Belytschko T. - Introduction (779635), страница 52

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Many materials (such as metals and ceramics) can be modeled asisotropic in the linear elastic range and the linear isotropic elastic constitutive relation isperhaps the most widely used material model in solid mechanics. There are many excellenttreatises on the theroy of elasticity and the reader is referred to (Timoshenko and Goodier,1975; Love, and Green and Zerna, ) for more a more detailed description than that givenhere.

As in equation (5.3.10) above the number of independent elastic constants for anisotropic linearly elastic material reduces to 2. The isotropic linear elastic law is written inVoigt notation asσ 1  λ +2µσ   2  σ3   = σ 4  σ 5    σ 6  λλ + 2µλ0λ0λ +2µ 0µsym0000µ0 ε 1 0  ε 2 0  ε3  0  ε4 0  ε 5  µ ε 6 (5.3.21)where λ and µ $ are the Lamé constants.The isotropic linear elastic relation (5.3.21) has been derived from the generalanisotropic material model (5.3.19) by considering material symmetry.

It is instructive tosee also how the relation (5.3.21) may be generalized from the particular by starting withthe case of a linearly elastic isotropic bar under uniaxial stress. For small strains, the axialstrain in the bar is given by the elongation per unit original length, i.e., ε11 = δ L 0 andfrom Hooke's law (5.2.5)ε11 =σ11E(5.3.22)The lateral strain in the bar is given by ε 22 =ε 33 =∆D D0 where ∆D is the change in theoriginal diameter D0 .

For an isotropic material, the lateral strain is related to the axial strainbyε 22 =ε 33 =− vε11 = −vσ11E(5.3.23)where v is Poisson's ratio. To generalize these relations to multiaxial stress states, considerthe stress state shown in Figure 5.8 where the primed coordinate axes are aligned with thedirections of principal stress. Because of the linearity of the material repsonse, the strainsdue to the indiviudal stresses may be superposed to giveσ11′−v (σ ′22 + σ 33′ )Eσ′ε ′22 = 22 −v (σ 11′ + σ ′33 )Eσ′ε ′33 = 33 −v (σ 11′ + σ ′22 )Eε11′ =(5.3.24)Referring the stresses and strains to an arbitrary set of (rectangular Cartesian) axes by usingthe relation (3.2.30) for transformation of tensor components givesε ij =(1+ v) σEij−vσ δE kk ij(5.3.25)Exercise 5.1.

Derive Eq. (5.3.25) from (5.3.24) and (3.2.30).The relation between shear stress and shear strain is given by (for example) σ12 = 2µε12where the shear modulus (or modulus of rigidity) µ is defined asµ=E2(1+ v)(E5.1.1)From Eq.(5.3.25) it follows thatε kk =(1− 2v) σEkk=σ kk3K(E5.1.2)whereK=E2µ=λ+3(1−2v)3(E5.1.3)is the bulk modulus. Introducing the Lamé constant λ , given byλ=vE(1+ v)(1− 2v)(E5.1.4)the bulk modulus is wrtten asK =λ +2µ3(E5.1.5)From (5.3.29) and (5.3.26), the quantity we obtain the relation v E = λ 2µ .

Using thisresult and (5.3.26) in (5.3.25), the stress strain relation is given byε ij =σ ij2µ−λσ δ2µ (3λ + 2µ ) kk ij(E5.1.6)Using (5.3.27) this expression may be inverted to give Eq.(5.3.11), the generalizedHooke's law.Writing the stress and strain tensors as the sum of deviatoric and hydrostatic orvolumetric parts, i.e.,1σ ij = sij + σ kkδ ij31ε ij = eij + ε kkδ ij3(E5.1.7)then using (5.3.11) and (5.3.26-27) the constitutive relation for an isotropic linearly elasticmaterial can be written asσ ij = 2µeij + Kε kkδ ij(E5.1.8)The strain energy (5.3.16) for an isotropic material is given by1W = σ ijεij2111=  sij + σ kkδij  eij + ε mmδij 23312= µeijeij + K (ε kk )2(E5.1.9)Positive definiteness of the strain energy W ≥ 0 , equality iff ε =0 imposesrestrictions on the elastic moduli (see Malvern, for example).

For the case of isotropiclinear elasticity positive definitness of W requiresK >0andE >0andµ> 0or1− 1< v <2(E5.1.10)Exercise 5.2. Derive these conditions by considering appropriate deformations. Forexample, to derive the condition on the shear modulus, µ , consider a purely deviatoricdeformation and the positive definiteness requirement.Incompressibility.The particular case of v =1 2 ( K =∞ ) corresponds to an incompressible material. In anincompressible material in small deformations, the trace of the strain tensor must vanish,i.e., ∈kk = 0 . Deformations for which this constraint is observed are called isochoric.From (5.3.33) it can be seen that, for an incompressible material, the pressure can not bedetermined from the constitutive relation.

