An implicit constitutive relation in which the stress and the linearized strain appear linearly, for describing the small displacement gradient response of elastic solids
aa r X i v : . [ phy s i c s . c l a ss - ph ] J a n An implicit constitutive relation in which thestress and the linearized strain appear linearly,for describing the small displacement gradientresponse of elastic solids
K. R. RajagopalDepartment of Mechancial EngineeringTexas A&M UniveristyCollege Station, Texas-77845December 2020
Abstract
In this short note we develop a constitutive relation that is linear inboth the Cauchy stress and the linearized strain, by linearizing implicitconstitutive relations between the stress and the deformation gradient thathave been put into place to describe the response of elastic bodies (seeRajagopal (2003)), by assuming that the displacement gradient is small.These implicit equations include the classical linearized elastic constitutiveapproximation as well as constitutive relations that imply limiting strain,as special subclasses.
The classical linearized theory of elasticity is derived as an approximation withinthe context of the Cauchy theory of elasticity (see Cauchy (1823), Cauchy(1828)) wherein the Cauchy stress T is expressed explicitly as a function ofthe deformation gradient F , under the assumption that the norm of the dis-placement gradient is sufficiently small. This linearization that stems fromthe Cauchy theory of elasticity, while astonishingly successful in describing theresponse of many elastic solids when they are sustaining small displacementgradients, has several shortcomings: (a) The theory is not in keeping with thedemands of causality in that it is the body and surface forces and consequentlythe stress in the body that cause the body to deform rather than the defor-mation causing the stresses, and thus it would be reasonable to expect thedeformation gradient or the strain to be specified in terms of the stress and1ot vice-versa. Interestingly the approximate constitutive relation describinglinearized elastic response is expressed by defining the stress in terms of thelinearized strain and also the linearized strain in terms of the stress. How-ever, the representation for the linearized strain in terms of the stress doesnot stem from an appropriate approximation of a nonlinear theory wherein aproper measure of the deformation is assumed to depend nonlinearly on thestress. (b) Many metallic alloys (see Saito et al. (2003), Sakaguch et al. (2004),Sakaguchi et al. (2005),Hao et al. (2005), Li et al. (2007), Talling et al. (2008),Withey et al. (2008), Zhang et al. (2009)) as well as ubiquitous and commonlyused materials like concrete (see Grasley et al. (2015)) exhibit nonlinear re-sponse even in the range of “strains” that are considered small wherein classicallinearized elasticity is supposed to be operative. The classical linearized theoryis incapable and impotent to describe the nonlinear response that is observed inthe aforementioned materials. (c) The material moduli that go into character-izing linearized elastic response, the Lame’ moduli (or equivalently the Young’smodulus, and Poisson’s ratio) have to be constants (they cannot depend on thedensity, strain, mean value of the stress, etc.) as the theory would not thenbe linear in the stress and the linearized strain (the density in virtue of thebalance of mass depends on the linearized strain). Often one sees ad hoc as-sumptions concerning properties such as the Young’s modulus depending on thepressure(the mean value of the stress) but this is not possible within the contextof the classical linearized theory of elasticity. (d) Since the stress and linearizedstrain are related linearly, if the stress becomes very large as in problems such asthe state of stress at a crack tip or due to a concentrated load (namely loadingthat leads to singularity in the stresses), the strains have to necessarily becomevery large, thereby contradicting the basic tenet under which the linearizationis established. (e) The linearized elastic response cannot capture the possibil-ity of limited small strain as the stress increases. The