CIRCUMFERENTIAL CRACK PROBLEM FOR AN FGM CYLINDER UNDER THERMAL STRESSES

The main objective of this study is to determine the stress intensity factors associated with a circumferential crack in a thin-walled cylinder subjected to quasi-static thermal loading. The cylinder is assumed to be a functionally graded material. In order to make the problem analytically tractable, the thin-walled cylinder is modeled as a layer on an elastic foundation whose thermal and mechanical properties are exponential functions of the thickness coordinate. Hence a plane strain crack problem is obtained. First temperature and thermal stress distributions for a crack-free layer are determined. Then using these solutions, the crack problem is reduced to a local perturbation problem where the only nonzero loads are the crack surface tractions. Both internal and edge cracks are considered. Stress intensity factors are computed as functions of crack geometry, material properties, and time.     

THERMAL STRESS IN AN ORTHOTROPIC PLATE CONTAINING A PAIR OF COPLANAR CENTRAL CRACKS

Abstract

The thermal stress problem for a pair of coplanar central cracks contained in an orthotropic plate is investigated using the techniques of Fourier transforms and finite Hilbert transforms. The crack surfaces are subjected to symmetrical thermal loadings. Exact expressions for the temperature field, the crack shapes, and the thermal stresses in the crack plane are obtained for the case of constant temperature. The opening-mode stress intensity factors at the inner and outer crack tips are also obtained in closed forms in terms of the material properties and the distance between the cracks. The properties of the stress intensity factors are shown to be different from the shear-mode stress intensity factors because of the disturbance of a uniform heat flow by a pair of central cracks.     

THERMAL FRACTURE IN A BIOMATERIAL DURING RAPID FREEZING

Thermal stresses leading to fracture during rapid freezing of a biomaterial are studied using potato tissues as an example. The thermal model includes the gradual freezing of the tissue, that is, the presence of a mushy zone. Water in the biomaterial expands almost 9% during freezing and develops transient stresses when the material is frozen from all sides. A viscoelastic model for the tissue is used with mechanical properties changing continuously during freezing. A comparison of stresses developed during freezing, with experimentally measured failure strength, is used as a criterion for initiation of a crack. The strain energy release rate G is calculated using the J integral. Experimental evidence of a catastrophic fracture during very rapid freezing (boundary temperature of 200 C) suggests that the G value is likely to exceed the critical rate (G) at this temperature but not so for slow freezing at a boundary temperature of c.40 C, for which no experimental fractures are observed. A multistep freezing process consisting of initial slow freezing followed by fast freezing to reach the final temperature of 200 C reduces the strain energy release rate, and experiments show that catastrophic failure can be avoided with this protocol.     

STEADY THERMAL STRESSES IN AN INFINITE NONHOMOGENEOUS ELASTIC SOLID CONTAINING A CRACK

This article considers the crack problem for an infinite nonhomogeneous elastic solid subjected to steady heat flux over the crack surfaces. The aim is to understand the effect of nonhomogeneities of materials on stress intensity factors. By using the Fourier transforms, the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. Results are presented illustrating the influence of the nonhomogeneity of the material on the stress intensity factors. For some groups of the material constants, there exist minimum stress intensity factors, which is very interesting for the understanding of compositions of advanced functionally gradient materials.     

EDGE CRACK IN A NONHOMOGENEOUS HALF PLANE UNDER THERMAL LOADING

This paper deals with the problem of an edge crack in a semi-infinite nonhomogeneous plate under steady heat flux loading conditions. The objective of the study is to assess the effect of material nonhomogeneity on the thermal stress intensity factor. All material properties are supposed to be exponentially dependent on the distance from the boundary of the plate. By using the Fourier transform, the problem is reduced to a singular integral equation that is solved numerically. The thermal stress intensity factors for various material constants are calculated. The results show that by selecting the material constants appropriately the stress intensity factor can be reduced.     

TWO CRACK GROWTHS IN A FUNCTIONALLY GRADED PLATE UNDER THERMAL SHOCK

This article examines the problem of two thermal cracks under a transient temperature field in a ceramic/metal functionally graded plate. When the functionally graded plate is subjected to thermal shock, multiple cracks often occur on the ceramic surface. It is shown that the crack paths are influenced by interaction between multiple cracks and a compositional profile of the functionally graded plate. Transient thermal stresses are treated as a linear quasi-static thermoelastic problem for a plane strain state. The crack paths of two cracks are obtained using the finite element method with mode I and mode II stress intensity factors.     

