Hybrid stress finite element analysis of bending of a plate with a through flaw

A hybrid stress finite element procedure for the solution of bending stress intensity factors of a plate with a through-the-thickness crack is presented. Reissner's sixth-order plate theory including the effects of transverse shear deformation is used. The dominant singular crack tip stress field is embedded in the crack tip singular elements and only regular polynomial functions are assumed in the far field elements. The stress intensity factors can be calculated directly from the crack tip singular stress solution functions. The effects of the plate thickness, the ratio between the crack size and the inplane dimension of the plate, and the singular element size on the stress intensity factor solution are investigated. The effects of the explicit enforcement of traction-free conditions along crack surfaces, which are the natural boundary conditions in the present hybrid stress finite element model, are also investigated. The numerical results of bending of a plate with a straight central crack compare favourably with analytical solutions. It is also found that the explicit enforcement of traction-free conditions along crack surfaces is mandatory to obtain meaningful results for the Mode I type of bending stress intensity factor.     

Stress intensity factors of a central interface crack in a bonded finite plate and periodic interface cracks under arbitrary material combinations

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary material combinations. This paper deals with a central interface crack in a bonded finite plate and periodic interface cracks. Then, the effects of material combination and relative crack length on the stress intensity factors are discussed. A useful method to calculate the stress intensity factor of interface crack is presented with focusing on the stress at the crack tip calculated by the finite element method.     

A hybrid finite element analysis of interface cracks

A versatile hybrid finite element scheme consisting of special crack-tip elements and crack face contact elements is developed to analyse a partially closed interface crack between two dissimilar anisotropic elastic materials. The crack-tip element incorporates higher-order asymptotic solutions for an interfacial crack tip. These solutions are obtained from complex variable methods in Stroh formalism. For a closed interfacial crack tip, a generalized contact model in which the crack-tip oscillation is eliminated is adopted in the calculation. The hybrid finite element modelling allows the stress singularity at an open and closed crack tip to be accurately treated. The accuracy and convergence of the developed scheme are tested with respect to the known interface crack solutions. Utilizing this numerical scheme, the stress intensity factors and contact zone are calculated for a finite interface crack between a laminated composite material.     

Stress solution for a crack at an arbitrary angle to an interface

Two-dimensional elasticity solution and the stress intensity factors are determined for a finite crack in one of the materials of a bimaterial composite. The crack has an arbitrary orientation and distance from the straight interface. The solution for general stress boundary conditions on the crack surface is presented in the form of coupled Fredholm integral equations of the second kind. Numerical values of the stress intensity factors are computed for various crack orientations, distances from the interface, and different combinations of material properties when the boundary conditions are uniform pressure and uniform shear stress.     

两种材料界面裂纹的应力强度因子

选用复变函数的幂级数作为应力函数,分析在两种材料界面上存在裂纹的情况。利用裂纹自由表面力的边界条件及材料界面上的连续条件求得特征值及待定系数之间的关系式,从而求得应力、位移及应力强度因子表达式。本文修改了Lin[1]和Mar[2]的应力函数表达式,使所得应力、位移表达式与Williams[6]及Sih—Rice[3,4]结果一致。基于上述解析解建立了裂纹尖端的奇异元。并以含有界面中心裂纹平板拉伸情况作为计算实例。      

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.     

Computation of stress intensity factors of interface cracks based on interaction energy release rates and BEM sensitivity analysis

