Evaluation of fracture parameters for functionally gradient materials J. Sladek, V. Sladek & M. Hrina Institute of Construction and Architecture, Slovak Academy of Sciences, 842 20 Bratislava, Slovakia Abstract Efficient numerical methods are presented to compute fracture parameters (stress intensity factor and T-term stress) for a cracked composite material body subjected to a stationary thermal load. The path independent integral representations for stress intensity factors and T-term stress in functionally gradient materials are derived. The thermomechanical fields along the integration path and in the enclosed domain are obtained by the boundary element method. The method is appropriate for computation of fracture parameters because the contour integral is evaluated over the path, which is far away from the crack tip. Numerical results are given for a thick-walled tube with edge cracks on the internal surface. 1 Introduction Cracking of structures due to thermal loading is an important phenomenon in many industrial applications, especially aerospace and nuclear engineering. Only composite materials can resist high temperatures effectively and simultaneously the thermal stresses could be relaxed significantly. Therefore, in recent years, the concept of so-called functionally gradient materials (FGM) has been introduced and applied to the development of structural component. In FGM the volume fractions are optimized to satisfy both operational and minimal stress requirements. In such materials, although the absence of sharp interfaces does largely reduce material property mismatch, cracks occur when they are subjected to external loading. Ceramics are brittle materials and microcracks play crucial roles in determining the strength and life of components under service conditions.
36 Damage and Fracture Mechanics VI Conventional fracture theories assume that the state of stress and strain in the vicinity of a crack tip is characterized by a single parameter. This single parameter, frequently called the first fracture parameter, can be the stress intensity factor (SEF) or J-integral. In literature there is a lot of original techniques for evaluation these fracture parameters. However, they are mostly restricted to homogeneous materials. Recently, some results have been published, where the SIF for a crack in the FGM is computed [1-3]. In all above mentioned papers the analytical methods have been used for solution of boundary value problems. In analytical methods the selection of a technique for evaluation of fracture parameters is not such significant as in a pure numerical approach. Namely, in numerical methods the most inaccurate quantities are obtained at the crack tip vicinity. Then, the results obtained directly from asymptotic expansion formulae are less accurate than those obtained from integral representations along a contour far away from the crack tip. Early theories of fracture mechanics assume that the stress and displacement states are controlled by the stress intensity factor. Significant quantitative changes could occur when the leading nonsingular terms of the crack tip series expansion are included. Recent numerical and experimental studies in elastostatics have attempted to describe fracture by using two parameters such that additional information could be gained. This second parameter is nonsingular corresponding to the uniform stress term in crack tip series expansion. It shall be referred to as the T-term stress [4]. Many possibilities exist for evaluation of the T-term [5-7]. Like to the SIF, all suggested methods are restricted to homogeneous bodies. The purpose of this paper is to present an accurate computational method for evaluation of the T-term stress in functionally gradient material bodies subjected to stationary thermal loading. The reciprocity theorem will be applied to derive a path independent integral for evaluating the T-term stresses. Suggested computational methods are applied to numerical examples for a thick-walled cracked tube. 2 Evaluation of thefirstfracture parameter In stationary thermoelasticity for non-homogeneous bodies the thermal and stress fields are described by the following governing equations [8] where 9 -g (1) 0 (2) is the temperature measured in the scale with its origin at the equilibrium state, %, and Q are body force vector and heat source, respectively and o-jj is stress tensor. The coefficient of heat conduction A is a continuous function of cartesian coordinates. A subscript preceded by a comma denotes a differentiation with respect to the corresponding cartesian coordinate. The strain tensor s^ is given by displacement gradients
