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2.8.6 Prescribed Pre-Stretch
An alternative approach to multigenerational growth is to prescribe as a user-defined function. For isotropic stretching we may define where is the stretch ratio. A value greater than unity causes swelling whereas a value less than unity causes contraction; produces , which recovers the simple solid mixture described in Section 2.8.4↑. In FEBio a load curve may be associated with to ramp up the prescribed deposition stretch.
For orthotropic stretching we define mutually orthogonal symmetry planes with unit normals ( and ). Then, where is the stretch ratio for the prescribed stretch along ; in general, , though this model can be specialized to transversely isotropic stretch by setting two of these stretch ratios equal to each other.
At lower symmetries the constitutive model for must necessarily combine stretch and rotation. For monoclinic materials, deformations may be prescribed along three unit vectors that satisfy , , and , such that defines the single plane of symmetry. In this case, where is the stretch along and . Under general conditions (i.e., when and ), the polar decomposition theorem shows that this deformation gradient is not a pure stretch, as it also involves a rotation.
The deformation gradient for expansion of a triclinic material may be similarly constructed by finding such that (no sum) for non-orthogonal and non-collinear unit vectors , where and is the cosine of the angle between and .
This type of elastic solid with prescribed pre-stretch is modeled in FEBio using the materials “prestretch elastic” and “prestretch uncoupled elastic”. In the uncoupled version each solid constituent in the mixture has a deformation gradient which produces nearly isochoric responses ( ), whereas no constraint is placed on the volumetric strain of the pre-stretch .