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.     

Finite element simulations of notch tip fields in magnesium single crystals

Recent experiments using three point bend specimens of Mg single crystals have revealed that tensile twins of \(\{10\bar{1}2\}\) -type form profusely near a notch tip and enhance the fracture toughness through large plastic dissipation. In this work, 3D finite element simulations of these experiments are carried out using a crystal plasticity framework which includes slip and twinning to gain insights on the mechanics of fracture. The predicted load–displacement curves, slip and tensile twinning activities from finite element analysis corroborate well with the experimental observations. The numerical results are used to explore the 3D nature of the crack tip stress, plastic slip and twin volume fraction distributions near the notch root. The occurrence of tensile twinning is rationalized from the variation of normal stress ahead of the notch tip. Further, deflection of the crack path at twin–twin intersections observed in the experiments is examined from an energy standpoint by modeling discrete twins close to the notch root.     

Wave-crack interaction in finite elastic bodies

This paper continues studies in Lalegname et al. (Int J Fract 152:97–125, 2008) on crack propagation in a bounded linear elastic body under the influence of incident waves. In Lalegname et al. (2008) we have considered shear waves, whereas in this paper we discuss the influence of plane elastic waves to a running crack. Actually, the time dependent problem is formulated in a two-dimensional current cracked configuration by a system of linear elasto-dynamic equations. In order to describe the behaviour of the elastic fields near the straight crack tip, we transform these equations to a reference configuration and derive the dynamic stress singularities. Furthermore, we assume that an energy balance law is valid. Exploiting the knowledge on the singular behaviour of the crack fields, we derive from the energy balance law a dynamic energy release rate. Comparing this energy release rate with an experimentally given fracture toughness we get an ordinary differential equation for the crack tip motion. We present first numerical simulations for a Mode I crack propagation.     

Assessment of energy dissipation during mixed-mode delamination growth using cohesive zone models

The use of cohesive elements to simulate delamination growth involves modeling the inelastic region existing ahead of the crack tip. Recent numerical and experimental findings indicate that the mixed-mode ratio varies at each material point within the inelastic region ahead of the crack tip during crack propagation, even for those specimens whose mixed-mode ratio is expected to be constant. Although the local variation of the mode mixity may adversely affect the predicted numerical results, most existing formulations do not take it into account. In this work, the mode-decomposed J-integral is implemented as a finite element post-processing tool to obtain the strain energy release rates and the mixed-mode ratio of the inelastic region as a whole, allowing the assessment of crack propagation in terms of energy dissipation and mixed-mode ratio computation. Different cohesive elements are assessed with this method.     

Crack tip fields in a viscoplastic solid: monotonic and cyclic loading

The asymptotic stress and strain field near the tip of a plane strain Mode I stationary crack in a viscoplastic material are investigated in this work, using a unified viscoplastic model based on Chaboche (Int J Plast 5(3):247–302, 1989). Asymptotic analysis shows that the near tip stress field is governed by the Hutchinson–Rice–Rosengren (HRR) field (Hutchinson in J Mech Phys Solids 16(1):13–31, 1968; Rice and Rosengren in J Mech Phys Solids 16(1):1–12, 1968) with a time dependent amplitude that depends on the loading history. Finite element analysis is carried out for a single edge crack specimen subjected to a constant applied load and a simple class of cyclic loading history. The focus is on small scale creep where the region of inelasticity is small in comparison with typical specimen dimensions. For the case of constant load, the amplitude of the HRR field is found to vanish at long times and the elastic K field dominates. For the case of cyclic loading, we study the effect of stress ratio on inelastic strain and find that the strain accumulated per cycle decreases with stress ratio.     

Recrystallization-etch approach to study the plastic energy absorbtion at crack initiation and extension of pressure-vessel steel A533B-1

To evaluate the elastic-plastic fracture toughness parameter of nuclear pressure-vessel steel A533B-1, a newly developed technique (the recrystallization-etch technique) for plastic strain measurement was applied to different sizes of compact tension specimens with a crack length/specimen width of 0.6–0.5 that were tested to generate resistance curves for stable crack extensions. By means of the recrystallization-etch technique, the plastic energy dissipation or work done within an intense strain region at the crack tip during crack initiation and extension was measured experimentally. Furthermore, the thickness effects on this crack tip energy dissipation rate were examined in comparison with other fracture-parameter J integrals. Thickness effects on critical energy dissipation and energy dissipation rate during crack extension were obtained and the energy dissipation rate dW _p/da in the mid-section shows a constant value irrespective of specimen geometry and size, which can be used as a fracture parameter or crack resistance property.     

