Elastic Potential or Strain Energy Function

16 Green defined an elastic material as one for which a strain-energy function exists. Such a material is called Green-elastic or hyperelastic if there exists an elastic potential function W or strain energy function, a scalar function of one of the strain or deformation tensors, whose derivative with respect to a strain component determines the corresponding stress component.

17 For the fully recoverable case of isothermal deformation with reversible heat conduction we have

hence W = p0^ is an elastic potential function for this case, while W = p0u is the potential for adiabatic isentropic case (s = constant).

is Hyperelasticity ignores thermal effects and assumes that the elastic potential function always exists, it is a function of the strains alone and is purely mechanical

and W(E) is the strain energy per unit undeformed volume. If the displacement gradients are small compared to unity, then we obtain

which is written in terms of Cauchy stress Tij and small strain Eij.

19 We assume that the elastic potential is represented by a power series expansion in the small-strain components.

'W c0 + cij Eij + 2 cijkmEij Ekm + 3 cijkmnpEijEkmEnp + ''' (7.21)

where c0 is a constant and cij, cijkm, cijkmnp denote tensorial properties required to maintain the invariant property of W. Physically, the second term represents the energy due to residual stresses, the third one refers to the strain energy which corresponds to linear elastic deformation, and the fourth one indicates nonlinear behavior.

20 Neglecting terms higher than the second degree in the series expansion, then W is quadratic in terms of the strains

W = co + C1E11 + C2E22 + C3E33 + 2c4 E23 + 2c5 E31 +2ce E12

+ 2 c1111E11 + c1122E11E22 + c1133E11E33 + 2c1123E11E23 + 2c1131E11E31 + 2c1112E11E12 + 2 c2222E22 + c2233E22E33 + 2c2223E22E23 + 2c2231E22E31 + 2c2212E22E12 + 2 c3333 E33 + 2c3323E33E23 + 2c3331E33E31 + 2c3312E33E12 +2c2323E23 + 4c2331E23E31 + 4c2312E23E12 + 2C3131 E31 + 4C31 12E31E12 +2c1212E22

we require that W vanish in the unstrained state, thus c0 = 0.

21 We next apply Eq. 7.20 to the quadratic expression of W and obtain for instance

T12 = TTF;— = 2C6 + cnnEn + cnnEn + c33nE33 + cnnEn + c1223E23 + c^31E31 (7.23) E12

if the stress must also be zero in the unstrained state, then c6 = 0, and similarly all the coefficients in the first row of the quadratic expansion of W. Thus the elastic potential function is a homogeneous quadratic function of the strains and we obtain Hooke's law

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