ABSTRACT
Consider the deformation of a thin, initially curved beam of length and constant cross-section of area A, which, in its undeformed reference configuration, occupies the region
Ω = {r =: r0(x1)+ x2e2(x1)+ x3e3(x1)| x1 ∈ [0, ], (x2, x3) := x2e2(x1)+ x3e3(x1) ∈ A}
“73648_C068” #2
where r0 : [0, ] → R3 is a smooth function representing the centerline, or the reference line, of the beam at rest. The orthonormal triads e1(·), e2(·), e3(·) are chosen as smooth functions of x1 so that e1 is the direction of the tangent vector to the centerline, i.e., e1(x1) = (dr0/dx1)(x1), and e2(x1), e3(x1) span the orthogonal cross section at x1. The meanings of the variables xi are as follows: x1 denotes arc length along the undeformed centerline, and x2 and x3 denote lengths along lines orthogonal to the reference line. The set Ω can then be viewed as obtained by translating the reference curve r0(x1) to the position x2e2 + x3e3 within the cross-section perpendicular to the tangent of r0.