THERMAL STRESS ANALYSIS IN THERMOPIEZOELASTIC STRIP WITH AN EDGE CRACK

The analysis of thermal stresses becomes important when the piezoelectric material has to be operated in either extremely cold or hot temperature environments. Hence, it is essential to know the interaction of mechanical defects with temperature changes. This investigation is concerned with a strip problem of transversely isotropic thermopiezoelastic material containing an edge crack under partial thermal and electric loading conditions. Thermopiezoelastic stresses are analyzed by introducing potential functions and Fourier transforms. The problem reduces to solving a singular integral equation, and the singular integral equation is solved. Numerical calculations of the thermal stress intensity factors are carried out for a cadmium selenide material.     

THERMAL SHOCK FRACTURE IN A W-Cu DIVERTOR PLATE WITH A FUNCTIONALLY GRADED NONHOMOGENEOUS INTERFACE

The plane elasticity solution is presented in this article for the crack problem of a W-Cu divertor plate under thermal shock. The material is made of a graded layer with exponentially varying thermomechanical properties bonded between a homogeneous substrate and a homogeneous coating and is subjected to a cycle of heating and cooling on the coating surface of the material. The surface layer contains an embedded or a surface crack perpendicular to the boundaries. Using superposition the problem is reduced to a perturbation problem in which the crack surface tractions are only external forces. The dimensions, geometry, and loading conditions of the original problem are such that the perturbation problem may be approximated by a plane strain mode I crack problem for an infinite divertor plate. Fourier transforms are used to formulate the crack problem in terms of a singular integral equation. After giving some sample results regarding the distribution of thermal stresses, stress intensity factors for embedded and surface cracks are presented. Also included are the results for a crack/contact problem in a divertor plate that is under compression near and at the surface and tension in the interior region.     

THERMAL SHOCK FRACTURE IN AN EDGE-CRACKED PLATE

In this paper the transient thermal stress problem for an elastic strip with an edge crack is investigated. The elastic medium is assumed to be insulated on one face and cooled by surface convection on the face contaning the edge crack. Using the principle of superposition, the formulation results in a mixed boundary value problem, with the thermal stresses calculated from the thermoelasticity solution for an uncracked strip utilized as the necessary crack surface tractions. The resulting singular integral equation is of a well-known type and is solved numerically. In this paper, inertia effects are assumed negligible and possible temperature dependence of thermoelastic constants is not considered. The numerical results presented, include the stress intensity factor as a function of nondimensional time (Fourier number) and crack length, for various values of the dimensionless Biot number. The temperature distribution and the thermal stresses in the uncracked strip are also included. The time lag, which occurs between the time at which the stress on the surface of the strip is a maximum and the time when a maximum occurs in the stress intensity factor, is clearly shown to be a function of the Biot number for any given ratio of crack length to strip thickness. A result of particular interest is the degree with which the maximum stress intensity factor decreases, as a function of crack length, for decreasing values of the Biot number.     

MODELING OF THERMAL CRACKING IN ELASTIC AND ELASTOPLASTIC TWO-PHASE SOLIDS

A review is given about fracture mechanical investigations concerning the thermal crack initiation and propagation in one of the segments or in the material interface of two-arid three-dimensional self-stressed two-phase compounds. The resulting boundary value problems of the stationary thermoelasticity and thermoplasticity for the cracked two- and three-dimensional bimaterial structures considered are solved using the finite element method. Furthermore, by applying an appropriate crack growth criterion based on the numerical calculation of the total energy release rate G of a quasistatic mixed-mode crack extension the further development of thermal crack paths starting at the intersection line of the material interface with the external stressfree surface of the two- and three-dimensional elastic bimaterials could be predicted. In the case of the disklike two-phase compounds, the theoretically predicted crack paths show a very good agreement with results gained by associated cooling experiments. Several specimen geometries consisting of different material combinations and subjected to uniform and nonuniform temperature distributions have been studied using the relevant methods of fracture mechanics. Thereby thermal cracks propagating in one segment of an elastic bimaterial only obey the condition G_II = 0, whereas for interface cracks a mixed-mode propagation is always existent where the G_II values play an important role. Moreover, by applying the proposed crack growth criterion the possible crack kinking direction ? of an interface crack tip out of the interface could be predicted by taking into consideration the finite thickness of an interlayer (interphase). In addition, an analysis of the stress and strain fields in the vicinity of thermal interface cracks in the discontinuity area of two- and three-dimensional elastoplastic two-phase compounds has been performed by using the FE-method. Thereby a heat source Q was assumed in one of the two materials in the neighborhood of an interface crack tip. The corresponding stress states in the bimaterial structuresand especiallyin the vicinity of the interface crack tip have been calculated by applying the incremental I₂-plasticity and using a bilinear hardening material law and based on a sequentially coupled solution of the heat transfer and the thermal stress boundary value problems. Finally, the failure assessment has been performed on the basis of the local J-integral which, for three-dimensional interface cracks, was recently generalized by two of the authors.     

