Folding Patterns in Compressed Thin-film Diaphragms

In recent years, a large number of novel applications have been proposed for relatively large,  microfabricated thin-film diaphragms (also called membranes). Although these applications have elicited a considerable interest in thin-film diaphragms, a thorough understanding of many aspects of the mechanical behavior of thin-film diaphragms remains unavailable.

Apply a compressive strain parallel to the surface of a diaphragm: driven by the attendant in-plane stresses, the diaphragm may buckle and develop folds. Our aim is to understand the resulting folding patterns. In particular, we would like to interpret them as minimizers of the elastic energy of the diaphragm.

Diaphragms on Substrates: Notation

Consider a thin film of constant thickness h bonded to the planar surface of a thick substrate. The substrate occupies the half-space x₃ <0. Suppose that the film is not bonded to the substrate over a certain portion of the surface of the substrate (denoted by a Greek letter omega in the figure): then, the part of the film which is not bonded is called a thin-film diaphragm. We prepared  diaphragms by gluing  paper sheets and polymeric films (h=0.01 to 0.1 mm) onto substrates (thickness~30 mm) of the shape  shown in the figure, made of a high-density Styrofoam. Then we compressed the substrates in two perpendicular directions, x ₁ and x₂, using screw-driven steel plates. When the applied strain components are equal to each other the strain is isotropic ; otherwise it is anisotropic.

The Isotropic Case: Experiments

The figure shows the foldings of two equally shaped diaphragms subject to isotropic strains. The photographs are top views of the folded diaphragms. The first is a paper diaphragm of d=15 cm and d/h=950. The second is a SiC diaphragm of d=200 µm and d/h=140 (this photo was published by Argon et al., J. Mat. Sci. 24, 1989). The folding pattern is the same in both diaphragms; the number of folds and the spatial arrangement of the folds depend exclusively on the shape of the diaphragm, being independent of d/h and of the strain.

The Anisotropic Case: Experiments

A completely different situation obtains when the  strains are anisotropic. Then, the folds are all parallel to each other and perpendicular to the direction of the higher compressive strain (vertical in the figure), regardless of the shape of the diaphragm. The number of folds depends on d/h, however, and also on the strain.

One-Dimensional Diaphragms: Analysis

For the analysis of these results we model the  film as a constrained Karman plate. A constrained Karman plate is a plate governed by the Karman theory of plates, except that the in-plane displacements are constrained to remain null. We start with a one-dimensional diaphragm. It is a straightforward exercise to show that in such a diaphragm the membraneous energy density is minimized when the absolute value of the slope of the diaphragm equals a certain characteristic slope, k, where k is the square root of the strain multiplied by a numerical constant. Mathematically, |w,₁ |=k, where w is the deflection of the diaphragm from the substrate, and (.), ₁ denotes the derivative with respect to x₁ (see the figure). We now seek minimizers of the total membraneous energy of the one-dimensional diaphragm. We can construct a minimizer by covering the domain of the diaphragm with any set of simple roofs of slopes ±k (such as R₁ , R₂ and R₃ in the figure), and then choosing their upper envelope. Infinitely many  such minimizers exist, all of which contain sharp folds. To minimize the bending energy associated with the sharp folds, we must minimize their number; this we effect by selecting the upper envelope of all the minimizers (see the dashed line in the figure). This simple construction can be readily generalized to solve the isotropic case in two dimensions.

The Isotropic Case: Analysis

In the two-dimensional case the membraneous energy density is minimized when the modulus of the gradient of the deflection equals the characteristic slope, i.e., when the following eikonal equation is satisfied, |(w, ₁)²+(w,₂)²|=k². Clearly, the eikonal equation is satisfied on any cone of slope k (such as C ₁ in the figure). Given the shape of the domain of a diaphragm, we can construct a minimizer of the total membraneous energy by covering the domain with a set of cones of slope k, and then choosing the upper envelope of the set. Infinitely many  such minimizers exist, all of which contain sharp folds. To minimize the bending energy associated with the sharp folds, we proceed by analogy with the one-dimensional case, and select the upper envelope of all the minimizers. We call the upper envelope of all the minimizers the preferred folding. In the following we show an example in which the preferred folding is in excellent agreement with the experimentally observed folding pattern.

The Isotropic Case: Results

The picture labeled Experimental shows a folding pattern observed byArgon et al. in a SiC diaphragm (op. cit., 1989). Using the shape of Argon's diapragm and the associated value of characteristic slope we obtained the preferred folding in the form discussed in the previous paragraph. The picture labeled Analytical shows the result. Note the shrap folds of the preferred folding.The preferred folding pattern is in excellent agreement with the experimentally observed one.  It bears emphasis that although we used a computer program to display the preferred folding, the solution is entirely analytical. 

The Anisotropic Case: Analysis

The Anisotropic Case: Results