Belytschko T. - Introduction (779635), страница 51

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Atsmall strains, the relation reduces to Hooke's Law (5.2.5).We could also express the elastic potential in terms of any other conjugate stressand strain measures. For example, it was pointed out in Chapter 3 that the quantityU = U − I is a valid strain measure (called the Biot strain), and that in one-dimension theconjugate stress is the nominal stress PX ,soPX =dΨdΨ=dU X dU X(5.2.10)We can write the second form in (5.2.10) because the unit tensor I is constant and hencedU X = dU X .

It is interesting to observe that linearity in the relationship between a certainpair of stress and strain measures does not imply linearity in other conjugate pairs. For(example if S X = EEX it follows that PX = E U X2 + 2UX) 2.A material for which the rate of Cauchy stress is related to the rate of deformation issaid to be hypoelastic. The relation is generally nonlinear and is given byσ˙ = f (σ x , Dx )(5.2.11)where a superposed dot denotes the material time derivative and Dx is the rate ofdeformation.

A particular linear hypoelastic relation is given byλ˙σ˙ x = EDx = E xλx(5.2.12)where E is Young's modulus and λ x is the stretch. Integrating, this relation we obtainσ x = E ln λx(5.2.13)orσx =d λxE ln ξdξdλ x ∫1(5.2.14)which is a hyperelastic relation and thus path-independent. However, for multiaxialproblems, hypoelastic relations can not in general be transformed to hyperelastic. Multixialconstitutive models for hypoelastic, elastic and hyperelastic materials are described inSections 5.3 and 5.4 below.A hypoelastic material is, in general, strictly path-independent only in the onedimensional case. (• check). However, if the elastic strains are small, the behavior is closeenough to path-independent to model elastic behavior. Because of the simplicity ofhypoelastic laws, a muti-axial generalization of (5.2.11) is often used in finite elementsoftware to model the elastic response of materials in large strain elastic-plastic problems(see Section 5.7 below).For the case of small strains, equation (9.2.12) above can be written asσ˙ x = Eε˙ xwhich is the rate form (material time derivative) of Hooke's law (5.2.5).(5.2.15)For the general elastic relation (5.2.1) above, the function s(ε x ) was assumed to bemonotonically increasing.

The corresponding strain energy is shown in Figure 5.6b andcan be seen to be a convex function of strain. Materials for which s(ε x ) first increases andthen decreases exhibit strain-softening or unstable material response (i.e., ds dε x < 0 ). Aspecial form of non-monotonic response is illustrated in Figure 5.7a.

Here, the functions(ε x ) increases monotonically again after the strain-softening stage. The correspondingenergy is shown in Figure 5.7b. This type of non-convex strain energy has been used innonlienar elastic models of phase transformations (Knowles). At a given stress σ belowσ M the material may exist in either of the two strained states ε a or ε b as depicted in thefigure. The reader is referred to (Knowles) for further details including such concepts asthe energetic force on a phase boundary (interface driving traction) and constitutiverelations for interface mobility.5.3.

Multiaxial Linear ElasticityIn many engineering applications involving small strains and rotations, the responseof the material may be considered to be linearly elastic. The most general way to representa {\em linear} relation between the stress and strain tensors is given byσ ij = Cijkl εklσ = C:ε(5.3.1)where Cijkl are components of the 4th-order tensor of elastic moduli. This represents thegeneralization of (5.2.5) to multiaxial states of stress and strain and is often referred to asthe generalized Hooke's law which incorporates fully anisotropic material response.The strain energy per unit volume, often called the elastic potential., as given by(5.2.4) is generalized to multixial states by:W = ∫ σ ij dε ij =11Cijklε ijε kl = ε:C:ε22(5.3.2)The stress is then given byσ ij =∂w,∂ε ijσ=∂w∂ε(5.3.3)which is the tensor equivalent of (5.2.2).

The strain energy is assumed to be positivedefinite, i.e.,W=11Cijklε ij ε kl ≡ ε:C:ε ≥ 022(5.3.4)with equality if and only if ε ij = 0 which implies that C is a positive-definite fourth-ordertensor. From the symmetries of the stress and strain tensors, the material coefficients havethe so-called minor symmetriesCijkl = C jikl = Cijlk(5.3.5)and from the existence of a strain energy potential (5.3.2) it follows thatCijkl =∂2W,∂εij∂ε klC=∂ 2W∂ε∂ε(5.3.6)( )If W is a smooth C1 function of ε , Eq.

(5.3.6) implies a property called majorsymmetry:Cijkl = Cklij(5.3.7)since smoothness implies∂2 W∂2 W=∂εijε kl ∂ε klε ij(5.3.8)The general fourth-order tensor Cijkl has 34 = 81 independent constants. These 81constants may also be interpreted as arising from the necessity to relate 9 components of thecomplete stress tensor to 9 components of the complete strain tensor, i.e., 81= 9× 9.

Thesymmetries of the stress and strain tensors require only that 6 independent components ofstress be related to 6 independent components of strain. The resulting minor symmetries ofthe elastic moduli therefore reduce the number of independent constants to 6 ×6 = 36.Major symmetry of the moduli, expressed through Eq. (5.3.7) reduces the number ofindependent elastic constants to n( n +1) 2 = 21 , for n = 6 , i.e., the number of independentcomponents of a 6 ×6 matrix.Considerations of material symmetry further reduce the number of independentmaterial constants.

