Small and large deformation of thick and thin-film multi-layers: Effects of layer geometry, plasticity and compositional gradients

Extensions of the Stoney formula for substrate curvature to configurations with thin substrates or large deformations

“…The formula (1) follows from an analysis of a model of the film-substrate system which is based on several assumptions, and the point of the present discussion is to examine the range of applicability of some of these assumptions in light of current practice [7,8]. The main assumptions are: ( i ) both the film and substrate thicknesses are small compared to the lateral dimensions; ( i i ) the film thickness is much less than the substrate thickness; (iii) the substrate material is homogeneous, isotropic and linearly elastic, and the film material is isotropic; (iv) edge effects near the periphery of the substrate are inconsequential and all physical quantities are invariant under change in position parallel to the interface; ( u ) all stress components in the thickness direction vanish throughout the material; (vi) the strains and rotations are infinitesimally small.…”

Applicability range of Stoney’s formula and modified formulas for a film/substrate bilayer

“…Stoney's formula has been modified many times for the purpose of evaluating bilayer and multilayer structures with arbitrary layer thickness ratios. [2][3][4][5][6][7][8][9][10][11][12][13] Klein made comments on the accuracy of some modified Stoney formulas mentioned above. 14 The assumption of a uniform curvature is explicitly or implicitly used in the context of bilayer or multilayer structures modeled as either beams or plates.…”

“…14 The assumption of a uniform curvature is explicitly or implicitly used in the context of bilayer or multilayer structures modeled as either beams or plates. [2][3][4][5][6][7][8][9][10][11][12][13]15,16 Among these studies, Freund et al, in particular, presented an excellent analysis of the film/substrate bilayer and multilayer deflection due to lattice mismatch. [7][8][9] Freund et al also noted that the assumption of a uniform curvature is not valid for large nonlinear deformations.…”

Dielectric Elastomer Based “Grippers” for Soft Robotics

“…(In the special case of where the sheets are circular, the actuated shape is in the form of a hemispherical cap). [18,19] The response to successively higher voltages is shown in the finite element simulations of a square sheet in Figure 2 for two cases, one where the electrodes extend all the way to the edges (Figure 2d) and the other where there is a border around the edges that is not electrode (Figure 2e-g). In both cases, the deformation exhibits the same four-fold symmetry as the square geometry of the sheets and with "dog-ear" deformation at the corners.…”