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.     

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.     

Rotation at the crack tip under uniform heat flow in an infinite medium

A complex analysis of rigid body rotation is presented. The crack-tip rotation for a line crack subjected to steady uniform heat flow is obtained in terms of thermal stress intensity factor in shear mode of the crack, the material and thermal parameters and coordinates of points close to the crack tip. The shear strip configuration is analysed on the basis of rotation and displacement at the end of the shear strip.     

Thermal fracture analysis of a functionally graded orthotropic strip with a crack

This paper is concerned with the thermal fracture problem of a functionally graded orthotropic strip, where the crack is situated parallel to the free edges. All the material properties are assumed to be dependent only on the coordinate y (perpendicular to the crack surfaces). By using Fourier transform, the thermoelastic problem is reduced to those that involve a system of singular integral equations. Numerical results are presented to show the effects of the crack position and the material distribution on the thermal stress intensity factors.     

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.     

An elastic strip with periodic surface cracks under thermal shock

The problem of two periodic edge cracks in an elastic infinite strip located symmetrically along the free boundaries under thermal shock is investigated. It is assumed that the infinite strip is initially at constant temperature. Suddenly the surfaces containing the edge cracks are quenched by a ramp function temperature change. Very high tensile transient thermal stresses arise near the cooled surface resulting in severe damage. The degree of the severity for a subcritical crack growth mode is measured by determining the stresses intensity factors. The thermoelastic problem is treated as uncoupled quasi-static. The superposition technique is used to solve the problem. The thermal stresses obtained from the uncracked strip with opposite sign are utilized as the only external loads to formulate the perturbation problem. By expressing the displacement components in terms of finite and infinite Fourier transforms, a hypersingular integral equation is derived with the crack surface displacement as the unknown function. Numerical results for stress intensity factors are carried out and presented as a function of time, cooling rate, crack length, and periodic crack spacing.     

Two parallel penny-shaped or annular cracks in a functionally graded piezoelectric strip under electric loading

In this paper, the mixed-mode fracture problem of a functionally graded piezoelectric material strip with two penny-shaped or annular cracks is considered. It is assumed that the electroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under electric loading. The problem is formulated in terms of a system of singular integral equations, which are solved numerically. Numerical calculations are carried out, and the stress and electric displacement intensity factors are presented for various values of dimensionless parameters representing the crack size, the crack location, and the material nonhomogeneity.     

Transient thermal stress analysis of an edge crack in a functionally graded material

An edge crack in a strip of a functionally graded material (FGM) is studied under transient thermal loading conditions. The FGM is assumed having constant Young's modulus and Poisson's ratio, but the thermal properties of the material vary along the thickness direction of the strip. Thus the material is elastically homogeneous but thermally nonhomogeneous. This kind of FGMs include some ceramic/ceramic FGMs such as TiC/SiC, MoSi₂/Al₂O₃ and MoSi₂/SiC, and also some ceramic/metal FGMs such as zirconia/nickel and zirconia/steel. A multi-layered material model is used to solve the temperature field. By using the Laplace transform and an asymptotic analysis, an analytical first order temperature solution for short times is obtained. Thermal stress intensity factors (TSIFs) are calculated for a TiC/SiC FGM with various volume fraction profiles of the constituent materials. It is found that the TSIF could be reduced if the thermally shocked cracked edge of the FGM strip is pure TiC, whereas the TSIF is increased if the thermally shocked edge is pure SiC.     

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.     

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.     

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.     

The stress intensity factor for a cracked stiffened sheet repaired with an adhesively bonded composite patch

