Some problems of viscoelastic-stress analysis

There are many stress-analysis problems involving viscoelastic materials that are of a statically determinate class, i.e., the stresses in the body depend only on the applied forces and moments and not specifically on the elastic properties of the body. Such problems can be solved by invoking the correspondence principle. Then, the time and temperature dependences of the strains and flexures in the body can be obtained through the time temperature-shift properties of the viscoelastic polymer. 

The correspondence principle states that for problems of a statically determinate nature involving bodies of viscoelastic materials subjected to boundary forces and moments, which are applied initially and then held constant, the distribution of stresses in the body can be obtained from corresponding linear elastic solutions for the same body subjected to the same sets of boundary forces and moments. This is because the equations of equilibrium and compatibility that are satisfied by the linear elastic solution subject to the same force and moment boundary conditions of the viscoelastic body will also be satisfied by the linear viscoelastic body. Then the displacement field and the strains derivable from the stresses in the linear elastic body would correspond to the velocity field and strain rates in the linear viscoelastic body derivable from the same stresses. The actual displacements and strains in the linear viscoelastic body at any given time after the application of the forces and moments can then be obtained through the use of the shift properties of the relaxation moduli of the viscoelastic body. Below we furnish a simple example. 

Consider a simply supported slender beam of PMMA of uniform cross-sectional dimensions, of width b, height h, and length L, loaded centrally with a dead load P. Of interest are the stress distributions in the beam and the time-dependent sag, vo(t), of its center at, say, 50 °C. It is noteworthy that this simple and trivial-appearing example actually is a useful approximation for a large number of applications where beam theory provides insight. 

The required solution of the problem, which is discussed in many elementary books such as Timoshenko (1930), gives, by geometry and the observation that, when the beam bends, plane sections remain plane along the beam (a statement of satisfaction of compatibility), the following distribution of axial strain s x across the thickness direction y 

50) E is Young s modulus and the stress Gxx must satisfy equilibrium, i.e., it changes along the beam only through the change in curvature along the beam. Since there are no apphed axial forces in the x direction, at any point x along the beam the distribution in the y direction needs to satisfy only a moment boundary condition, i.e.. 

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