Piola kirchhoff stress tensor pdf

The preceding formulas assume that the material has some perhaps small. Determine the 1st and 2nd piolakirchhoff stress tensors pj and s if the cauchy stress tensor. Second piola kirchhoff stress tensor the second piola stress tensor was concocted to be a symmetric tensor. Stress balance principles 03 the cauchy stress tensor. We do not derive this functional here, but rather see what are the implications of this form. Basically, p is chosen such that it satisfies eq i. First and second piolakirchhoff stress tensors definitions. Since it gives us the actual stress of the body, it is called the true stress in engineering. The 1st piolakirchhoff stress is the 3d generalization of the 1d concept of engineering stress. Two often used for large deformation mechanics are the 1st piolakirchoff pk stress tensor and the 2nd piolakirchoff pk stress tensor. Continuum mechanics and nonlinear elasticity stefano giordano. The first piolakirchhoff stress tensor p which will be called the pk1 stress for brevity is defined by. The main objectives are to introduce the hypothesis of infinitesimal deformation theory and particularizing.

The principles of balance of linear and angular momentum. The cauchy stress tensor defined previously, related area vectors to traction vectors in the current state of deformation of a material object. The pk1 stress is given in terms of the cauchy stress as. Note that detf appears in the elastic energy functional due to the incompressibility condition. Cauchy stress tensors and first piolakirchhoff stress tensor piola transformation and piola identity kirchhoff and second piolakirchhoff stress tensors description. For infinitesimal deformations and rotations, the cauchy and piola.

The most commonly used measure of stress is the cauchy stress tensor, often called simply the. Difference between first and second piolakirchhoff stress. Piolakirchhoff stress tensor by the reference cauchy theorem t. Energy momentum tensor, eshelby stress tensor, meshless local petrov galerkin, mlpg 1 balance laws for the cauchy stress s, the. In the case of finite deformations, the piola kirchhoff stress tensors express the stress relative to the reference configuration. Kyuichiro washizu on the occasion of his 60th birthday and the completion of a distinguished academic career at the university of. Difference between first and second piolakirchhoff stress vector.

Piola kirchhoff stress tensor by the reference cauchy theorem t. It may be shown that the 2nd piolakirchhoff pk stress tensor s that gives the. Made with explain everything for continuum mechanics. The 2nd piola kirchhoff stress tensor and the greenlagrange strain tensor are energetically conjugate.

Therefore, we need to develop alternative stress tensors. If the material rotates without a change in stress state rigid rotation, the components of the 1st piolakirchhoff stress tensor will vary with material orientation. Work conjugate pair of stress and strain in molecular dynamics. Topic 3 lagrangian continuum mechanics variables for. Nonlinear constitutive models and applications in forming. Introduction this page introduces several definitions of stress. Continuum mechanicsobjective stress rates wikiversity. It is interesting that most, perhaps even all, stress definitions can be paired with.

Derivation of cauchy stress pge 334 reservoir geomechanics. Piolakirchhoff stress tensor according to the linearized finite theory of elasticity for hyperelastic constrained materials. Since the 1st piola kirchhoff stress tensor is notsymmetric, one can create a symmetric tensor as. This tensor results from a complete transformation of the cauchy stress to the. Notice that p is a twopoint tensor possessing the geometrical mapping. This is in contrast to the cauchy stress tensor which expresses the stress relative to the present configuration. This definition is motivated by the following argument. Including the definitions of cauchy stress, the nominal stress and the second piola kirchhoff pk2 stress tensor. Symmetric version is the second piola kirchhoff stress. This is sometimes useful in doing computations for instance using the finite element method for large deformation problems. The 1st piolakirchhoff stress is energy conjugate to the deformation gradient. The first piola kirchoff stress tensor relates the cauchy stress tensor to the stress. J a stress tensor work conjugate to the green strain tensor g must be referred to the initial undeformed configuration as is the green strain tensor.

The following conjugate pairs, 1 secondorder piolakirchho. Tensors 9 of 45 stress in tensor in 2dimensions duration. This is not a symmetric tensor and for computational ease, this we use a the second piola kirchoff stress which is symmetric in nature as it is a. Hello, here are 2 pages from my disseration explaining why p the first piolakirchhoff stress has the form that it does. A key discriminator among the different stress tensors is whether they report stress in a materials undeformed, and especially unrotated state, the reference configuration, or in its deformed and rotated state, the current configuration. Cauchy stress tensor kirchhoff stress tensor first piolakirchhoff stress tensor ylhis paper is presented to my good friend, prof. The 2nd piolakirchhoff stress tensor and the greenlagrange strain tensor are energetically conjugate. The first and second piolakirchhoff stress tensors extend the concept of true and engineering stress to the threedimensional case and operate on area.

It is worthwhile to note that cauchy stress tensor is not conjugate to any strain tensor. The terms j and f1 come in to play when you are comparing areas in the reference configuration to those in the current configuration. Topic 3 lagrangian continuum mechanics variables for analysis. To recall, it is the limiting value of the ratio of force over area. The stress measure could be any of the stress measures discussed earlier, such as the cauchy stress tensor, the second piolakirchhoff stress tensor.