Consider the case of incompressible elastic materials. For such materials, there will be a constraint on all deformations such that the change in volume must be zero, thus implying (see Exercise 2.11) that e_kk = 0. First show that, under this constraint, Poisson’s ratio will become 12 and the bulk modulus and Lame´’s constant will become unbounded. Next show that the usual form of Hooke’s law σ_ij =λe_kkδ_ij + 2μe_ij will now contain an indeterminate term.

For such cases, Hooke’s law is commonly rewritten in the form σ_ij =–pδ_ij + 2μe_ij, where p is referred to as the hydrostatic pressure, which cannot be determined directly from the strain field but is normally found by solving the boundary-value problem. Finally justify that p =–σ_kk/3.

Data from exercise 2.11

A rectangular parallelepiped with original volume Vo is oriented such that its edges are parallel to the principal directions of strain as shown in the following figure. For small strains, show that the dilatation is given by:

Transcribed Image Text:

d = ekk change in volume original volume AV Vo