Minimum Principle of Complementary Energy for Nonlinear Elastic Cable Networks with Geometrical Nonlinearities

In a companion paper [1], the authors presented a total Lagrangean rate formulation for a hybrid stress finite-element method, based on a rate complementary energy principle which involves both the rates of Piola-Lagrange stress and rotation as variables, for finite strain analysis of nonlinear elastic compressible solids. In this paper the method is extended to the case of precisely incompressible rubber-like materials. Two plane stress problems, one corresponding to a biaxial strip test and, the other, a sheet with a circular hole, both involving strains in excess of 100 percent, are solved and the numerical results are discussed.

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On the Shrinkage and Stiffening of a Cellulose Sponge Upon Drying

Journal of Applied Mechanics ◽

10.1115/1.4007906 ◽

2013 ◽

Vol 80 (2) ◽

Cited By ~ 4

Author(s):

Justine Rey ◽

Matthieu Vandamme

Keyword(s):

Experimental Data ◽

Bulk Modulus ◽

Capillary Pressure ◽

The Other ◽

Partially Saturated ◽

Geometrical Nonlinearities ◽

Cellulose Material ◽

Cellulose Sponge ◽

Nonlinear Elastic ◽

Water Contents

Everyone can observe the peculiar effect of water on a sponge: upon drying, a sponge shrinks and stiffens; it swells and softens upon wetting. In this work, we aim to explain and model this behavior by using the Biot–Coussy poromechanical framework. We measure the volume and the bulk modulus of sponges at different water contents. Upon drying, the volume of the sponge decreases by more than half and its bulk modulus increases by almost two orders of magnitude. We develop a partially saturated microporomechanical model of the sponge undergoing finite transformations. The model compares well with the experimental data. We show that about half of the stiffening of the sponge upon drying is due to geometrical nonlinearities induced by a closing of the pores under the action of capillary pressure. The other half of the stiffening can be explained by the nonlinear elastic properties of the cellulose material itself.

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Finite Elasticity Solutions Using Hybrid Finite Elements Based on a Complementary Energy Principle

Journal of Applied Mechanics ◽

10.1115/1.3424358 ◽

1978 ◽

Vol 45 (3) ◽

pp. 539-547 ◽

Cited By ~ 36

Author(s):

H. Murakawa ◽

S. N. Atluri

Keyword(s):

Finite Elasticity ◽

Strain Analysis ◽

Elastic Solid ◽

Element Model ◽

Equilibrium Equations ◽

Complementary Energy ◽

Rotation Tensor ◽

Energy Principle ◽

Complementary Energy Principle ◽

Nonlinear Elastic

The possibility of deriving a complementary energy principle, for the incremental analysis of finite deformations of nonlinear-elastic solids, in terms of incremental Piola-Lagrange (unsymmetric) stress alone, is examined. A new incremental hybrid stress finite-element model, based on an incremental complementary energy principle involving both the incremental Piola-Lagrange stress, and an incremental rotation tensor which leads to discretization of rotational equilibrium equations, is presented. An application of this new method to the finite strain analysis of a compressible nonlinear-elastic solid is included, and the numerical results are discussed.

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