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.     

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.     

A pseudo-elastic material damage model, part II: Application to the scaling of specimen size, material behavior and loading rate

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.     

Crack growth characteristics influenced by load time record

Crack growth patterns are predicted by application of the strain energy density theory. The direction and amount of crack growth are dictated by the relative minima of the strain energy density function, dW/dV, whose critical value is material dependent. In large structures, the energy dissipated by crack growth can dominate while plastic deformation plays a minor role. Because loading in service can often vary in an unpredictable manner, energy released during crack growth can be a highly nonlinear process. The precise dependency of crack growth on load time history is analyzed in detail for a centrally cracked panel. Load history are shown to effect the critical failure loads in ways that are not intuitively obvious.     

Crack initiation and growth in contact problems

The problem of crack initiation and growth in contact problems is studied by the strain energy density theory. This is accomplished by considering each point of the body as a possible source of crack initiation. The direction of crack initiation is determined by calculating the minimum value of the strain energy density function along a circular core area surrounding the point. The crack initiates from the point with the maximum of the local minimum values of the strain energy density function. Considered is a circular disc subjected to two equal and opposite forces and a circular cylindrical body pressed by a cylindrical punch. The stress field for the first case is obtained from the classical elasticity solution, while a finite element code is used for the second case. The locations of fracture initiation and the subsequent fracture trajectories for fast unstable fracture are determined.     

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.     

Elastoplasticity of crack loaded by two pairs of isolated shear forces in finite width plate

The failure behavior of an elastic-perfectly plastic body with a crack loaded by two pairs of concentrated shear forces is discussed. The analytical solutions of an eccentric crack in a finite plate loaded by two pairs of point shear forces are obtained. It includes the unit normal vector of the elastic-plastic boundary near the crack line, the elastic-plastic stress fields near crack line and the law of the plastic zone along the crack line with external loads. The solutions of this paper are sufficiently precise near the crack line in elastic-perfectly plastic materials. Subsequently, the present results are compared with solutions based on the minimum strain energy density theory and elastic-plastic solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

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.     

Elastoplastic solution for an eccentric crack loaded by two pairs of point tensile forces

The strain energy density theory and the near crack line analysis method are applied to investigate an eccentric crack loaded by two pairs of tensile point forces in a finite plate. The minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. Obtained is the elastic-plastic solution near the crack line of an eccentric crack loaded by two pairs of point tensile forces under large scale yielding condition. More specifically, the near field solution contains the unit normal vector of the elastic-plastic boundary and the elastic-plastic stress field. The length of the plastic zone along the crack line is found to vary with the external load and the bearing capacity of a finite plate with an eccentric crack loaded by two pairs of tensile point forces. Compared with small scale yielding condition, the normalized load obtained is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

Damage accumulation for growth of anti-plane crack in work-hardening material

Elastic–plastic solutions of an anti-plane crack in an infinite body are used in conjunction with a continuum damage model to describe the conditions necessary for the onset of crack instability, fatigue crack propagation due to cyclic loading, and rates of crack growth due to time dependent events. A power law relates the stress to the strain of the material. The damage, which invokes nucleation, growth and coalescence of microvoids due to elevated strain, is confined to the plastic zone surrounding the crack tip. For applied loading below the yield stress, the small-scale and large-scale yielding solutions are used to determine the influence of strain hardening on crack instability and failure. Crack growth due to cyclic loading and time-dependent deformations are studied using the small-scale yielding solution of the deformation theory of plasticity.     

Elastic–plastic near field solution of an eccentric crack under shear in a finite width plate

The near crack line analysis method is used to investigate an eccentric crack loaded by shear forces in a finite width plate, and the analytical solution is obtained in this paper. The solution includes: the unit normal vector of the elastic–plastic boundary near the crack line, the elastic–plastic stress fields near crack line, variations of the length of the plastic zone along the crack line with an external loads, and the bearing capacity of a finite plate with a centric crack loaded by shear stress in the far field. The results obtained in this paper are sufficiently precise near the crack line because the assumptions of small scale yielding theory have not been made and no other assumptions have been taken. Subsequently, the present results are compared with the traditional line elastic fracture mechanical solutions and elastoplastic near field solutions under small scale yielding condition. On the basis of the minimum strain energy density (SED) theory, the minimum values of SED in the vicinity of the crack tip are determined, the initial growth orientation of crack are determined. It is found that the normalized load under large scale yielding condition is higher than those under small scale yielding condition when the length of the plastic zone is the same.     

Growth of an arc-crack in bonded dissimilar materials

The problem of growth of a crack lying along the interface of a circular inclusion embedded in an infinite plate is studied within the framework of linear elasticity. The plate is subjected to a uniform uniaxial stress at infinity at any angle of inclination relatively to the crack. The critical load for unstable crack growth, the angle of initial crack extension and the subsequent crack path are investigated using the strain energy density fracture criterion. The combined effect of crack length and orientation on the fracture stress is considered for the case of an aluminum-epoxy composite.     

