Some Notes on the Linear Viscoelasticity of Functionally Graded

Some Notes on the Linear Viscoelasticity of Functionally Graded Materials12342567School of Aeronautics and Astronautics,Purdue University,W est Lafayette,IN47907-1282, USA(Received10June2002;accepted2July2003)Abstract:This work discusses the elastic–viscoelastic correspondence principle and the related issues in lin-ear viscoelasticity of functionally graded materials(FGMs).It is clarified that the correspondence principle is directly applicable to certain boundary value problems of viscoelastic FGMs with closed-form solutions, and for general boundary value problems,the correspondence principle holds only for FGMs with variables separable relaxation moduli(moduli in separable form in space and time).For the relaxation moduli of an FGM to be variables separable,the time-dependence of the relaxation moduli in shear and dilatation should be identical.This imposes a rather severe restriction on the application of the correspondence principle to FGMs because engineering materials usually exhibit significantly different relaxation behavior in shear and dilatation.As an application of the correspondence principle,the stress and displacement fields near a sta-tionary crack tip in a viscoelastic FGM are obtained.The paper also presents general stress intensity factor formulae for the crack problems of FGMs with separation of variables solutions.Key W ords:Functionally graded material;viscoelasticity;correspondence principle;relaxation function;crack tip field;stress intensity factor123456789 5374The interest in functionally graded materials(FGMs)has been steadily growing in recent years.An FGM is usually a multi-phased material with the volume fractions of its constituents varied gradually thereby giving a non-uniform microstructure with continuously graded pared with conventional material systems,FGMs possess a number of advantages such as reduced residual and thermal stresses,improved bonding strength between dissimilar materials,enhanced resistance to environmental attacks and optimized strength [1,2].Under elevated temperature conditions,FGMs will exhibit creep and relaxation behavior.For polymer-based FGMs,such behavior may be studied by viscoelasticity in the framework of conventional micromechanics/continuum theory.For a homogeneous material, the viscoelastic response may be determined from the corresponding elastic solution via the elastic–viscoelastic correspondence principle.When there exists a separation of variables solution(a solution in separable form in space and time),the space-dependent part of the viscoelastic solution may be obtained directly from the corresponding elastic solution andMathematics and Mechanics of Solids11:216–224,2006DOI:10.1177/1081286504040401 112006SAGE PublicationsSOME NOTES ON LINEAR VISCOELASTICITY217 the time-dependent part is determined by the external loading and the relaxation behavior of the material.When applying these procedures to FGMs,however,some restrictions may have to be imposed on the relaxation moduli.Reference[3]showed that the correspondence principle is applicable to nonhomogeneous viscoelastic media.However,Ref.[4]proved that the correspondence principle may be used to obtain viscoelastic solutions only for a class of FGMs with variables separable relaxation moduli(moduli in separable form in space and time).It seems that there is confusion as to whether the correspondence principle is directly applicable to viscoelastic FGMs.In addition,the conditions for the existence of variables separable relaxation moduli are not clear.The main purpose of this paper is to discuss the elastic–viscoelastic correspondence principle and the related issues in linear viscoelasticity of FGMs.Section2discusses in detail the correspondence principle and obtains the stress and displacement fields near a stationary crack tip using the correspondence principle.Section3employs a simple micromechanics model to derive the conditions for the relaxation moduli of an FGM to be variables separable. Section4describes separation of variables viscoelastic solutions and provides general stress intensity factor expressions for the crack problems with separation of variables solutions. Finally,Section5provides some concluding remarks.2 53 3 7 53 766 7484 634 3The elastic–viscoelastic correspondence principle is one of the most useful tools in viscoelasticity because the Laplace transform of a viscoelastic solution can be directly obtained from the corresponding elastic solution.While Ref.