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A viscoelastic functionally graded strip containing a crack subjected to in-plane loading

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A viscoelastic functionally graded strip containing a crack subjected to in-plane loading
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  A viscoelastic functionally graded strip containing acrack subjected to in-plane loading Z.-H. Jin, Glaucio H. Paulino * Department of Civil and Environmental Engineering, Newmark Laboratory, University of Illinois at Urbana-Champaign, 205 North Mathews Avenue, Urbana, IL 61801, USA Received 20 October 2000; received in revised form 13 April 2001; accepted 17 April 2001 Abstract In this paper, a crack in a viscoelastic strip of a functionally graded material (FGM) is studied under tensile loadingconditions. The extensional relaxation modulus is assumed as E  ¼  E  0 exp ð b  y  = h Þ  f  ð t  Þ , where h is a scale length and f  ð t  Þ isa nondimensional function of time t either having the form f  ð t  Þ¼  E  1 =  E  0 þð 1 À  E  1 =  E  0 Þ exp ðÀ t  = t  0 Þ for a linear stan-dard solid or f  ð t  Þ¼ð t  0 = t  Þ q for a power law material model, where E  0 , E  1 , b , t  0 and q are material constants. An ex-tensional relaxation function in the form E  ¼  E  0 exp ð b  y  = h Þ½ t  0 exp ð d  y  = h Þ = t   q is also considered, in which the relaxationtime depends on the Cartesian coordinate y exponentially with d being a material constant describing the gradation of the relaxation time. The Poisson’s ratio is assumed to have the form m ¼ m 0 ð 1 þ c  y  = h Þ exp ð b  y  = h Þ  g  ð t  Þ , where m 0 and c arematerial constants, and g  ð t  Þ is a nondimensional function of time t . An elastic FGM crack problem is first solved andthe ‘‘correspondence principle’’ is used to obtain both mode I and mode II stress intensity factors, and the crackopening/sliding displacements for the viscoelastic FGM considering various material models. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Crack; Stress intensity factor; Viscoelasticity; Standard linear solid; Power law material; Functionally graded material 1. Introduction Materials exhibit creep and relaxation behavior at elevated temperatures. Upon such conditions, de-formation creep occurs under constant stress state, while stress relaxation takes place under constant strainstate. In the framework of linear continuum theory, such behavior can be studied by viscoelasticity.Generally speaking, the viscoelastic response may be obtained from the elastic solution via the corre-spondence principle [1], or the analogy between the Laplace transform of the viscoelastic solution and thecorresponding elastic solution. Fracture behavior of homogeneous materials has been investigated byBroberg [2] who has given some examples of stress intensity factors (SIFs) for stationary cracks in visco-elastic solids. Atkinson and Chen [3,4] have investigated cracks in layered viscoelastic materials. Crackgrowth problems have been studied, for example, by Knauss [5] and Shapery [6–8]. Engineering Fracture Mechanics 69 (2002) 1769–1790www.elsevier.com/locate/engfracmech * Corresponding author. Fax: +1-217-265-8041. E-mail address: paulino@uiuc.edu(G.H. Paulino).0013-7944/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved.PII: S0013-7944(02)00049-8  Functionally graded materials (FGMs) are an alternative to homogeneous materials or layered com-posites, and are promising candidates for future advanced technological applications [9,10]. The materialproperties of FGMs are continuously graded with gradual change in microstructural details over pre-determined geometrical orientations and distances, such as composition, morphology, and crystal structure[11,12]. In applications involving severe thermal gradients (e.g. thermal protection systems), FGM systemstake advantage of heat and corrosion resistance typical of ceramics, and