Stress intensity factors of bimaterial interface cracks are evaluated based on the interaction energy release rates. The interaction energy release rate is defined based on the energy release rates of a cracked body, corresponding to two independent loading conditions, actual field and an auxiliary field, and is related to the sensitivities of the potential energies for crack extensions. The potential energy of a cracked body is expressed with a domain integral, which is converted to a boundary integral expression by applying the divergence theorem. By differentiating this expression with the crack length, a boundary integral expression for the interaction energy release rate is obtained. The boundary integral representation for the interaction energy release rate involves the displacement, the traction, and their sensitivity coefficients with respect to the crack length. The boundary element sensitivity analyses are used to calculate these quantities accurately. A regularized boundary integral equation relating the boundary displacement and traction is differentiated with respect to an arbitrary shape parameter to derive the regularized boundary integral equation for the sensitivity coefficients of the boundary displacement and traction. The proposed approach is applied to several cracks in dissimilar media and the results are compared with those obtained by the conventional approach based on the extrapolation method. The analytical displacement and stress solutions for an interface crack between two infinite dissimilar media subjected to uniform stresses at infinity are used to give the auxiliary field, in which the values of the stress intensity factors are known. It is demonstrated that the present method can give accurate results for the stress intensity factors of various bimaterial interface cracks under coarse mesh discretizations.     

Electroelastic analysis of an electrically dielectric Griffith crack in a piezoelectric layer

The fracture analysis of an electrically dielectric Griffith crack embedded in a piezoelectric layer is made under in-plane electro-mechanical loadings. To simulate an opening crack full of a dielectric interior, the energetically consistent crack-face boundary conditions are utilized. Applying the Fourier transform technique, the boundary-value problem is reduced to solving two coupling singular integral equations. The intensity factors of stress, electric displacement, crack opening displacement (COD) and electric potential are further determined by the Lobatto-Chebyshev collocation method. The variations of the electric displacement at the crack surfaces are investigated by using the energetically consistent and semi-permeable boundary conditions respectively. The observations show that the electric displacement inside the crack is decreasing with an increase of the ratio between the crack length and piezoelectric layer width. Numerical computations are further carried out to compare the intensity factors of stress and electric potential, and the energy release rate using the energetically consistent boundary conditions with those using the semi-permeable boundary conditions. The obtained results reveal that the stress induced by a dielectric inside a crack has great effects on the stress intensity factor and energy release rate, but little influence on the electric potential difference across the crack.     

Weight function for an external circumferential semielliptical crack in a cylinder

In this paper, the weight function was extracted at the deepest point of a semielliptical circumferential crack. The crack is assumed to exist on the outer surface of the cylinder. For this purpose, the three‐dimensional finite element method was accomplished to specify two reference loads, which are indispensable for determining the weight function. The verification study confirms the accuracy of the derived weight function under prescribed mechanical loading on the crack surfaces. There is consistency among the solution results compared with those in the literature. The second part describes the application of the weight function for the thermal boundary conditions. Steady‐state thermal stress intensity factors are demonstrated using the weight function and presented as a closed‐form solution. The results were compared with the finite element data on the special case of thermal loading, and good agreement is obtained.     

Determination of stress intensity factors for bimaterial interface stationary rigid line inclusions by boundary element method

The stress intensity factors for a rigid line inclusion lying along a bimaterial interface are calculated by the boundary element method with the multiregion and the discontinuous traction singular elements. The relationships between the stress intensity factors and the inclusion surface stresses are derived. The numerically computed stress intensity factors for the bimaterial interface rigid line inclusion in the infinite body are proved to be in good agreement within 3% when compared with the previous exact solutions. In the finite bimaterial models, the stress intensity factors for the center and edge rigid line inclusions at the interface are computed with the variation of the rigid line inclusion length and the shear modulus ratio under the uniaxial and biaxial loading conditions.     

A complex variable boundary element method for solving interface crack problems

The problem of finite bimaterial plates with an edge crack along the interface is studied. A complex variable boundary element method is presented and applied to determine the stress intensity factor for finite bimaterial plates. Using the pseudo-orthogonal characteristic of the eigenfunction expansion forms and the well-known Bueckner work conjugate integral and taking the different complex potentials as auxiliary fields, the interfacial stress intensity factors associated with the physical stress-displacement fields are evaluated. The effects of material properties and crack geometry on stress intensity factors are investigated. The numerical examples for three typical specimens with six different combinations of the bimaterial are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

A direct determination of weight functions for arbitrary three-dimensional cracks – a combined analytical and numerical approach