Damage and Fracture Mechanics VI 37 In the presence of temperature gradient, the total strain tensor can be decomposed into its elastic part e^ and another one accounting for the free thermal expansion of medium. Thus, f,y=f*+a^0 (4) where a is the coefficient of thermal expansion and 5^ is the Kronecker delta. According to the Neumann hypothesis the stress tensor is related to the elastic strain in the usual way, viz 0",y=%,4 (5) where c^ is the tensor of material constants. For an isotropic continuum it is given by where // is the Lame constant (shear modulus) and v is Poisson's ratio. Note that the material is non-homogeneous in general, since the elasticity tensor and the coefficient of thermal expansion depend on position. The elastic strain energy density W = W(e^,x^ can be written in the following form Then, gradient of strain energy density is given as w -. + -*4,2 (6) "" &V*» l^l, ^^ UJexp, where "explicit" derivation of the strain energy density for non-homogeneous material and temperature gradient fields becomes ah " Y^K"^ "" ^'^^ ^ "" ^ ^y^^ (^) \ /exp/ Utilizing eqs. (3) and (5) the gradient of strain energy density can be rewritten into the form ^,m = k^w))-^.y^ +KJ^ (9) Then, from eqs. (9) and (2) it follows that W/m -^^wjly = ^^,m +K^Lpy (10) An integral form of equation (10) may be obtained upon application of the divergence theorem. If Q is a regular bounded region enclosed by a surface T whose unit outward normal vector is /*, it follows that
38 Damage and Fracture Mechanics VI (11) Integral identity (11) is valid in a region where no field irregularities prevail. In a crack problem the stresses at the crack tip are singular and displacements are discontinuous on both crack surfaces. Therefore, a small region Q^ has to be excluded at the vicinity of a crack tip. This region is surrounded by F^ as shown in Fig. 1. Fig. 1. Integration paths and coordinate definitions. Both fields cr^ and %, are regular in the region Q-Q^. Contour F = FQ + F^ - T^ + r~ is a closed integration path in the anticlokwise direction. The radius s is considered to be very small and shrunk to zero in the limiting process. The crack surfaces F*,F~ are assumed to be free of tractions, tj = (Tijtij = 0, and crack is parallel to the x\ - axis. Then, eq. (11) can be written as -lim "o-o., (12) The left hand side of eq. (12) is equal to the definition of J - integral [9] for m = 1. Then, o-og (13) According to Sih et al. [10] the local character of thermal stresses at the crack tip vicinity is the same nature as problems with static mechanical stresses. Noda and Jin [11] analysed the asymptotic distribution of stresses for cracks in a nonhomogeneous body. It is found that the crack tip fields are the same as those in
Damage and Fracture Mechanics VI 39 the homogeneous material provided that the material properties of FGM are continuous. Finally, the stress intensity factors can be evaluated from the wellknown relation between Kj, K^ and the path independent J - integral [9]. The advantage of the presented method for computation of stress intensity factors through the J - integral technique consists in high accuracy because the contour integral is evaluated over the path far away from the crack tip. Inaccuracies of numerical calculations at the crack tip vicinity are avoided on this approach. Moreover, the discretization mesh is not required to be so fine as in a direct computation of SIF from the asymptotic expansion formulae. In such case results are sensitive to the distance of a selected point from the crack tip. 3 Evaluation of the second fracture parameter The T-term stress can be computed directly from the asymptotic expansion of stresses or displacements, if stress intensity factors and stress or displacement values at nodal points close to the crack tip (the validity of asymptotic expansions is assumed) are obtained from a numerical analysis. A drawback of this method is substantial dependence of results on the distance of a selected point from the crack tip. Therefore, it is required to suggest a method in which the T-term stress is expressed in terms of the solution at points far away from the crack tip. In the following an integral representation of the T-term stress will be found for a cracked body analysed in the framework of stationary thermoelasticity. Let { cr,y,%,,0 } and { a*-,w* } be two systems of fields (stresses, displacements and temperature) governed respectively by equations (l)-(5) and equations * = 0 (14) where c^ can be obtained from c..^ by replacing ju with //Q = const. In view of eqs (4), (5) and (15), one may write *^ - cjy^jin =J Iju-^-aOi nl_ i-^y (16) Having been interested in two-dimensional problems, one can rewrite the last identity, after some manipulations into the form
