Belytschko T. - Introduction (779635), страница 54

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This behavior isapproximately observed in many rubber-like materials. To illustrate the independence ofwork on deformation path, consider the stored energy per unit reference volume in goingfrom deformation state C1 to C2 . Since the second Piola-Kirchhoff stress tensor S and(C − I )the Green strain E =are work conjugates,2C1 2S:dC = Ψ (C1 ) − Ψ (C2 )2 C∫(5.4.25)1which depends only on the initial and final states of deformation and is thereforeindependent of the deformation (or load) path.

(Contrast this with the behavior of theCauchy elastic material in Example 5.1 above.)The rate forms of consitutive equations for hyperelastic materials and thecorresponding moduli can be obtained by taking the material time derivative of Eq. (5.4.22)as follows:˜˙S = ∂h(C) :C˙∂C∂ 2 Ψ (C ) ˙:C∂C∂C˙C= C SC :2=4(5.4.26)where∂2 Ψ (C)C =4∂C∂C(5.4.27)is the tangent modulus.

It follows that the tangent modulus for a hyperelastic material hasSCSCthe major symmetry C ijkl= C klij, in addition to the minor symmetries shown already forthe Cauchy elastic material.It is often desirable (particularly in the linearization of the weak form of thegoverning equations (Chapter 6) to express the stress rate in terms of an Eulerian stresstensor such as the Kirchhoff stress. To this end we recall the Lie Derivative (also referredto as the convected rate) of the Kirchhoff stress introduced earlier in this Chapter, i.e.,L v τ = F⋅()d −1F ⋅τ ⋅ F− T ⋅F T = F ⋅ ˙S ⋅F Tdt= τ˙ − L ⋅τ −⋅τ ⋅LT(5.4.28)d= φ x  φ ∗ ( τ) dtNote that the right Cauchy Green deformation tensor can be written asC = F ⋅F = F T ⋅g ⋅F where g is the spatial metric tensor.

In Euclidean space, we have˙g = I the identity tensor. Noting also that C 2 = FT ⋅ D ⋅F it follows that the rate ofdeformation tensor can be written asTD = F− T ⋅d T1 d  g F ⋅ g⋅F ⋅F −1 = Lv g = φ x  φ ∗    dt  2  dt2()(5.4.29)where Lv g is the Lie derivative of the spatial metric tensor. Using Eqs. (5.4.29) and(5.4.26) in (5.4.28) givesLv τ = C τD :DwhereτDSCC ijkl= Fim Fjn Fkp FlqC mnpq(5.4.30)are referred to as the spatial tangent moduli.

It can be seen from the above that the Liederivative of the Kirchhoff stress arises naturally as a stress rate in finite strain elasticity.•Issues of uniqueness and stability of solutions in finite strain elasticity aremathematically complex. The reader is referred to [Ogden] and [Marsden and Hughes]for a detailed description.It can be shown that, using the representation theorem (Malvern, 1969), the stored(strain) energy for a hyperelastic material which is isotropic with respect to the initial,unstressed configuration, can be written as a function of the principal invariants ( I1 , I2 , I3 )of the right Cauchy-Green deformation tensor, i.e., W = W(C) .

The principal invariants ofa second order tensor and their derivatives figure prominently in elastic and elastic-plasticconstitutive relations. For reference, Box 5.1 summarizes key relations involving princpalinvariants.Box 5.1Principal InvariantsThe principal invariants of a second order tensor A are given byI1 (A) = Trace A1I2 ( A) = (Trace A)2 − Trace A2(B5.1.1)2I3 (A) = det AWhen the tensor in question is clear from the context, the argument A is omitted and theprincipal invariants denoted simply as I1 , I2 , and I3 .{}If A is symmetric, then A = AT and a set of 3 real eigenvalues (or principal values) of Amay be formed and written as λ1 , λ 2 , λ 3 .

