A pseudo-elastic material damage model,part I: Strain energy density criterion

A failure criterion is presented which relates the strain energy density of the material to both yielding and fracture. Cumulative material damage throughout a structural component may be monitored and the relative influence of yielding and stable crack growth assessed. The criterion is demonstrated, using finite element analysis, for center cracked panel specimens differing by material toughness values. From crack growth increment predictions using the uniaxial stress-strain behavior of the material, the criterion predicts the critical value of the strain energy density factor S_c governing crack instability.     

Non-self-similar crack growth in elastic-plastic finite thickness plate

This work is concerned with non-self-similar crack growth in medium strength metal plates while the loading step, plate thickness and material properties are altered. The three-dimensional elastic-plastic finite element stress analysis is combined with the strain energy density criterion for modeling the material damage process from crack initiation to final global instability including the intervening stage of slow crack growth. Both inelastic deformation and crack growth are accounted for each increment of loading such that the redistribution of stresses and strains are made for each new crack profile. Numerical results are obtained for the center cracked plate configuration under uniform extension with twenty-seven (27) different combinations of specimen thickness, loading step and material type. The fracture toughness S_c being related to K_1c for three different materials are predicted analytically from the corresponding uniaxial tensile test data. Effective strain energy density factor and half crack length are defined so that the results can be compared with their two-dimensional counterparts. Crack growth resistance curves (R-curves) are constructed by plotting as a function of . The condition is found to prevail during slow crack growth. Translation and/or rotation of the lines can yield results other than those calculated and serve a useful purpose for scaling component size and test time. The minimum thickness requirement for the ASTM valid K_1c test is also discussed in connection with predictions based on the strain energy density criterion. The corresponding K_1c for smaller specimens that exhibit moderate ductility and nonlinearity can also be obtained analytically. In such cases, the influence of loading step can be significant and should not be neglected. Notwithstanding the shortcomings of the theory of plasticity, the qualitative features of non-self-similar crack growth are predicted by the strain energy density criterion. Any refinements on the analytical modeling of the material damage process would only affect the results qualitatively, a subject that is left for future investigation.     

A damage model for crack initiation and growth

A damage accumulation model is presented for the study of the problem of crack initiation and stable growth in an elastic-plastic material. A centre-cracked specimen subjected to a uniform stress perpendicular to the crack plane is considered. A coupled stress and failure analysis is performed by using a finite element computer program based on J₂-plasticity theory in conjunction with the strain energy density theory. After initial yielding, each material element follows a different equivalent uniaxial stress-strain behavior depending on the amount of energy dissipation by permanent deformation. A host of uniaxial stress-strain curves constituting parts of the same stress-strain curve were assigned to material elements for each increment of loading. The path-dependent nature of the onset of crack initiation and growth was revealed. The proposed model predicts faster crack growth rates than those obtained on the basis of a single uniaxial stress-strain curve and is closer to experimental observation.     

Prediction of mixed-mode fracture based on an elasto–plastic energy balance criterion

The maximum energy release rate criterion, i.e., G _max criterion, is commonly used for crack propagation analysis. This fracture criterion is based on the elastic macroscopic strength of materials. In the present investigation, however, the G _max criterion has been modified in order to accommodate the consideration of plastic strain energy. This modified criterion is extended to study the fatigue crack growth characteristics of mixed-mode cracks. To predict crack propagation due to fatigue loads, a new elasto–plastic energy model is presented. This new model includes the effects of material properties such as strain hardening exponent n, yield strength σ _y , and fracture toughness and stress intensity factor ranges. The results obtained are compared with those obtained using the commonly employed crack growth law and the experimental data.     

