Verification of Frame Indifference for Complicated Numerical Constitutive Models

The principle of material frame indifference requires spatial stresses to rotate with the material, whereas reference stresses must be insensitive to rotation. Testing of a classical uniaxial strain problem with superimposed rotation reveals that a very common approach to strong incremental objectivity taken in finite element codes to satisfy frame indifference (namely working in an approximate un-rotated frame) fails this simplistic test. A more complicated verification example is constructed based on the method of manufactured solutions (MMS) which involves the same character of loading at all points, providing a means to test any nonlinear-elastic arbitrarily anisotropic constitutive model. INTRODUCTION The principle of material frame indifference (PMFI) requires that if a deformed material is rotated, then the spatial tractions and stresses should rotate along with it whereas the reference stresses must be insensitive to the rotation. A review of material frame indifference is already presented in [1]. This concept is different from basis indifference. The difference between basis indifference and frame indifference is the initial configuration. Figure 1 represents a simple shear under basis rotation and superimposed rotation. Under basis rotation, both the initial configuration and the deformed configuration are rotated whereas under superimposed rotation, only the deformed configuration is rotated. The PMFI demands that the stress for deformation in Fig. 1.(b) should be same as Fig. 1.(a) except rotated appropriately. PMFI does not guarantee the accuracy of the material model. It is one of the physical principles tested to check material models for consistency under superimposed rotations and translations. Verification & Validation of development codes with complicated numerical constitutive models is important for testing the accuracy and robustness of these methods [3]. Verification is a process that is used to evaluate the correctness of the solution of the governing equations in the code. Validation is a quality control process of establishing evidence that the equations themselves provide an acceptable description of reality with respect to intended requirements, and is done by comparing with the experimental data. The MMS is an accepted standard of verification testing in the scientific community that has been used extensively in fluid mechanics [4], but is rarely demonstrated in solid mechanics because of the increased mathematical complexity. MMS is a process of determining the external body force required to achieve a pre-decided deformation analytically. Then the code is verified by running it with the computed body force and demonstrating that the pre-decided deformation is achieved. This paper will first focus on PMFI where a uniaxial strain problem with superimposed rotation is tested with and without a frame indifferent constitutive model. An analytical solution for the simplest possible constitutive model (linear elasticity) is sufficient to demonstrate that a common approximation used in implementations of strong incremental objectivity [6] results in failure to satisfy the PMFI. Subsequently a MMS approach will be used to construct a more complicated verification example of large deformation and large rotation of a thick vertical beam that will serve as a means to test arbitrarily nonlinear anisotropic elastic constitutive model for simultaneous basis and frame indifference. Figure 1: (a) Basis rotation (b) Superimposed rotation.