40 Damage and Fracture Mechanics VI where E 3-/ 2VK~\ for an isotropic material with E = 2//(l + v) and 13-4v for plane strain f v for plane strain 3-v i/=< v - for plane stress - for plane stress 1+v ^ U+v Applying the Gauss divergence theorem to eq. (17) and utilizing the governing equations (2) and (14), we get r with Q (18) J_ 2 _1-2F "'" The integral identity (18) is valid in a region, which is free of any irregularities. To avoid crack tip irregularities it is necessary to consider the region Q- O.^ bounded with the integration contour r - YQ + F* -F^ + F~ shown in Fig. 1. Then, eq. (18) can be written as -lim }F(0^^,^^,)dn (19) In the section 2 of this paper it is mentioned that the thermal stresses at the crack tip vicinity in material with a continuous non-homogeneity have the same singularity and the angular distributions as the mechanical stresses in a homogeneous material [12]. Let us try to find an appropriate auxiliary solution (u *,er *^ ), which enables us to obtain the integral expression of the T-term stress. In order to have a nonvanishing contribution of this term in the l.h.s of eq. (19), the auxiliary displacement w* must necessarily vary as r~'. Then, the traction vector /* is
Damage and Fracture Mechanics VI 41 proportional to r~*. Now, the products of auxiliary fields with the singular terms in the asymptotic expansions yield singular integrands on T^. These singular terms have to be cancelled out only due to the angular variation of the auxiliary fields. The asymptotic displacements and stresses can be split into two parts %*=<+%! and <?*/, =0-^+0-^, (20) where the superscript s denotes the terms which contain the stress intensity factors in the asymptotic expansion [12] and "a = -fki (1-1/2 )cosp - <^y(i +,/) sinp] (21) The auxiliary fields utilized in Kfouri's integral expression [5] of T-stresses have one order lower singularity than it is required for our auxiliary fields (u *, er*^ ). The first derivative of Kfouri's auxiliary fields (^, cr*/? ) with respect to coordinate component xi will satisfy above mentioned singularity of displacements and stresses. Substituting requirements on the %*=%,, and cr*=<7,y,i (22) are substituted into the l.h.s. of eq. (19), one can verify identity = 0 (23) Thus, the auxiliary fields given by eq. (22) satisfy requirements for cancellation of the singular terms. Their explicit expressions are given in [7]. On the other hand, the T-terms (u^, a^ ) give a finite contribution lim f /,"y,%i - o^%;fr = 19" (24) *-+Ur, ///o Using eqs. (20), (23) and (24), the integral representation of the T-term stress from eq. (19) becomes where (25) Unknown displacements, traction vector and temperature, required along the integration path and within of domain enclosed by the integration contour in eq. (25) can be received from a numerical or experimental analysis. In this paper the boundary element method is used.
42 Damage and Fracture Mechanics VI 4 Numerical results An infinitely long thick-walled tube with two radial cracks in opposite directions is considered. The cracks are situated on the internal surface of the tube. A permanent temperature gradient is prescribed with temperature T% = 30 C on the internal surface (radius R% = 8cm) and T: = 200 C on the external surface (% = 10cm). The shear modulus is assumed to be dependent on temperature linearly where 0 = 7-7,, ju^ = 310^ MPa and Poisson's ratio is a constant, y =0.3. The coefficient of thermal expansion a is assumed to be independent on temperature, a = a^ = 0.12510~"(deg~^). Three various relaxation parameters #1 are considered in the numerical analysis, B, =15, 40 and 63MPa/deg. The functionally gradient material with prescribed material behaviour is fabricated in such a way that the volume fractions of ceramics and metals are varied continuously in a predetermined composition profile. For material #3 (#1 = 63MPa/deg) the ratio of shear moduli on internal and external surface of tube is given as //, / ju^ = 2.3 16 at the prescribed temperature field. Plane strain conditions are considered in the analysis. Because of the symmetry of the problem only a quarter of the cross section is discretized. The boundary