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Note that both the wavelength and one of the Lamé coefficients are denoted by (formula presented) To avoid ambiguity we write (formula presented) (formula presented) or (formula presented) for the wavelength where necessary
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Note that both the wavelength and one of the Lamé coefficients are denoted by (formula presented) To avoid ambiguity we write (formula presented) (formula presented) or (formula presented) for the wavelength where necessary.
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Note that in 2D, the upper bound for the Poisson ratio is 1 and not 1/2 as in 3D
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It is instructive to consider the simple one-dimensional analogy, i.e., a long string of connected springs with force constants (formula presented) to appreciate the approximation made in the above equations. We recall that the effective macroscopic Young modulus is (formula presented) rather than (formula presented) This is due to the fact that the tension along the chain is constant but not the displacement field. Hence, Eqs. (9) are in general only good approximations if the dispersion of the effective spring constants is small and an external macroscopic strain is locally transmitted in an affine way. (formula presented) is then indeed given by the mean coupling constant with a negative quadratic dispersion correction in leading order.)
-
It is instructive to consider the simple one-dimensional analogy, i.e., a long string of connected springs with force constants (formula presented) to appreciate the approximation made in the above equations. We recall that the effective macroscopic Young modulus is (formula presented) rather than (formula presented) This is due to the fact that the tension along the chain is constant but not the displacement field. Hence, Eqs. (9) are in general only good approximations if the dispersion of the effective spring constants is small and an external macroscopic strain is locally transmitted in an affine way. (formula presented) is then indeed given by the mean coupling constant with a negative quadratic dispersion correction in leading order.)
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