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Hello,
I am trying and failing to derive that the shear modulus ##G## is equal to the Lame parameter ##\mu##. I start with the linear, symmetric, isotropic stress-strain formula: $$\sigma = \lambda \mathrm{tr}(\epsilon) \mathrm{I} + 2\mu \epsilon$$ I then substitute a simple (symmetric) shear strain: $$\epsilon = \epsilon_{xy} (\hat{x} \otimes \hat{y} + \hat{y} \otimes \hat{x} )$$ But then I end up with $$\sigma_{xy} = 2\mu \epsilon_{xy}$$ or equivalently $$G = 2\mu$$ What did I goof up?
Thanks!
I am trying and failing to derive that the shear modulus ##G## is equal to the Lame parameter ##\mu##. I start with the linear, symmetric, isotropic stress-strain formula: $$\sigma = \lambda \mathrm{tr}(\epsilon) \mathrm{I} + 2\mu \epsilon$$ I then substitute a simple (symmetric) shear strain: $$\epsilon = \epsilon_{xy} (\hat{x} \otimes \hat{y} + \hat{y} \otimes \hat{x} )$$ But then I end up with $$\sigma_{xy} = 2\mu \epsilon_{xy}$$ or equivalently $$G = 2\mu$$ What did I goof up?
Thanks!