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Finite Strain Elasticity

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Finite Strain Elasticity Part 2 THEORY OF ELASTICITY CHAPTER 5 FINITE STRAIN ELASTICITY 5.1. Green-Elasticity Elastic deformation does not cause irreversible rearrangement of internal structure, and the corresponding Helmholtz free energy is a function of stress and temperature only. Restricting consideration to isothermal elastic defor- mation (θ˙ = 0), Eqs. (4.3.7) and (4.3.9) give ˙ ∂ψ ˙ 1 ˙ ψ = : E(n) = 0 T(n) : E(n), (5.1.1) ∂E(n) ρ i.e., ∂Ψ 0 T(n) = , Ψ=ρ ψ E(n) . (5.1.2) ∂E(n) Alternatively, Eq. (5.1.2) can be deduced by adopting an experimentally observed property that there is no net work left in a body upon any closed cycle of elastic strain, i.e., T(n) :dE(n) =0. (5.1.3) This implies that T(n) :dE(n) = dΨ (5.1.4) is a total differential, which leads to Eq. (5.1.2). The function Ψ = Ψ E(n) is the strain energy function per unit initial volume. It represents the work done to isothermally deform a unit of initial volume to the state of strain E(n). The explicit representation of the function Ψ E(n) depends on the selected strain measure E(n) and the material properties. Since the material and spatial strain tensors (see Section 2.3) are related by ˆ T E(n) = E (n) = R ·EE (n) · R, (5.1.5) the strain energy per unit mass can be written as ˆ ψ = ψ E(n) = ψ E (n) . (5.1.6) It can be easily verified that • ∂ψ ∂ψ T ˆ˙ T = R · · R , E (n) = R · E (n) · R, (5.1.7) ∂E (n) ∂E(n) and the rate of ψ becomes • ˙ ∂ψ ˙ ∂ψ ψ = : E(n) = : E (n). (5.1.8) ∂E(n) ∂E (n) The stress tensor T (n) conjugate to spatial strain tensor E (n) is defined in Section 3.6 by • ˙ T T(n) : E(n) = T (n) : E (n), T (n) = R · T(n) · R . (5.1.9) Consequently, in addition to (5.1.2), from Eq. (5.1.7) we deduce that ∂Ψ 0 ˆ T (n) = , Ψ=ρ ψ E (n) . (5.1.10) ∂E (n) In view of the expressions for the conjugate stress and strain tensors corresponding to n = ±1, given in Section 3.6, the following expressions for the Kirchhoff stress τ = (det F)σ are obtained from Eqs. (5.1.2) and (5.1.10) ∂Ψ ∂Ψ τ = F · · FT = F−T · · F−1, (5.1.11) ∂E(1) ∂E(−1) ∂Ψ ∂Ψ τ = V · · V = V−1 · · V−1. (5.1.12) ∂E (1) ∂E (−1) If the conjugate pair associated with n =1/2 is used, from Eq. (3.6.3) and Eq. (5.1.2) there follows 1 ∂Ψ ∂Ψ 1 τˆ = U · + · U − U · Kˆ − Kˆ · U . (5.1.13) 2 ∂E(1/2) ∂E(1/2) 2 Here, τˆ = RT · τ · R and 1 ∂Ψ ∂Ψ Kˆ = J I0 − U−1 · U−1 · − · U−1 J J + J 1 ∂E ∂E 1 2 3 (1/2) (1/2) 0 −1 · J1I − U . (5.1.14) −1 The invariants of U are denoted by Ji. In the derivation, the results from Subsection 1.12.1 were used to solve the matrix equation of the type A · X + X · A = B. If Eq. (3.6.25) is used, Eq. (5.1.10) gives 1 ∂Ψ ∂Ψ 1 τ = V · + · V − (V · K − K · V) , (5.1.15) 2 ∂E (1/2) ∂E (1/2) 2 where 1 ∂Ψ ∂Ψ K = J I − V−1 · V−1 · − · V−1 J J + J 1 ∂E ∂E 1 2 3 (1/2) (1/2) (5.1.16) −1 · J1I − V . The invariants of V−1 are equal to those of U−1 and are again denoted by Ji. The transition from Eq. (5.1.13) to (5.1.15) is straightforward by noting that Kˆ = RT · K · R. (5.1.17) For elastically isotropic materials, considered in the next section, the tensors −1 V and ∂Ψ/∂E (1/2) are coaxial, hence commutative, and K = 0. Similar expressions are obtained when Eqs. (3.6.6) and (3.6.26) are used to specify the conjugate stress and strain measures corresponding to n = −1/2. With a properly specified strain energy function Ψ E(n) for a given material, Eqs. (5.1.11) and (5.1.12), or (5.1.13) and (5.1.15), define the stress response at any state of finite elastic deformation. Since stress is derived from the strain energy function, the equations are referred to as the constitutive equations of hyperelasticity or Green-elasticity (Doyle and Ericksen, 1956; Truesdell and Noll, 1965). The nominal stress is ∂Ψ P = , (5.1.18) ∂F which follows from Ψ=˙ P ··F˙ , (5.1.19) and Ψ = Ψ(F). Since Ψ is unaffected by rotation of the deformed configu- ration, Ψ(F)=Ψ(Q · F). (5.1.20) By choosing Q = RT , it follows that Ψ depends on F only through U,or C = U2, i.e., Ψ=Ψ(C), C = FT · F. (5.1.21) The functional dependences of Ψ on different tensor arguments such as F, U or C are, of course, different. 