CRACK PROBLEMS FOR FUNCTIONALLY GRADED MATERIALS UNDER TRANSIENT THERMAL LOADING

The problem considered in this article is the response of a graded composite material plate containing some noncollinear cracks subjected to dynamic thermal loading. It is assumed that all the material properties depend only on the coordinates y (along the thickness direction) . In the analysis, graded regions are treated as a series of perfectly bonded composite layers, each layer being assigned slightly different material properties. Utilizing the Laplace transform and Fourier transform techniques, the general solution for each layer is derived. The complete solution of the entire medium is then obtained by introducing the mechanical boundary and layer interface conditions. The main features of the proposed method are: (1) the material may be orthotropic, (2) multiple crack problem, (3) the material properties may vary arbitrarily along the thickness direction, and (4) with the inertial terms taken into account, the present algorithm can be applied to a fracture problem under dynamic mechanical loading. Numerical examples are provided for a FGM and a substrate FGM coating structure under a nonuniform heating condition. Transient and steady-state thermal stress intensity factors are calculated and their variation due to a change of the material gradient and the location of the crack are studied.     

MULTIPLE SURFACE CRACKING IN GLASS-FIBER-REINFORCED PLASTICS DUE TO A THERMAL SHOCK AT A LOW TEMPERATURE

We consider the transient thermal singular stress problem of multiple surface cracking in glass-fiber-reinforced plastics due to a thermal shock at a low temperature. The layered composite is made of a layer bonded between two layers of different physical properties, and it is suddenly cooled on the surfaces. The surface layers contain parallel arrays of the embedded or edge cracks perpendicular to the boundaries. The thermal and elastic properties of the material are dependent on the temperature. For the case of the crack that ends at the interface between orthotropic elastic materials, the order of stress singularity around the tip of the crack is obtained. Finite element calculations are carried out, and the transient thermal stress intensity factors are shown graphically.     

Thermal Stress Analysis for Octagonal Quasicrystals

The complex variable method for solving the two-dimensional thermal stress problem of octagonal quasicrystals is stated. The closed-form solutions for octagonal quasicrystals containing an elliptical hole subjected to a remote uniform heat flow are obtained. When the hole degenerates into a crack, the explicit solutions for the stress intensity factors and energy release rate are presented.     

THERMAL STRESSES IN LAMINATED BEAMS

This paper considers laminated beams consisting of layers of different materials fastened together by thin adhesives. The stresses that result from subjecting the beam to temperature stimuli are calculated. The problem is treated by two-dimensional elasticity theory in conjunction with the variational theorem of complementary energy. A pair of governing differential equations is developed, and boundary conditions concerning stress-free surfaces and ends of the beam are satisfied. The calculation of the distributions of interlaminar normal and shear stresses shows that high stress intensity occurs in the end zones of the beam. Thus, the satisfaction of end conditions is of prime importance in the analysis of laminated structural elements. Delamination failure-when it occurs - will probably start at the ends of the beam, This agrees with observed failures of laminated structural elements subjected to stress-free end conditions.     

CRACK KINKING IN FUNCTIONALLY GRADED MATERIALS DUE TO AN INITIAL STRAIN RESULTING FROM STRESS RELAXATION

Small crack kinking in functionally graded materials (FGMs) subjected to a constant initial strain resulting from stress relaxation is studied. The FGMs are modeled as simply nonhomogeneous materials; that is, the effect of microstructure is neglected and the material property variation is considered to be continuous. The material gradation in the FGMs is taken to be of a power-law type. Systematic finite element calculations are made for the energy release rate of the crack. With regard to the local homogeneity, the toughness is taken to be independent of direction; therefore, the crack propagates along the direction of the maximum energy release rate. It is shown that the crack length and the thickness of the surface zone subjected to the initial strain have a strong effect on the kink direction, whereas the material gradients have little effect.     

Thermal Fracture Analysis Of Nonhomogeneous Plate with Interfaces Under Uniform Heat Flow

The mixed-mode thermomechanical fracture problem in a nonhomogeneous material plate with two interfaces is studied in this research. Uniform heat flow conditions are considered. The interaction energy integral method for the thermal fracture problem is developed to calculate the thermal stress intensity factors (TSIFs) in nonhomogeneous materials. This method is proved to be domain independent for nonhomogeneous materials even when the integral domain is cut by one interface or many interfaces. Combining the interaction energy integral method with the extended Finite Element Method (XFEM), the temperature fields, the displacement fields, the thermal stress fields, and the TSIFs are calculated. In this article, both the edge crack and the internal crack are considered. Some examples are presented to study the influence of the material properties on the TSIFs. It can be found that the mismatch of the elastic modulus and thermal expansion coefficient can affect the TSIFs dramatically; however, the thermal conductivity interface will not arouse a kinking behavior of the TSIFs. It can be concluded that the existence of an interface (especially for elastic modulus and thermal expansion coefficient) affects the TSIFs greatly.     