Modeling of fatigue crack growth threshold development for a microstructurally small crack

A method for predicting the fatigue crack growth threshold using finite element analysis is investigated. The proposed method consists of monitoring the plastic strain hysteresis energy dissipation in the crack tip plastic zone, with the threshold being defined in terms of a critical value of this dissipated energy. Two-dimensional plane-strain elastic-plastic finite element analyses are conducted to model fatigue crack growth in a middle-crack tension M(T) specimen. A single-crystal constitutive relationship is employed to simulate the anisotropic plastic deformation near the tip of a microstructurally small crack without grain boundary interactions. Variable amplitude loading with a continual load reduction is used to generate the load history associated with fatigue crack growth threshold measurement. Load reductions with both constant load ratio R and constant maximum stress intensity K_max are simulated. In comparison with a fixed K_max load reduction, a fixed R load reduction is predicted to generate a 35% to 110% larger fatigue crack growth threshold value.     

Numerical analysis of dynamic crack propagation in rubber

In the present paper, dynamic crack propagation in rubber is analyzed numerically using the finite element method. The problem of a suddenly initiated crack at the center of stretched sheet is studied under plane stress conditions. A nonlinear finite element analysis using implicit time integration scheme is used. The bulk material behavior is described by finite-viscoelasticity theory and the fracture separation process is characterized using a cohesive zone model with a bilinear traction-separation law. Hence, the numerical model is able to model and predict the different contributions to the fracture toughness, i.e. the surface energy, viscoelastic dissipation, and inertia effects. The separation work per unit area and the strength of the cohesive zone have been parameterized, and their influence on the separation process has been investigated. A steadily propagating crack is obtained and the corresponding crack tip position and velocity history as well as the steady crack propagation velocity are evaluated and compared with experimental data. A minimum threshold stretch of 3.0 is required for crack propagation. The numerical model is able to predict the dynamic crack growth. It appears that the strength and the surface energy vary with the crack speed. Finally, the maximum principal stretch and stress distribution around steadily propagation crack tip suggest that crystallization and cavity formation may take place.     

Crack tip deformation fields and fatigue crack growth rates in Ti–6Al–4V

In this paper we present an overview of experimental and modelling studies of fatigue crack growth rates in aerospace titanium alloy Ti–6Al–4V. We review work done on the subject since the 1980s to the present day, identifying test programmes and procedures and their results, as well as predictive approaches developed over this period. We then present the results of some of our recent experiments and simulations. Fatigue crack growth rates (FCGRs) under constant applied load were evaluated as a function of crack length, and the effect of overload (retardation) was considered. Crack opening was measured during cycling using digital image correlation, and residual stress intensity factor was determined using synchrotron X-ray diffraction mapping. Modelling techniques used for the prediction of FCGRs are then reviewed, and an approach based on the analysis of energy dissipation at the crack tip is proposed. Finally, directions for further research are identified.     

Evaluating the heat energy dissipated in a small volume surrounding the tip of a fatigue crack

Fatigue crack initiation and propagation involve plastic strains that require some work to be done on the material. Most part of this irreversible energy is dissipated as heat and consequently the material temperature increases. The heat being an indicator of the intense plastic strains occurring at the tip of a propagating fatigue crack, the hypothesis is formed that it can be used to assess the fatigue damage accumulation rate of cracked components. Moreover, the heat energy at the crack tip is averaged according to Neuber’s finite particle concept. The aim of the present paper is to present the theoretical framework and the corresponding experimental technique to evaluate the heat energy dissipated in a structural volume surrounding the crack tip. The shape and size of the structural volume have been assumed according to the literature, even though the definition of the structural volume size of the analysed material in a fatigue sense is not the scope of the present paper. The proposed experimental technique to evaluate the averaged heat energy is based on the radial temperature profiles measured around the crack tip by means of an infrared camera. The temperature fields measured within few millimetres from the crack tip have been compared successfully with existing analytical solutions.     

On the Mechanism of Energy Dissipation in a Fatigue Crack

Experimental studies of specimens with a Mode-I edge crack in bending vibration have demonstrated that energy dissipation in the fatigue crack is mainly due to an elastoplastic zone at the crack tip. The absolute level of energy dissipation in the crack is unambiguously governed by the stress intensity factor range and is independent of specimen dimensions and crack location.     

On J-integral and potential energy variation

The divergence theorem has been used in a region containing the crack tip to derive the J-integral from the potential energy variation in most fracture mechanics books. Such a derivation is flawed because of the crack tip stress singularity. The present study describes a rigorous and straightforward derivation of the J-integral from the potential energy variation with crack extension by carefully addressing the effect of the crack tip singularity.     

On the dependence of crack surface morphology and energy dissipation on microstructure in ductile plate tearing

The two distinct tearing mechanisms observed in ductile metal plates are the void-by-void advance of the crack tip, and the simultaneous interaction of multiple voids on the plane ahead of the crack tip. Void-by-void crack advance, which leads to a cup-cup crack surface morphology, is the dominant mechanism if the plate contains a low number of small void nucleation sites (i.e., second phase particles). Conversely, a large number and/or size of nucleation sites trigger the simultaneous interaction of multiple voids resulting in a slanted crack. The present work aims to provide further insight into the parameters controlling the mechanisms and energy dissipation of plate tearing by focusing on the shape and, thereby, the orientation of the nucleation sites. The study uses a two-dimensional plane strain finite element domain to model the cross section of a plate, subject to mode I tearing, with discretely modeled, randomly distributed, finite-sized elliptic void nucleation sites. The developed finite element setup can capture the dependence of the crack surface morphology on the microstructure of the plate. The simulation results confirm that cup-cup crack propagation develops by intense plastic straining throughout the thinning region of the plate. Conversely, slanted and cup-cone cracks propagate in thin localized shear deformation bands. The energy dissipation is, therefore, greater for cup-cup cracks. The study demonstrates that the damage-related microstructure has a significant role in determining the overall hardening capacity of a plate, which in turn dictates the tearing mode and energy.