[3]showed that the correspondence principle is applicable to nonhomogeneous viscoelastic media,Ref.[4] proposed that the correspondence principle may be used only for a class of FGMs with variables separable relaxation moduli.It seems that there is confusion as to whether the correspondence principle is directly applicable to viscoelastic FGMs.This section thus discusses the correspondence principle in further detail to clarify the situation.Description of the correspondence principle involves the Laplace transformed viscoelasticity equations.For quasi-static,isotropic,linear viscoelastic problems of general nonhomogeneous materials including FGMs,we only need to examine the constitutive law and its Laplace transform1121231123134154671255384416311931341546 44553(1)711212 72313 477125784416 79313 47 445(2)1121812186844 1237121 1218644 123(3)where812are stresses, 12are the infinitesimal strains,112and712are the deviatoric stress and strain components,respectively, 12is the Kronecker delta,231344and931344 are the relaxation moduli,or relaxation functions,in shear and dilatation,respectively,218Z.-H.JIN113 23 33 44is the Cartesian coordinate,4is time,the indices and take values1,2 and3,a repeated index implies summation over the range of the index,and a bar over a variable represents its Laplace transform and is the transform variable.Here we assume 1219for4 9.The relaxation functions231344and931344are explicit functions of spatial coordinate1and time4.Throughout the paper,we assume that the boundary of the concerned body does not vary with time.Equation(2)has the same form as that of linear elasticity of nonhomogeneous materials if the transformed viscoelastic variables are associated with the corresponding elastic variables, and if 72313 4and 79313 4are associated with the elastic moduli26314and96314, respectively.Hence,the Laplace transform of the viscoelastic solution can be obtained directly from the solution of the corresponding elastic problem with the transformed boundary conditions by replacing26314and96314with 72313 4and 79313 4,respectively.This is the so-called elastic–viscoelastic correspondence principle for nonhomogeneous media proposed in Ref.[3].It is a natural extension of the correspondence principle for homo-geneous viscoelastic materials[5].W e note that the above-extended correspondence principle may be applied only to a boundary value problem for which the closed-form solution of the corresponding elastic problem is available.In general,the correspondence principle may not hold for viscoelastic problems of FGMs.This is because the transform variable is generally coupled with the spatial coordinate1in the transformed relaxation functions.Thus,a non-closed-form elastic solution is not ready to be used to construct the corresponding transformed viscoelastic solution.Of course,one can still use the Laplace transform method to solve viscoelastic problems of FGMs,which is a general solution approach without resorting to pre-existing elastic solutions.In fact,Refs.[6,7]solved a number of viscoelastic crack problems of nonhomogeneous materials using this approach.For the correspondence principle to be applicable to general boundary value problems of viscoelastic FGMs,the transform variable and the spatial coordinate1in the transformed shear and dilatational relaxation functions must be uncoupled,i.e.72313 41223147 3 43179313 41993147 3 43(4)1where21and91are constants.T aking the inverse Laplace transform of Equation(4),we obtain the variables separable shear and dilatational relaxation functions231344121 2314 3443931344191 9314 344 (5) Substituting Equation(4)into Equation(2)yields711212 21 23147 3 477125784416 91 9314 3 47 447(6)Now the Laplace-transformed solution of the viscoelastic FGM with the relaxation functions (5)can be directly obtained from the solution of the corresponding elastic FGM with the shear and bulk moduli21 2314and91 9314by replacing21and91in the elastic solution with 7 3 421and 7 3 491,respectively.The final viscoelastic solution is obtained by inverting the transformed solution.This conclusion was reached in Ref.