Crack-extension resistance of polycarbonate material

Crack-extension resistance for the polycarbonate material is examined by application of the strain energy density criterion and the incremental theory of plasticity. The energy state ahead of a slow moving crack in a three-point bend specimen is obtained for each load increment and used to determine the crack growth characteristics. The analytical results are displayed by plotting the strain energy density factor S as a function of crack length and compared with available experimental data on the polycarbonate material. Standard deviations and mean errors are computed for the experimentally measured and analytically determined values of S and are shown to be much lower than those based on the J-integral parameter. Modeling of the polycarbonate material by the theory of plasticity still remains much to be desired. Crack growth calculations are performed for a strain hardening parameter α = 0.85 that controls the proportion of isotropic and kinematic hardening. Nevertheless, the criterion dS/da = const. is shown to collate well with the experimental crack growth data.     

Evaluation of crack initiation angle under mixed mode loading at diverse strain rates

Crack initiation angle, under mixed mode loading at several strain rates, is analysed using an experimental–numerical approach. The physical phenomenon for the problem at hand is influenced by the local and global conditions. One of such factors is the strain rate at the crack tip. For this purpose, PMMA plates with centred angled cracks under mixed mode loading were tested. The strain rate at the neighbourhood of the crack tip before crack propagation was evaluated. Considering that this material is strain rate sensitive, the numerical models were calibrated with the modulus of elasticity measured in tension tests at the observed strain rates. Numerical evaluations were performed with the finite element method in conjunction with the volume energy density criterion. An improvement in the evaluation of the crack propagation angle was observed. In order to complete the analysis, the crack initiation angle was also evaluated with the strain energy density factor S, considering the mechanical properties of PMMA, as evaluated at the observed strain rates, and the stress intensity factors k₁ and k₂. Results are in agreement with those observed experimentally.     

Growth of crack caused by temperature gradients with change in surface insulation

This work is concerned with thermoelastic stress and failure analysis of a centrally cracked panel subjected to temperature gradients while the insulation on the crack surface is varied. The corresponding temperature and thermoelastic stress fields are obtained by application of the finite element method. According to the strain energy density criterion, the crack grows incrementally when the maximum of the minimum strain energy density function reaches a critical value for a given material. Crack growth resistance curves involving plots of the strain energy density factor S versus the half crack length a are developed for crack surfaces with varying degree of heat resistance. The resulting curves are straight lines satisfying the condition dS/da = const. and useful for determining combined influence of thermal loading and structural geometry that lead to global instability.     

Damage accumulation and crack growth in bilinear materials with softening: Application of strain energy density theory

A pseudo-elastic damage-accumulation model is developed by application of the strain energy density theory. The three-point bending specimen is analyzed to illustrate the crack growth characteristics according to a linear elastic softening constitutive law that is typical of concrete materials. Damage accumulation is accounted for by the decrease of elastic modulus and fracture toughness. Both of these effects are assessed by means of the strain energy density functions in the elements around a slowly moving crack. The rate of change of the strain energy density factor S with crack growth as expressed by the relation dS/da = constant is shown to describe the failure behavior of concrete. Results are obtained for different loading steps that yield different slopes of lines in an S versus a (crack length) plot. The lines rotate about the common intersect in an anti-clockwise direction as the load steps are increased. The intersect shifts upward according to increase in the specimen size. In this way, the combined interaction of material properties, load steps and specimen geometry and size are easily analyzed in terms of the failure mode or behavior that can change from the very brittle to the ductile involving stable crack growth. An upper limit on specimen or structural size is established beyond which stable crack growth ceases to occur and failure corresponds to unstable crack propagation or catastrophic fracture. The parameters that control the failure mode are the threshold values of the strain energy density function (dW/dV)_c and the strain energy density factor S_c.     

The crack resistance of a dilatation-sensitive material

The entire history of crack propagation in high-strength steel with dilatation-sensitivity is investigated in this work. Based on the experimentally determined stress-strain curves varying in the strain history according to the mean stress level, a central cracked panel made of unaged maraging steel subject to a remote tension is considered in the analysis. Due to tremendous amount of hydrostatic stress established in the vicinity of the crack tip, the effect of mean stress is accounted for in the onset for plastic flow. The J₂−I₁ non-associate flow rule is incorporated in the incremental analysis employing the isoparametric finite element method with displacement formulation. When the stage of global instability is approached, the possible formation of a secondary crack in front of the primary crack is demonstrated by the use of strain energy density criterion. Also, the size effect on the crack resistance curve is established at the end for the unaged maraging steel.     

