= Nonnegative isotropic elastic strain energy
{title2=$\mu\ge0,\quad3\lambda+2\mu\ge0$}
In three-dimensional isotropic <linear elasticity>, decompose the symmetric strain as $e=pI+d$ with $p=\operatorname{tr}(e)/3$ and $\operatorname{tr}d=0$. The stored energy is $E=\mu\sum_{ij}d_{ij}^2+\tfrac32(3\lambda+2\mu)p^2$. Its nonnegativity for every strain is equivalent to the displayed inequalities: the traceless and purely volumetric test strains prove necessity, and the sum of squares proves sufficiency.
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