mechanical strength and toughnesstypical of metals. Several aspects of fracture mechanics of FGMs have been studied, for example, basictheory and review [13], crack tip fields [14,15], crack growth resistance curve [16], crack deflection [17],conservation laws [18], strain gradient theory [19], fracture testing [20], and statistical models [21].Under elevated temperature conditions, FGMs also exhibit creep and relaxation behavior. For polymer-based FGMs (see, for example, [22–24]), their creep and relaxation behavior may be studied by visco-elasticity. However, in general, the correspondence principle, does not hold for FGMs. To avoid thisproblem, Paulino and Jin [25] have shown that the correspondence principle can still be used to obtain theviscoelastic solution for a class of FGMs exhibiting relaxation ( or creep ) functions with separable kernels inspace and time . By using such revisited correspondence principle for FGMs, they have subsequently studiedcrack problems of FGM strips subjected to antiplane shear conditions [26,27]. Other studies on crackproblems of nonhomogeneous viscoelastic materials directly solve the viscoelastic governing equations.For example, Schovanec and co-workers have considered stationary cracks [28], quasi-static crack prop-agation [29] and dynamic crack propagation [30] in nonhomogeneous viscoelastic media under anti-plane shear conditions. Schovanec and Walton also considered quasi-static propagation of a plane strainmode I crack in a power-law inhomogeneous linearly viscoelastic body [31] and calculated the corre-sponding energy release rate [32]. Although a separable form for the relaxation functions was employed inRefs. [28–32], no use of the correspondence principle was made. Recently, Yang [33] performed stressanalysis in FGM cylinders where steady-state creep conditions are considered only for the homogeneousmaterial.In the present study, a stationary crack in a viscoelastic FGM strip is investigated under tensile loadingconditions. The extensional relaxation function of the material is assumed as  E  ¼  E  0 exp ð b  y  = h Þ  f  ð t  Þ ; where h is a scale length and f  ð t  Þ is a nondimensional function of time t either having the form  f  ð t  Þ¼  E  1 =  E  0 þð 1 À  E  1 =  E  0 Þ exp ðÀ t  = t  0 Þ : linear standard solidor  f  ð t  Þ¼ð t  0 = t  Þ q : power law material : We also consider the following variant form of the power law material model  E  ¼  E  0 exp ð b  y  = h Þ½ t  0 exp ð d  y  = h Þ = t   q ; in which the relaxation time depends on y exponentially. In the above expressions, the parameters E  0 , E  1 , b , t  0 ; d , q are material constants. The Poisson’s ratio is assumed to take the form (also separable in space andtime) m ¼ m 0 ð 1 þ c  y  = h Þ exp ð b  y  = h Þ  g  ð t  Þ : where m 0 and c are material constants, and g  ð t  Þ is a nondimensional function of time t . The material modelsconsidered above may be suitable, for example, for two phase polymeric/polymeric FGMs with their basicconstitutents having different Young’s moduli and Poisson’s ratios but having approximately the sameviscoelastic relaxation behavior. Since an FGM is a special composite of its constitutents, the viscoelasticrelaxation behavior may remain unchanged if its constitutents have the same relaxation behavior. Thus, the 1770 Z.-H. Jin, G.H. Paulino / Engineering Fracture Mechanics 69 (2002) 1769–1790  relaxation moduli of the FGM would have separable forms in space and time. Further, the independentmaterial constants b and c describe the spatial gradation in Young’s modulus and Poisson’s ratio.According to the correspondence principle, we first consider a crack in an elastic strip of an FGM withthe following properties:  E  ¼  E  0 exp ð b  y  = h Þ ; m ¼ m 0 ð 1 þ c  y  = h Þ exp ð b  y  = h Þ : The Laplace transform of the viscoelastic solution is directly obtained from the elastic solution. For thetraction boundary value problem, the stress