consequences of abovedrawbacks are discussed in detail in Rajagopal (2014).The above comments con-cerning the linearization that stems from Cauchy elasticity are also true of thelinearization that stems from Green elasticity (see Green (1837), Green (1839))as Green elasticity is a sub-class of Cauchy elasticity wherein one assumes theexistence of a stored energy function that depends on the deformation gradient.The question which then confronts us is whether we can develop a consti-tutive relation for elastic bodies which when linearized under the assumptionthat the displacement gradients are small does not present the shortcomingsdiscussed above, and the answer to that question is a resounding yes. Recently,Rajagopal (see Rajagopal (2003), Rajagopal (2007), Rajagopal (2011)) showedthat the class of bodies that are incapable of dissipation in the sense that theycannot convert mechanical working into heat (energy in thermal form) is muchlarger than the class of Cauchy elastic bodies, and Rajagopal and Srinivasa(2007) provided a thermodynamic basis for the development of implicit con-stitutive relations for such bodies. These implicit constitutive relations includeamongst other subclasses, constitutive relations for classical Cauchy elastic bod-ies as well constitutive relations wherein the deformation gradient is expressed2s a function of the stress. It transpires that linearization of the the latterclass of constitutive relations leads to an approximation wherein the linearizedelastic strain is a nonlinear function of the Cauchy stress (see Devendiran et al.(2017), Sandeep et al. (2016) and Kulvait et al. (2017) for experimental corrob-oration of such approximations of the nonlinear constitutive relations). Includedamongst this class are constitutive relations wherein the Cauchy stress and thelinearized strain appear linearly.The equation that will be considered in this short note are not bilinear pair-ings or bilinear functions as usually defined (see MacLane and Birkhoff (1967))as this requires the pairing f ( x, y ) to be both left linear in x and right linearin y , since the constitutive relations that we consider also include linear termsincluding bilinear terms.2. Implicit constitutive relations in which both the stress and linearizedstrain appear linearly.The starting point for the study of elastic bodies described by implicit con-stitutive relations is (see Rajagopal (2003)) f ( ρ, T , F , X ) = (1)where ρ is the density, T is the Cauchy stress, F is the deformation gradientand X is a point in the reference configuration. This point just acts as a sur-rogate for a particle belonging to the real body. The body is inhomogeneous ifthe response is different at different points belonging to the real body and it ishomogeneous if the response is the same at every particle of the body((see Noll(1958) and Truesdell and Noll (1992) for a detailed and careful discussion ofmaterial isomorphism, material uniformity, and homogeneity)). Henceforth, weshall not indicate the dependence of the function of X for the sake of notationalsimplicity.In the case of isotropic elastic bodies, the above implicit relation reduces to f ( ρ, T , B ) = , (2)where B is the right Cauchy-Green tensor. One can use representation theoremsand obtain the relationship between the stress and the Cauchy-Green tensor B .In the case of the constitutive relation (2), standard representation theory leadsto (see Spencer (1975)) α I + α T + α B + α B + α T + α ( T B + BT ) + α ( T B + B T )+ α ( T B + BT ) + α ( B T + T B ) = . (3)where α i , i = 1 , − − − − , depend on ρ, tr T , tr B , tr T , tr B , tr ( T ) , tr B ,tr T B , tr T B , tr B T , tr T B . X ∈ B,t ∈ R (cid:13)(cid:13)(cid:13)(cid:13) ∂ u ∂x (cid:13)(cid:13)(cid:13)(cid:13) = O ( δ ) , δ << . (4)A special sub-class of such elastic bodies in given by β ǫ + β I + β T + β T + β [ T ǫ + ǫT ] + β [ T ǫ + ǫT ] = (5)where the β i , i = 1 , , ǫ , but arbitrarily on the invariants of T , while β i , i = 0 , , T . Because the density depends on the trace of epsilon wecannot have the above constants depend in a random fashion on the density ifwe want the model to depend linearly on ǫ .We notice that equation (5) in general provides an implicit nonlinear rela-tionship between the linearized strain and Cauchy stress. A special subclassof such models is given by constitutive relations that provide an explicit non-linear expression for the linearized strain as a function of the Cauchy stress.It is possible that some of these nonlinear relations can be inverted to obtainthe stress as a nonlinear function of the linearized strain. Such an expressiondoes not have the status of an approximate constitutive relation, as discussedlater, it is a purely mathematical expression that can be manipulated, but ulti-mately it has to be expressed as the linearized strain as a function of stress (seeRajagopal (2018) for a detailed discussion of linearization of implicit constitu-tive relations). In fact, it is best not to invert the expression for ǫ as a functionof σ even if it were possible as it might mislead one in the misapplication andmisinterpretation of the procedure and results.If we require that the implicit constitutive relation in which both the stressand the linearized strain appear linearly, then (5) reduces to E ǫ + E T + E ( tr ǫ ) T + E ( tr T ) I + E ( tr ǫ ) I + E ( tr T ) ǫ + E ( tr T )( tr ǫ ) I + E ( ǫT + T ǫ ) + E ( tr ( ǫT ) I ) = . (6)In the above equation, E i , i = 1 − − E is not zero and divide through by E to express the above equation as ǫ + A T + A ( tr ǫ ) T + A ( tr T ) I + A ( tr ǫ ) I + A ( tr T ) ǫ + A ( tr T )( tr ǫ ) I + A ( ǫT + T ǫ ) + A tr ( ǫT ) I = . (7)4n the above equation A i , i = 1 − − ρ R = ρ ( det F ) , (8)where ρ R is the density in the reference configuration and ρ the density inthe deformed current configuration, can be approximated in the case of smalldisplacement gradients , within the context of (4), as ρ R = ρ (1 + tr ǫ ) , (9)we can replace tr ǫ by the density, That is, in virtue of the balance of mass,we can think of tr ǫ as a substitute for the density ρ , and thus one can view,for instance, E ( tr ǫ ) as a density dependent material moduli. Of course, since tr ǫ can only occur linearly, the dependence of the material moduli on the den-sity has to be in a special manner. An interesting consequence of recognizingthat material moduli depending linearly on the tr ǫ allows one to consider theresponse of inhomogeneous bodies wherein the material moduli can depend onthe density, which in turn can depend on the particle in the reference configura-tion. Using such an approach, Murru et al. (2020a), Murru et al. (2020b) haverecently studied damage in concrete.When A , A , A , A , A and A are zero, the model reduces to ǫ = − A T − A ( tr T ) I , (10)and we can then identify A and A as A = − (1 + ν ) E ; A = νE , (11)where E is the Young’s modulus and ν is the Poisson’s ratio.3. A Strain Limiting Constitutive Relation.In order to illustrate some interesting features of the constitutive relation(7), let us consider the one dimensional static response wherein the stress andthe linearized strain take the form: T = σ ( e ⊗ e ) , ǫ = ǫ ( e ⊗ e ) , (12)where e i , i=1,2,3 denote unit vectors in a Cartesian co-ordinate system.5n substituting (12) into (7) and simplifying, we obtain ǫ = − ( A + A ) σ [(1 + A ) + ( A + A + A + 2 A + A ) σ ] . (13)First, let us suppose that ( A + A + A + 2 A + A ) is not zero. Then, when σ = 0, we find ǫ = 0 and when ǫ = 0, we find that σ = 0, which is to beexpected. Moreover, we note that when σ tends to infinity, we notice that ǫ = − [ A + A ]( A + A + A + 2 A + A ) , (14)that is there is a limit to the strain. We need this limit to be positive. Since A + A = − /E , which is negative, we need the denominator ( A + A + A +2 A + A ) to be positive. Next, we note that when the stress is compressive, if σ = − (1+ A )( A + A + A +2 A + A ) then the linearized strain blows up, but this is notallowed. Hence we need to ensure that the compressive stress is always greaterthan − (1+ A )( A + A + A +2 A + A ) . One could view limiting the range of stresses thatare allowable as a tremendous drawback invalidating the use of such a constitu-tive relation. To the contrary, the constitutive relation that we are dealing withis an approximation