Stress-intensity factor and crack opening for a linearly viscoelastic strip with a slowly propagating central crack

The stress-intensity factor and the size of the crack opening have been calculated for a linearly viscoelastic strip with a slowly propagating central crack. The edges of the infinitely long strip are displaced normal to the crack and both clamped and shear-free strip edges have been investigated. The results are based on the solution to the problem of a suddenly loaded strip with a stationary crack. The resulting integral equation has been solved numerically for arbitrary crack length and analytical solutions in form of asymptotic series are given for crack length up to about half the strip width. The response to a propagating crack is found by superposition.This work represents part of a Ph.D. Thesis submitted to the California Institute of Technology. The author gratefully acknowledges the support of this work by the National Aeronautics and Space Administration under Research Grant NGL-05-002-005, GALCIT 120.     

Edge crack in a strip of an elastic solid

The problem of determining the distribution of stress and the deformation of a long strip of an elastic material, damaged by a crack normal to an edge of the strip, is investigated. The strip is deformed by pressure applied to the faces of the crack. The stress intensity factor is calculated and its variation with the depth of the crack, relative to the width of the strip, in the special case of uniform pressure, is illustrated.     

Crack propagation in a strip of material under plane extension

The problem of a uniformly propagating finite crack in a strip of elastic material is solved using the dynamic equations of elasticity in two-dimensions. Two specific conditions of loading on the strip with finite width are discussed. In the first case, the rigidly clamped edges are pulled apart in the opposite directions. The second case considers equal and opposite tractions applied to the crack surface. By varying the strip width to the crack length ratio, the amplitude of the dynamic stresses ahead of the running crack is determined as a function of the crack velocity. The local dynamic stresses are found to be lower than the corresponding static values for the displacement loading condition and higher for the stress loading condition. This effect becomes increasingly more important as the crack length to strip width ratio is enlarged. Numerical results for the dynamic crack opening displacement are also presented.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Transient thermoelectroelastic response of a functionally graded piezoelectric strip with a penny-shaped crack

Transient response of a penny-shaped crack in a plate of a functionally graded piezoelectric material (FGPM) is studied under thermal shock loading conditions. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the crack faces are completely insulated. By using both the Laplace and Hankel transforms, the thermal and electromechanical problems are reduced to a singular integral equation and a system of singular integral equations which are solved numerically. The intensity factors vs. time for various crack size, crack position and material nonhomogeneity are obtained.     

Free end effect on the stress and displacement distributions in a cracked composite strip

Taking into account the marked anisotropic character of carbon fibre composite materials, a small parameter related to material properties is introduced. The stress field in a semi-infinite strip with a semi-infinite crack whose tip is close to the strip free end is investigated through a singular perturbation method. For slow crack propagation, the quasi-static character of the stress field is established. An accurate asymptotic solution is derived, which allows the interaction between the strip free end and the stress field around the crack tip to be studied. For the carbon–epoxy material investigated, this interaction becomes negligible when the distance between the crack tip and the strip free end is greater than 1.5h, where h is the strip half-height.     

Effects of electric boundary conditions on electroelastic field in a cracked piezoelectric strip

Effects of electric boundary conditions on electroelastic field in a cracked piezoelectric strip are examined. Attention is focussed on an antiplane shear central crack normal to the strip surfaces. By decoupling equations and using the conformal mapping technique, expressions for electroelastic field in the piezoelectric strip are determined under the assumptions of an impermeable, permeable, or conducting crack, respectively. Comparison for the singularity near the crack tips among the obtained electroelastic fields is made.     

On the use of the finite element method for the solution of a cracked strip under thermal shock

The finite element method is used to solve the problem of a strip containing a crack and subjected to a thermal shock on one edge. The solution of the edge crack problem is compared to the exact solution and it is shown that the FEM yields results with less than 1% error. The dependence of the accuracy of the solution on the time increment is examined. The problem of a strip containing an internal crack is then solved using the FEM. The effects of the Biot number, the length of the crack and the distance of the crack to the edge of the strip, on the transient stress intensity factors are analyzed.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Stress intensities in a strip reinforced by stiffeners at the edges

The problem of a cracked or perforated strip, reinforced by stiffeners, is considered. Since each defect (crack or hole) is simulated by a continuous distribution of dislocations, the problem of a single dislocation, lying in the reinforced strip, has been solved via Fourier transform. Consequently, a singular integral equation is set up for any crack or hole in the strip. Numerical results, regarding stress intensities at straight crack tips, are presented.     

Analysis of a unidirectional, symmetric buffer strip laminate with damage

A method of analysis capable of predicting accurately the fracture behavior of a unidirectional composite laminate containing symmetrically placed buffer strips is presented. The analysis is based on a materials modeling approach using the classical shear-lag assumption to describe the stress transfer between fibers. Explicit fiber and matrix properties of the three regions are retained and changes in the laminate behavior as a function of the relative material properties, buffer strip width and initial crack length are discussed. As an example, for a notch (broken fibers) in a graphite/epoxy laminate, the results show clearly the manner in which to select the most efficient combination of buffer strip properties necessary to arrest the crack. Ultimate failure of the laminate after crack arrest can occur under increasing load, either by continued crack extension through the buffer strip, or by fiber breakage in the undamaged half-plane. That is, for certain choices of relative material properties and width, the crack can jump the buffer strip. For some typical hybrid laminates it is found that a buffer strip spacing to width ratio of about four to one is the most efficient.     