5.2. Cauchy-Elasticity Constitutive equations of finite elasticity can be derived without assuming the existence of the strain energy function. Suppose that at any state of elastic deformation, the stress is a single-valued function of strain, regardless of the history or deformation path along which the state has been reached. Since no strain energy is assumed to exist, the work done by the stress could in general be different for different deformation paths. This type of elasticity is known as Cauchy-elasticity, although experimental evidence does not indicate existence of any Cauchy-elastic material that is also not Green- elastic. In any case, we write T(n) = f E(n) , (5.2.1) where f is a second-order tensor function, whose representation depends on the selected strain measure E(n) (relative to an undeformed configuration and its orientation), and on elastic properties of the material. In terms of the spatial stress and strain measures, Eq. (5.2.1) can be rewritten as ˆ ˆ ˆ T T (n) = f(E (n)), T (n) = R ·TT (n) · R. (5.2.2) The rotated Kirchhoff stress can be expressed from these equations by using any of the conjugate stress and strain measures. For example, τˆ = g E(1) , (5.2.3) 0 1/2 0 1/2 g E(1) = I +2E(1) · f E(1) · I +2E(1) , or τˆ = g E(−1) , (5.2.4) 0 1/2 0 1/2 g E(−1) = I − 2E(−1) · f E(−1) · I − 2E(−1) . Note that (det F) can be cast in terms of the invariants of E(n), since 2n 2 3 (det F) =1+2nIE − 4n IIE +8n IIIE. (5.2.5) T Thus, Eqs. (5.2.3) and (5.2.4) also define σˆ = R · σ · R in terms of E(1) and E(−1). All constitutive equations given in this section are objective under rigid- body rotation of the deformed configuration. The material tensors are un- ∗ · ∗ ∗ affected by the transformation F = Q F, since E(n) = E(n) and T(n) = E ∗ ·EE · T T(n). The spatial tensors transform according to (n) = Q (n) Q and T ∗ ·TT · T (n) = Q (n) Q , preserving the physical structure of the constitutive equations such as Eq. (5.2.2). 5.3. Isotropic Green-Elasticity If the strain energy does not depend along which material directions the principal strains are applied, so that · · T Ψ Q0 E(n) Q0 =Ψ E(n) (5.3.1) for any rotation tensor Q0, the material is elastically isotropic. A scalar function which satisfies Eq. (5.3.1) is said to be an isotropic function of its second-order tensor argument. Such a function can be expressed in terms of the principal invariants of the strain tensor E(n), defined according to Eqs. (1.3.3)–(1.3.5), i.e., Ψ=Ψ(IE,IIE,IIIE) . (5.3.2) Since ∂IE 0 ∂IIE 0 = I , = E(n) − IEI , ∂E(n) ∂E(n) (5.3.3) ∂III E 2 − − 0 = E(n) IEE(n) IIEI , ∂E(n) Equation (5.1.2) yields, by partial differentiation, 0 2 T(n) = c0I + c1E(n) + c2E(n). (5.3.4) The parameters are ∂Ψ ∂Ψ ∂Ψ ∂Ψ ∂Ψ c = − I − II ,c= − I , 0 ∂I E ∂II E ∂III 1 ∂II E ∂III E E E E E (5.3.5) ∂Ψ c2 = . ∂IIIE For example, if it is assumed that (Saint-Venant–Kirchhoff assumption) 1 Ψ= (λ +2µ)I2 +2µII , (5.3.6) 2 E E a generalized Hooke’s law for finite strain is obtained as 0 T(n) = λIEI +2µE(n). (5.3.7) The Lam´ematerial constants λ and µ should be specified for each selected strain measure E(n). If a cubic representation of Ψ is assumed (Murnaghan, 1951), i.e., 1 l +2m Ψ= (λ +2µ)I2 +2µII + I3 +2mI II + nIII , (5.3.8) 2 E E 3 E E E E the stress response is 2 − 0 T(n) =[λIE + lIE +(2m n)IIE]I (5.3.9) − 2 +[2µ +(2m n)IE]E(n) + nE(n). The constants l, m, and n are the Murnaghan’s constants. By choosing Q0 = R, Eq. (5.3.1) gives Ψ E (n) =Ψ E(n) , (5.3.10) ˆ so that Ψ is also an isotropic function of E (n).
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