A combined analytical and numerical method is proposed for the determination of the weight functions of stress intensity factors of cracks in an arbitrary three-dimensional elastic body. Having defined the weight functions for a given geometry of a structure, the stress intensity factors for arbitrary loading conditions can be obtained by a simple inner product of the weight function and a traction vector. Traditionally weight functions are defined in the two ways; the one is defined by the hyper-singular term of the eigen-function expansion of the displacement field of a cracked body, and the other is defined by the variation of displacement field with respect to a virtual extension of a crack. In the present paper, the weight functions for stress intensity factors are defined by applying the Maxwell-Betti's reciprocal theorem to an original problem and the auxiliary problems subjected to three kinds of force-couples acting on the crack surfaces near the limiting periphery of an arbitrary three-dimensional crack. In the present formulation, weight functions can be calculated by using a general-purpose finite element code combined with analytical expressions near the condensation point, where hyper-singularities exist. The validity of the method is confirmed by two- and three-dimensional illustrative examples.     

Evaluation of fracture parameters of composites subjected to thermal shock using the boundary element method and sensitivity analysis techniques

Quasi-static crack extension in fiber-reinforced composites subjected to thermal shock is analyzed using the boundary integral equation method, in combination with sensitivity analysis techniques. Buekner's formulation is employed to evaluate the stress intensity factor in a cracked body. This method eliminates the need for special element types to model the crack tip, as well as the use of a large number of elements near the cracked zone of interest. A numerical procedure involving sensitivity analysis techniques based on the adjoint structure approach has been developed to evaluate the energy integrals in the cracked body. Gradients of the functionals of response quantities with respect to variables such as the crack length, necessary for the evaluation of fracture parameters, are determined directly by this method. The numerical differentiation used in other numerical methods, such as the finite element method, which requires the repeated solution of the equations for different crack sizes is avoided. Results for stress intensity factors as a function of crack length are presented for various composite systems. These results are in good agreement with analytical results and results from the finite element method. The present approach results in significantly improved computational efficiency.     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.     

The effect of material combinations and relative crack size to the stress intensity factors at the crack tip of a bi-material bonded strip

Several types of singular stress fields may appear at the corner where an interface between two bonded materials intersects a traction-free edge depending on the material combinations. Since the failure of the multi-layer systems often originates at the free-edge corner, the analysis of the edge interface crack is the most fundamental to simulate crack extension. In this study, the stress intensity factors for an edge interfacial crack in a bi-material bonded strip subjected to longitudinal tensile stress are evaluated for various combinations of materials using the finite element method. Then, the stress intensity factors are calculated systematically with varying the relative crack sizes from shallow to very deep cracks. Finally, the variations of stress intensity factors of a bi-material bonded strip are discussed with varying the relative crack size and material combinations. This investigation may contribute to a better understanding of the initiation and propagation of the interfacial cracks.     

Determination of thermal stress intensity factors for an interface crack under vertical uniform heat flow

In the case where an interface crack exists in an infinite two-dimensional elastic bimaterial, the crack surface is insulated under traction-free conditions and the uniform heat flow vertical to the crack from an infinite boundary is given, temperature and stress potentials are obtained by using the complex variable approach to solve Hubert problems, and the results are used to obtain thermal stress intensity factors. The mode II thermal stress intensity factor only occurs if both the shear moduli, as well as the Poisson's ratios in the upper and lower material, are the same. Otherwise, mode I and II thermal stress intensity factors exist but the value of the mode I thermal stress intensity factor is much smaller than that of mode II.     