     

A multiscale parametric study of mode I fracture in metal-to-metal low-toughness adhesive joints

The failure of adhesively-bonded joints, consisting of metallic adherends and epoxy-based structural adhesive with a relatively low toughness ~200 J/m², has been studied. The failure was via quasi-static mode I, steady-state crack propagation and has been modelled numerically. The model implements a ‘top-down approach’ to fracture using a dedicated steady-state, finite-element formulation. The damage mechanisms responsible for fracture are condensed onto a row of cohesive zone elements with zero thickness, and the responses of the bulk adhesive and of the adherends are represented by continuum elements spanning the full geometry of the joint. The material parameters employed in the model are first quantitatively identified for the particular epoxy adhesive of interest, and their validity is verified by comparison with experimental results. The model is then used to conduct a detailed study on the effects of (a) large variations in the geometrical configuration of the different types of specimens and (b) the adherend stiffness on the predicted value of the adhesive fracture energy, G _a . These numerical modelling results reveal that the adhesive fracture energy is a strong nonlinear function of the thickness of the adhesive layer, the other variables being of secondary importance in influencing the value of G _a providing the adhesive does not contribute significantly to the bending stiffness of the joint. These results which fully agree with experimental observations are explained in detail by identifying, and quantifying, the different sources of energy dissipation in the bulk adhesive contributing to the value of G _a . These sources are the locked-in elastic energy, crack tip plasticity, reverse plastic loading and plastic shear deformation at the adhesive/adherend interface. Further, the magnitudes of these sources of energy dissipation are correlated to the degree of constraint at the crack tip, which is quantified by considering the opening angle of the cohesive zone at the crack tip.     

Modeling of crack tip high inertia zone in dynamic brittle fracture

A phenomenological modification is proposed to the existing cohesive constitutive law of Roy and Dodds to model the crack tip high inertia region proposed by Gao. The modification involves addition of a term which is attributed to fracture mechanisms that result in high energy dissipation around the crack tip. This term is assumed to be a function of external energy per unit volume input into the system. Finite element analysis is performed on PMMA with constant velocity boundary conditions and mesh discretizations based on the work of Xu and Needleman. The cohesive model with the proposed dissipative term is only applied in the high inertia zone and the traditional Roy and Dodds model is applied on cohesive elements in the rest of the domain. The results show that crack propagates in three phases with a speed of 0.35c_R before branching, confirming experimental observations. The modeling of high inertia zone is one of the key aspects to understanding brittle fracture.     

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.     

Morphological aspects of fatigue crack propagation Part I—Computational procedure

In the present paper a simulation method is proposed for the evaluation of paths and lives of fatigue cracks. The simulation is based on an incremental crack extension procedure. At each increment the stress analysis ahead of a crack tip is carried out by the finite element method, and the next incremental crack-growth path is predicted by the first order perturbation method with the use of the local symmetry criterion. From the computational viewpoint, the step-by-step rezoning of finite element mesh subdivision is one of the most difficult processes of the simulation procedure. In order to overcome this difficulty, we shall use the modified quadtree method as an automatic mesh generation technique. Considerations are made for the proper mesh arrangement in the vicinity of a crack tip, where a special fine mesh pattern is embedded so that mixed mode stress intensity factors and the higher order coefficients of the near tip stress field parameters can accurately be obtained. Using the proposed method, we simulate the branched and curved fatigue crack growth in three-point-bending specimens. They show fairly good agreement with the experimental results. The simulation procedure is also applied to biaxially loaded cruciform joints.     

Numerical modeling of the nucleation of facets ahead of a primary crack under mode I + III loading

A numerical study of crack front segmentation under mode I + III loading is proposed. Facets initiation ahead of a parent crack is predicted through a tridimensional application of the coupled criterion. Crack initiation shape, orientation and spacing are determined for any mode mixity ratio by coupling a stress and an energy criterion using matched asymptotic expansions. The stress and the energy conditions are computed through a 3D finite element modeling of a periodic network of facets ahead of the parent crack. The initiation shape, loading and spacing of facets depend on the blunted parent crack tip radius. A good estimate of facet orientations is obtained based on the direction of maximum tensile stress. The facet shapes, determined using the stress isocontours, are qualitatively similar to those observed experimentally. The order of magnitude of numerical predictions of facets spacing is very close to experimental measurements.     

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.