Anisotropic Elasticity
Rubber-like materials are usually isotropic. It is possible, of course, to strengthen them by embedding fibers in prescribed directions and creating the fiber-reinforced composites . Nature does so with the soft biological tissues which usually consist of
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Anisotropic Elasticity
Rubber-like materials are usually isotropic. It is possible, of course, to strengthen them by embedding fibers in prescribed directions and creating the fiber-reinforced composites. Nature does so with the soft biological tissues which usually consist of an isotropic matrix with the embedded and oriented collagen fibers. The collagen fibers are aligned with the axes of ligaments and tendons forming one characteristic direction or they can form two and more characteristic directions in the case of blood vessels, heart etc.
5.1 On Material Symmetry In the case of isotropy, material properties are equivalent in all directions. This equivalence can be formalized with the help of the rotated reference configuration. Let point x be rotated to point x . Then, the relative deformation gradient is a proper-orthogonal tensor: ∂x /∂x = Q, where QT = Q−1 and det Q = 1. The full deformation gradient can be calculated by using the chain rule as follows ∂ yi ei ⊗ e j ∂x j ∂ yi ∂xm = ei ⊗ e j ∂xm ∂x j ∂ yi ∂xn = δmn ei ⊗ e j ∂xm ∂x j ∂xn ∂ yi = e ⊗ e e ⊗ e i m n j ∂xm ∂x j
F=
© Springer Science+Business Media Singapore 2016 K. Volokh, Mechanics of Soft Materials, DOI 10.1007/978-981-10-1599-1_5
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5 Anisotropic Elasticity
∂y ∂x , ∂x ∂x = F Q, =
(5.1)
where ∂y , ∂x
(5.2)
F = FQT .
(5.3)
F = and, consequently,
Now, the symmetry property of the strain energy can be written in the form ψ(F ) = ψ(FQT ) = ψ(F).
(5.4)
The set of the proper-orthogonal tensors Q, for which the invariance of the strain energy holds, generates the symmetry group of the material with respect to the reference configuration. In the case of isotropy the symmetry group is the full proper orthogonal group, i.e. all rotations.
5.2 Materials with One Characteristic Direction Materials enjoying one characteristic direction are also called materials with transverse isotropy, i.e. isotropy in the planes perpendicular to the preferred direction. Let us designate the preferred direction by unit vector m0 in the reference configuration. Then, the symmetry group includes rotations obeying condition: Qm0 = ±m0 ; and the strain energy should be invariant in the form ψ(FQT , Qm0 ) = ψ(F, m0 ).
(5.5)
To meet the latter condition, the strain energy function ψ(I1 , I2 , I3 , I4 , I5 ) should depend on two more invariants, as compared to the isotropic case, I4 = m · m = Fm0 · Fm0 = m0 · FT Fm0 = C : m0 ⊗ m0 , I5 = C2 : m0 ⊗ m0 ,
(5.6)
where m = Fm0 is not a unit vector.
(5.7)
5.2 Materials with One Characteristic Direction
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The fourth invariant, I4 , has a clear physical meaning of the squared stretch in the characteristic direction. Tensor product m0 ⊗ m0 is often called the structural or structure tensor, which characterizes the internal design of material. Differentiating these new invariants with respect to C we get ∂ I4 = m0 ⊗ m0 , ∂C ∂ I5 = Cm0 ⊗ m0 + m0 ⊗ Cm0 . ∂C
(5.8)
Now, the hyperelastic constitutive law takes form S=2
5 ∂ψ ∂ Ia ∂ Ia ∂C a=1
= 2{(ψ1 + I1 ψ2 )1 − ψ2 C + I3 ψ3 C−1 + ψ4 m0
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