The problem of a cracked, stiffened metallic sheet adhesively bonded by a composite patch is analyzed. The composite patch is assumed to be either an infinite orthotropic sheet or an infinite orthotropic strip normal to the crack. Due to the high stress concentration around the crack and on the interface, an elliptical disbond is assumed to exist around the crack. The crack is asymmetric with respect to the stiffener's locations as well as to the patch's center. The effect of thermal stresses in curing process is also considered. The fracture problem is solved by the displacement compatibility method, using the complex variable approach and the Fourier integral transform method.The problem is dealt with in two steps. First, starting with an uncracked, patched stiffened sheet, the stress at the prospective location of the crack is determined in a closed-form solution. The second step is to introduce a crack into the stiffened patched sheet. The multivalue of the analytical formulation is treated in detail to ensure proper implement in the computer. The results show that the effect of the stiffeners on the stress intensity factor is not significant for a crack fully covered by a patch.For the repairs by Boron/Epoxy patches, the difference in KI between the infinite sheet patch and the infinite strip model is only minor (less than 5 percent) in the absence of the curing thermal stresses and it becomes more pronounced when these stresses are taken into consideration. The stress intensity factor for a crack repaired by an infinite composite strip also can be estimated with a good or reasonable accuracy via a simplified analysis in which the patch is considered as an infinite strip in the first step and is treated as an infinite sheet in the second step of the solution procedure mentioned above.The latter simplified analysis is based on the approach originally proposed by Rose for a relatively simple repair configuration. For most cases, that approach seems to work well for the repair of a stiffened sheet by an infinite composite strip with the effects of thermal stresses and a disbond included. It should be emphasized that the present methodology can apply to the problem of a crack in a metallic stiffened sheet growing beyond the patch's boundary and also to the repairs by an infinite adhesively bonded composite strip parallel to the crack.     

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.     

On the estimation of the toughness of surface microcracks

Hardness testing is a method frequently used for evaluating the resistance of body surfaces to the influence of contact loading. The Vickers hardness test is applied for this purpose in a case when the material is sufficiently ductile and no cracks occur in the corners of the indents. Testing of the surface hardness of such materials as glass and ceramics on the basis of the Vickers testing method must take into account the energy that is spent on crack spreading. This paper describes a more exact and accurate method for evaluating resistance to microcrack formation on the surface of a material. To evaluate the conditions for crack spreading, it is necessary to test a specimen under loading in bulk. The suggested procedure involves a bent strip. As bending stresses are known, from the differences in crack length along and across the strip, the resistance of the material surface to crack propagation is determined.     

On the estimation of the toughness of surface microcracks

Hardness testing is a method frequently used for evaluating the resistance of body surfaces to the influence of contact loading. The Vickers hardness test is applied for this purpose in a case when the material is sufficiently ductile and no cracks occur in the corners of the indents. Testing of the surface hardness of such materials as glass and ceramics on the basis of the Vickers testing method must take into account the energy that is spent on crack spreading. This paper describes a more exact and accurate method for evaluating resistance to microcrack formation on the surface of a material. To evaluate the conditions for crack spreading, it is necessary to test a specimen under loading in bulk. The suggested procedure involves a bent strip. As bending stresses are known, from the differences in crack length along and across the strip, the resistance of the material surface to crack propagation is determined.     

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).     

Thermoelastic analysis of a partially insulated crack in a strip under thermal impact loading using the hyperbolic heat conduction theory

In this paper, the transient temperature and thermal stresses around a partially insulated crack in a thermoelastic strip under a temperature impact are obtained using the hyperbolic heat conduction theory. Fourier and Laplace transforms are applied and the thermal and mechanical problems are reduced to solving singular integral equations. Numerical results show that the hyperbolic heat conduction parameters, the thermal conductivity of crack faces, and the geometric size of the strip have significant influence on the dynamic temperature and stress field. The results based on hyperbolic heat conduction show much higher temperature and much more dynamic thermal stress concentrations in the very early stage of impact loading comparing to the Fourier heat conduction model. It is suggested that to design materials and structures against fracture under transient thermal loading, the hyperbolic model is more appropriate than the Fourier heat conduction model.     

On the dynamic propagation of an anti-plane shear crack in a functionally graded piezoelectric strip

Summary. In this paper, a finite crack propagating at constant speed in a functionally graded piezoelectric material (FGPM) is studied. It is assumed that the electroelastic material properties of the FGPM vary continuously according to exponential gradients along the thickness of the strip, and that the strip is under anti-plane shear mechanical and in-plane electrical loads. The analysis is conducted on the electrically unified (natural) crack boundary condition, which is related to the ellipsoidal crack parameters. By using the Fourier transform, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, crack propagation speed, electric field, FGPM gradation, crack length, and electromechanical coupling coefficient. It reveals that there are considerable differences between traditional electric crack models and the present unified crack model.     

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.     

The collinear cracks in an inhomogeneous medium subjected to transient load

Summary A laminate model is employed to solve the elastodynamic problem of a collinear crack in an inhomogeneous material. The inhomogeneous material is treated as a series of thinner layer. The Laplace and Fourier transforms are used to reduce the problem to a set of singular integral equations that is solved numerically. Numerical results of two collinear cracks in a functionally graded material strip are obtained to show the influence of material inhomogeneity and crack position on crack tip field intensities.