Reference Work Entry

Encyclopedia of Thermal Stresses

pp 1243-1247

Ellipticity Condition and Acceleration Waves in Nonlinear Thermoelastic Solids

  • Victor A. EremeyevAffiliated withOtto–von–Guericke University MagdeburgSouth Scientific Center of RASci and South Federal University Email author 


Acceleration waves; Ellipticity; Nonlinear thermoelasticity​


By the term “ellipticity”, we mean the property of partial differential equations describing deformations of nonlinear elastic solids. Elliptic operator is a generalization of the well-known Laplace operator. The ellipticity is defined by the condition that the coefficients of the highest-order (second-order) derivatives constitute a positive definite quadratic form. In the nonlinear mechanics of solids, ellipticity plays a role of so-called constitutive restriction. Ellipticity is closely related with the condition of propagation of acceleration waves in nonlinear solids. Acceleration wave is a propagating nonmaterial surface across which the acceleration (but not the velocity) is discontinuous. Acceleration waves are similar, but not the same, to sound waves in prestressed solids.


Ellipticity of equilibrium equations of the nonlinear elasticity belongs to the class of so-call ...

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