4 Constitutive Theory 4.9 Transversely Isotropic Elasticity This section is devoted to the coordinate-free representation of transversely isotropic elasticity, which is probably the sim- plest kind of anisotropy one can conceive. The coordinate- independent framework is obtained in terms of structural tensors that describe the microstructure of the material un- ME338A der consideration. CONTINUUM MECHANICS a X lecture notes 17 NX B Thursday, May 29, 2008 The microstructure of a transversely isotropic material can be considered as a continuum reinforced by micro-fibers. Common examples cover man-made fiber-reinforced com- posites such as carbon nanotube loaded materials, automo- bile tires, and natural collagenous biological tissues, to men- tion a few. The fibrous microstructure of a transversely isotropic ma- terial possesses a well-defined preferred direction, which we denote with the unit vector a, ||a||=1. This is the key in- formation that helps us to construct the symmetry group of a transversely isotropic matter. As we mentioned in Sec- tion 4.7, a symmetry group is defined by a set of rotations that preserve the structural characteristics of the material on 143 4 Constitutive Theory 4 Constitutive Theory micro-level. In the case of transverse isotropy, the symmetry Therefore, the new isotropic free energy also fulfills group contains all rotations that do not alter the preferred ψˆ(Q · C · QT, M) = ψˆ(C, M) ∀Q ∈ G ⊂ SO(3) . orientation a. These include the rotations of an arbitrary transiso (4.9.5) amount about the axis a, i.e. Q!a and the rotations about π the axis perpendicular to a by an amount of π, i.e. Q⊥a, that The representation theorem of general isotropic tensor func- flip a horizontally. We have then the transversely isotropic tions of several arguments is based on the concept of in- symmetry group defined as tegrity bases, which defines a minimum number of invariants for a particular set of arguments of the energy storage func- π tion. For two generic symmetric second order tensors, say A1 Gtransiso := Q!a, Q⊥a . (4.9.1) ! " and A2, the irreducible integrity bases are given by The underlying key idea of coordinate-free representation of anisotropic materials is to introduce isotropic tensor func- I = { J1(A1), J2(A1), J3(A1), J1(A2), J2(A2), J3(A2) (4.9.6) tions with extended set of arguments such that they remain J4(A1, A2), J5(A1, A2), J6(A1, A2), J7(A1, A2)} invariant under arbitrary rotations Q ∈ SO(3). This opens where up a possibility to recast an anisotropic constitutive function 2 3 into an isotropic one by means of the so-called structural ten- J1(Ai) := tr(Ai), J2(Ai) := tr(Ai ), J3(Ai) := tr(Ai ) sors. For the case of transverse isotropy, we have are the basic invariants of respective tensors for i = 1, 2 and J := tr(A · A ), J := tr(A2 · A ), ψ = ψˆ(C, M) with M := a ⊗ a . (4.9.2) 4 1 2 5 1 2 J := tr(A · A2), J := tr(A2 · A2) denoting the constant rank-one structural tensor. The free 6 1 2 7 1 2 energy function with the extended list of arguments is then denote the mixed invariants. required to be an isotropic function, i.e. invariant under ar- In transversely isotropic elasticity we have the free energy bitrary rotations Q ∈ SO(3) function depending upon the symmetric tensors A1 = C and A = M. Since the structural tensor M is also constant ψˆ(Q · C · QT, Q · M · QT) = ψˆ(C, M) ∀Q ∈ SO(3) . (4.9.3) 3 and Mn = M, the following identities Since the rotations belonging to the symmetric group Gtransiso 3 2 ! preserve the direction a, the structural tensor M remains in- J3(M) = tr(M ) = tr(M ) = tr(M) J3(M) = J2(M) = J1(M) 2 variant under these rotations J6 = tr(C · M ) = tr(C · M) ! J6 = J4 T 2 2 2 ! Q · M · Q = M ∀Q ∈ Gtransiso . (4.9.4) J7 = tr(C · M ) = tr(C · M) J7 = J5 144 145 4 Constitutive Theory 4 Constitutive Theory reduce the integrity bases of transverse isotropy to The isotropic free energy function is taken to be identical to the compressible neo-Hookean model we discussed in Sec- I = {J (C), J (C), J (C), J (C, M), J (C, M); J (M)} (4.9.7) 1 2 3 4 5 1 tion 4.8.1 Since we also know that ||a|| = 1, the basic invariant(s) of Λ 2 Λ µ ψˆ(I1, J) = (J − 1) − (µ + ) ln J + (I1 − 3) . (4.9.12) the structural tensor are none other than J1(M) = 1. There- 4 2 2 fore, it has no effect on the energy storage function. The ef- We then retrieve the stress expressions for the isotropic part fective list of deformation-dependent invariants then boils ∂ψˆ Λ S = 2 I = µ(1 − C−1) + J2 − 1 C−1 , down to I ∂C 2 # $ Λ I = {J1(C), J2(C), J3(C), J4(C, M), J5(C, M)} . (4.9.8) P = F · S = µ(F − F−T) + J2 − 1 F−T . (4.9.13) I I 2 # $ Since the basic invariants of C can be expressed in terms of For the anisotropic part of the free energy we consider the the principal invariants following form 2 3 J1 = I1, J2 = I1 − 2I2, J3 = I1 − 3I1 I2 + 3I3 , 2 ψˆ A(I1, I2, J4, J5) = α1(J4 − 1) + α2[J5 − I1 J4 + I2 − 1] the set in (4.9.8) may be recast into + α2(J4 − I1 + 2) , (4.9.14) I = {I1(C), I2(C), I3(C), J4(C, M), J5(C, M)} . (4.9.9) where α1 and α2 are the additional material parameters. Ge- 4.9.1 Extended Neo-Hookean Model of Transverse Isotropy ometrical interpretation of the kinematic terms appearing in this free energy will make the meaning of these additional We consider a model problem of transverse isotropty. The material parameters more transparent. The square of the free energy is assumed to be given as stretch in the direction a is given by J4, i.e. ψ = ψˆ I(I1, J) + ψˆ A(I1, I2, J4, J5) , (4.9.10) 2 2 λ := ||F · a|| = (F · a) · (F · a) = C : (a ⊗ a) = C : M ≡ J4 . which is split into the purely isotropic ψI and the anisotropic a ψA parts. This leads us to the additive form of the stresses Thus, the first non-linear term of the free energy governs the amount energy stored due to the stretching of fibers, and the material parameter α1 is closely related to the stiffness of S = 2∂ ψ = SI + SA and P = PI + PA , (4.9.11) C these fibers. The second term in square brackets, however, ˆ ˆ where SI:=2∂CψI, SA:=2∂CψA and PI:=F · SI, PA:=F · SA. measures the area stretch νa in the direction a . We compute 146 147 4 Constitutive Theory 4 Constitutive Theory the square of the area stretch through the area map cof(F) := The anisotropic part of the second Piola-Kirchhoff stresses is JF−T shown to be 2 2 2 −T −T 2 −1 νa := || cof(F) · a|| = J (F · a) · (F · a) = J C : M . SA = 4 α1(J4 − 1)M In order to see the relation between this result and the sec- + 2 α2[(M · C + C · M) − C + (I1 − J4 − 1)1 − (I1 − 1)M] , ond term of ψˆ A, we recall the Cayley-Hamilton theorem 3 2 from which the first Piola-Kirchhoff stresses can readily be C − I1C + I2C − I31 = 0 , computed through the push forward operation PA = F · SA. which states that every second order tensor satisfies its char- acteristic equation obtained in the eigenvalue problem. Mul- Uniaxial stress-stretch response of the transversely isotropic −1 tiplying this equation with C and solving the result for model in two perpendicular directions is depicted below. −1 I3C , we obtain Observe the fairly stiffer behavior in the direction coincid- −1 2 −1 2 I3C = J C = C − I1C + I21 . ing with the preferred direction a. Contraction of this result with the structural tensor M then 30 gives the square of the area stretch in the preferred direction Λ = 0.3 MPa µ = 0.8 MPa a 2 2 −1 2 25 νa = J C : M = C : M − I1C : M + I21 : M α1 = 0.1 MPa α2 = 0.1 MPa = J5 − I1 J4 + I2 . 20 ] a P Therefore, the material parameter α2 can be conceived as the M [ 15 2 2 a stiffness of the isotropic matrix that surrounds the fibers. P , 1 1 ( − + ) ˆ P The third term α2 J4 I1 2 in ψA is introduced to have 10 a stress-free state in the undeformed configuration. Compu- tation of the stress contributions from the anisotropic part 5 requires the knowledge of the following derivatives 0 1 1.5 2 2.5 3 3.5 4 λ [−] 2∂C I1 = 2∂C(C : 1) = 2 1 , 1 2 2 2∂C I2 = 2∂C 2 (I1 − C : 1) = 2 (I11 − C) , 2∂C J4 = 2∂C(C : M) = 2 M , 2 2∂C J5 = 2∂C(C : M) = 2 (M · C + C · M) . 148 149.