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Let me describe the physical scenario first. Suppose that one grabs the boundary of a flat piece of material, stretching it in some direction that lies in the same plane with the material. And suppose this piece of material has varying, discontinuous "elastic coefficients".

Let's assume the original material is a unit square, and after deformation it is parametrised by $$u(x,y):[0,1]^2\to \mathbb{R}^2,$$ Assume that $S\subset[0,1]^2$ is a proper subset where $u$ can only be deformed via rigid motion, while the remaining part can be distorted. The effect of stretch is described as certain boundary condition.

What equations describe the deformation of this material?

Syl.Qiu
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  • Not sure on the real physics but in the case of uniform elasticity this sounds like the harmonic mapping problem, i.e. the minimization problem for the energy $\int |Du|^2$. For varying elasticity you'd study an energy of the form $\int e(x,Du)$, which would give you some nonlinear elliptic equation. – Anthony Carapetis Jun 29 '17 at 06:31
  • @AnthonyCarapetis Thanks for the comment! Can you give more details, or reference, about the latter energy you mentioned? – Syl.Qiu Jun 29 '17 at 06:55
  • It's a bit more complicated because even linear elastic isotropic materials have two Lamé parameters. Furthermore, continuum mechanics of solids uses material coordinates instead of spatial coordinates, that is, Lagrangian instead of Eulerian specs. Thus, the strain-displacement relations follow a concept similar to that of a metric tensor and turn out to be nonlinear, although they can be linearized under the assumption of small deformations and small strains. – ccorn Jun 29 '17 at 08:22
  • In tensor notation, the theory looks very much like that of the 3D continuum, except for the choice of physical dimensions for stress resultants and adaptations of the constitutive equation. The reason is that in 3D there is no out-of-the-continuum direction, and in flat elastic sheets there may be transverse strain, but no transverse stress, thus no energy flow in that direction. – ccorn Jun 29 '17 at 08:31

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