| Summary: | Kinematics of membrane analysis are described, including deformation gradients, Cauchy-Green tensors, and strain invariants developed from Cauchy-Green tensors. The membrane is described in terms of three configurations: initial, reference, and current. The membrane problem is solved using the finite element method. A strain energy functional is defined and then minimized. The resulting equilibrium equations are solved using a standard Newton-Raphson technique. This method is applied to various membrane situations. Strain energy functional for two incompressible materials (neo-Hookean and Ogden) are developed and then used to analyze membrane problems. The tangent stiffness matrix, and subsequently the Newton-Raphson solution technique, are modified to account for a constant mass of gas in the case of inflated membranes under external loads. Form finding is a mathematical exercise in which one finds the surface of minimal area connecting given boundaries. Form finding is accomplished using a modification of the updated reference strategy. Two pseudo-strain energy functionals are developed, one representing the area of the membrane, and the other representing its shear stiffness. As form finding proceeds, shear stiffness is manipulated to ensure that the finite element mesh resulting from the analysis is not distorted. Membranes are sometimes made out of textiles, which are orthotropic, and can be composite in nature, consisting of fibers embedded in some sort of matrix. In this case, the matrix has been assumed to be an Ogden material. Two fiber types have been described, exhibiting neo-Hookean and logarithmic strain. New strain energy functionals are developed which use fiber stretches which are orthotropic in the reference configuration. Appropriate tests validated the response of the materials. In many loading situations, the membrane stresses are isotropic in nature, meaning that the tensions are equal (or nearly so) in each of the principal directions. Occasionally loading may occur in which the tension in one principal direction is accompanied by compression in the other. In such a case, the membrane will wrinkle. A new strain energy functional describing the wrinkling of an incompressible neo-Hookean material is described. Two inflated membranes demonstrate the detection of wrinkling.
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