A.4 Hydrostatic Compression

In many cases, it is preferable to interrogate the pressure-dependent response of various models independently of any deviatoric deformations. To consider such purely volumetric loadings, hydrostatic (almost always compression) problems are invoked. Such loadings are often also referred to as uniform dilation as the volumetric change is the same in all three directions. Specifically, in these cases a purely volumetric response is investigated by applying a deformation of the form \(u_i=\lambda(t)\). In a finite element problem, such a deformation field is reproduced by applying the displacement components onto the corresponding edges. With such applied displacement fields, the resulting logarithmic strain tensor is simply,

\[\varepsilon_{ij}=\ln\left[1+\lambda\left(t\right)\right]\delta_{ij},\]

and the corresponding (elastic) stress field is simply \(\sigma_{ij}=-p\delta_{ij}\) where,

\[p=-3K\ln\left(1+\lambda\right).\]

Note, in the preceding relation \(p\) is defined as positive in compression.