contour is divided into 34 conforming elements with a quadratic approximation. The domain Q is discretized by 30 quadratic quadrilateral cells. The crack length is denoted by a. The normalized stress intensity factor is defined as Variations of fj with the crack length ratio a/\v are given in Fig. 2 for homogeneous and non-homogeneous materials. One can see that the character of all the curves is practically the same. Small differences are observed only for larger sizes of cracks, since at the vicinity of external surface of tube a substantial influence of the parameter B\ on the hoop stresses has been observed for a thickwalled tube without a crack. The normalized T-term stress is frequently called in literature as the biaxiality parameter. It is denoted by B = T/<JQ, where the stress OQ = {p'^j/naf EC, = E^/(l-v^) for plane strain conditions and E^ = 2//Q (1 + v).the dependence of the biaxial parameter B on the crack length ratio a/\v is given in Fig. 3. The similar dependence can be observed also for a single edge cracked pure bending specimen. In a pure bending specimen it is a linear variation of normal stresses over the cross section, similarly to a thick-walled tube under a thermal gradient loading, where normal stresses are almost linear. with
Damage and Fracture Mechanics VI 43 * homogeneous material non-homog. material #1 * non-homog. material #2 * non-homog. material #3 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 a/w Fig. 2. Variations of the SIF with the length of crack in a thick-walled tube 0,8 0,6 0,4 non-homog. material #3 homogeneous material CQ 0,2 0,0 0,70-0,65-0,60-0,55-0,50-0,45-0,35- -0,2- -0,4- -0,6- -0,8- r i i OJ 0,2 0,3 0,4 1 1 0,5 1 1 0,6, -, 0,7 P- 0,8 a/w Fig. 3. Values of the biaxiality parameter B for a cracked thick-walled tube Then, the similarity of biaxial parameter variations for both cases is not surprising. One can see in Fig. 3 a moderate dependence of B on the relaxation parameter B,, which governs the linearity of the Young modulus on the thickness of tube.
44 Damage and Fracture Mechanics VI 5 Conclusions This paper presents the numerical methods for evaluation of stress intensity factors and T-term stresses for crack problems in functionally gradient materials under a stationary thermal loading. The path independent integral representations for both fracture parameters are derived. The present integral methods are numerically more expedient than those based on the direct computation of fracture parameters from the asymptotic expansion of the stresses and/or displacements. The integral approach is well suited for elastic boundary element type analyses. An advanced BEM formulation is employed for stationary thermoelasticity crack problems in a functionally gradient material where material parameters are varied continuously. References [1] Noda, N. & Jin, Z.H. Thermal stress intensity factors for a crack in a functionally gradient material. Int. J. Solids Structures, 30, pp. 1039-1056, 1993. [2] Erdogan, F. & Wu, B. Analysis of FGM specimens for fracture toughness testing. Trans. American Ceramic Soc. Functionally Gradient Materials, 34, pp.39-46, 1993. [3] Nemat-Alla, M. & Noda, N. Thermal stress intensity factor for functionally gradient half space with an edge crack under thermal load. Archive Appl. Mech., 66, pp. 569-580, 1996. [4] Rice, J.R. Limitations to the small scale yielding approximation for crack tip plasticity. J. Mech. Phys. Solids, 22, pp. 17-26, 1974. [5] Kfouri, A.P. Some evaluations of the elastic T-term using Eshelby's method. Int. J. Fracture, 30, pp.301-315, 1986. [6] Sherry, A.H., France, C.C. & Goldthorpe, M.R. Compendium of T-stress solutions for two and three dimensional cracked geometries. Fatigue Fract. Engn. Mater. Struct, 18, pp. 141-155, 1995. [7] Sladek, J., Sladek, V. & Fedelinski, P. Contour integrals for mixed-mode crack analysis; effect of nonsingular terms. Theoret. Appl. Fracture Mech., 27, pp. 115-127, 1997. [8] Nowacki, W. Dynamic Problems of Thermoelasticity, PWN, Warsaw, 1975. [9] Kishimoto, K., Aoki, S. & Sakata, M. On the path independent integral - J. J. Engn. Fracture Mech., 13, pp.841-850, 1980. [10] Sih, G.C. On the singular character of thermal stresses near a crack tip. J. ^X Afgc/,., 29, pp. 587-589, 1962. [11] Noda, N. & Jin, Z.H. Crack tip singularity fields in non-homogeneous body under thermal stress fields. JSMEInt. Jour. ser. A, 38, pp. 364-369, 1995. [12] Williams, M.L. On the stress distribution at the base of stationary crack. J. Appl. Mech., 24, pp. 109-114, 1957.