and the constitutive relation is valid only if we ensure thatthe basic approximation that we have made is in place, namely that the strainshave to be sufficiently small. Thus, depending on the values of the materialparameters only a certain range of values for the compressive stresses would beallowable. When ( A + A + A + 2 A + A ) = 0, the constitutive relation re-duces to the approximate classical linearized elastic constitutive relation and aswe know such a constitutive relation does not exhibit limited strain (Of course,the linearized constitutive relation ought not to be used when the strains arelarger than the requirement expressed by (4). However,ignoring such a caveatthe linearized constitutive relation is used to study problems wherein singulari-ties in stresses, which implies singularities in the linearized strains, arise).It is important to recognize that one can also obtain an expression for thestress σ in terms of ǫ by manipulating equation (14), but one ought to recognizethat this does not lead to an appropriate approximate constitutive relation. Ingeneral an expression for the Cauchy-Green stretch as a nonlinear function of theCauchy stress when inverted and then linearized will not lead to the same con-stitutive relation that is obtained by linearizing and then inverting. The processof linearization just with respect to the displacement gradient and inversion arenot commutative operations when dealing with these implicit equations. Thatis, an expression for the Cauchy Green tensor B as a nonlinear function of theCauchy stress T , when linearized and then inverted will not give rise to thesame model as the process of inverting and then linearizing, when the lineariza-tion is carried out just with respect to the displacement gradient. Thus, even ifone can obtain a nonlinear expression for the Cauchy stress T in terms of ǫ itcan never arise from a linearization of the Cauchy stress within the context ofthe Cauchy theory of elasticity as on linearization within the Cauchy theory of6lasticity, one is inexorably and inescapably led to the linearized approximateconstitutive equation. Thus, if one is going to use the linearization of implicitequations wherein one obtains a nonlinear expression for the linearized strainin terms of the stress, it is best not to invert it and obtain the expression forthe stress in terms of the linearized strain. However, if one does that, one hasto then remember that such an expression is valid only for strains wherein theoriginal expression is valid.Let us now turn our attention to a special sub-class of constitutive relationsof (7) that take the form:(1 + λ ( tr T )) ǫ = B (1 + λ ( tr ǫ )) T + B (1 + λ ( tr ǫ ) )( tr T ) I . (15)When λ i , i = 1 , , B = (1 + ν ) E , B = − νE , (16)where E is the Young’s modulus and ν is the Poisson’s ratio, we recover theclassical linearized elastic model.Also, when λ is zero in (15), the constitutive relation reduces to ǫ = B (1 + λ ( tr ǫ )) T + B (1 + λ ( tr ǫ ))( tr T ) I . (17)Next, we can express the constitutive relation (15) in a manner in whichthe dependence on density is brought out explicitly. In virtue of (9), and (4) itfollows that ρ = ρ R (1 − tr ǫ ) , (18)and thus (1 + λ tr ǫ ) = 1 + λ ( ρ R − ρ ) ρ R (19)(1 + λ tr ǫ ) = 1 + λ ( ρ R − ρ ) ρ R . (20)Hence, equation (17) can be rewritten as ǫ = B [(1 + λ ) − ( λ ) ρρ R ] T + B [(1 + λ ) − λ ρρ R )]( tr T ) I . (21)The above simple constitutive equation in which the stress, the linearized strainand the density occur linearly accords an opportunity to study several interest-ing and important classes of problems.First, let us consider the problem of damage which is essentially a conse-quence of the inhomogeneity of the body when the material properties deterio-rate as the body is deformed. The constitutive relation (22) provides a simple7ethod to incorporate the deterioration of material properties in a body and theresulting degradation and damage that ensues. When λ and λ are negative,we notice that the product of B and B with the term adjacent to them in thesquare parenthesis can be viewed as material parameters that decrease as thedensity decreases. Of