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In the modeling of biological tissues there are various reasons why the model needs to start from a prestressed configuration. For example, the biological tissue may exhibit residual stress or in situ stress. Or the geometry is taken from in vivo data and is subjected to loads in the reference configuration. In these cases, the reference configuration cannot be considered stress-free and this so-called prestress must somehow be accounted for. In large strain analysis, stresses are usually not additive and in addition, it must somehow be ensured that these prestresses are in equilibrium in the reference configuration. In general, this makes prestressing the reference configuration challenging in large strain analyses.
Progress can be made by assuming that the material is hyperelastic and that a prestrain can be found that defines the prestress via the hyperelastic constitutive formulation. Usually this prestrain is characterized by a second-order tensor, termed the prestrain gradient and here denoted by . The interpretation of this tensor is that its inverse, applied to a small neighborhood of a point in the reference configuration, would render this neighborhood stress-free. The total elastic response of the material is then defined via the elastic deformation gradient.
Here, is the deformation gradient that is the result of subsequent loading of the reference configuration. Finally, the Cauchy stress and spatial elasticity tangent are given by
Here, and are the same response functions from the “natural” material, i.e. the response of the material that starts from a stress-free configuration.
It must be noted that although is termed the prestrain gradient, in general it will not be the derivative of a deformation map. Consequently, a global stress-free reference configuration may not exist and the mapping of a neighborhood in the reference configuration to a stress-free state can at best be defined only locally.
In addition, it must also be recognized that the prestrain gradient is not unique. Due to the requirements of objectivity in the presence of material symmetries, the prestrain gradients and , where is any orthogonal tensor, must result in the same material response. This implies that the prestrain can only be dependent on the right stretch tensor . However, in general even the right-stretch tensor cannot be defined uniquely. For instance, an isotropic material would only depend on the eigenvalues of this tensor and thus any tensor of the form , where is any orthogonal tensor and is a diagonal tensor with the three eigenvalues of on its diagonal, would render the same material response. In the presence of material symmetries, the situation is similar although will only be part of the symmetry group.
Despite the fact that is not unique, it is far from arbitrary. The most important restriction is that the resulting prestress field must be in equilibrium with the prestressed reference configuration. In other words it must hold that
in the interior of the reference domain (in the absence of body forces) and that on the free boundaries of the domain
In a finite element simulation, when these conditions are not satisfied, the mesh will distort and a new reference state is obtained as well as an altered prestrain field. We will denote the deformation gradient between the original (incompatible) reference configuration and prestressed reference configuration by . When the mesh distorts the effectively applied prestrain field is also altered. In general, this is not desired and a correction to the prestrain field is necessary that either eliminates the distortion or finds a new reference geometry in which the desired prestrain field is supported.
The prestrain framework as currently implemented assumes that the total effective prestrain that is compatible with the possibly altered prestressed reference configuration is given by the following multiplicative decomposition.
Both the prestrain gradient and the distortion are in general unknown. The framework implements an iterative algorithm that updates and until an effective prestrain gradient is found that is compatible with the possibly distorted reference configuration. Without loss of generality the framework assumes that the altered prestrain gradient is itself given by the multiplicative decomposition
where is an initial guess for the prestrain gradient and is a correction factor. The algorithm starts by initializing to a user-defined value and to the identity. This initial value can be specified in a variety of ways and the code that generates this initial guess is called the prestrain generator. The framework provides several prestrain generators and users can easily add new ones. Then, a forward analysis is executed keeping fixed. Unless the initial guess was compatible with the reference configuration, the mesh will distort. This distortion will define the new value for which can be calculated directly from the nodal displacements in the usual manner. Next, the framework will update using a user-defined update rule. How is updated depends on the particular application and the framework expects the user do provide the desired update rule. The current implementation provides two particular update rules: one to eliminate the distortion and one to enforce a particular form of the prestrain gradient. See section ?? on how to use these update rules. In addition, users can easily implement new update rules. After this update, a new forward analysis is executed. This process is repeated until converges.