ThenI1 = λ1 + λ 2 + λ 3I2 = λ1λ 2 + λ 2 λ 3 + λ3 λ1(B5.1.2)I3 = λ1λ 2 λ3The derivatives of the principal invariants of a second order tensor with respect to thetensor itself are often required in constitutive equations and in the linearization of the weakform (Chapter 6). For reference:∂I1∂I1= I;= δij(B5.1.3)∂A∂Aij∂I2∂I2= I1 I − AT ;= Akk δij − A ji(B5.1.4)∂A∂Aij∂I3∂I3= I3 A − T ;= I3 A−1(B5.1.5)ji∂A∂AijThe second Piola-Kirchhoff stress tensor is given by ( ). Thus, for an isotropic materialwe haveS=2 ∂w∂w∂w ∂w∂w −1= 2+ I1C + 2I3CI −2∂C∂I2 ∂I2I3 ∂I1(5.4.32)The Kirchhoff stress tensor is given by ∂w∂w ∂w 2∂wτ = F⋅S ⋅F T = 2+ I1B + 2I3I B− 2∂I2 ∂I2I3 ∂I1where B = F⋅ FT is the left Cauchy-Green deformation tensor. Note that S is co-axial hasthe same principal directions) with C while τ is co-axial with B.

These results will beused below in deriving expressions for the stress tensors for specific hyperelastic models.In the remainder of this section, examples of hyperelastic materials which arefrequently used to model the behavior of rubber-like materials are presented.Neo-Hookean Material. The stored energy function for a compressible Neo-Hookeanmaterial [Ref] (isotropic with respect to the initial, unstressed configuration) is written asw(C) =112λ 0 (log J ) − µ0 log J + µ 0 (trace C − 3)22(5.4.34)From Eq. (5.4.32), the stresses are given by(S = λ 0 log JC−1 + µ0 I − C−1)(5.4.35)τ = λ 0 log JI + µ0 (B − I)Lettingλ = λ0 ,µ = µ0 − λ log J(5.4.36)and using Eqs. (5.4.27) and (5.4.31), the elasticity tensors (tangent moduli) are written incomponent form on Ω 0 as(SC−1 −1Cijkl= λCij−1Ckl−1 + µ Cik−1 C−1jl + Cil Ckj)(5.4.37)and on Ω as(CτDijkl = λδijδ kl + µ δ ikδ jl + δilδ kj)(5.4.38)The elasticity tensor in Eq.

(5.4.38) has the same form as in Hooke's Law for small strainelasticity, except for the dependence of the shear modulus $\mu$ on the deformation (seeEq. 5.4.36). Here λ 0 and µ0 are the Lamé constants of the linearized theory. Nearincompressible behavior is obtained for λ 0 >> µ0 .Saint Venant - Kirchhoff Model. A wide class of engineering problems can bestudied by linear elastic material behavior. If the effects of large deformation are primarilydue to rotations (such as in the bending of a marine riser or a fishing rod, for example) astraightforward generalization of Hooke's law to finite strains is often adequate. The SaintVenant-Kirchhoff model accomplishes this through the use of the Green strain measure Eas follows. Let1w(C) = W (E ) = E:C SE :E2(5.4.39)where(SECijkl= λ 0δ ijδ kl + µ0 δ ikδ jl + δilδ kj)and where λ 0 and µ0 are Lamé constants. Noting thatµ0and that\begin{equation}{\bf S} = 2{\partial \Psi({\bf C})\over \partial {\bf C}} ={\partial W({\bf E})\over\partial {\bf E}}\end{equation}the components $S_{ij}$ of the second Piola-Kirchhoff stress are given by\begin{equation}S_{ij} = \lambda_0 E_{kk}\delta_{ij} +2\mu_0 E_{ij}\end{equation}or\begin{equation}{\bf S}=\lambda_0\, {\rm trace}\,({\bf E}){\bf I}+ 2\mu_0 {\bf E} = \mbox{\boldmath ${\cal D}$}:{\bf E}\end{equation}Because the Green strain tensor is symmetric, it follows that thestress tensor ${\bf S}$ is also symmetric.

