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Chapter 6 Strain Energy and Related Principles

T n. F. Work Done by Surface and Body Forces on Elastic Solids Is Stored Inside the Body in the Form of Strain Energy. Chapter 6 Strain Energy and Related Principles. U. . y. . dy. . dz. dx.  x. u. x. U = Area Under Curve. z. e. e x. Uniaxial Extensional Deformation.

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Chapter 6 Strain Energy and Related Principles

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  1. Tn F Work Done by Surface and Body Forces on Elastic Solids Is Stored Inside the Body in the Form of Strain Energy Chapter 6 Strain Energy and Related Principles U

  2. y  dy  dz dx x u x U = Area Under Curve z e ex Uniaxial Extensional Deformation . . . Strain Energy Density

  3. y yx xy dy  dx x xy U = Area Under Curve  xy Shear Deformation

  4. General Deformation Case In Terms of Strain In Terms of Stress Note Strain Energy Is Positive Definite Quadratic Form

  5. Example Problem

  6. Derivative Operations on Strain Energy For the Uniaxial Deformation Case: For the General Deformation Case: ThereforeCij = Cji, and thus there are only 21 independent elastic constants for general anisotropic elastic materials

  7. Decomposition of Strain Energy Strain Energy May Be Decomposed into Two Parts Associated With Volumetric DeformationUv , and Distortional Deformation, Ud Failure Theories of Solids Incorporate Strain Energy of Distortion by Proposing That Material Failure or Yielding Will Initiate When Ud Reaches a Critical Value

  8. Bounds on Elastic Constants Simple Tension Pure Shear Hydrostatic Compression

  9. Related Integral Theorems Clapeyron’s Theorem The strain energy of an elastic solid in static equilibrium is equal to one-half the work done by the external body forces Fi and surface tractions Betti/Rayleigh Reciprocal TheoremIf an elastic body is subject to two body and surface force systems, then the work done by the first system of forces {T(1), F(1)} acting through the displacements u(2) of the second system is equal to the work done by the second system of forces {T(2), F(2)} acting through the displacements u(1) of the first system Integral Formulation of Elasticity - Somigliana’s IdentityRepresents an integral statement of the elasticity problem. Result is used in development of boundary integral equation methods (BIE) in elasticity, and leads to computational technique of boundary element methods (BEM) Gij is the displacement Green’s function to the elasticity equations and

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