[4].SOME NOTES ON LINEAR VISCOELASTICITY219 In summary,the correspondence principle is directly applicable to certain boundary value problems of viscoelastic FGMs with closed-form solutions.For general boundary value problems,however,the correspondence principle may hold only for FGMs with variables separable relaxation functions.As a simple application of the correspondence principle,we consider the asymptotic stress and displacement fields near a stationary crack tip in a viscoelastic FGM.When the Laplace transforms of the shear and dilatational relaxation functions are continuous and piecewise differentiable,the transformed crack tip stress and displacement fields under plane strain(or plane stress)conditions may be obtained from the result for elastic FGMs[8]as follows:78 893 3 3 418222793 4 8526893 4 79 3 4 8536893 433(7)7 83 3 3 4187271 3 442579 3 4 52683 3 7 31 3 44 79 3 4 53683 3 7 31 3 4463(8)where79 3 4and79 3 4are the transformed viscoelastic mode I and mode II stress intensity factors(SIFs),respectively, 852689and 853689are the well-known angular variations of the elastic crack tip stress field, 5268and 5368are the angular variations of the elastic crack tip displacement field,7231 3 4and7 31 3 4are the transformed shear modulus and Poisson’s ratio at the crack tip,respectively,3 3 )are the polar coordinates centered at the crack tip,and 3 take values1and2.T aking the inverse Laplace transform of Equations(7)and(8)yields the crack tip stress and displacement fields in the time domain8893 3 344122 29 344 8526893 4 9 344 8536893 433(9)83 3 344142712879 3 47231 3 452683 3 7 31 3 44912879 3 47231 3 453683 3 7 31 3 4493(10)where9 344and9 344are the mode I and mode II SIFs,respectively,and 12()denotes the inverse Laplace transform.The crack tip fields(9)and(10)hold true for viscoelastic FGMs with general relaxation functions and have the same forms as those for homogeneous viscoelastic materials[9].Because the closed-form crack tip elastic solution is available,the relaxation functions are not required to be variables separable to validate the correspondence principle.26 5374 94 5374 34 6 76 34 4853FGMs are generally multi-phased materials.The material properties of an FGM can be obtained from the properties and the volume fractions of its constituents via an appropriate220Z.-H.JINmicromechanics model in the conventional micromechanics/continuum approach,which contrasts with the coupled higher-order theory [10].Reference [11]proposed that,among various micromechanics models for conventional composites,Mori–T anaka and self-consistent type models may be used to estimate the effective elastic properties of FGMs with reasonable accuracy .These models,however,may not be appropriate for rigorously analyzing the characteristics of relaxation functions of viscoelastic FGMs.Because we are only concerned with the conditions of existence of variables separable relaxation functions,we may first consider a sub-region of a two-phase FGM where the volume fraction of the inclusion phase is much smaller than that of the matrix phase so that the dilute suspension model is applicable.The relaxation functions must be variables separable in this sub-region if they are variables separable in the entire FGM.Under dilute suspension conditions,the Laplace transforms of the shear and dilatational relaxation functions of a two-phase composite may be obtained as follows [12,13]:72313 4172 18 81548 1138 163548 198 16 372 17214 1314 1 54 1138 163548 198 16 2 1 54 1138 163548 198 16 8 28 1 3(11)79313 4179 791179 1314 717 7 7 3(12)where 1314is the volume fraction of the inclusion phase and subscripts and denote the properties of the matrix and inclusion phases,respectively .Since 1314is a function of spatial position,the relaxation functions of FGMs become position dependent.The shear and dilatational relaxation functions of the matrix and inclusion phases take the following forms:2121 34439 191 3443211211 134******** 13443(13)where 21 391 3211and 911are constants,and 344, 344, 1344and 1344are dimensionlessfunctions of time,respectively .For the standard linear solid,for example,2 is given by 2 122 321 122 4 13 331121 822 1 8122 113 3319 (14)Thus, 344in Equation (13)is 3441822 1 8122 1 13 3319 (15)Equations (11)–(13)imply that for the relaxation functions of the FGM to be in separable form in space and time,the following relations must be satisfied without loss of generality:3441 3441 13441 13443(16)which means thatSOME NOTES ON LINEAR VISCOELASTICITY2213441 344(17) in Equation(5).Hence,for the relaxation functions of a two-phase FGM to be variable separable,the relaxation behavior in both shear and dilatation of its constituent materials must be identical.This,in turn,leads to the conclusion that if an FGM has variable separable relaxation functions,the time-dependent parts of the relaxation functions in shear and dilatation must be identical.The above conclusion contrasts sharply with the conventional