Abstract
Materials respond to external load by deforming and straining, and by developing stresses. The internal stresses corresponding to a given set of strains depend on the constitution of the material itself. For this reason, the rules that permit calculation of internal stresses from known strains, or vice versa, are called constitutive laws or constitutive equations. There are two equivalent ways to describe the mathematical relationships between stresses and strains for viscoelastic materials. One form uses integrals to define the constitutive relations, while the other relates stresses and strains by means of differential equations. Starting from Boltzmann’s superposition principle, this chapter develops the integral form of the one-dimensional constitutive equations for linearly viscoelastic materials. This is followed by a discussion of the principle of fading memory, which helps to define the acceptable analytical forms of the material property functions. It is then shown that the closed-cycle condition (i.e., that the steady-state response of a non-aging viscoelastic material to a periodic excitation be periodic) requires that the material property functions depend only on the difference of their arguments. The chapter also examines the relationships between the relaxation modulus and creep compliance functions in the physical time domain as well as in Laplace-transformed space. Various alternative forms of the integral constitutive equations often encountered in practice are discussed as well.
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Notes
- 1.
On this assumption, for instance, M(t) and C(t) will be used for M(t,T) and C(t,T), respectively.
- 2.
The terms “load” and “loading” are used in their broader sense to include tractions, or stresses, as well as displacements, or strains. The exact meaning should be clear from the context in which the term is used.
- 3.
The notation x + is used to signify a value of x that is just larger than x. Similarly, x − means a value of x just less than x.
- 4.
The mean value theorem of integral calculus states that \( \int_{a}^{b} {f(x)dx} = (b - a)f[a + \lambda (b - a)];\quad 0 < \lambda < 1 . \)
- 5.
Here, use is made of the total derivative: \( \frac{d}{dp}f\left( {x,y} \right) = \frac{\partial }{\partial x}f\left( {x,y} \right)\frac{dx}{dp} + \frac{\partial }{\partial y}f\left( {x,y} \right)\frac{dy}{dp}. \)
- 6.
Material property functions which depend on the difference between current and loading time are known as “difference” kernels.
- 7.
The 3 × 3 stress and strain matrices—indeed any square matrix of any order—may be split into a spherical and a deviatoric part. The spherical part is a diagonal matrix with each of its three non-zero entries equal to the average of the diagonal elements of the original matrix. Therefore, any one of its non-zero entries may be used to represent it. The deviatoric part of the matrix is, by definition, the matrix that is left over from such decomposition. This decomposition is discussed fully in Appendix B.
- 8.
The s-multiplied Laplace transform of a function is simply called the Carson transform of the function.
References
A.D. Drosdov, Finite elasticity and viscoelasticity; a course in the nonlinear mechanics of solids, World Scientific, pp. 267–271, 279–283 (1996)
A.C. Pipkin, Lectures on viscoelasticity theory (Springer, New York, 1986), pp. 14–16
H.-P. Hsu, Fourier analysis (Simon and Schuster, New York, 1970) pp. 88–92
R.M. Christensen, Theory of viscoelasticity, 2nd edn. (Dover, New York, 2003), pp. 3–9
L.E. Malvern, Introduction to the mechanics of a continuous medium (Prentice-Hall, Englewood, 1963) pp. 278–290, 282–285
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Gutierrez-Lemini, D. (2014). Constitutive Equations in Hereditary Integral Form. In: Engineering Viscoelasticity. Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8139-3_2
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DOI: https://doi.org/10.1007/978-1-4614-8139-3_2
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