Page 330 - Introduction to Continuum Mechanics

314 The Elastic Solid

Thus Gg is the shear modulus in the plane of jcjj^ and G$ is the shear modulus in the plane
ofjci^. Note also that the shear stresses in the jei*2 plane produce shear strain in the xj^
plane and vice versa with p^ representing the coupling coefficients.
Finally if only 723 = 732 are nonzero,

We see that G4 is the shear modulus in the plane of X2*3 plane, and the shear stresses in this
plane produces normal strains in the three coordinate directions, with ijij representing normal
stress-shear stress coupling.
Obviously, due to the positive definiteness of the compliance matrix, all the Young's moduli
and the shear moduli are positive. Other restrictions regarding the engineering constants can
be obtained in the same way as in the previous section.

Part C Constitutive Equation for Isotropic Elastic Solid Under Large Deformation

5.32 Change of Frame

In classical mechanics, an observer is defined as a rigid body with a clock. In the theory of
continuum mechanics, an observer is often referred to as a frame. One then speaks of "a change
of frame" to mean the transformation between the pair {x,t} in one frame to the pair {xV* }
of a different frame, where x is the position vector of a material point as observed by the
un-starred frame and x * is that observed by the starred frame and t and t* are times in the two
frames. Since the two frames are rigid bodies, the most general change of frame is given by
[See Section 3.61

where c (f) represents the relative displacement of the base point x^,, Q(t) is a time-dependent
orthogonal tensor, representing a rotation and possibly reflection also (the reflection is
included to allow for the observers to use different handed coordinate systems), a is a constant.
It is important to note that a change of frame is different from a change of coordinate system.
Each frame can perform any number of coordinate transformations within itself, whereas a
transformation between two frames is given by Eqs. (5.32).