Simple mathematical models of energy dissipation in materials I am interested in simple mathematical models of energy dissipation due to impact or pressure on a material;
that is, how such occurrences usually represented (e.g. as stress, strain, or something else as a function of space and time in the material) and computed, preferably for simple models with analytic solutions.
For instance, an explicit analytic model of how it propagates in linearly elastic, transversely isotropic 3D materials (after some simple impulse of force) would be very helpful. 
I am mostly hoping for some pointers to books and/or literature, if possible.
 A: Generically speaking, you are looking for viscoelasticity or viscoplasticity depending on whether the material undergoes permanent deformation due to the stresses. The linked pages contain plenty of models to choose from depending on the behavior, but I will call out the viscoelastic models because I have used them previously.
For most of the models, some network of springs and dampers are created. The Maxwell model puts a spring and damper in series while the Kelvin-Voigt model puts them in parallel. The Maxwell model is good for melting polymers and metals while the Kelvin-Voigt is better for polymer and rubbers away from melting points. 
More complex networks can be composed by connecting Maxwell or KV sub-elements in series or parallel as needed, but such networks would need to be correlated to experimental data. 
For simple cases, these models are analytically tractable. For more complex cases, they can be treated numerically -- this is my experience with them. Additionally, they can be isotropic or anisotropic depending on how you connect a lattice of them together. There is considerable flexibility. 
For viscoplastic materials, the Johnson-Cook is the go-to standard. Unfortunately, it is based on empirical data and requires correlations to existing observed behavior. However, for many materials this is already done for you. The viscoplastic models are more complex though, and I would be surprised if they are analytically tractable -- I have only ever worked with them numerically. 
