T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms (original) (raw)
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International journal of solids and structures, 2003
The interaction integral is an accurate and robust scheme for evaluating mixed-mode stress intensity factors. This paper extends the concept to orthotropic functionally graded materials and addresses fracture mechanics problems with arbitrarily oriented straight and/or curved cracks. The gradation of orthotropic material properties are smooth functions of spatial coordinates, which are integrated into the element stiffness matrix using the so-called ''generalized isoparametric formulation''. The types of orthotropic material gradation considered include exponential, radial, and hyperbolic-tangent functions. Stress intensity factors for mode I and mixed-mode two-dimensional problems are evaluated by means of the interaction integral and the finite element method. Extensive computational experiments have been performed to validate the proposed formulation. The accuracy of numerical results is discussed by comparison with available analytical, semi-analytical, or numerical solutions. Im imaginary part of the complex function J path-independent J -integral for the actual field J aux J -integral for the auxiliary field J s J -integral for the superimposed fields (actual and auxiliary) J Jacobian matrix J À1 inverse of the Jacobian matrix K I mode I stress intensity factor K II mode II stress intensity factor K 0 normalizing factor for stress intensity factors,
Engineering fracture mechanics, 2002
A finite element methodology is developed for fracture analysis of orthotropic functionally graded materials (FGMs) where cracks are arbitrarily oriented with respect to the principal axes of material orthotropy. The graded and orthotropic material properties are smooth functions of spatial coordinates, which are integrated into the element stiffness matrix using the isoparametric concept and special graded finite elements. Stress intensity factors (SIFs) for mode I and mixed-mode two-dimensional problems are evaluated and compared by means of the modified crack closure (MCC) and the displacement correlation technique (DCT) especially tailored for orthotropic FGMs. An accurate technique to evaluate SIFs by means of the MCC is presented using a simple two-step (predictor-corrector) process in which the SIFs are first predicted (e.g. by the DCT) and then corrected by Newton iterations. The effects of boundary conditions, crack tip mesh discretization and material properties on fracture behavior are investigated in detail. Many numerical examples are given to validate the proposed methodology. The accuracy of results is discussed by comparison with available (semi-) analytical or numerical solutions.
Mechanics of Materials, 2008
The plane strain problems of semi-infinite cracks in an infinite functionally graded orthotropic material are studied. Two uniform impact loading modes are considered, i.e. opening and in-plane shear. Laplace and Fourier transforms along with the Winner-Hopf technique are employed to solve the displacement formulation of the equations of motion. Closedform solutions of the dynamic stress intensity factors are obtained. It is observed that the stress intensity factors are not all proportional to the square root of time as expected. The results can be reduced to the known solutions derived independently for orthotropic or isotropic materials.
Computer Methods in Applied Mechanics and …, 2003
For linear elastic functionally graded materials (FGMs), the fracture parameters describing the crack tip fields include not only stress intensity factors (SIFs) but also T-stress (nonsingular stress). These two fracture parameters are important for determining the crack initiation angle under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). In this paper, the mixed-mode SIFs and T-stress are evaluated by means of the interaction integral, in the form of an equivalent domain integral, in combination with the finite element method. In order to predict the crack initiation angle in brittle FGMs, this paper makes use of a fracture criterion which incorporates the T-stress effect. This type of criterion involves the mixed-mode SIFs, the T-stress, and a physical length scale r c (representative of the fracture process zone size). Various types of material gradations are considered such as continuum models (e.g. exponentially graded material) and micromechanics models (e.g. self-consistent model). Several examples are given to show the accuracy and efficiency of the interaction integral scheme for evaluating mixed-mode SIFs, T-stress, and crack initiation angle. The techniques developed provide a basic framework for quasi-static crack propagation in FGMs.
Engineering Fracture Mechanics, 2004
A ''non-equilibrium'' formulation is developed for evaluating T -stress in functionally graded materials with mixedmode cracks. The T -stress is evaluated by means of the interaction integral (conservation integral) method in conjunction with the finite element method. The gradation of material properties is integrated into the element stiffness matrix using the so-called ''generalized isoparametric formulation''. The types of material gradation considered include exponential, linear, and radially graded exponential functions; however, the present formulation is not limited to specific functions and can be readily extended to micromechanics models. This paper investigates several fracture problems (including both homogeneous and functionally graded materials) to verify the proposed formulation, and also provides numerical solutions to various benchmark problems. The accuracy of numerical results is discussed by comparison with available analytical, semi-analytical, or numerical solutions. The revisited interaction integral method is shown to be an accurate and robust scheme for evaluating T -stress in functionally graded materials.
Mechanics of materials, 2003
The path-independent J Ã k -integral, in conjunction with the finite element method (FEM), is presented for mode I and mixed-mode crack problems in orthotropic functionally graded materials (FGMs) considering plane elasticity. A general procedure is presented where the crack is arbitrarily oriented, i.e. it does not need to be aligned with the principal orthotropy directions. Smooth spatial variations of the independent engineering material properties are incorporated into the element stiffness matrix using a ''generalized isoparametric formulation'', which is natural to the FEM. Both exponential and linear variations of the material properties are considered. Stress intensity factors and energy release rates for pure mode I and mixed-mode boundary value problems are numerically evaluated by means of the equivalent domain integral especially tailored for orthotropic FGMs. Numerical results are discussed and validated against available theoretical and numerical solutions.
Mathematical Problems in Engineering, 2013
This paper puts forward two different-integral-based methods, which can be used to perform mixed-mode fracture analysis of orthotropic functionally graded materials subjected to hygrothermal stresses. The first method requires the evaluation of both components of-integral, whereas the second method employs the first component 1 and the asymptotic crack tip displacement fields. Plane orthotropic hygrothermoelasticity is the basic theory behind the-integral formulation, which is carried out by assuming that all material properties are functions of the spatial coordinates. Developed procedures are implemented by means of the finite element method and integrated into a general purpose finite element analysis software. Temperature and specific moisture concentration fields needed in the fracture analyses are also computed through finite element analysis. Each of the developed methods is utilized in conjunction with the superposition technique to calculate the hygrothermal fracture parameters. An inclined crack located in a hygrothermally loaded orthotropic functionally graded layer is examined in parametric analyses. Comparisons of the results generated by the proposed methods do indicate that both methods lead to numerical results of high accuracy and that the developed form of the-integral is domain independent. Further results are presented so as to illustrate the influences of crack inclination angle, crack length, and crack location upon the modes I and II stress intensity factors.
International journal for numerical …, 2003
The interaction integral is a conservation integral that relies on two admissible mechanical states for evaluating mixed-mode stress intensity factors (SIFs). The present paper extends this integral to functionally graded materials in which the material properties are determined by means of either continuum functions (e.g. exponentially graded materials) or micromechanics models (e.g. self-consistent, Mori-Tanaka, or three-phase model). In the latter case, there is no closed-form expression for the material-property variation, and thus several quantities, such as the explicit derivative of the strain energy density, need to be evaluated numerically (this leads to several implications in the numerical implementation). The SIFs are determined using conservation integrals involving known auxiliary solutions. The choice of such auxiliary ÿelds and their implications on the solution procedure are discussed in detail. The computational implementation is done using the ÿnite element method and thus the interaction energy contour integral is converted to an equivalent domain integral over a ÿnite region surrounding the crack tip. Several examples are given which show that the proposed method is convenient, accurate, and computationally e cient. integral for two admissible states (actual and auxiliary ÿelds) of an elastic solid. The analysis requires evaluation of the integral along a suitably selected path surrounding the crack tip (far-ÿeld).