Differential Form for the Constitutive Stress-Strain Relationship

3 DIFFERENTIAL FORM FOR THE CONSTITUTIVE STRESS-STRAIN RELATIONSHIP 

The integral form is not the only possible form in which stress-strain relationships can be written. The constitutive equation for an isotropic material whose response is sensitive to the derivatives of stress and strain can be written as (5) 

Though the preceding equations represent a convenient way to express the relationship between stress and strain, it is necessary that they be consistent with the integral formulation of Section 16.2 (6). Consequently, taking the Laplace transform of both formulations and identifying them, we obtain 

Appropriate combinations of the coefficients of Eq. (16.20) can reproduce determined idealized behavior of viscoelastic materials such as those corresponding to the Maxwell and Kelvin-Voigt models. Thus, for the Maxwell model in shear, 

Additional terms in the general equation, Eq. (16.18), give a better account of the actual behavior of the material. However, the resulting equations are not easily solved unless Laplace transforms are used. In this way, the expression obtained is 

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Differential Form for the Constitutive Stress-Strain Relationship 701... 

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