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.     

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 growth prediction of an inclined crack in a half-plane thermopiezoelectric solid

The strain energy density theory is used to determine the direction of crack propagation in the case of an inclined crack in a half-plane thermopiezoelectric sheet subjected to uniform heat flow. Based on the Stroh's formulation and the thermoelectroelastic Green's function, a system of singular integral equations for the unknown thermal analog of dislocation density defined on the crack faces is presented and used to calculate the corresponding strain energy density. The direction of crack extension is then determined by way of the minimum strain energy density criterion. The study shows that the criterion is easy-to-use for thermal fracture problems.     

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.     

Non-local theory solution for in-plane shear of through crack

A non-local theory of elasticity is applied to obtain the plane strain stress and displacement field for a through crack under in-plane shear by using Schmidt's method. Unlike the classical elasticity solution, a lattice parameter enters into the problem that make the stresses finite at crack tip. Both the angular variations of the circumferential stress and strain energy density function are examined to associate their stationary value with locations of possible fracture initiation. The former criterion predicted a crack initiation angle of 54° from the plane of shear for the non-local solution as compared with about 75° for the classical elasticity solution. The latter criterion based on energy density yields a crack initiation angle of 80° for a Poisson's ratio of 0.28. This is much closer to the value that is predicted by the classical crack tips solution of elasticity.     

Fracture instability of notched cracked plates characterized by the strain energy density theory

The fracture instability of a mechanical system is analyzed by the strain energy density theory. The local relative minima of the strain energy density function dW/dV referred to local coordinate systems at each point of the body are distinguished from the global minimum of dW/dV, G, which is referred to a fixed global coordinate system. Failure by fracture starts from the maximum of the local minima of dW/dV, L, and passes from point G. The distance l between L and G along the fracture trajectory is introduced as a length parameter to characterized the fracture instability of the system. Numerical results are obtained and discussed for a cracked plate with two symmetrical notches subjected to a monotonically rising tensile stress perpendicular to the crack axis.     

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.     

Crack initiation direction from interface of bonded dissimilar media

Debonded region of an interface between two dissimilar materials are modeled as a line crack that tends to enhance the initiation of failure by fracture. Depending on the load that interacts with dissimilar materials, no a priori knowledge of how failure would initiate from an existing interface crack is assumed. By application of the strain energy density criterion, potential crack initiation sites are obtained for different biaxial loading states and materials with dissimilar properties.Numerical results are obtained for an epoxy/aluminum medium. In each case, a finite line segment of debonding is assumed. Uniform stresses are applied normal and parallel to the interface so that a biaxial load factor k determines the relative magnitude of biaxiality. Positive and negative k correspond, respectively, to applied tension and compression parallel to the interface. For a fixed ratio of the elastic moduli, crack initiation angles measured from the interface would increase with positive k and decrease with an increase of negative k. These findings are presented for different values of k. The direction of maximum yield initiation could also be determined from the stationary values of the strain density function. These locations are identified with elements that undergo excessive distortion while the possible fracture sites are assumed to coincide with regions where dilatational effects would dominate.     

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.     

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.     

Stress multiaxiality factor for crack growth prediction using the strain energy density theory

The multiaxiality factor defined as the ratio of the von-Misses equivalent stress to the volumetric stress has been reported to be related to the initiation and progression of failure in structures. It is demonstrated in the present paper that the location around the crack tip where the multiaxiality factor obtains minimum value is an indicator of the direction of minimum material fracture resistance for crack propagation. It is also proposed that the location along the direction of crack propagation path where multiaxiality factor obtains minimum value is considered as the critical distance away from the crack tip, where the strain energy density should be evaluated and compared to its critical value. Theoretical predictions correlate well with the test results for the investigated cases.     

Theoretical investigation of an elliptical crack in thermopiezoelectric material. Part II: Crack propagation

Propagation behavior of an elliptical crack in thermopiezoelectric material subjected to a uniform temperature is investigated in this paper. The three-dimensional strain energy density formulation is used to determine the direction of crack propagation and the shape of the initial fracture increment. It is found that the elliptical crack grows coplanarly under this particular load case but not normal to the crack front. The elliptical crack tends to become a circular one when thermal loading is applied.     

Strain energy density prediction of crack initiation and direction in cracked T-beams and pipes

In this paper, the S-theory is applied to determine crack initiation and direction for cracked T-beams and circumferentially cracked pipes. It makes use of a parameter called strain energy density factor, S, which is a function of the stress intensity factors. The strain energy density theory provides a more general treatment of fracture mechanics problems by virtue of its ability in describing the multiscale feature of material damage and in dealing with mixed mode crack propagation problem. A simple method for obtaining approximate stress intensity factors is also applied. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Some basic loading conditions in beams and pipes are studied.     

