Theory Manual Version 3.4
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Subsection 5.2.10: Large Poisson's Ratio Ligament Up Section 5.2: Compressible Materials Subsection 5.2.12: Cell Growth

### 5.2.11 Porous Neo-Hookean Material

Consider a porous neo-Hookean material with referential porosity . The pores are compressible but the skeleton is intrinsically incompressible. Thus, upon pore closure, the material behavior needs to switch from compressible to incompressible.
In the current configuration, the porosity is given by We may define a new variable, which represents the pore volume ratio. It is equal to when and is equal to when (or ). Now, and Pore closure occurs when , which corresponds to and .
Let us also define a modified deformation gradient, such that . Let the corresponding modified right Cauchy-Green tensor be given by so that
The constitutive relation for the strain energy density of the compressible porous neo-Hookean material may be given by where . This relation shows that the material develops an infinite strain energy density as approaches zero. From this expression, the 2nd Piola-Kirchhoff stress is given by When we can verify that . The corresponding Cauchy stress is where is the left Cauchy-Green tensor.
The material elasticity tensor is given by where and Then, the spatial elasticity tensor may be evaluated as
In the limit of infinitesimal strains and rotations, when and , we find that and Thus, by comparison to a standard neo-Hookean material, this porous neo-Hookean material has an effective Young's modulus equal to and an effective Poisson's ratio equal to The two material properties that need to be provided are and the referential porosity (or referential solid volume fraction ). Poisson's ratio in the limit of infinitesimal strains is dictated by the porosity according to the above formula. In particular, a highly porous material () has an effective (infinitesimal strain) Poisson ratio that approaches zero () and . A low porosity material () has and , which is the expected behavior of an incompressible neo-Hookean solid. Note that setting would not produce good numerical behavior, since the Cauchy stress in an incompressible material would need to be supplemented by a pressure term (a Lagrange multiplier that enforces the incompressibility constraint). Nevertheless, this compressible porous neo-Hookean material behaves well even for values of as low as .
Subsection 5.2.10: Large Poisson's Ratio Ligament Up Section 5.2: Compressible Materials Subsection 5.2.12: Cell Growth