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Simulation of Crystalline Nanoporous Materials and the Computation of Adsorption/Diffusion Properties
Published in T. Grant Glover, Bin Mu, Gas Adsorption in Metal-Organic Frameworks, 2018
Stress was introduced into the theory of elasticity by Cauchy and is a measure of the average amount of force exerted per unit area (N/m2 or Pa or energy density J/m3). The Cauchy stress tensor relates forces in the present (deformed) configuration to areas in the present configuration, and another name for this tensor is true stress in engineering. The Piola–Kirchhoff stress tensor of the first kind relates forces in the present deformed configurations in terms of the surface area in the reference configuration, while the Piola–Kirchhoff stress tensor of the second kind relates forces in the reference configuration to areas in the reference configuration [202]. The symmetric Cauchy stress is used in the Eulerian equations of motion, the first nonsymmetric Piola–Kirchhoff stress tensor is used in the Lagrangian equation of motion. The second Piola–Kirchhoff stress tensor is symmetric and, therefore, more suitable than the first Piola–Kirchhoff stress tensor to use in stress–strain relations. Pressure is a special case of stress and has the same units. Stress is more general and complex because it varies with direction and the surface it acts on. Many types of stress are known: compressional stress (acts to shorten an object), tensional stress (acts to lengthen an object), normal stress (acts perpendicular to a surface), shear stress (acts parallel to a surface), hydrostatic stress (uniform in all directions), and directed stress (varies with direction). Stress is generally not directly observable, but we can observe the deformation of an object when stress acts on it.
Review of Governing Equations
Published in J.N. Reddy, A. Miravete, Practical Analysis of COMPOSITE LAMINATES, 2018
A suitable strain measure to use with the Cauchy stress tensor σ would be the infinitesimal (or the Euler) strain tensor ε. Since we defined strain tensor E as a function of the material point X in a reference state, the stress must also be expressed as a function of the material point. The stress measure that is used in nonlinear analysis of solid bodies is the second Piola–Kirchhoff stress tensorS (Sij), which is measured in the deformed body but referred to the material coordinates, Xi. The second Piola–Kirchhoff stress tensor is symmetric whenever the Cauchy stress tensor is symmetric. For small deformation problems, the difference between the two measures of stress disappears. Since the scope of this book is limited to linear analyses, we work with stress tensor σ.
Mechanics Preliminaries
Published in J. N. Reddy, Theories and Analyses of Beams and Axisymmetric Circular Plates, 2022
In this chapter, beginning with a short discussion of vectors and tensors and the introduction of the Cauchy stress vector and Cauchy stress tensor, measures of Green strain tensor, infinitesimal strain tensor and the von Kármán strain tensor components are reviewed. The definition of the second Piola–Kirchhoff stress tensor is introduced, but for small strains (as is the case with the present study), it is indistinguishable from the Cauchy stress tensor. Then the equations of motion of a deformable solid are presented, and stress–strain relations for a linear elastic material are summarized.
A symmetric tangent stiffness approach to cohesive mechanical interfaces in large displacements
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2022
Guido Borino, Francesco Parrinello
The principle of virtual work for solid body, occupying the region with an embedded cohesive interface written in the reference configuration reads where P is the First Piola–Kirchhoff stress tensor, is a virtual deformation gradient, is a virtual separation displacement vector and is the external virtual work.
Topology optimization of geometrically nonlinear structures using an evolutionary optimization method
Published in Engineering Optimization, 2018
Meisam Abdi, Ian Ashcroft, Ricky Wildman
In this study, the assumption is that the structures undergo large deformation with small strain. To model this nonlinear behaviour, the total Lagrangian (TL) formulation is utilized, in which all static and kinematic variables are referred to the initial undeformed configuration of the structure and the integrals are calculated with respect to that configuration. Because of the transformations, a new measure for stress, the second Piola–Kirchhoff stress tensor, has to be introduced with the Green–Lagrange strain tensor. Considering TL formulation for a general body subjected to applied body forces and surface tractions on the surface S and displacement field , the equation of motion is given by (Gea and Luo 2001) where denote the Cartesian components of the second Piola–Kirchhoff stress tensor, are the components of the Green–Lagrange strain tensor corresponding to the virtual displacement field , and denotes the body volume at initial configuration. The Green–Lagrange strain tensor, which is defined with respect to the initial configuration of the body, is given by (Gea and Luo 2001) Considering reasonably small strains, the general elastic constitutive equation can still be used: where C is the elasticity tensor. Equations (8)–(10) are the basic equations for calculating the response of a continuum body using the TL formulation. However, to solve these equations for strongly nonlinear problems, one may need to use an incremental–iterative approach, such as Newton–Raphson, as discussed in Section 3.1.