Modeling of Networks of Thermoelastic Beams

Abstract

We consider the deformation of a thin beam of length ℓ with a given initial curvature and torsion, and with variable doubly symmetric cross section. To be precise, the undeformed beam, in its initial reference configuration, occupies the region

$$\Omega :\left\{ {r: = {{r}_{0}}\left( {{{x}_{1}}} \right) + {{x}_{2}}{{e}_{2}}\left( {{{x}_{1}}} \right) + {{x}_{3}}\left( {{{x}_{1}}} \right)\left. {{{e}_{3}}} \right|{\text{ }}{{x}_{1}} \in \left[ {0,\ell } \right],\left( {{{x}_{2}},{{x}_{3}}} \right): = {{x}_{2}}{{e}_{2}}\left( {{{x}_{1}}} \right) + {{x}_{3}}{{e}_{3}}\left( {{{x}_{1}}} \right) \in A\left( {{{x}_{1}}} \right)} \right\},$$

where r₀: [0,ℓ] ↦ ℝ₃ is a smooth function representing the centerline of the beam at rest, and the orthonormal triads e₁(•), e₂(•), e₃(•) are chosen to be smooth functions of x₁ such that e₁ is the direction of the tangent of the centerline with respect to the variable x₁, i.e. $\frac{d}{{d{{x}_{1}}}}{{r}_{0}}\left( {{{x}_{1}}} \right) = {{e}_{1}}\left( {{{x}_{1}}} \right)$, and such that e₂(x₁), e₃(x₁) span the orthogonal cross section at x₁. The meaning of the variables x _i are as follows: x₁ denotes the length along the undeformed centerline, x₂ and x₃ denote the 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 vertical to the tangent of r₀. The cross section at x₁ is defined as

$$A\left( {{{x}_{1}}} \right) = \left\{ {{{x}_{2}}{{e}_{2}} + \left. {{{x}_{3}}{{e}_{3}}} \right|{{x}_{1}}{{e}_{1}} + {{x}_{2}}{{e}_{2}} + {{x}_{3}}{{e}_{3}} \in \Omega } \right\}.$$