Rather, it is determined from the momentumequation. Thus, the constitutive relation for an incompressible, isotropic linear elasticmaterial is written asσ ij =− pδ ij + 2µεij(5.3.26)where the pressure p =− σ kk 3 is unspecified and is determined as part of the solution.Plane StrainFor plane problems, the stress-strain relation (5.3.21) can be even further simplified. Inplane strain, ε i3 = 0, i.e., ε 3 = ε 4 = ε5 = 0 . In finite element coding, the standard Voigtnotation used above is often modified to accommodate a reduction in dimension of thematrices. Letting 12 → 3, the stress-strain relation for plane strain is written as σ  λ + 2µ 11   σ 22  =  λσ   0 12  λ0  ε11 λ +2µ 0  ε 22 0µ 2ε12 v0  11− v  ε11 E (1− v)  v=10   ε 22 1+v1−2v1−v( )()1− 2v 2ε12 002(1 − v) (5.3.27)and in additionσ 33 = λ ( ε11 + ε22 ) = v (σ11 + σ 22 )(5.3.28)Plane StressFor plane stress, σ i3 = 0.ε 33 =−The condition σ 33 = 0 gives the relationλ(ε + ε ) =− v(ε11 + ε22 )λ + 2µ 11 22(5.3.29)Letting λ = 2µλ (λ + 2µ ) and using (5.3.21), the stress-strain relation for plane stress isgiven by σ 11   λ + 2µλ0   ε11 1 v0   ε11  E  σ 22  =  λλ + 2µ 0   ε22  =v 10  ε 22 21 −v 1− v σ   00µ 2ε 12  12  2ε12  0 02 (5.3.30)AxisymmetryFor problems with an axis of symmetry (using a cylindrical polar coordinatesystem) the constitutive relation is given by σ rr  λσθθ  = σ zz   σ rz  +2µλλ0   ε rr λλ + 2µλ0  ε θθ λλλ + 2µ 0   ε zz 000µ 2ε rz vv0  11− v 1− v v ε rr v10  ε E (1− v) 1− v1− v  θθ =vv(1 +v )(1−2v)10 ε zz 1− v 1− v2ε 1−2v   rz 00 02(1− v) (5.3.31)whereε rr = ∂u ∂u ∂uru∂u, εθθ = r , ε zz = z , ε rz =  r + z  ∂z∂rr∂z∂r (5.3.32)5.4.

Multiaxial Nonlinear ElasticityIn this section, the small strain linear elasticity constitutive relations presented above will beextended to the case of finite strain. As will be seen, the extension to finite strains can becarried out in different ways andmany different constitutive relations can be developed formultiaxial elasticity at large strains. In addition, because of the many different stress anddeformation measures for finite strain, the same constitutive relation can be written inseveral different ways. It is important to distinguish between these two situations.

The firstcase gives different material models while in the second, the same material model isrepresented by different mathematical expressions. In the latter, it is always possible tomathematically transform from one form of the constitutive relation to another.The constitutive models for large strain elasticity are presented in order of increasing degreeof what is commonly thought of as elasticity, i.e., hypoleasticity is presented first,followed by elasticity and finally hyperelasticity.5.4.1 Hypoelasticity. One of the simplest ways to represent elasticity at large strains,is to write the increments in stress as a function of the incremental deformation. Asdiscussed in Section 3.7.2, in order to satisify the principle of material fame indifference,the stress increments (or stress rate) should be objective and should be related to anobjective measure of the increment in deformation. A more detailed treatment of materialframe indifference is given in the Appendix to this chapter and we will draw on thatmaterial as needed in the remainder of the chapter.

Truesdell [ ] presented a generalhypoelastic relation of the form∇σ = f (σ, D)(5.4.1)∇where σ represents any objective rate of the Cauchy stress and D is the rate of deformationtensor which is an objective tensor (see Equation (A.x)).A large class of hypoelastic constitutive relations can be written in the form of alinear relation between the objective measure of stress and the rate of deformation tensor,i.e.,∇σ = C :D(5.4.2)In general, the fourth order tensor C is a function of the stress state.

As noted byPrager ( ), the relation (5.4.2) is rate-independent and incrementally linear and reversible.This means that for small increments about a finitely deformed state, the increments instress and strain are linearly related and are recovered upon unloading. However, for largedeformations, energy is not necessarily conserved and the work done in a closeddeformation path is not-necessarily zero. It should be noted that the primary use ofhypoelastic constitutive relations is in the representation of the elastic response inphenomenological elastic-plastic constitutive relations where the elastic deformations aresmaal.

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