# Modeling of Networks of Thermoelastic Beams

• J. E. Lagnese
• Günter Leugering
• E. J. P. G. Schmidt
Part of the Systems & Control: Foundations & Applications book series (SCFA)

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

## Authors and Affiliations

• J. E. Lagnese
• 1
• Günter Leugering
• 2
• E. J. P. G. Schmidt
• 3
1. 1.Department of MathematicsGeorgetown UniversityUSA
2. 2.Fakultät für Mathematik und PhysikUniversity of BayreuthBayreuthGermany
3. 3.Dept. of Mathematics and StatisticsMcGill UniversityMontrealCanada