Piola-Kirchoff Stress Research Articles

Nonlinear Analysis of RC Shell Structures Using Laminated Element. II

An incremental variational formulation and a curved-shell element were proposed in the companion paper. In this part, the material model and solution techniques are discussed in detail. The strain-hardening plastic approach is employed to model the compressive behavior of the concrete. A dual criterion is considered for yielding and crushing in terms of stresses and strains, which is completed with a tension cut-off representation. Both the crack interface effects and dowel action are accounted for by using an average shear modulus, and a full bond is assumed at the steel-concrete interface. A total Lagrangian approach making use of the second Piola-Kirchoff stress tensor is employed in this analysis. An incremental and iterative modified Newton-Raphson scheme is used for the solution of nonlinear problem. An energy criterion in terms of both forces and displacements is implemented. The applicability and validity of this analytical model incorporating both geometrical and material nonlinearities is amply demonstrated through several numerical examples. The solution technique proposed in this paper has been shown to be successful in preventing the drifing of the solution in incremental process.

Applications of nonlinear finite element method to contact problems and paper handling problems

A model that predicts the paper behavior during the printing process at various time intervals is proposed. The tendency of wrinkling can therefore be observed. This model will accommodate any paper topology induced by moisture effects or otherwise, and can analyze virtually any roll geometry and roll combinations. Nonlinear Finite Element Method is applied to the contact problem of two layered cylinders compressing a paper with arbitrary initial shape. The analysis assumed plane strain condition, the paper can therefore be treated as a beam. The result from the contact problem is applied to predict the behavior of the paper as it is feeding through the roll system. The Total Lagrange Formulation with Piola-Kirchoff Stress and Green-Lagrange strained is used. The quasi-static solution at different time intervals is presented. It was found that they agreed favorably with experiments.

Notes on a duality of stress and deformation fields in plane finite elasticity

In [ 1] for a given plane deformation of a homogeneous isotropic compressible elastic material with strain-energy function X, the first Piola-Kirchoff stress potentials are shown to be a possible deformation for another material of the same type having a strain-energy function 22*. A formula for 27* in terms of 27 is given as well as formulae connecting the strain invariants of the two deformations. In these notes we show that the same function 27* arises as a Legendre transformation when inverting Kirchoffs form of the stress-strain relations and that the duality principle established in [11 can be deduced by means of this Legendre transformation. In addition we analyse the formulae given in [1] in terms of principal stretches. We find that if 21 and 22 are the principal stretches of the given deformation then the principal stretches of the "stress 'potential" deformation are essentially the partial derivatives of 2; with respect to 21 and 2z. Although the results of [11 were derived for homogeneous isotropic elastic materials, we emphasize that the method of derivation of the duality principle as given in section 3 also applies to anisotropic homogeneous elastic materials provided they have enough symmetry to allow finite plane strain. The reader is referred to Green and Adkins [2] for details of such materials. These notes are expected to be read in conjunction with [-11. The notation and terminology used here is that given in [-11 except that here for convenience material and spatial rectangular cartesian co-ordinates are denoted by Z K ( K 1, 2, 3) and z i (i = 1, 2, 3) respectively, and the first Piola-Kirchoff stress potentials are denoted by u a (a = 1, 2). We use the convention that equation numbers with a dagger, that is ( )t, refer to equations of [11.

On a duality of stress and deformation fields in finite elasticity

For plane deformations of an isotropic compressible hyperelastic material, the first Piola-Kirchoff stress potentials are shown to be a possible plane deformation for another material of the same type. An explicit formula connecting the strain-energy functions of the two materials is derived.