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When we change our coordinate axes, the strain tensor

Chapter 16, Problem 16.29

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QUESTION:

When we change our coordinate axes, the strain tensor changes in accordance with Equation (15.132), which we can rewrite as ER = RER, where R is the (3 x 3) orthogonal rotation matrix. Use the property (15.129) of orthogonal matrices to show that tr ER = tr E; that is, the trace of any tensor is rotationally invariant. Use this result to show that the decomposition E = el + E' is rotationally invariant, in the sense described below Equation (16.92).

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QUESTION:

When we change our coordinate axes, the strain tensor changes in accordance with Equation (15.132), which we can rewrite as ER = RER, where R is the (3 x 3) orthogonal rotation matrix. Use the property (15.129) of orthogonal matrices to show that tr ER = tr E; that is, the trace of any tensor is rotationally invariant. Use this result to show that the decomposition E = el + E' is rotationally invariant, in the sense described below Equation (16.92).

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