Stress intensity factors for curvilinear cracks loaded by bending and torsional moments

The geometry of a thin flat plate containing a curvilinear crack of finite size is investigated. The plate is loaded by uniform bending and torsional moments at infinite distance from the crack contour, while distributed normal bending moments and torques exist at the upper and lower crack surfaces. The bending stress intensity factors for the curvilinear crack are calculated on the basis of classical plate theory. The displacements and internal moments are represented by two complex analytic functions. Extra conditions are imposed to ensure the univalence of the displacements, which is not evident because the plate comprises a multiply connected domain due to the presence of the crack. A linearization with respect to the crack-curvature function has been performed and the bending stress intensity factors are calculated as the first-order solutions for slightly curved cracks. The results are illustrated with a few examples, such as uniform loading configurations and the geometry of a crack along a circular arc. The loading of thin flat plates by a combination of tensile forces and bending moments is also investigated. In analogy with the variation of the stress components over the cross section of the plate, two combined stress intensity factors are introduced having the same dependence on the perpendicular coordinate and being related to the symmetric and anti-symmetric stress intensity factors of the separate plane-stress and bending problems. The resulting energy release rate is shown to be in full agreement with known results in the literature.     

Coupled transient thermoelastic contact problems for axial cracks in hollow cylinders

A coupled transient thermoelastic behaviour of an axial-cracked hollow circular cylinder subjected to a sudden heating is investigated in this study. It is shown that surface heating may induce the 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, we can divide this problem into two parts and solve it by the principle of superposition. First, the temperature and transient thermal stress distributions along the axisymmetric surface of the imaginary cylinder without crack are obtained by finite element implicit time integration method Secondly, the opposite sense of the stress distributions along the cracked surfaces, which is obtained previously, is treated as the traction boundary conditions; the contact length and contact pressure of the real cracked cylinder are obtained by modified elimination finite element scheme. Finally, we also obtained the normalized stress intensity factor for the crack tip of the cylinder. It is concluded that the effect due to thermoelastic coupling term on stress intensity factor becomes more important for higher coupling coefficient, and this coupling term also results in a small time lag in temperature, thermal stress and stress intensity factor.     

Thermal fracture interference: a two-dimensional boundary element approach

Thermal loading of fractured structures is associated with the development of differential deformations along crack surfaces which result in the closure of the crack. Inherent non-linearities demand application of numerical procedures to resolve this problem. In this paper, a boundary element procedure is formulated to treat crack surface interference imposed under thermal steady-state or transient loadings. An iterative-incremental procedure is developed to deal with the non-linearity produced by the frictional contact of the crack surfaces. The open, adhesion and slip contact conditions are modeled through the utilization of the multi-domain technique. Two approaches are followed regarding the thermal boundary contact conditions along the crack region. In the first, crack surfaces are assumed to be thermally insulated. This assumption simplifies the formulation significantly. In the second, the crack surfaces are assumed to provide perfect thermal contact. Thermal stress intensity factors are evaluated from traction nodal results that adopt singular elements in the crack tip region. Numerical examples are illustrated, discussed and compared with analytical solutions, where possible. Fracture characteristics are predicted in terms of the involved parameters. As a general conclusion, peak values of thermal stress intensity factors depend on the friction conditions existing between crack faces.*Author for correspondence. (E-mail: nanif@mech.upatras.gr, Phone: +30-2610-997-197, Fax: +30-2610-997-207)     

Fatigue crack growth simulations of interfacial cracks in bi-layered FGMs using XFEM

An investigation of fatigue crack growth of interfacial cracks in bi-layered materials using the extended finite element method is presented. The bi-material consists of two layers of dissimilar materials. The bottom layer is made of aluminium alloy while the upper one is made of functionally graded material (FGM). The FGM layer consists of 100 % aluminium alloy on the left side and 100 % ceramic (alumina) on the right side. The gradation in material property of the FGM layer is assumed to be exponential from the alloy side to the ceramic side. The domain based interaction integral approach is extended to obtain the stress intensity factors for an interfacial crack under thermo-mechanical load. The edge and centre cracks are taken at the interface of bi-layered material. The fatigue life of the interface crack plate is obtained using the Paris law of fatigue crack growth under cyclic mode-I, mixed-mode and thermal loads. This study reveals that the crack propagates into the FGM layer under all types of loads.