Transient thermal contact problem for axial crack in a hollow circular cylinder

The transient behavior of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated. It is shown that surface heating may induce compressive thermal stress near the inner surface of the cylinder which in turn may force the cracked surfaces to close together. Assuming that the existence of the crack does not alter the temperature distribution, this problem can be divided into two parts and solved by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without a crack are obtained by finite element/implicit time integration method. The calculated temperature and thermal stress distributions are in good agreement with the values predicted by the analytical method. Secondly, the opposite senses of the stress distributions along the cracked surfaces, which are obtained previously, are treated as the traction boundary conditions, and the contact length and contact pressure of the real cracked cylinder are obtained by a modified elimination finite element scheme. In this scheme, the concepts of contact-node-pairs' penetration, contact-double-forces and compliance matrix are introduced. The calculated results indicate that the contact length ratio becomes smaller when the crack length ratio increases, and becomes larger as the radius ratio increases. Finally, the normalized stress intensity factor for the crack tip of the cylinder is obtained. It is shown that the larger the crack length ratio the higher the stress intensity factor.     

Hybrid finite element analysis of transient thermoelastic fracture problems subjected to general heat transfer conditions

This paper presents the singular characteristics of heat flux in the vicinity of the crack-tip for two dimensional transient thermoelastic fracture problems subjected to general heat transfer conditions at crack surfaces. Based on a restricted variational principle, a rigorous hybrid finite element procedure is then developed to perfectly describe the singularities of heat flux and thermal stress induced at the crack-tip. For verification purposes, the examples of transient thermoelastic problems with insulated crack surfaces are first analyzed. Excellent agreements between the computed results and referenced solutions can be drawn. To evaluate the influence of heat convection and radiation on the computation of temperature distributions and thermal stress intensity factors, several numerical examples are also presented.     

Thermal shock in an edge-cracked plate subjected to uniform surface heating

Thermal shock due to sudden surface heating of an edge-cracked plate is examined and compared with the opposite thermal shock condition that is associated with surface cooling. The plate is assumed to be insulated on one face with convective thermal boundary conditions existing on the side of the plate containing the crack. It is shown that surface heating results in compressive transient thermal stresses close to the plate surface which force the crack surfaces together over a certain contact length. The resulting nonlinear crack contact problem is formulated in terms of a singular integral equation and solved numerically. Calculated results include the transient stress intensity factors for various crack lengths at different values of the Biot number. A result of particular interest is the crack length at which the maximum stress intensity factor during heating exceeds the maximum stress intensity factor for cooling with otherwise identical heat transfer conditions.     

Calculation of stress intensity factors based on mode I crack opening integral parameters

We present stress intensity factor assessment using nodal displacements of the crack surfaces determined by the finite element method for cracked bodies. The equation is solved by expanding the crack opening displacement in the Chebyshev function, where crack front asymptotic behavior corresponds to the regulations of the linear elastic fracture mechanics. Results of the stress intensity factor calculations are obtained for test problems with analytical solution. Crack opening displacements are defined with the help of the 3D SPACE software package designed to model mixed variational formulation of the finite element method for displacements and strains of the thermoelastic boundary value problems. Translated from Problemy Prochnosti, No. 6, pp. 122–127, November–December, 2008.     

Transient thermoelastic analysis for a multilayered hollow cylinder with piecewise power law nonhomogeneity due to asymmetric surface heating

This paper is concerned with the theoretical treatment of transient thermoelastic problems involving a multilayered hollow cylinder with piecewise power law nonhomogeneity due to the asymmetrical heating of its surfaces. The thermal and thermoelastic constants of each layer are expressed as power functions of the radial coordinate, and their values continue on the interfaces. The exact solution for the two-dimensional temperature change in a transient state is obtained using the Laplace transformation and separation-of-variables method. The exact solution for the thermoelastic response of a multilayered hollow cylinder under the state of generalized plane strain, where the strain is not bound, is obtained herein. Some numerical results for the temperature change and the stress distributions are presented in figures. The influence of the functional grading on the thermal stresses is investigated. Furthermore, the influence on the axial stress of the restraint condition in the axial direction is investigated.     

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.     