THERMAL STRESS ANALYSIS OF A RECTANGULAR PLATE AND ITS THERMAL STRESS INTENSITY FACTOR FOR COMPRESSIVE STRESS FIELD

Abstract

In this study, a transient thermal stress problem of a rectangular plate due to a nonuniform heat supply is treated theoretically and, thereafter, fracture behaviors of the plate with a crack are examined for compressive stress states. Assuming that a crack located on an arbitrary position, with an arbitrary direction, is sufficiently small and is closed because of the compressive stress field, a temperature field, in a transient state, is analyzed by taking into account the effect of relative heat transfer on both surfaces of the plate. Thereafter, the corresponding thermal stress analysis is developed on the basis of the two-dimensional plane stress problem using Airy's stress function method, and the stress intensity factor is analyzed for the biaxial stress state. As an analytical model, we consider mechanical boundary conditions of prescribed displacement and estimate the stress intensity factor of a crack tip using parameters of the crack configuration such as the location, direction, length, and coefficient of friction. These numerical results are shown in graphical form.     

IMPORTANCE OF INERTIA TERM IN DYNAMIC CRACK PROBLEMS CONSIDERING LORD–SHULMAN THEORY OF THERMOELASTICITY

A numerical technique is presented for the accurate calculation of stress intensity factors as a function of time for generalized coupled thermoelastic problems. In this task, the effect of the inertia term is investigated, considering different theories of thermoelasticity, and its importance is shown.

A boundary element method using the Laplace transform in time domain is developed for the analysis of fracture mechanics; dynamic coupled thermoelasticity problems with relaxation time are considered in the two-dimensional finite domain. The Laplace transform method is applied to the time domain and the resulting equations in the transformed field are discretized using the boundary element method. Actual physical quantities in the time domain are obtained using the numerical inversion of the Laplace transform method.

The singular behavior of the temperature and displacement fields in the vicinity of the crack tip is modeled by quarter-point elements. The thermal dynamic stress intensity factor for mode I is evaluated using the J-integral method. The accuracy of the method is investigated through comparison of the results with the data available in literature.

The J integral, which represents the dynamic energy release rate for propagating cracks, contains a boundary integral and a domain integral. The boundary integral contains strain energy, tractions, and strains whereas the domain integral contains inertia and strains. The J-integral method allows these two terms to be calculated separately. In this way, the importance of each term may be investigated by considering different theories of dynamic thermoelasticity.     

Optimum material distributions for prescribed apparent fracture toughness in thick-walled FGM circular pipes

This study treats the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in thick-walled functionally graded material (FGM) circular pipes. The incompatible eigenstrain induced in the pipes after cooling from the sintering temperature due to the nonhomogeneous coefficient of thermal expansion is taken into consideration. An approximation method of finding stress intensity factors for a crack in the FGM pipes is introduced in which the nonhomogeneous material properties are simulated by a distribution of equivalent eigenstrain. A radial edge crack emanating from the inner surface of the homogenized pipes is considered for the case of a uniform internal pressure applied to the surfaces of the pipes and the crack. The stress intensity factors determined for the crack in the homogenized pipes represent the approximate values of the stress intensity factors for the same crack in the FGM pipes, and are used in the inverse problem of evaluating optimum material distributions intending to realize prescribed apparent fracture toughness in the FGM pipes. Numerical results obtained for a thick-walled TiC/Al₂O₃ FGM pipe reveal that the apparent fracture toughness significantly depends on the material distributions, and can be controlled within possible limits by choosing an optimum material distribution profile.     

EFFECT OF THERMAL RESIDUAL STRESS ON THE INTERLAMINAR CRACK EXTENSION BEHAVIOR IN CROSS-PLY LAMINATES DUE TO TRANSVERSE LOADING

The effect of thermal residual stress on the two-dimensional interlaminar crack extension behavior in a cross-ply laminate subjected to transverse loading is estimated. Attention is focused on the contribution of thermal residual stress to the local energy release rate along the interlaminar crack front. Computational simulations are carried out on the basis of constant critical energy release rates in order to examine implicitly how the two-dimensional size and shape would be changed by the presence of thermal residual stress. A considerably large amount of energy is found to be supplied for the interlaminar crack extension by thermal residual stress, while little influence is perceived with respect to the apparent extension behavior including the shape of the interlaminar crack.