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.     

Controlled Crack Propagation Experiments with a Novel Alumina‐Based Refractory

Innovative, carbon‐reduced and carbon‐free refractory materials are currently under development within the framework of the DFG priority program “FIRE”. Among various novel material solutions an alumina‐based refractory with titania and zirconia additives (AZT) has gained special interest for application in high temperature processes under thermal shock conditions. The resistance against fracture of the AZT material and, for comparison, of a pure alumina refractory was examined by controlled crack propagation experiments. Wedge splitting and compact tension tests with in situ crack growth observation, partially on microstructural level, have been performed for both materials. Based on the measured room temperature values of dissipated energy, refractory stiffness and fracture stress, the Hasselman thermal shock parameter R″″ was determined. The results allow to predict that AZT is less prone to scale thermal shock damage than pure alumina. The microstructural observations reveal that growth and opening displacement of the main crack is accompanied in AZT by pronounced microcracking, branching and bridging processes. First efforts are also directed towards a mechanical quantification of this fracture behavior in terms of an R‐curve representation (fracture resistance as a function of apparent crack length). The specific problems of R‐curve evaluation that exist in AZT due to nonlinear deformation behavior are addressed and the influence of the observed crack growth mechanisms is discussed.     

Generalized plane problem of a cracked piezoelectric layer bonded to dissimilar layers

Summary In this paper, we proposed a model to study the electro-elastic crack problem for a cracked piezoelectic layer bonded to two elastic layers of finite thickness. The crack is assumed to be through thez-direction and the crack faces perpendicular to they-direction. Fourier transforms technique is used to reduce the problem to the solution of singular integral equations. The model is general enough to account for arbitrary electrical polarized direction and material anisotropy, for any mechanical or electrical mode of loading. Numerical results are plotted to illustrate the influence of the crack face electrical boundary condition on crack tip fields for different layer thickness.     

A thermal-medium crack model

In analyzing the fracture behavior of a cracked thermoelastic material, of much importance are the effects of thermal loadings on the crack growth. Under the consideration of a medium in an opening crack, a thermal-medium crack model is proposed in this paper. The heat flux at the crack surfaces is assumed to depend on the jumps of the temperature and the elastic displacement across the crack. The thermally permeable and impermeable crack models are the limiting cases of a thermal-medium one. The proposed crack model is applied to solve the problem of a Griffith crack in a transversely isotropic material under thermal and mechanical loadings. Using two introduced displacement functions and the Fourier transform technique, the thermoelastic field and the elastic T-stress are determined in explicit forms by using elementary functions. Numerical results are presented to show the effects of the thermal conductivity inside a crack and applied mechanical loadings on the heat flux at the crack faces, the jumps of temperature across the crack and mode-II stress intensity factor in graphics respectively. The obtained results reveal that the mode-II stress intensity factor for a thermal-medium crack in a thermoelastic material depends not only on applied thermal loadings but also on applied mechanical ones.     

Thermal stress intensity factors of an infinite orthotropic layer with a crack

Stress intensity factors are determined for a crack in an infinite orthotropic layer. The crack is situated parallel to the plane surfaces of the layer. Stresses are solved for two kinds of the boundary conditions with respect to temperature field. In the first problem, the upper surface of the layer is heated to maintain a constant temperature T ₀, while the lower surface is cooled to maintain a constant temperature –T ₀. In the other problem, uniform heat flows perpendicular to the crack. The surfaces of the crack are assumed to be insulated. The boundary conditions are reduced to dual integral equations using the Fourier transform technique. To satisfy the boundary conditions outside the crack, the difference in temperature at the crack surfaces and differences in displacements are expanded in a series of functions that vanish outside the crack. The unknown coefficients in each series are evaluated using the Schmidt method. Stress intensity factors are then calculated numerically for a steel layer that behaves as an isotropic material and for a tyrannohex layer that behaves as an orthotropic material.     

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.     

Thermoelastic analysis for an opening crack in an orthotropic material

The thermoelastic analysis of an opening crack embedded in an orthotropic material is made under applied uniform heat flux and mechanical loadings. To simulate the case of an opening crack filled with a medium, a thermal-medium crack model is proposed. The thermally permeable and impermeable cracks are the limiting ones of the proposed thermal-medium one. The crack-tip thermoelastic fields induced by a crack in an orthotropic material are determined in closed forms. The elastic T-stress can be also obtained explicitly. The effects of applied mechanical loadings and the thermal conductivity of crack interior on the heat flux at the crack surfaces and the mode-II stress intensity factor are investigated through numerical computations. The obtained results reveal that an increase of the thermal conductivity of crack interior decreases the mode-II stress intensity factor. And when an applied mechanical loading is increasing, the mode-II stress intensity factor is rising.     