含半椭圆表面裂纹圆柱壳体的三维热弹性动态断裂

研究了含轴向半椭圆表面裂纹的圆柱壳体在热应力与冲击载荷作用下的动态断裂情况,并应用所研制的三维动态断裂有限元程序进行了大规模的数值计算,确定了圆柱壳体的三维温度分布及热-力耦合下的动态应力强度因子,所得结果在一定程度上揭示了热-力作用下圆柱壳体的边界表面、裂纹面、物质惯性和弹性波的相互作用在结构动态断裂中的重要性。     

Stress intensity factors and crack opening displacements for slanted axial through-wall cracks in pressurized pipes

This paper proposes elastic stress intensity factors and crack opening displacements (CODs) for a slanted axial through-wall cracked cylinder under an internal pressure based on detailed three-dimensional (3D) elastic finite element (FE) analyses. The FE model and analysis procedure were validated against existing solutions for both elastic stress intensity factor and COD of an idealized axial through-wall cracked cylinder. To cover a practical range, four different values of the ratio of the mean radius of cylinder to the thickness ( R _m/ t ) were selected. Furthermore, four different values of the normalized crack length and five different values of the ratio of the crack length at the inner surface to the crack length at the outer surface representing the slant angle were selected. Based on the elastic FE results, the stress intensity factors along the crack front and CODs through the thickness at the centre of the crack were provided. These values were also tabulated for three selected points, that is, the inner and outer surfaces and at the mid-thickness. The present results can be used to evaluate the crack growth rate and leak rate of a slanted axial through-wall crack due to stress corrosion cracking and fatigue. Moreover, the present results can be used to perform a detailed leak-before-break analysis considering more realistic crack shape development.     

The application of a displacement controlled weight function for a single edge cracked plate to thermal fatigue damage assessment

A weight function based on displacement boundary conditions for the single edge cracked plate is extended and tested against finite element analysis results for the stress intensity factor under a linear end displacement. The stress intensity factor fluctuation, caused by sinusoidal thermal striping on the surface of the plate, is predicted as a function of crack depth and plate aspect ratio by the frequency response model with the new weight function and by the finite element method. Good agreement is found. Sample calculations are presented for the threshold surface temperature fluctuation for propagation of existing surface defects.     

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.     

Transient thermal stress intensity factors for an internal longitudinal semi-elliptical crack in a thick-walled cylinder

In this paper, the stress intensity factors are derived for an internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. First, the problem of transient thermal stresses in a thick-walled cylinder is solved analytically. Thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. The quasi-static solution of the thermoelasticity problem is derived analytically using the finite Hankel transform and then, the stress intensity factors are extracted for the deepest point and the surface points of the semi-elliptical crack using the weight function method. The results show to be in accordance with those cited in the literature in the special case of steady-state problem. Using the closed-form relations extracted for the transient thermal stress intensity factors, some conclusive results are drawn.     

Thermoelastic transient response of an infinitely long annular cylinder composed of two different materials

This paper deals with the transient response of one-dimensional axisymmetric quasistatic coupled thermoelastic problems. The governing equations, taking into account of the thermome-chanical term, are expressed in terms of temperature increment and displacement. Using the Laplace transform with respect to time, the general solutions of the governing equations are obtained in the transform domain. Also presented are the numerically transient distributions of stress and temperature increments in the real domain for the case of an infinitely long annular cylinder composed of two different materials. The inversion to the real domain is obtained by using a Fourier series technique and matrix operations simultaneously; therefore, no thermoelastic potentials are introduced in the solution process. It shows that the coupling effect behaves as a clear lag in both the stress and the temperature distributions.     

Weight function for a single edge cracked geometry with clamped ends

A single edge cracked geometry with clamped ends is well suited for fracture toughness and fatigue crack growth testing of composites and thin materials. Analysis of fiber bridging phenomenon in the composites and determination of stress intensity factors due to non-uniform stress distributions such as residual and thermal stresses generally require the use of a weight function. This paper describes the development and verification of a weight function for the single edge cracked geometry with clamped ends. Finite element analyses were conducted to determine the stress intensity factors (K) and crack opening displacements (COD) due to different types of stress distributions. The weight function was developed using the K and COD solution for a constant stress distribution. K and COD predicted using this weight function correlated well with the finite element results for non-uniform crack surface stress distributions.     