Thermoelectroelastic response of a functionally graded piezoelectric strip with two parallel axisymmetric cracks

A mixed-mode thermoelectroelastic fracture problem of a functionally graded piezoelectric material strip containing two parallel axisymmetric cracks, such as penny-shaped or annular cracks, is considered in this study. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. Using integral transform techniques, the problem is reduced to that of solving two systems of singular integral equations. Systematic numerical calculations are carried out, and the variations of the stress and electric displacement intensity factors are plotted for various values of dimensionless parameters representing the crack size, the crack location and the material non-homogeneity.     

Steady-State Crack Velocity in a Viscoelastic Composite Material

The velocity of a semi-infinite crack slowly propagating in aninfinitely long strip made of a viscoelastic composite material isdetermined according to Christensen's fracture criterion. The edges ofthe strip are subjected to uniform opposite displacements normal to thecrack plane. The crack velocity is obtained from an energy balanceequation involving the energy dissipated in the whole strip; the latteris evaluated using an approximate, but sufficiently accurate, expressionof the stress field in the structure obtained by taking into account thestrong anisotropy of the long fibre composite material of interest. Thisnew version of Christensen's criterion compares favourably with theoriginal one and gives crack velocity predictions very close to thoseprovided by Schapery's criterion.     

The linear thermoelastic problem of uniform heat flow disturbed by a two-dimensional crack in a strip

Under the assumption of plane strain, a solution for a thermoelastic problem concerning a strip is obtained by the method of dual integral equations. It is assumed that the crack is parallel to the edges of the strip. The variation with the strip width of stress-intensity factor is shown graphically.     

On the Energy Release Rate at the Crack Tips in a Finite Pre-Strained Strip

The influence of the initial finite stretching or compressing of the strip containing a single crack on the Energy Release Rate (ERR) and on the SIF of mode I at the crack tips is studied by the use of the Three-Dimensional Linearized Theory of Elasticity. It is assumed that the edges of the crack are parallel to the face planes of the strip and the ends of the strip are simply supported. The initial finite strain state arises by the uniformly distributed normal forces acting at the ends of the strip. The additional normal forces act on the edges of the crack. The elasticity relations for the strip material are given by the harmonic type potential. The corresponding boundary-value problem is solved by employing FEM. The numerical results on the influence of the initial finite strain state the values of the ERR and of the SIF of mode I are presented. In particular, it is established that the values of the ERR and of the SIF of mode I decrease (increase) monotonically with an increase (decrease) in the initial stretching (compression).     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semi-infinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Closed-form solution for an eccentric anti-plane shear crack normal to the edges of a magnetoelectroelastic strip

Summary The problem of an anti-plane shear crack embedded in a magnetoelectroelastic strip is investigated. The crack is assumed to be normal to the strip edges. By using the finite Fourier transform, the associated mixed boundary-value problem is reduced to triple series equations, then to singular integral equations. Solving the resulting equations analytically, the field intensity factors and energy release rates at the crack tips can be determined in explicit form. The influences of applied electric and magnetic loadings on the normalized energy release rate and mechanical strain energy release rate are presented graphically. Obtained results reveal that applied electric and magnetic loadings affect crack growth, depending on their directions and adopted fracture criteria. The derived solution is applicable to other cases including two collinear cracks distributed symmetrically in a magnetoelectroelastic strip, and a periodic array of collinear cracks in a magnetoelectroelastic plane.     

Delamination of thin film strips

Delamination of residually stressed thin film strips is analyzed to expose the dependence on strip width and film/substrate elastic mismatch. Isotropic films and substrates are assumed. The residual stress in the film is tensile and assumed to originate from mismatch due to thermal expansion or epitaxial deposition. Full and partial delamination modes are explored. In full delamination, the interface crack extends across the entire width of the strip and releases all the elastic energy stored in the strip as the crack propagates along the interface. The energy release rate available to propagate the interface crack is a strong function of the strip width and the elastic modulus of the film relative to that of the substrate. The energy release rate associated with full delamination is determined as a function of the interface crack length from initiation to steady-state, revealing a progression of behavior depending in an essential way on the three dimensionality of the strip. The dependence of the energy release rate on the remaining ligament as the interface crack converges with the strip end has also been calculated, and the results provide an effective means for inferring interface toughness from crack arrest position. A partial delamination propagates along the strip leaving a narrow width of strip attached to the substrate. In this case, the entire elastic energy stored in the strip is not released because the strain component parallel to the strip is not relaxed. A special application is also considered, in which a residually stressed metal superlayer is deposited onto a polymer strip. The energy release rate for an interface crack propagating along the interface between the polymer and the substrate is determined in closed form.