### Tuesday, 12 October 2004 - 2:40 PM

### This presentation is part of : Rajagopal Symposium

### Novel Kinematics Framework for High Strain Materials with 2 Fiber Families

**John C. Criscione**, Department of Biomedical Engineering, Department of Biomedical Engineering, Tecas A&M University, College Station, TX 77843-3120
Materials with two families of fibers are an important class that is used to model the behavior of anisotropic soft tissues (e.g. intervertebral disks) and high strain composites (e.g. fiber reinforced tubes). Not surprising that there are existing theoretical frameworks for defining the constitutive behavior of materials with two fiber families that are orthogonal and/or mechanically equivalent or neither. Yet, the traditional approach utilizes invariants with response functions that have a high covariance ratio. We define the covariance ratio of two response terms as the absolute value of their inner product divided by their magnitudes. Such a ratio has a range [0,1]. The covariance ratio is null iff the response terms are orthogonal and is 1 iff the response terms are collinear. The covariance ratio is important because the system of equations to calculate the response functions from data has a condition number that depends inversely on 1 minus the square of the covariance ratio. When the covariance ratio is high (i.e., nearly unity as with the traditional invariant approaches) then the condition number is so large that it is experimentally ill-conceived to determine response functions because error obscures any trends. Using the bisectors of the fibers as the material directions, we developed a novel set of six strain parameters with response terms that are mostly orthogonal (14 of the 15 possible inner products are null). In the small strain range the response terms are entirely orthogonal.

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