given to the evolution of the deformation energy density to quantify the energy of tear of a real material identified by hyper-elastic material models.     

Ductile crack propagation with the strain energy density criterion

The strain energy density criterion is used to characterize subcritical crack growth in a thin aluminum alloy sheet undergoing general yielding. A finite element analysis which incorporates both material and geometrical nonlinear behaviors of the cracked sheets is developed to predict fracture loads at varying crack growth increments. The predicted results are in excellent agreement with those measured experimentally, thus confirming the validity of the strain energy density criterion for characterizing ductile crack propagation.     

A thermodynamic study of materials representable by integral expansions

A thermodynamic formulation is developed for nonlinear compressible viscoelastic materials and is used to quantitatively study the thermodynamic states associated with fracture. The Helmholtz free energy is assumed to be approximated by a fourth order multiple integral expansion on the histories of the Lagrangian strain tensor and temperature, and the first and second laws are then utilized to develop a constitutive equation and dissipation function for the material. Simplified expressions are obtained for the special cases of slow motions and long term steady flows. An experimental study is then made on equi-temperature metamorphosed snow, a nonlinear viscoelastic material, to determine the states associated with fracture for a variety of deformation paths. In view of the experimental study, the author is able to conclude that this approach to fracture investigation can produce much insight on the strength properties and may be instrumental in the formulation of a useful fracture criterion of materials for which the fracture condition is history dependent. Additionally, due to the thermodynamic nature of this approach, the information gathered could be useful in investigations of crack growth in nonlinear viscoelastic materials such as snow or even in the formulation of structural constitutive equations.     

Physics‐based multiscale damage criterion for fatigue crack prediction in aluminium alloy

In this paper, a physics‐based multiscale approach is introduced to predict the fatigue life of crystalline metallic materials. An energy‐based and slip‐based damage criterion is developed to model two important stages of fatigue crack initiation: the nucleation and the coalescence of microcracks. At the microscale, a damage index is developed on the basis of plastic strain energy to represent the growing rate of a nucleated microcrack. A statistical volume element model with high computational efficiency is developed at the mesoscale to represent the microstructure of the material. Also, the formation of a major crack is captured by a coalescence criterion at mesoscale. At the macroscale, a finite element analysis of selected test articles including lug joint and cruciform is conducted with the statistical volume element model bridging two scale meshes. A comparison between experimental and simulation results shows that the multiscale damage criterion is capable of capturing crack initiation and predicting fatigue life.     

A Finite Element Analysis of Creep-Crack Growth in Viscoelastic Media

In this paper, the effects of viscoelastic characteristics, on the creep-crack growth process are studied through a finite element approach. The general approach of an independent path integral is extended to crack propagation. Afterwards, fracture parameters are computed through a coupling process with an incremental viscoelastic formulation. Finally, numerical examples are presented in order to demonstrate the independence of the integration domain and the possibility of evaluating fracture characteristics which can be energetic (energy release rate) and local in the vicinity of the crack tip (stress and crack opening intensity factors).     

A new M-integral parameter for mixed-mode crack growth in orthotropic viscoelastic material

This paper deals with a new independent path integral which provides the mixed-mode during a creep crack growth process in viscoelastic orthotropic media. The developments are based on an energetic approach using conservative laws. The mixed-mode fracture separation is introduced according to the generalization of the virtual work principle. The fracture algorithm is implemented in a finite element software and coupled with an incremental viscoelastic formulation and an automatic crack growth simulation. This M-integral provides the computation of stress intensity factors and energy release rate for each fracture mode. A numerical validation, in terms of energy release rate and stress intensity factors, is carried out on a CTS specimen under mixed-mode loading for different crack growth speeds.     

Coupled hygro-thermo-viscoelastic fracture theory

A coupled hygro-thermo-viscoelastic fracture theory is developed for quasi-static and dynamic crack propagation in viscoelastic materials subject to combined mechanical loading and hygrothermal environmental exposure based on fundamental principles of thermodynamics. The Helmholtz free energy is taken to be a functional of the histories of strain, temperature and fluid concentration with the crack parameter being introduced as an internal state variable. A thermodynamically consistent time-dependent fracture criterion for crack propagation in the presence of thermally and mechanically assisted fluid transport is obtained from the global energy balance equation and the requirement of non-negativity of the global energy dissipation rate, which is generally applicable to both quasi-static and dynamic loading and both isothermal/isohumidity and non-isothermal/non-isohumidity conditions with classic fracture criteria as special cases. On the basis of the developed theory, the generalized energy release rate method, the generalized contour integral method and the extended essential work of fracture method are proposed for fracture characterization of load-carrying viscoelastic materials in hygrothermal environments, and the interrelation of these methods and their correlation with conventional methods and existing models, simulations and experiments are discussed.     

An engineering approach to fatigue analysis based on elastic‐plastic fracture mechanics

Nowadays cast iron components are widely used in highly stressed structures. Component lifetime is strongly influenced by inhomogeneities caused by the material's microstructure and the manufacturing process (graphite particles, (micro‐)shrinkage pores, inclusions). Inhomogeneities often act as a fatigue crack starter. Lifetime until failure may be divided into stages for crack initiation, short and long crack growth. Initiation of a crack of technical size (a ≈ 1mm) is often dominated by the growth of short cracks. The paper presents an approach to analyse the mechanically short fatigue crack growth based on elastic‐plastic fracture mechanics considering the closure behaviour of short cracks. The effective J‐integral range is used as a crack driving force. Finite element analysis results as well as analytical solutions to approximate the crack driving force are presented. The application of the approach is successfully demonstrated for cast iron material EN‐GJS‐400‐18‐LT using data from fatigue tests, microstructure and fracture surface analyses to assess the fatigue life.     

Fracture under compression: the direction of initiation

The preferential orientation for the initiation of a crack is the one which gives the maximum strain energy reduction for a given crack length. This proposed criterion is a logical extension of the maximum energy release rate criterion. It makes no assumptions on the configuration, the homogeneity, the stress condition on the crack faces, or the material response, consequently it is applicable to the usual engineering cases as well as to cases under compression and/or high confining pressures such as obtain inside the Earth. Numerical results for brittle materials (rocks) agree with laboratory and field data, and show that the criterion is an improvement over the empirical and approximate Coulomb-Mohr criterion which has been used for compressive fracture problems for more than 200 years. They also show that our method can be used in cases where it is not a priori evident whether the fracture will remain closed or will open.The mathematical formulation of the criterion is approached by way of constrained optimization, and the solution is proven to exist uniquely. The numerical implementation is based on a finite element scheme. An iterative method is employed to handle the material and geometric non-linearities.     

Application of energy density theory on finite cracked bodies

Abstract

A new concept of the energy release rate of a finite cracked body is proposed. Considering the global view of the strain energy density field, the new fracture parameter presented here is different from the conventional energy release rate that only depends on the stress field around the crack tip but neglects the influences induced by the boundary conditions on the far field. Based on the hypothesis of the energy density theory, fracture initiation and termination, respectively can be predicted by the local and global relative minima of the strain energy density function. The new energy release rate is then defined as the integration of the strain energy density along the fracture trajectory from the initiation point to the destination point. The results show that the difference between the new and the conventional energy release rate becomes more pronounced if the material has a large core region (or the material is more ductile) and if the height‐width ratio of a finite cracked plate is comparatively small.