Belytschko T. - Introduction (779635), страница 53
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In this case, dissipative effects are usually small also.Some commonly used forms of hypoelastic constitutive relations are∇Jσ = C J: D(5.4.3)∇Jwhere σ is the Jaumann rate of Cauchy stress given in equation (3.7.9) andL v τ = JC T : D(5.4.4)where L v τ is the Lie-Derivative of the Kirchhoff stress. Note thatL v τ = ˙τ − L⋅τ − τ⋅LT()˙ − L⋅σ − σ ⋅LT + (trace L )σ=J σ(5.4.5)∇T=J σ∇Twhere J = det F and σ is the Truesdell rate of Cauchy stress. Thus the Lie-derivative ofthe Kirchoff stress is simply the weighted Truesdell rate of the Cauchy stress.
A moredetailed discussion of Lie derivatives in the context of pull-back and push-forard operationsis given in the Appendix. We will use the concept of the Lie derivatives more extensively inour treatment of hyperelasticity (Section 5.4.3) and hyperelastic-plastic constitutiverelations (Section 5.7.4).Other forms of hypoelastic relations are based on the Green-Nagdhi (also called the∇GDienes) rate which is denoted here by σ and is given by∇G˙ ⋅σ − σ⋅ ΩT˙ −Ωσ =σd= R⋅RT ⋅σ ⋅R ⋅RTdt(5.4.6)˙ ⋅RTΩ= R(5.4.7)()whereis the spin associated with the rotation tensor R. The hypoelastic relation is given by∇Gσ = C G :D(5.4.8)Note that the Green-Naghdi rate is a form of Lie Derivative (Appendix A.x) in thatthe Cauchy stress is pulled back by the rotation R to the unrotated configuration where thematerial time derivative is taken with impunity and the result pushed forward by R again tothe current configuration.
The quantityσ = RT ⋅ σ⋅R(5.4.9)is the co-rotational Cauchy stress (Equation 3.7.18) discussed in Chapter 3.In the consitutive equations (5.4.3), (5.4.4) and (5.4.11) above, the fourth-ordertensors of elastic moduli C J , C T and C G are often taken to be constant and isotropic,e.g.,()JCijkl= λδijδ kl + µ δ ikδ jl + δilδ jk ,C J = λI ⊗ I + 2µI(5.4.10)Given a constitutive equation∇Jσ = C J: D(5.4.11)with constant moduli C J then, using the defintion of the Jaumann stress rate (3.7.9) andthe co-rotational rate (6.4.6), this relation can be written as∇Rσ = C J :D + (Ω − W ) ⋅σ + σ ⋅ (Ω − W )Twhich is a different constitutive equation to (5.4.8) with constant moduli C G .(5.4.12)5.4.2.Cauchy Elastic Material.
As previously mentioned, an elastic material maybe characterized as one which has no dependence on the history of the motion. Theconstitutive relation for a Cauchy elastic material is given by a special form of (A.y)written asσ = G ( F)(5.4.13)where G is called the material response function and the explicit dependence on position Xand time t has been suppressed for notational convenience.
Applying the restriction (A.z)due to material objectivity gives the formσ = R ⋅G (U) ⋅RT(5.4.14)Alternative forms of the same constitutive relation for other representations of stress andstrain follow from the stress transformation relations in Box (3.2), e.g., the first PiolaKirchhoff stress for a Cauchy elastic material is given byP = J −1σ ⋅F −T= J −1R ⋅G ( U) ⋅RT ⋅R ⋅U −1(5.4.15)= J −1R ⋅G( U) ⋅U−1while the relationship for the second Piola-Kirchhoff stress takes the formS = J −1 F −1 ⋅σ ⋅F −T= J −1U−1 ⋅RT ⋅R ⋅G ( U) ⋅RT ⋅R ⋅ U−1= J −1U−1 ⋅RT ⋅G( U) ⋅ U−1 = h(U) = ˜h(C )(5.4.16)where C = F T ⋅F = U2 is the right Cauchy Green deformation tensor. For a given themotion, the deformation gradient is always known by its definition F = ∂ x ∂X (Equation3.2.14). The stresses can therefore be computed for a Cauchy elastic material by (5.4.13)or one of the specialized forms (5.4.14-5.4.16) independent of the history of thedeformation.
However, the work done may depend on the deformation history or loadpath. Thus, while the material is history independent, it is in a sensepath dependent. Thisapparent anomaly arises from the complications of large strain theory (see Example 5.1)below. In material models for small deformations, the work done in history-independentmaterials is always path-independent.To account for material symmetry, we note that following Noll ( ) (see Appendixfor further discussions of material symmetry) the stress field remains unchanged if thematerial is initially rotated by a rotation which belongs to the symmetry group of thematerial, i.e., if the deformation gradient, F is replaced by F ⋅Q where Q is an element ofthe symmetry group.
