Math 2.0

Mechanics of deformable solids Nonlinear elasticity Problem

Ericksen's problem of universal deformations

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Problem

Fix a class of homogeneous isotropic hyperelastic materials. A deformation of a body is universal for the class if every material in the class can be held in this deformation in equilibrium. The equilibrium must hold without body forces, with surface tractions alone. Find all universal deformations, separately for the class of compressible materials and for the class of incompressible materials, where only deformations with det⁡F=1\det F = 1 are allowed.

Remarks

Ericksen showed in 1955 that for compressible materials the universal deformations are exactly the homogeneous ones, those with constant deformation gradient. For incompressible materials, the homogeneous isochoric deformations are universal. Five families of inhomogeneous universal deformations are also known, including the bending of a block into a sector of a cylindrical tube and the inflation of spherical shells. Whether this list is complete for incompressible materials is still open. Universal deformations are important because they allow tests that work without knowing the strain energy function in advance, and they are the basis of the semi-inverse solutions of finite elasticity.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Hyperelastic_material
About this page
TagVH55PZZYKS
URIm2://VH55PZZYKS
Kindproblem
WritingW0 stub
MSC74B20, 74G05
StatusThis copy of the site has no build data. Run m2 status VH55PZZYKS.
AboutHyperelastic material
UsesHyperelastic material, Deformation
Changed3 November 2026

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