An analysis of ductile rupture modes at a crack tip

An elastic-Viscoplastic model of a ductile, porous solid is used to study the influence of the nucleation and growth of micro-voids in the material near the tip of a crack. Conditions of small scale yielding are assumed, and the numerical analyses of the stress and strain fields are based on finite strain theory, so that crack tip blunting is fully accounted for. An array of large inclusions or inclusion colonies, with a relatively low strength, results in large voids near the crack tip at a rather early stage, whereas small second phase particles in the matrix material between the inclusions require large strains before cavities nucleate. Various distributions of the large inclusions, and various critical strains for nucleation of the small scale voids between the inclusions, are considered. Localization of plastic flow plays an important role in determining the failure path between the crack tip and the nearest larger void, and the path is strongly sensitive to the distribution of the large inclusions. Values of the J-integral and the crack opening displacement at fracture initiation are estimated, together with values of the tearing modulus during crack growth, and these values are related to experimental results.     

The effect of load biaxiality on crack growth in non-linear materials

The influence of load biaxiality on the stress field and fracture behavior of a cracked plate is investigated. Considered is a square plate containing a central through the thickness crack and subjected to a biaxial loading perpendicular and parallel to the crack plane. The stress field of the plate is analyzed by a finite element code based on incremental plasticity and the von Mises yield condition. A method based on the strain energy density theory is used to determine the critical stress for crack initiation. It was found that the equi-biaxial loading mode induces the smallest plastic zones, while the critical applied stress for crack initiation becomes maximum. Quite the contrary happens for the shear loading system which causes the largest plastic zones and the minimum applied stress values fro crack growth. Results showing the dependence of the above quantities on the biaxiality of the applied stress are presented in graphical form.     

Strain energy release rate for interfacial cracks in hybrid beams

The finite element modeling and fracture mechanics concept were used to study the interfacial fracture of a FRP-concrete hybrid structure. The strain energy release rate of the interfacial crack was calculated by the virtual crack extension method. It is shown that the crack growth has three phases, namely, cracking initiation, stable crack growth and unstable crack propagation. The effects of geometric and physical parameters of the hybrid beam on the energy release rate were considered. These parameters include Young’s moduli of the FRP, the concrete and the adhesive, thickness of the FRP plate and adhesive, and the distance of FRP plate end from the beam end. The numerical results show that the energy release rate of the interfacial crack is influenced considerably by these parameters. The present investigation can contribute to the mechanism understanding and engineering design of the hybrid structures.     

Initiation and growth characterization of corner cracks near circular hole

The finite element analysis of crack problems often incorporates the asymptotic character of the local solution into the formulation. Embedment of stress or strain singularities can impose serious restrictions on the outcome and inconsistencies in predicting crack and/or growth. These restrictions are discussed in connection with the problem of two diametrically opposite corner cracks near a circular hole subjected to remote uniform tension. Enforced in the numerical treatment is the 1/r character of the strain energy density function local to the corner crack border where r is the radial distance measured from the crack front. The tendency for the corner crack to become a through crack is predicted by assuming that each point of the crack border extends by an amount proportional to the strain energy density factor. The path would correspond to the loci of minimum strain energy density function. Numerical results are displayed graphically and discussed in connection with crack initiation and non-self-similar crack growth.     

Crack initiation angles for systems of two interacting cracks in media with rectilinear preferred directions

The direction of initial crack growth of two interacting cracks in solids with rectilinear preferred directions is predicted by using the strain energy density theory. The minimum local strain energy density factors are presented; they correspond to the directions along which the material elements attain maximum dilatation and hence are assumed to coincide with the direction of crack initiation. Numerical results are obtained for different crack systems and presented in graphical forms.     

Size effect of cylindrical specimens with fatigue cracks

The rate at which energy is accumulated within a unit volume of material in fatigue is assumed to depend not only on load-time history but also on the specimen size and geometry in addition to material type. A threshold level for the hysteresis strain energy density function accumulated in the material is used for predicting macrocrack growth. This is accomplished by application of the incremental theory of plasticity for each increment of crack growth. The accumulated hysteresis strain energy density factor ΔS to crack growth increment Δa ratio is found to be constant for fixed specimen size and loading, i.e., $\text{ΔS}\text{Δa}\text{=}\text{const}$. Results are obtained for the cylindrical bar specimens with a penny-shaped defect at the center subjected to a constant amplitude and frequency loading. The resistance curves in the ΔS versus Δa plot are parallel lines as specimen size is altered. This information provides a rational means for predicting the influence of specimen size on fatigue lifetime.The results are also compared with those found for geometrically similar plate specimens with line cracks. Cylinder bar specimens of the same material are found to sustain more load cycles prior to catastrophic failure.     

各向异性对平板裂纹扩展路径的影响研究

近场动力学理论克服了经典连续介质力学在模拟裂纹扩展方面的不足。本文基于微梁键构建了一般各向异性近场动力学平面模型,并探讨了各向异性对于裂纹扩展路径的影响。首先,建立一般各向异性的微梁键,并基于插值法建立了一般各向异性近场动力学平面模型的应变能密度表达式;然后,利用应变能密度互等原理求解出近场动力学平面本构模型的一般各向异性参数;最后,通过带初始切口的平板断裂实验研究了不同各向异性参数对于平板裂纹扩展路径的影响。