Thus (5.4.13) is written asσ = G ( F⋅ Q)(5.4.17)For an initially isotropic material, all rotations belong to the symmetry group (5.4.17) musttherefore hold for the special case Q = RT , i.e.,()σ = G F⋅ RT = G (V)(5.4.18)where the right polar decomposition (3.7.7) of the deformation gradient has been used.It can be shown (Malvern, ) that for an initially isotropic material, the Cauchy stressfor a Cauchy elastic material is given byσ = α 0 I + α 1V + α 2 V2(5.4.19)where α 0 , α 1, and α 2 are functions of the scalar invariants of V. For further discussion ofthe invariants of a second order tensor, see Box 5.x below. The expression (5.4.19) is aspecial case of the general relation for an isotropic material given in (5.4.18).Example 5.1.
Consider a Cauchy Elastic material with consitutive relation given byσ = α (V − I), α = α 0 J, J = det V(E5.1.1)Let the motion be given by3R = I, F = V = ∑ λ i e i ⊗ e i(E5.1.2)i =1with λ3 = 1 and λ1 = λ1 (t ), λ 2 = λ 2 (t ) .The principle stretches for two deformation paths 0AB and 0B are shown in Figure5.y below:λ2B ( λ1 , λ 2)0(1, 1)A(λ1, 1)λ1Figure 5.y. Deformation paths 0AB and 0B.Show thatthe work done in deforming the material along paths 0AB and 0B is different,i.e., path-dependent. λ1V= λ2 1λ1 − 1σ =α λ 2 −1 1(E5.1.3)Here, λ˙ 1 λ1˙ V−1 = D= Vλ˙ 2 λ20 J = det V = λ1λ 2 λ 3 = λ1λ 2(E5.1.4)(E5.1.5)The stress power is given byW˙ = σ:Dλ˙ 1λ˙+ α 0 λ1λ 2 (λ 2 −1) 2λ1λ2= α 0 λ 2 (λ1 −1)λ˙ 1 + α 0 λ1 (λ 2 −1)λ˙ 2= α 0 λ1 λ2 (λ1 − 1)Path 0AB:(E5.1.6)dW = α 0 λ2 (λ1 − 1)dλ1 + α 0 λ1 (λ 2 − 1)dλ 2(E5.1.7)On 0A, λ 2 = 1, constant.
On AB λ1 = λ1 , constant. ThusλλW = α 0 λ1 λ 2 1 − 1 + α 0 λ1λ 2 −1 2 2(E5.1.8)Path 0AB:λ 2 = mλ1m = λ 2 λ1dW = αmλ 1( λ1 −1) dλ1 + α 1 λ2 (λ 2 −1)dλ2(E5.1.9)(E5.1.10)m λ3 λ2 λ3 λ2 W = αm 1 − 1 + α 1 2 − 2 dλ 22 32 3m λ 1λ1= α 0 λ1 λ2 1 − + α 0 λ1λ 2 2 − 3 2 3 2(E5.1.11)which differs from Eq. (E5.1.8), i.e., the work done is path-dependent.Exercise 5.2. Show that the consitutive relation σ = α 0 (V − I) gives a path-independentresult for the two paths considered in Example 5.1 above.Rate (or incremental) forms of the constitutive relation are required in the treatmentof linearization (Chapter 6).
A useful starting point for derivation of the rate form of theconstitutive relation is, where possible, to take the material time derivative of the expressionfor the second Piola-Kirchhoff stress S . Thus, for a Cauchy elastic material˜˙S = ∂h(C) :C˙∂C(5.4.20)The fourth order tensor C SC = ∂ ˜h(C) ∂(C) is called the instantaneous tangent modulus.From the symmetries of S and C, the tangent modulus possesses the minor symmetries,SCSCi.e, C ijkl= C SCjikl = C ijlk .5.4.3.
Hyperelastic MaterialsElastic materials for which the work done on the material is independent of the loadpath are said to be hyperelastic (or Green elastic materials). In this section, some generalfeatures of hyperelastic materials are considered and then examples of hyperelasticconstitutive models which are widely used in practice are given. Hyperelastic materials arecharacterized by the existence of a stored (or strain) energy function which is a potential forthe stress.
Note that from Eq. (5.4.16) the second Piola-Kirchhoff stress for a Cauchyelastic material can be written as˙S = ˜h(C)(5.4.21)where C = F T ⋅F = U2 is the right Cauchy Green deformation tensor. For the case of ahyperelastic material, the second-order tensor ˜h is derived from a potential, i.e.,∂Ψ( C)S = ˜h(C) = 2∂C(5.4.22)where Ψ is called the stored energy function. Expressions for different stress measuresare obtained through the appropriate transformations (given in Box (3.2)), e.g.,τ = Jσ = F⋅S ⋅F T = 2F⋅∂Ψ(C) T⋅F∂C(5.4.23)It can be shown (Marsden and Huges) that, given (5.4.22), the Kirchhoff stress isalso derivable from a potential, i.e.,τ=2∂Ψ( g)∂g(5.4.24)where g is the spatial metric tensor (which is equivalent to the identity tensor for Euclideanspaces).A consequence of the existence of a stored energy function is that the work done ona hyperelastic material is independent of the deformation path.
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