Samer Adeeb

Constitutive Laws: Frame Invariance

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Introduction

Objectivity or Frame Invariance is an importance concept in continuum mechanics. Simply put, the concept of frame invariance is the idea that constitutive laws describing the behaviour of a material (for example the relationship between stress and strain) should be independent of the motion of the observer. For example, Young’s modulus of a material should be the same to an observer whether the observer is standing or moving. In this section we will first introduce the idea of the motion of the observer. Then, frame invariant vectors and tensors will be defined. Following that, the various stress and strain measures will be investigated under the umbrella of frame invariance.

Motion of the Observer and Frame Invariance Tensors

Assuming an embedding of a three dimensional object in \mathbb{R}^3 as described in the description of motion section, we will assume that the observer in the spacial configuration is moving with a rigid body motion described by a rotation matrix Q(t) and a speed c(t) where t is time. Let B=\{e_1,e_2,e_3\} be an arbitrary fixed basis set in the spatial configuration and B'=\{e'_1,e'_2,e'_3\} be the spatial basis set that is associated with the motion of the observer. If x and x' are the position vectors with respect to B and B' respectively, then the relationship between them is described as:

    \[ x'(t)=Q(t)x(t)+c'(t) \]

In this case, Q_{ij}=e'_i\cdot e_j. It is important to note that Q is dependent on time through the vectors e'_i while the vectors e_i are fixed. The motion of the observer here can be viewed similar to the change of basis described earlier in which different basis sets are used to describe vector and tensor quantities. A vector or a tensor field is said to be frame invariant if it is unaffected by the motion (velocity or acceleration) of the observer except through the natural change of basis associated with Q(t). In other words, a vector field u:\mathbb{R}^3\rightarrow \mathbb{R}^3 is said to be frame invariant if under the motion of the observer we have:

    \[ u'=Qu \]

Similarly, a second order tensor T:\mathbb{R}^3\rightarrow \mathbb{M}^3 is frame invariant if under the motion of the observer we have:

    \[ T'=QTQ^T \]

Similarly, higher order tensors are frame invariant if under the motion of the observer their components change according to their rank as described in the higher order tensors section.

View Mathematica Code:

cc = {-Sqrt[5], 0}
e1 = {2/Sqrt[5], 1/Sqrt[5]}
e2 = {-1/Sqrt[5], 2/Sqrt[5]}
Q = {e1, e2}
x = {2, 1}
Q.x + cc