intensity factors are the same as those for the nonhomogeneouselastic strip. The crack opening/sliding displacements, however, are functions of time.The remainder of this paper is organized as follows. The basic equations of viscoelasticity are providedin the next section and some viscoelastic constitutive models for FGMs are discussed in Section 3. Theboundary value problem of a crack in a viscoelastic FGM strip is presented in Section 4. SIFs and crackopening/sliding displacements are discussed in Sections 5 and 6, respectively. Relevant numerical aspectsfor solving the governing system of integral equations are given in Section 7. Numerical results of SIFs andcrack opening/sliding displacements are given in Section 8. Finally, some concluding remarks and exten-sions are given in Section 9. Appendix A, containing the explicit form of the Fredholm kernels in the in-tegral equations, supplements the paper. 2. Basic equations The basic equations of quasi-static viscoelasticity of FGMs are the equilibrium equation (in the absenceof body forces) r ij ;  j ¼ 0 ; ð 1 Þ the strain–displacement relationship e ij ¼ 12 ð u i ;  j þ u  j ; i Þ ; ð 2 Þ and the viscoelastic constitutive law e ij ¼ Z  t  0  J  1 ð x ; t  À s Þ d  s ij d s d s ; e kk  ¼ Z  t  0  J  2 ð x ; t  À s Þ d r kk  d s d s ð 3 Þ with  s ij ¼ r ij À 13 r kk  d ij ; e ij ¼ e ij À 13 e kk  d ij ; ð 4 Þ where r ij are stresses, e ij are strains, s ij and e ij are deviatoric components of the stress and strain tensors,respectively, u i are displacements, d ij is the Kronecker delta, x ¼ð  x 1 ;  x 2 ;  x 3 Þ , J  1 ð x ; t  Þ and J  2 ð x ; t  Þ are the creepfunctions, t denotes time, and the Latin indices have the range 1–3 with repeated indices implying thesummation convention. Note that for FGMs the creep functions also depend on spatial positions , whereas inhomogeneous viscoelasticity, they are only functions of time, i.e. J  1   J  1 ð t  Þ and J  2   J  2 ð t  Þ [1].The creep functions J  1 ð x ; t  Þ and J  2 ð x ; t  Þ are related to the relaxation function in extension, E  ð x ; t  Þ , andthe relaxation function in Poisson’s ratio, m ð x ; t  Þ , by the following equations [1]:  J  1 ¼ 1 þ  p   mm  p  2  E  ; J  2 ¼ 1 À 2  p   mm  p  2  E  ; ð 5 Þ where a bar over a variable means the Laplace transform, and p is the Laplace transform variable. Z.-H. Jin, G.H. Paulino / Engineering Fracture Mechanics 69 (2002) 1769–1790 1771  Under plane stress conditions, the Laplace transforms of the basic equations (1)–(3) are reduced to o  rr  xx o  x þ o  rr  xy  o  y  ¼ 0 ; o  rr  xy  o  x þ o  rr  yy  o  y  ¼ 0 ; ð 6 Þ    xx ¼ o  uu o  x ;    yy  ¼ o  vv o  y  ;    xy  ¼ 12 o  uu o  y   þ o  vv o  x ! ; ð 7 Þ    xx ¼ 1  pE   rr  xx  À p   mm  rr  yy   ;    yy  ¼ 1  pE   rr  yy   À p   mm  rr  xx  ;    xy  ¼ 2 ð 1 þ  p   mm Þ  pE   rr  xy  ; ð 8 Þ where x   x 1 , y    x 2 , and u and v are displacements in the x and y  directions, respectively. 3. Viscoelastic models for FGMs This section describes three viscoelastic material models considered in this investigation. The first is a standard linear solid  with constant relaxation time  E  ¼  E  0 exp ð b  y  = h Þ E  1  E  0  þ 1  À  E  1  E  0  exp  À t t  0  ; ð 9 Þ where b , E  0 , E  1 and t  0 are material constants and h is a scale length (e.g., the strip thickness). The secondmodel is a power law material  with constant relaxation time  E  ¼  E  0 exp ð b  y  = h Þ t  0 t    q ; 0 < q < 1 ; ð 10 Þ where q is a material constant. The third model is still a power law material, but with position-dependentrelaxation time  E  ¼  E  0 exp ð b  y  = h Þ t  0 exp ð d  y  = h Þ t    q ¼  E  0 exp ½ð b þ d q Þ  y  = h  t  0 t    q ; ð 11 Þ where d is a material constant. For all three models, the Poisson’s ratio is assumed as m ¼ m 0 ð 1 þ c  y  = h Þ exp ð b  y  = h Þ  g  ð t  Þ ; ð 12 Þ where m 0 and c are material constants, and g  ð t  Þ is a nondimensional function of time t . The