course, depending on whether λ and λ are positive ornegative and the deformation in question and thus the density, locally materialproperties may increase or decrease in value. Using a similar model, Murruet al. (2020a), (2020b) studied damage that takes place in cement concrete.Their results are in qualitative agreement with experimental observation. Thisis just one of many problems wherein damage occurs even within small strains,in inhomogeneous bodies, that can be studied within the context of such sim-ple models in which the stress, linearized strain and density occur linearly byassuming that properties vary with X .Finally, it is worth considering unsteady deformations of a body describedby the constitutive relation (6), especially with regard to the possibility of un-steady motions. For the sake of illustration, let us consider one-dimensionalproblems.First, let us record the equations that govern the motion of a body describedby the one-dimensional constitutive relation corresponding to (15):(1 + λ σ ) ǫ = B (1 + λ ǫ ) σ + B (1 + λ ǫ ) σ, (22)where we have assumed that the stress T and ǫ are of the form given by (12),but now under the assumption that σ = σ ( x, t ) , ǫ = ǫ ( x, t ) . (23)The balance of linear momentum in one dimensions, in the absence of bodyforces, reduces to ρ ( ∂ u∂t ) = ∂σ∂x . (24)Next, let us define α = ( B λ + B λ ) . (25)Then, equation (17) can be expressed as ǫ = σE + αǫσ. (26)However, one could also express (17) as σ = Eǫ (1 + Eαǫ ) = Eǫ (1 − Eαǫ ) , (27)8he last equality is a consequence of (4) which implies ǫ is small provided weassume that Eα is of order 1. As we shall see later, using the above expressionfor the stress σ in terms of ǫ needs to be interpreted carefully though it accordssome ease with manipulating the equations as it will be seem below. We no-tice that the stress σ is a nonlinear function of ǫ and this is not tenable as anappropriate constitutive relation even as an approximation. As mentioned ear-lier, we can use the expression for mathematical manipulations but we then haveto re-invert to interpret the results within the context of the approximation (15).Now, on inserting the expression for σ given in (27) into the right hand sideof the equation of motion, we obtain ρ ( ∂ u∂t ) = E ∂∂x [ ǫ (1 − Eαǫ )] (28)and since ǫ = ∂u∂x , (29)we obtain ρ ( ∂ u∂t ) = E ∂∂x [ ∂u∂x (1 − Eα ∂u∂x )] (30)which can be expressed as ρ ( ∂ u∂t ) = E ∂ u∂x − E α ∂∂x [( ∂u∂x ) ]and when α is zero we recover the one dimensional wave equation. We need tosupply the initial and boundary conditions in order to solve a specific problemof interest.Instead of choosing to express the stress σ as a function of the linearized strain ǫ the ideal way to address the problem is to solve equation (24) and the followingsimultaneously: ∂u∂x = σE (1 − ασ ) . (31)Let us now consider the possibility of unsteady motion in a one-dimensionalbody described by the constitutive relation (6). Let us once again assume thatthe stress and strain are given by (12) with σ and ǫ given by (23). In this casewe obtain an expression for the stress σ in terms of ǫ and substitute the sameinto the balance of linear momentum. However, as mentioned before, it is betterto not invert the expression for the linearized strain in terms of the stress butexpress the stress as a funtion of the linearized strain (also, it might not alwaysbe possible to invert the expression for the linearized strain as a function of thestress).We now find that σ = − (1 + A ) ǫ [( A + A ) + ( A + A + A + 2 A + A ) ǫ ] = Eǫ (1 + A )[1 + Eβǫ ]= Eǫ (1 + A )(1 − Eβǫ ) (32)9here we have used − ( A + A ) = E , that ǫ is small and β defined through β = A + A + A + 2 A + A E (33)and Eβ is of order 1. Substituting (32) into (24),and using (29) we obtain ρ ( ∂ u∂ t ) = E ∂∂x [(1 + A ) ∂u∂x (1 − Eβ ∂u∂x )] = E (1 + A )[ ∂ u∂x − Eβ ∂∂x [( ∂u∂x ) ] . (34)Once again, it is better to solve (24) simultaneously with ∂u∂x = − ( A + A ) σ (1 + A ) + ( A @ + A + A + 2 A + A ) σ (35)simultaneously.AcknowledgementK. R. Rajagopal thanks the Office of Naval Research for support of this work. References
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