This will be discussed below after the introduction of Voigt notation.An isotropic material is one which has no preferred orientations or directions, so that thestress-strain relation is identical when expressed in component form in any recatngularCartesion coordinate system. The most general constant isotropic fourth-order tensor canbe shown to be a linear combination of terms comprised of Kronecker deltas, i.e., for anisotropic linearly elastic material()(Cijkl = λδijδ kl + µ δ ikδ jl + δilδ jk + µ ′ δ ikδ jl + δ ilδ jk)(5.3.9)Because of the symmetry of the strain and the associated minor symmetry Cijkl = Cijlk itfollows that µ′ = 0 . Thus Eq.

(6.3.9) is written()Cijkl = λδijδ kl + µ δ ikδ jl + δilδ jk ,C = λI ⊗I +2µI(5.3.10)and the two independent material constants λ and µ are called the Lamé constants.The stress strain relation for an isotropic linear elastic material may therefore bewritten asσ ij = λε kkδij + 2µε ij = Cijkl εkl ,σ = λtrace( ε)I +2µε(5.3.11)Voigt NotationVoigt notation employs the following mapping of indices to represent thecomponents of stress, strain and the elastic moduli in convenient matrix form:11→1 22 → 2 33 → 323 → 4 13 → 5 12 → 6(5.3.12)Thus, stress can be written as a column matrix {σ } withσ11 σ 12σ 22symσ11 σ 22 σ13   σ33 σ 23  →  σ 23 σ 33  σ 13 σ 12 {σ }T = [ σ1,σ 2 , σ 3, σ 4 , σ 5 , σ 6 ](5.3.13)or= [σ1 , σ 22 , σ33 , σ 23, σ13 , σ 12 ](5.3.14)Strain issimilarly written in matrix form with the exception that a factor of 2 is introduced on theshear terms, i.e.,{ε }T = [ε1 ,ε 2, ε 3, ε 4 , ε5 , ε 6 ]= [ε1, ε 22 , ε 33,2ε 23, 2ε13 , 2ε12 ](5.3.15)The factor of 2 is included in the shear strain terms to render the stress and strain columnmatrices work conjugates, i.e.,111W = σ T ε = σ ij εij = σ: ε222(5.3.16)The matrix of elastic constants is obtained from the tensor components by mappingthe first and second pairs of indices according to (5.3.12).

For example, C11 = C1111 ,C12 = C1122 , C14 = C1123C56 = C1312 etc. For example, the stress strain relation for σ11 isgiven byσ11 = C1111ε11 + C1112ε12 + C1113ε 13+ C1121ε21 + C1122 ε22 + C1123ε 23+C1131ε 31 + C1132 ε32 +C11331ε 33111= C11ε1 + C16ε 6 + C15ε 5 + C16ε 622211111+ C12ε 2 + C14ε 4 + C15ε 5 + C14ε 4 + C13ε 322222= C11ε1 +C12ε 2 + C13ε 3 + C14ε4 +C15ε 5 + C16 ε6= C1 jε j(5.3.17)and similarly for the remaining components of stress.

The constitutive relation may then bewritten in matrix form asσ = Cε,σ i = Cij ε j(5.3.18)Major symmetry (5.3.7) implies that the matrix [C], of elastic constants is symmetric with21 independent entries, i.e.,σ 1  C11σ   2  σ3   = σ 4  σ 5    σ 6  C12C22symC13C23C33C14C24C34C44C15 C16   ε1 C25 C26  ε 2 C35 C36 ε 3  C45 C46  ε 4 C55 C56  ε5  C66 ε 6 (5.3.19)The relation (5.3.19) holds for arbitrary anisotropic linearly elastic materials.

Thenumber of independent material constants is further reduced by considerations of materialsymmetry (see Nye (1985) for example). For example, if the material has a plane ofsymmetry, say the x1 -plane, the elastic moduli must remain unchanged when thecoordinate system is changed to one in which the x1-axis is reflected through the x1-plane.Such a reflection introduces a factor of -1 for each term in the moduli Cijkl in which theindex 1 appears. Because the x1 plane is a plane of symmetry, the moduli must remainunchanged under this reflection and therefore any term in which the index 1 appears an oddnumber of times must vanish. This occurs for the terms Cα 5 and Cα 6 for α =1,2,3.

Foran orthotropic material (e.g., wood or aligned fiber reinforced composites) for which thereare three mutually orthogonal planes of symmetry, this procedure can be repated for allthree planes to show that there are only 9 independent elastic constants and the constitutivematrix is written asσ 1  C11σ   2  σ3   = σ 4  σ 5    σ 6  C12C22C13C23C33sym000C440000C550   ε1 0  ε 2 0 ε 3  0  ε 4 0  ε5  C66  ε 6 (5.3.20)An isotropic material is one for which there are no preferred orientations. Recall that anisotropic tensor is one which has the same components in any (rectangular Cartesian)coordinate system.

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