spatial positiondependent part of the Possion ratio (12) was first proposed by Delale and Erdogan [34] to study non-homogeneous elastic crack problems. Noda and Jin [35] later used it to investigate thermal crack problemsin nonhomogeneous solids. The Poisson’s ratio given by Eq. (12) is subjected to the condition that À 1  m  0 : 5 [36].With the assumptions (9)–(12) on the relaxation modulus and the Poisson’s ratio, the correspondenceprinciple for viscoelastic FGMs [25] can be used to study crack problems, i.e. the Laplace transformed viscoelastic FGM solution can be obtained directly from the solution of the corresponding elastic FGM solutionby replacing  E  0 and  m 0 with pE  0   f  f  ð  p  Þ and  p  m 0   g  g  ð  p  Þ , respectively. The final solution is realized upon inverting thetransformed solution , where   f  f  ð  p  Þ and   g  g  ð  p  Þ are the Laplace transforms of  f  ð t  Þ and g  ð t  Þ , respectively. For the standard linear solid  (9), f  ð t  Þ is given by  f  ð t  Þ¼ E  1  E  0  þ 1  À  E  1  E  0  exp  À t t  0  : ð 13 Þ 1772 Z.-H. Jin, G.H. Paulino / Engineering Fracture Mechanics 69 (2002) 1769–1790  For the power law material  (10) or (11), f  ð t  Þ is  f  ð t  Þ¼ t  0 t    q : ð 14 Þ Since the function g  ð t  Þ does not affect the final governing equations and boundary conditions considered(see Section 4), no specific functional form is required.It can be clearly seen from (9) to (12) that the relaxation moduli and the Poisson’s ratio are separablefunctions in space and time. This is a necessary condition for applying the revisited correspondence principle [25]. This kind of relaxation functions may be appropriate for an FGM with its constituent materialshaving different Young’s moduli and Poisson’s ratios but having approximately the same viscoelastic re-laxation behavior. Since the FGM is a special composite of its constitutents, the viscoelastic relaxationbehavior may remain unchanged if its constitutents have the same relaxation behavior. Thus, the relaxationmoduli of the FGM would have separable forms in space and time. Further, the material constants b and c describe the spatial gradation in Young’s modulus and Poisson’s ratio. For model (9), the separable form of the extensional relaxation modulus means that the constituents should have the same ratio E  1 =  E  0 andrelaxation time t  0 . For model (10), this implies that the constituents should have the same relaxation time t  0 and parameter q . For model (11), however, it is only required that the constituents have the same parameter q . The constituents may have different relaxation times. Potentially, this kind of FGMs may include somepolymeric/polymeric materials such as propylene-homopolymer/Acetal-copolymer. The relaxation behav-ior of propylene homopolymer and Acetal copolymer are found to be similar  ––  see Figs. 7.5 and 10.3,respectively, of Ogorkiewicz [37]. 4. A crack in a viscoelastic FGM strip Consider an infinite nonhomogeneous viscoelastic strip containing a crack of length 2 a , as shown in Fig.1. The strip is subjected to a uniform tension r 0 R ð t  Þ in the y -direction along both the lower boundary ð  y  ¼À h 1 Þ and the upper boundary ð  y  ¼ h 2 Þ , where r 0 is a constant and R ð t  Þ is a nondimensional function of time t . It is assumed that the crack lies on the x -axis, from À a to a . The crack surfaces remain traction free.The boundary conditions of the crack problem, therefore, are r  xy  ¼ 0 ; r  yy  ¼ r 0 R ð t  Þ ; y  ¼À h 1 ; j  x j < 1 ; ð 15 Þ r  xy  ¼ 0 ; r  yy  ¼ r 0 R ð t  Þ ; y  ¼ h 2 ; j  x j < 1 ; ð 16 Þ   Fig. 1. A viscoelastic FGM strip occupying the region j  x j < 1 and À h 1 6  y  6 h 2 with a crack at j  x j 6 a and y  ¼ 0. The boundaries of thestrip (  y  ¼À h 1 ; h 2 ) are subjected to uniform traction r 0 R ð t  Þ . Z.-H. Jin, G.H. Paulino / Engineering Fracture Mechanics 69 (2002) 1769–1790 1773
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