knowledge that an FGM possesses variable separable relaxation functions if its constituent materials have identical relaxation behavior in shear and dilatation,respectively(the shear and dilatational relaxation behavior of each constituent may not be necessarily identical).The Laplace transform of the viscoelastic Poisson’s ratio, 31344,can be expressed by the Laplace transforms of the shear and dilatational relaxation functions[5]7 313 41679313 41272313 42 723134 679313 4(18)T aking the inverse Laplace transform of Equation(18)for FGMs with variables separable relaxation functions(5)and(17),we obtain the viscoelastic Poisson ratio313441 3141691 93141221 23142 21 2314 691 9314(19)Now,the viscoelastic Poisson ratio is a function of spatial position only.Engineering materials usually exhibit significantly different viscoelastic behavior in shear and dilatation.Hence,FGMs as graded composites usually do not possess variable separable relaxation functions.As a result,correspondence principle may be only applicable to certain boundary value problems of viscoelastic FGMs with closed-form solutions.2 6 53747 63 3 7 53 7 95374Section3shows that FGMs as graded composites usually do not possess variable separable relaxation functions.Thus discussions of viscoelastic solutions based on the existence of variable separable relaxation functions do not relate to real FGMs in general.However,it may be worthwhile discussing general separation of variables solutions for calibration purposes. This section thus describes some general characteristics of separation of variables solutions and the stress intensity factors.Assume that a viscoelastic solution takes the following variables separable form:131**** **** 344312313441 12314 3443(20)812313441 812314 3443222Z.-H.JINand the variable separable relaxation functions are given in Equations (5)and (17).Substituting Equation (20)into Equation (1)yields112 3441221 2314 712311 341545 35455553 844 3441691 9314 44311341545 3545555 (21)It follows from the above equation that 3441311 341545 3545555(22)without loss of generality .When a viscoelastic FGM has variable separable relaxation functions (5)and (17),we may be able to construct general SIF formulae for the viscoelastic crack problems from the SIF solutions of the corresponding elastic FGM with the shear and bulk moduli 21 2314and 91 9314,respectively .Generally speaking,crack problems are typical mixed boundary value problems.The displacement boundary conditions on the crack extended line are homogeneous.Therefore,if only external tractions with spatial parts 1314and a temporal part 344are prescribed,the SIFs for the viscoelastic FGM may be expressed as follows:39 34439 34439 344415 9 3 9 3 9 6 3443(23)where 9 3 9 and 9 are the SIFs for the corresponding nonhomogeneous elastic material subjected to the external tractions1314.The temporal part of the displacement field is 311 341545 3545555 where 344is given by3441 1288379 (24)When only external displacements with spatial parts 131)and a temporal part 344are prescribed,the SIFs are given by 39 39 39 415 9 3 9 3 9 6311341545 35455553(25)where 9 3 9 and 9 are the SIFs for the corresponding nonhomogeneous elastic material subjected to the external displacements1314.SOME NOTES ON LINEAR VISCOELASTICITY223 The general SIF expressions(23)and(25)indicate that for the crack problem of a viscoelastic FGM with separation of variables solutions,the SIFs differ from those of the corresponding elastic FGM only by a temporal factor which can be completely determined from the loading history and the relaxation behavior of the FGM.It thus becomes unnecessary to calculate the SIFs for such kinds of viscoelastic crack problems as,for example,in Ref.[14].2 74 9834 6 6It is clarified in this paper that the correspondence principle is directly applicable to certain boundary value problems of viscoelastic FGMs with closed-form solutions and for general boundary value problems without closed-form solutions,the correspondence principle holds only for FGMs with variable separable relaxation functions.For the relaxation functions of an FGM to be variable separable,the time-dependence of the shear and dilatational relaxation functions must be identical.Engineering materials generally exhibit significantly different relaxation behavior in shear and dilatation.FGMs as graded composites thus do not exhibit variable separable relaxation functions in general.For crack problems of a viscoelastic FGM with separation 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