Stress intensity factors for an interface crack between a functionally graded coating and a homogeneous substrate

In this paper we consider the problem of a functionally graded coating bonded to a homogeneous substrate with a partially insulated interface crack between the two materials subject to both thermal and mechanical loading. The problem is solved under the assumption of plane strain and generalized plane stress conditions. The heat conduction and the plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the temperature and the displacement fields in the medium as well as the crack tip stress intensity factors. A crack-closure algorithm recently developed by the authors is applied to handle the problem of having negative mode I stress intensity factors. The Finite Element Method was additionally used to model the crack problem and to compute the crack-tip stress intensity factors. The main objective of the paper is to study the effect of the material nonhomogeneity parameters, partial insulation of the crack surfaces and crack-closure on the crack tip stress intensity factors for the purpose of gaining better understanding of the thermo-mechanical behavior of graded coatings.     

An analytical method for thermal stresses of a functionally graded material cylindrical shell under a thermal shock

In this paper, the thermal stresses of a thin functionally graded material (FGM) cylindrical shell subjected to a thermal shock are studied. An analytical method is developed. The studied problem for an FGM cylindrical shell is reduced to a plane problem. A perturbation method is used to solve the thermal diffusion equation for FGMs with general thermal properties. Then, the transient thermal stresses are obtained. The results show that the thermal shock is much easier to result in failure than the steady thermal loading. The present method can also be used to solve the crack problem of an FGM cylindrical shell with general thermal properties.     

The value of the J-integral at the onset of crack extension from a weld defect: weld material is harder than parent material

The paper addresses the problem of crack extension in a weld in an engineering structure for the situation where the crack is parallel to the plane of the weld. An earlier analysis, for the case where the weld material is softer than the parent material, has demonstrated the extent to which the value of the J-integral at the onset of crack extension depends on the flow properties of the weld and parent materials, the crack size and the weld thickness. The present paper extends these earlier considerations to the case where the weld material is harder than the parent material, and again demonstrates the non-uniqueness of the value of J at the onset of crack extension.     

A periodic array of cracks in functional graded materials subjected to thermo-mechanical loading

A periodic array of cracks in an infinite functionally graded material under mechanical and/or thermal loading is investigated. Due to non-uniform heating or cooling, compressive stresses occur causing the crack surfaces to come into contact at a certain contact length. The mixed boundary value problem is reduced to a singular integral equation with the crack contact length as an additional unknown variable. Numerical results for stress intensity factors and the crack contact length are obtained as a function of crack spacing. Effect of the material non-homogeneity on the crack tip intensity factors is discussed. Some suggestions are made for the design of thermal resistive functionally graded materials.     

A model of crack extension from thermal shock by surface contact

The growth of cracks in glass arising from thermal shock on contact with a second material is an important problem in the volume production of glass articles. The thermal properties of the contacting material are controlling factors in producing cracking and a simple model of crack extension from thermal shock was developed to investigate the process. Given a pre-existing microscopic crack at the surface of the glass, it is shown that one dimensionless parameter, termed the thermal index of the materials combination, controls the process of crack extension and crack arrest. If the index is very much less than unity, the probability of crack extension can be made very small. The consequence for practical applications of new materials in handling hot glass is discussed.     

The Effect of Magnetic Field on 2-D Problem for a Mode-I Crack of a Fiber-Reinforced in Generalized Thermoelasticity

In the present paper, the coupled theory, Lord–?hulman theory, and Green–Lindsay theory are introduced to study the influence of a magnetic field on the 2-D problem of a fiber-reinforced thermoelastic. These theories are also applied to study the influence of reinforcement on the total deformation of an infinite space weakened by a finite linear opening Mode-I crack. The material is homogeneous and an isotropic elastic half-space. The crack is subjected to a prescribed temperature and stress distribution. Normal mode analysis is used to solve the problem of a Mode-I crack. Numerical results for the temperature, the displacement, and thermal stress components are given and illustrated graphically in the absence and the presence of the magnetic field. A comparison between the three theories is also made for different depths.     