A wide range weight function for a single edge cracked geometry with clamped ends

A single edge cracked geometry with clamped ends is well suited for fracture toughness and fatigue crack growth testing of composites and thin materials. Stress intensity factors may be determined by the weight function method. A weight function for the single edge cracked geometry with clamped ends is developed and verified in this paper. It is based on analytical forms for the reference stress intensity factor and crack mouth opening displacement. The analytical forms are shown to be valid, by comparison with finite element results, over a wide range of crack depths and plate aspect ratios. Use of the analytical form enables the weight function to be calculated for any plate aspect ratio without the need for preliminary finite element analysis. Stress intensity factors and crack mouth opening displacements, predicted using this weight function, correlated well with finite element results for non-uniform crack surface stress distributions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Boundary element analysis of thermal stress intensity factors for interface griffith and cusp cracks

The thermal stress intensity factors for interface cracks of Griffith and symmetric lip cusp types under vertical uniform heat flow in a finite body are calculated by the boundary element method. The boundary conditions on the crack surfaces are insulated or fixed to constant temperature. The relationship between the stress intensity factors and the displacements on the nodal point of a crack-tip element is derived. The numerical values of the thermal stress intensity factors for an interface Griffith crack in an infinite body are compared with the previous solutions. The thermal stress intensity factors for a symmetric lip cusp interface crack in a finite body are calculated with respect to various effective crack lengths, configuration parameters, material property ratios and the thermal boundary conditions on the crack surfaces. Under the same outer boundary conditions, there are no appreciable differences in the distribution of thermal stress intensity factors with respect to each material property. However, the effect of crack surface thermal boundary conditions on the thermal stress intensity factors is considerable.     

Transient thermal stresses of work roll by coupled thermoelasticity

A numerical method, based on a two-dimensional plane strain model, is developed to predict the transient responses (that include distributions of temperature, thermal deformation, and thermal stress) of work roll during strip rolling by coupled thermoelasticity. The method consists of discretizing the space domain of the problem by finite element method first, and then treating the time domain by implicit time integration techniques. In order to avoid the difficulty in analysis due to relative movement between work roll and its thermal boundary, the energy equation is formulated with respect to a fixed Eulerian reference frame. The effect of thermoelastic coupling term, that is generally disregarded in strip rolling, can be considered and assessed. The influences of some important process parameters, such as rotational speed of the roll and intensity of heat flux, on transient solutions are also included and discussed. Furthermore, since the stress history at any point of the roll in both transient and steady state could be accurately evaluated, it is available to perform the analysis of thermal fatigue for the roll by means of previous data.     

Thermal shock and thermal fatigue study of ceramic materials on a newly developed ascending thermal shock test equipment

A test equipment was designed to study thermal shock and thermal fatigue of ceramic materials subjected to fast heating (ascending). The equipment was designed to generate thermal stress in a test specimen by heating one surface of it by an oxy-hydrogen flame while cooling the opposite surface. The sample cracked when thermal stress exceeded its mechanical strength. The in situ crack formation was detected by an acoustic emission system coupled to the set up. The hot zone temperature was measured by an infra red pyrometer. The equipment was also designed to run thermal fatigue test cycles in automatic mode between two selected temperatures. The temperature and thermal stress distribution in the test specimen were modelled using finite element software. The effect of temperature distribution of the top and bottom surfaces on thermal stresses was studied. It was observed that the thermal stress is very sensitive to the temperature distribution on the top surface and maximum near the periphery of the top surface. This was in agreement with the experimental results in which the cracks were originated from the periphery of top surface. It was also observed that the failure temperature was higher for thicker samples.     

Elastodynamic response of a penny-shaped crack in a cylinder of finite radius

The elastodynamic response of a penny-shaped crack in a cylinder of finite radius is investigated in this study. A step stress is applied to the crack surface resulting in transient behavior. The stress field near the crack front and the dynamic stress intensity factor are determined. Numerical resifits on the dynamic stress intensity factor are obtained to show the influence of inertia, geometry and their interactions on the load transfer to the crack.     

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.     

Calculation of stress intensity factors for a longitudinal semi-elliptical crack in a finite-length thick-walled cylinder

The purpose of this paper is to present the effect of finite boundary on the stress intensity factor of an internal semi-elliptical crack in a pressurized finite-length thick-walled cylinder  ( R _i/ t = 4)  . The three-dimensional finite element method, in conjunction with the weight function method, is used for computing the stress intensity factor at the deepest and surface points of an axial semi-elliptical crack in a cylinder. The transition aspect ratios, the aspect ratios in which the maximum stress intensity factor translates from the deepest to the surface points of the crack, are calculated for different relative depths and cylinder lengths. The results show that the stress intensity factor increases as the cylinder length decreases, especially at the corner point of the crack compared with the deepest point. The major advantage of this paper is that a closed-form expression is extracted for the stress intensity factor at the surface point of a semi-elliptical crack, which experiences higher changes due to the effect of the finite boundary of the cylinder.