Moving crack for functionally grated material in an infinite length strip under antiplane shear

Considered is a Yoffe crack in an infinite strip of functionally grated material (FGM) subjected to antiplane shear. The shear moduli in two directions of FGM are assumed to be of exponential form. The dynamic stress intensity factor and strain energy density factor at the crack tip are obtained by using integral transforms and dual-integral equations. The numerical results show that the decrease of the strain energy density factor varies with the shear moduli gradient, and the increase of the strain energy density factor varies with the increase of the moving crack speed. The ratio of shear moduli in material vertical orientation has a great influence on the strain energy density factor.     

Strain energy density criteria for dynamic fracture and dynamic crack branching

Dynamic extension of Sih's fracture criterion based on strain energy density factor, r_c (dW/dV), is used to analyze dynamic crack propagation and branching. Influence of the nonsingular components, which are known as the higher order terms (HOT) in the crack tip stress field, on the strain energy density distribution at a critical distance surrounding the crack tip moving at constant crack velocity is examined. This r_c (dW/dV) fracture criterion is then used to analyze available dynamic photoelastic results of crack branching and of engineering materials.     

Stress intensity factor range and propagation mode of surface cracks under rolling–sliding contact

Finite element analyses were conducted in order to evaluate the mode I and mode II stress intensity factors for inclined edge cracks under cyclic contact load under rolling and rolling–sliding condition. The SIF range depends on crack orientation, crack length to Hertzian contact zone half-width ratio, friction between the crack faces and friction on the contact surface. The results were combined in two compact functions that determine the ΔK_I and ΔK_II values. The crack propagation mode and direction were investigated using both the maximum stress criterion and the minimum strain energy density criterion. The results are displayed in graph form, which allows a fast evaluation of the crack growth condition.     

Stable growth of a crack interacting with a circular inclusion

The problem of failure of a plate containing a circular inclusion and a crack is studied. The crack is oriented along a diameter of the inclusion and the plate is subjected to a remote uniaxial stress perpendicular to the crack axis. The process of slow stable crack growth from initiation to termination is studied by the strain energy density theory. The crack growth is simulated by predicting finite increments of crack extension when material elements near the crack tip absorb a critical amount of strain energy density level, . Unstable crack growth occurs when the strain energy density factor S reaches a critical value where r_c is the critical size of the final crack increment prior to instability. The stress at crack initiation and the critical stress and crack length at failure are determined. The influence of the mechanical properties of the plate and the inclusion, the relative position of the inclusion and the crack and the crack length on the characteristic quantities of stable crack growth is analyzed. The dependence of the stable crack growth process on the loading rate is also investigated. Results are displayed in graphical form.     

Crack growth instability studied by the strain energy density theory

The strain energy density theory has successfully been used to address the problem of material damage and structural failure in problems of engineering interest. The theory makes use of the strain energy density function, dW/dV, and focuses attention in its stationary values. The directions of crack growth and yielding are determined from the minimum and maximum values of dW/dV, respectively, along the circumference of a circle centered at the point of failure initiation. Failure by crack growth or yielding takes place when these values of dW/dV become equal to their critical values which are material constants. In the present work the basic principles of the strain energy density theory were reviewed. Furthermore, this theory was used to study three problems of structural failure, namely the problem of slow stable growth of an inclined crack in a plate subjected to uniaxial tension, the problem of fracture instability of a plate with a central crack and two notches, and the problem of unstable crack growth in a circular disc subjected to two equal and opposite forces. The results of stress analysis were combined with the strain energy density theory to obtain the whole history of crack growth from initiation to instability. A length parameter was introduced to define the fracture instability of a mechanical system. Fracture trajectories were obtained for fast unstable crack propagation.     