Time-harmonic P-waves engulfing a rectangular limited-permeable crack in piezoelectric medium: Energy density and energy release

The dynamic behavior of a limited-permeable rectangular crack in a transversely isotropic piezoelectric material is impinged by to a P-wave. The generalized Almansi theorem and the Schmidt method are used to determine the stress intensity factor and energy density factor as the primary fracture criterion of failure. The mixed boundary value problem entails the evaluation of the appropriate crack edge stress singularities that are characteristics of the fundamental functions. The stress and electric displacement intensity factors are also used to find the energy release rate that can be computed numerically and compared with the results corresponding to those of the stress intensity factor, and energy density factor. Graphical presentation shows that the energy release rate is always negative for the boundary conditions considered while the energy density factors always remain positive. Under certain conditions, the stress and electric displacement intensity factors can be negative and subject to physical limitations. Piezoelectric material boundary value problem solutions should therefore be qualified by the application of failure criteria by fracture of otherwise, particularly when the mechanical and electrical energy can release by creating free surface at the macroscopic and microscopic scales. Negative energy release rate found for the piezoelectric medium in this work can be a case in point.Positive definiteness of the energy density factor can be applied to mutliscale fracture. This is not true for the stress intensity factor nor the energy release rate. Hence, crack initiation behavior for the permittivity of a rectangular crack due to the wave propagation effects may be studied. In particular, the initiation of micro-cracks may be identified with certain critical stress wave frequency band. Negative stress intensity factor may not enhance macrocracking but it does not exclude microcrack initiation.     

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.     

Failure initiation in unnotched specimens subjected to monotonic and cyclic loading

Failure initiation in unnotched cylindrical bar specimens is predicted by application of the strain energy density theory. Maximum value of the local minimum strain energy density function is calculated, the critical value of which is assumed to coincide with failure by monotonic as well as cyclic uniaxial loading. Damage is accumulated in the specimen for each increment of monotonically rising load and each cycle of repeatedly applied load. Use is made of the incremental theory of plasticity to account for permanent deformation that is nonuniformly distributed throughout the cylindrical bar. Failure initiation site is found to occur at the center of the bar for monotonic loading where dilatation is dominant and near the specimen surface for fatigue loading where distortion is more significant. The results are consistent with the experimental observations without including microstructural effects. Nonhomogeneity caused by macro-dilatation and macro-distortion is also shown to play an important role in failure initation.     

Three-dimensional transverse fatigue crack growth in rail head

Results are given in terms of crack growth area and tonnage of train load. A three-dimensional finite element procedure is developed for analyzing multiple-mode transverse fatigue crack growth in a rail section. Stress and failure analysis are performed for each increment of non-self-similar crack growth up to the point of global instability that is assumed to be governed by the fracture toughness of the rail steel. The strain energy density criterion is adopted to predict the crack profiles developed from a two-stage fatigue loading cycle where both Mode I and III crack extension are present. Use is made of the material data obtained from the past and present TSC programs for predicting the remaining life of a 132 lb/ft rail head with an initial transverse circular crack of 0.50 in. in diameter. The number of cycles to failure are estimated for four different vertical load and initial crack positions.     

Growth predictions of two interacting cracks

The elastic fracture behavior of a plate subjected to uniform stress surrounding two equal cracks inclined at an angle is investigated. The orientation of the crack plane with applied stress can be varied. Among the cases are: (1) two cracks inclined symmetrically with respect to the vertical and horizontal applied stress, and (2) one crack is horizontal while the other is inclined to the vertical applied stress. The strain energy density criterion is used for determining the combined crack and load arrangement that correspond to the lowest critical load at global instability. The direction of crack initiation is also determined. Quantitative results pertaining to the fracture characteristics are given in graphical forms.     

Stable growth of a central crack

The elastic fracture behavior of a plate subjected to uniform stress surrounding two equal cracks inclined at an angle is investigated. The orientation of the crack plane with applied stress can be varied. Among the cases are: (1) two cracks inclined symmetrically with respect to the vertical and horizontal applied stress, and (2) one crack is horizontal while the other is inclined to the vertical applied stress. The strain energy density criterion is used for determining the combined crack and load arrangement that correspond to the lowest critical load at global instability. The direction of crack initiation is also determined. Quantitative results pertaining to the fracture characteristics are given in graphical forms.     

Influence of indenter geometry on half-plane with edge crack subjected to fretting condition

An edge crack is analyzed to study fretting failure. A flat punch with rounded corners and a half-plane are regarded as an indenter and a substrate, respectively. Plane strain condition is considered. Contact shear traction in the case of partial slip is evaluated numerically. It is assumed that an initial crack is extended to the point of minimum strain energy density in the half-plane from the trailing edge of contact. Dislocation density function method is used to evaluate K_I and K_II. The variations of K_I and K_II during crack growth are examined in the case of indentation by a punch with different ratio of the flat region (l) to the punch width (L). Sih's minimum strain energy density theory [1] is also applied to predict the propagation direction of the initial crack. The direction evaluated is similar to that found in the experiment. Stress intensity factor ranges (ΔK_I and ΔK_II) are examined during cyclic shear on the contact. For the design of contacting bodies, a suggestible geometry of punch for alleviating cracking failure is studied.