From Eq. () it is apparent thatthefourth order material response tensor possesses major and minorsymmetries. Because ${\bf E}$, ${\bf F}$(5.4.40)and ${\bf C}$ are related ( ), it can also be shown that the components ofthe nominal stress tensor is given by\begin{equation}{\bf P} = {\partial W\over \partial {\bf F}^T}, \qquadP_{ij} = {\partial W\over \partial F_{ji}}\end{equation}As the deformation gradient tensor ${\bf F}$ is not necessarily symmetric,the 9 componentsof the nominal stress tensor ${\bf P}$ do not necessarily possess symmetry.Employing Eq.

( ), the Cauchy stress tensor $\mbox{\boldmath ${\sigma}$}$is related to W by:\begin{equation}\mbox{\boldmath ${\sigma}$} = {1\over J} {\bf F}\cdot{\partial W\over\partial {\bf F}^T} = {1\over J} {\bf F}\cdot {\partial W\over \partial{\bf E}}\cdot {\bfF}^T\end{equation}(Exercise: Show this.)\noindent {\bf Modified Mooney-Rivlin Material}\parIn 1951, Rivlin and Saunders [ ] published their experimental results on thelarge elastic deformations of vulcanized rubber - an incompressiblehomogeneous isotropic elastic solid, in the Journal of Phil.

Trans. A.,Vol. 243, pp. 251-288. This material model with a few refinements isstill the most commonly used model for rubber materials. It is assumedin the model that behavior of the material is initially isotropic andpath-independent, i.e., a stored energy function exists. The storedenergy function is written\begin{equation}\Psi = \Psi({\bf C}) = W(I_1, I_2, I_3)\end{equation}where $I_1$, $I_2$ and $I_3$ are the three scalar invariants of ${\bf C}$.Rivlin and Saunders considered an initially isotropic nonlinearelastic incompressible material ($I_3=1$) then\begin{equation}\Psi = \Psi(I_1,I_2) = \sum_{i=0}^\infty\sum_{j=0}^\infty\bar{c}_{ij}(I_1-3)^i(I_2-3)^j,\qquad \bar{c}_{00}=0\end{equation}where $\bar{c}_{ij}$ are constants.They performed a number of experiments on differenttypes of rubbers and discovered that Eq.

( ) may be reduced to\begin{equation}\Psi = c(I_1 - 3) + f(I_2 - 3)\end{equation}where $c$ is a constant and$f$ is a function of $I_2 - 3$. For a Mooney-Rivlin material, $W$ can bereducedfurther to\begin{equation}\Psi = \Psi(I_1, I_2) = c_1 (I_1 - 3) + c_2 (I_2 - 3)\end{equation}An example of the set of $c_1$ and $c_2$ is: $c_1 = 18.35 {\rm psi}$ and$ c_2 = 1.468 {\rm psi}.$ Equation ( ) is also an example of aNeo-Hookean material, and the components of the second Piola-Kirchhoff stresscan be obtained bydifferentiating Eq. ( ) with respect to the components of the rightCauchy Green deformation tensortensor; however, the deformation is constrained such that\begin{equation}{\bf S} = 2{\partial \Psi\over \partial {\bf C}}, \qquad{\rm with}\I_3 ={\rm det}\,{\bf C} = 1\end{equation}The condition $I_3 = 1$ simply implies that $J=1$ and there is no volumechange.

The condition can be written as\begin{equation}{\rm ln}I_3 = 0\end{equation}which represents a constraint on the deformation. One way in which theconstraint ( ) can be enforced is through the use of a constrainedpotential, or stored energy, function [Ref]. Alternatively, a penaltyfunction formulation (Hughes, 1987) can be used. In this case,the modified strain energy functionand the constitutive equation become:\begin{eqnarray}\bar{\Psi} &=& \Psi + p_0\,{\rm ln}I_3 + {1\over 2} \lambda({\rm ln}I_3)^2 \\{\bf S} &=& 2{\partial \Psi\over \partial {\bf C}} + 2(p_0 + \lambda({\rmln}I_3)){\bf C}^{-1}\end{eqnarray}respectively.

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