Fracture initiation at a crack or notch in a viscoplastic material under high rate loading

The two-dimensional problem of an edge crack in a half space or plate is considered. The body is loaded by a suddenly applied, spatially uniform normal velocity imposed on the plane boundary of the body on one side of the edge crack. Otherwise, the boundary of the body, including the crack faces, is traction free. Both cases of an initially sharp crack tip and a narrow notch with small but nonzero notch root radius are considered. The material is modeled as elastic viscoplastic, including strain hardening, rate sensitivity and thermal softening. The applied loading produces predominantly mode II loading in the crack tip region. Under these conditions it is possible to nucleate an adiabatic shear band at the crack tip as a precursor to a mode II fracture. On the other hand, because of the rate sensitivity of the material and the high rate of loading, it may be possible under certain conditions to generate tensile stresses in the crack tip region sufficiently large to nucleate brittle tensile fracture. The problem is solved numerically by means of the finite element method in order to investigate the competition between these two possible fracture initiation mechanisms. The magnitude of the impact velocity imposed on the edge of the plate and the notch tip acuity have an effect on processes near the crack tip. For given material, the inception of crack growth is determined by the competition between a stress-based brittle fracture condition, associated with rate sensitivity and strain hardening, and a strain based criterion, associated with high strain rate and thermal softening.     

The penny-shaped crack problem in micropolar thermoelasticity

The axisymmetry problem of a penny-shaped crack opened out by thermal loads is studied. The linear theory of micropolar elasticity is employed. Two types of thermal loads are considered—prescribed temperature on the crack faces and prescribed heat flux across the faces. It is shown that, in both the cases, the problem is equivalent to the isothermal problem of the crack opened out by suitable normal tractions on the crack faces. The stress intensity factors are found to depend on, in addition to the usual parameters, two parameters N and M; N is a number characterising the coupling of the displacement field with the microtation field and M is the ratio N/τ where τ is a non-dimensional material characteristic length. The classical values of the stress intensity factors are recovered as a limiting case. Numerical results are presented for the case of constant heat flux across the crack faces. These results show that the presence of couple stresses elevates the values of the stress intensity factors.     

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.     

An analysis of dynamic,ductile crack growth in a double edge cracked specimen

Dynamic crack growth is analysed numerically for a plane strain double edge cracked specimen subject to symmetric impulsive tensile loading at the two ends. The material behavior is described in terms of an elastic-viscoplastic constitutive model that accounts for ductile fracture by the nucleation and subsequent growth of voids to coalescence. Two populations of second phase particles are represented, including large inclusions or inclusion colonies with low strength, which result in large voids near the crack tip at an early stage, and small second phase particles, which require large strains before cavities nucleate. The crack growth velocities determined here are entirely based on the ductile failure predictions of the material model, and thus the present study is free from ad hoc assumptions regarding appropriate dynamic crack growth criteria. Adiabatic heating due to plastic dissipation and the resulting thermal softening are accounted for in the analyses. Different prescribed impact velocities, inclusion spacings and values of the inclusion nucleation stress are considered. Predictions for the dynamic crack growth behavior and for the time variation of crack tip characterizing parameters are obtained for each case analyzed.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

A two-dimensional problem of a mode-I crack in a rotating fibre-reinforced isotropic thermoelastic medium under dual-phase-lag model

In the present work, a general model of the equations of generalized thermoelasticity for a homogeneous isotropic elastic half-space solid whose surface is subjected to a mode-I crack problem under the effect of rotation is investigated. The normal mode analyses are used to obtain the expressions for the temperature distribution, the displacement component and thermal stresses in the context of the dual-phase-lag theory of thermoelasticity proposed by Tzou. The boundary of the crack is subjected to a prescribed stress distribution and temperature. Some particular cases are also discussed in the context of the problem. The numerical values of the temperature distribution, the displacement components and thermal stresses are also computed for a suitable material and the results are presented graphically. The effects of rotation, reinforcement and the phase lags parameters are discussed in detail in the light of earlier works.     

Inverse problems of material distributions for prescribed apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media

This study is concerned with the inverse problem of calculating material distributions intending to realize prescribed apparent fracture toughness in functionally graded material (FGM) coatings around a circular hole in infinite elastic media. The incompatible eigenstrain induced in the FGM coatings after cooling from the sintering temperature, due to mismatch in the coefficients of thermal expansion, is taken into consideration. An approximation method of determining stress intensity factors is introduced for a crack in the FGM coatings in which the FGM coatings are homogenized simulating the nonhomogeneous material properties by a distribution of equivalent eigenstrain. A radial edge crack emanating from the circular hole in the homogenized coatings is considered for the case of a uniform pressure applied to the surfaces of the hole and the crack. The stress intensity factors determined for the crack in the homogenized coatings represent the approximate values of the stress intensity factors for the same crack in the FGM coatings, and are used in the inverse problem of calculating material distributions in the FGM coatings intending to realize prescribed apparent fracture toughness in the coatings. Numerical results are obtained for a TiC/Al₂O₃ FGM coating, which reveal that the apparent fracture toughness in FGM coatings around a circular hole in infinite elastic media can be controlled within possible limits by choosing an appropriate material distribution profile in the coatings.