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Relativistic and nonrelativistic elastodynamics with small shear strains. (English) Zbl 0930.74008

Summary: We present a new variational formulation for relativistic dynamics of isotropic hyperelastic solids. We introduce the shear strain tensor, study the geometry of characteristics in the cotangent bundle for the relativistic equations under the assumption of small shear strains, and obtain a result on the stability of the double characteristic manifold. Then we focus on the nonrelativistic limit of the above formulation and compare it to the classical formulation of elastodynamics via displacements. We obtain a global existence result for small-amplitude elastic waves in materials under a constant isotropic deformation, and a result on the formation of singularities for large data.

MSC:

74B20 Nonlinear elasticity
83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
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References:

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