Material plasticity dependence of mixed mode fatigue crack growth in commonly used engineering materials

In this paper, an investigation of fatigue crack propagation in rectangular plates containing an inclined surface crack is presented. A criterion for the three-dimensional stress state is proposed to predict fatigue crack initiation angles. It is assumed that the direction of crack initiation coincides with the direction of the minimum radius of the plastic zone defined by the von Mises yield criterion. The maximum energy release rate criterion, i.e., G_max criterion, is extended to study the fatigue crack growth characteristics of mixed mode cracks. A modification has been made to this criterion to implement the consideration of the plastic strain energy. Subsequently, this concept is applied to predict crack growth due to fatigue loads. Experiments for checking the theoretical predictions from the proposed criterion have been conducted. The results obtained are compared with those obtained using the commonly employed fracture criteria and the test data.     

Analysis of dynamic growth of a tensile crack in an elastic-plastic material

The influence of inertia on the stress and deformation fields near the tip of a crack growing in an elastic-plastic material is studied. The material is characterized by the von Mises yield criterion and J₂ flow theory of plasticity. The crack grows steadily under plane strain conditions in the tensile opening mode. Features of the stress and deformation state at points near the moving crack tip are described for elastic-perfectly plastic response and for several crack propagation speeds. It is found that inertia has a significant effect on the elastic-plastic response of material particles near the crack tip, and that elastic unloading may occur behind the crack tip for higher speeds. The relationship between the applied crack driving force, represented by a remote stress intensity factor, and the crack tip speed is examined on the basis of a critical crack tip opening angle growth criterion. The calculated result is compared with dynamic fracture toughness versus crack speed data for a 4340 steel.     

Mode-I crack projection procedure using the strain energy density criterion

A practice used in linear elastic fracture mechanics is the projection of a crack onto a plane normal to the principal tensile stress axes for computing the stress intensity factor K_I. The minimum strain-energy criterion is applied for different crack configurations with the introduction of a safety factor S_i which is the ratio of the strain energy density factor of the projected crack and that of the original crack. Numerous crack configurations are investigated to illustrate the degree of conservativeness of the crack projection procedure.     

An extended strain energy density failure criterion by differentiating volumetric and distortional deformation

We extend Sih’s strain energy density criterion (Sih, 1974) for crack kinks and material failure by weighting differently the volumetric and distortional parts in the extended strain energy density factor. The work is inspired by the factor that failure by microscopic shearing governed by distortion and microscopic separation controlled by hydrostatic tension represent distinct deformation processes, and should be treated differently as we count their influences to material failure. With the weight parameter introduced to the extended strain energy density factor criterion, we explain satisfactorily several critical experiments which reported crack kink in samples subjected to mixed-mode loading. The extended strain energy density idea is also used to derive a generalized pressure-dependent yielding criterion, which supplies a theoretical basis for those novel strength criteria for materials like bulk metallic glasses. Corresponding methods to determine the two material parameters, the critical strain energy density factor and the weight parameter quantifying the relative contribution by distortion over volumetric deformation, are discussed.     

A line plastic-zone model for steady mode III crack growth in an elastic-plastic material

Steady-state quasi-static growth of a crack in the anti-plane shear mode through an elastic-plastic material is analyzed. The material is non-hardening and small-scale yielding conditions are assumed. The essential feature of the model is that the active plastic-zone is assumed to be a pair of discrete lines emanating from the crack tip out of the crack plane on which a suitable yield condition is satisfied. An exact solution is obtained for the plastic strain left in the wake of this active line plastic-zone. The extent of the plastic zone from the tip is determined to be 0.071 $\text{(}\text{k}\text{τ}{}_0\text{)}{}^2$ where k and τ₀ are the remote elastic stress intensity factor and the shear flow stress, respectively, and it is found that 36% of the elastic energy flowing into the crack-tip region during growth is dissipated through plastic work and 64% is trapped as residual elastic energy in the plastic-zone wake.