In this chapter, we study the deformation of nonhomogeneous Cosserat elastic cylinders, when the constitutive coefficients are independent of the axial coordinate. In the first part of the chapter, we consider the case of isotropic bodies and assume that

λ = λ(x1, x2), μ = (x1, x2), . . . , γ = γ(x1, x2), (x1, x2) ∈ Σ1 (6.1.1)

We suppose that the domain Σ1 is C∞-smooth [88], and that the elastic coefficients belong to C∞. The basic equations of the plane strain, parallel to the x1, x2-plane, consist of the equations of equilibrium

tβα,β + fα = 0, mα3,α + εαβtαβ + g3 = 0 (6.1.2)

the constitutive equations

tαβ = λeρρδαβ + (μ + κ)eαβ + μeβα, mα3 = γκα3 (6.1.3)

and the geometrical equations

eαβ = uβ,α + εβαϕ3, κα3 = ϕ3,α (6.1.4)

on Σ1. We restrict our attention to the second boundary-value problem, so that we consider the boundary conditions

tβαnβ = t˜α, mα3nα = m˜3 on Γ (6.1.5)

We assume that fα, g3, t˜α, and m˜3 are functions of class C∞, and that the elastic potential W˜ is a positive definite quadratic form in the variables eαβ and κα3.