Plates with Incompatible Prestrain

Abstract

We study effective elastic behavior of the incompatibly prestrained thin plates, where the prestrain is independent of thickness and uniform through the plate’s thickness h. We model such plates as three-dimensional elastic bodies with a prescribed pointwise stress-free state characterized by a Riemannian metric G, and seek the limiting behavior as ${h \to 0}$. We first establish that when the energy per volume scales as the second power of h, the resulting ${\Gamma}$ -limit is a Kirchhoff-type bending theory. We then show the somewhat surprising result that there exist non-immersible metrics G for whom the infimum energy (per volume) scales smaller than h ². This implies that the minimizing sequence of deformations carries nontrivial residual three-dimensional energy but it has zero bending energy as seen from the limit Kirchhoff theory perspective. Another implication is that other asymptotic scenarios are valid in appropriate smaller scaling regimes of energy. We characterize the metrics G with the above property, showing that the zero bending energy in the Kirchhoff limit occurs if and only if the Riemann curvatures R ₁₂₁₃, R ₁₂₂₃ and R ₁₂₁₂ of G vanish identically. We illustrate our findings with examples; of particular interest is an example where ${G_{2 \times 2}}$, the two-dimensional restriction of G, is flat but the plate still exhibits the energy scaling of the Föppl–von Kármán type. Finally, we apply these results to a model of nematic glass, including a characterization of the condition when the metric is immersible, for ${G = Id_{3} + \gamma n \otimes n}$ given in terms of the inhomogeneous unit director field distribution ${ n \in \mathbb{R}^3}$.