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Theory of Stress
Published in Prasun Kumar Nayak, Mijanur Rahaman Seikh, Continuum Mechanics, 2022
Prasun Kumar Nayak, Mijanur Rahaman Seikh
The Cauchy stress tensor is used for stress analysis of material bodies experiencing small deformations: It is a central concept in the linear theory of elasticity. For large deformations, also called finite deformations, other measures of stress are required, such as Piola-Kirchhoff stress tensor, the Biot stress tensor, and the Kirchhoff stress tensor.
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.
Analysis of the passive biomechanical behavior of a sheep-specific aortic artery in pulsatile flow conditions
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2021
Claudio M. García–Herrera, Álvaro A. Cuevas, Diego J. Celentano, Álvaro Navarrete, Pedro Aranda, Emilio Herrera, Sergio Uribe
In a general way, anisotropic models consider fiber-reinforced nature of the arterial tissue (Rivera et al. 2020), expressed through to the incorporation of pseudo-invariants related to presence of fibers (Huh et al. 2019). In the specific case where stress-strain behavior is similar in both directions, the effect of anisotropic material model should not be so relevant, and isotropic model can be used. According to specific characteristic observed in this work (detailed in Section 3), an elastic, isotropic and rate-independent material response is considered to describe mechanical behavior of the aortic wall. Moreover, its behaviour is taken as incompressible due to the large amount of water present in it (Oijen 2003). To this end, hyperelastic constitutive models can be used to describe its mechanical response (Ogden 1984; Fung 1993; Holzapfel 2000; Raghavan and Vorp 2000; Prendergast et al. 2003; Masson et al. 2008; Kroon and Holzapfel 2009). In this context, a deformation energy function W, assumed to describe the isothermal material behaviour under any loading conditions, is usually defined in terms of the right Cauchy deformation tensor where F is the deformation gradient tensor and is the transpose symbol (note that in this case). Invoking classical arguments of continuum mechanics, the Cauchy stress tensor is defined as
A finite deformation, finite strain nonclassical internal polar continuum theory for solids
Published in Mechanics of Advanced Materials and Structures, 2019
K. S. Surana, A. D. Joy, J. N. Reddy
For a deforming volume of matter the rate of change of linear momenta must be equal to the sum of all other forces acting on it. This is Newton’s second law applied to a volume of matter. This derivation is same as that in classical continuum theory. Thus, following reference [61], we can write the following using first and second Piola–Kirchhoff stress tensors and derived using contravariant Cauchy stress tensor. In the Lagrangian description holds. are velocities and are body forces per unit mass. Equation (35) is momentum equation in x1, x2 and x3 directions.
Non-classical continuum theories for solid and fluent continua and some applications
Published in International Journal of Smart and Nano Materials, 2019
K.S. Surana, D. Mysore, J.N. Reddy
Consider the deformed tetrahedron . Let be the average stress per unit area on plane , be the average moment per unit area on plane henceforth referred to as moment for short, and be the unit normal to the face . , , and all have different directions when the deformation is finite. Based on the small deformation assumption, the deformed coordinates are approximately same as undeformed coordinates , thus the deformed tetrahedron in the current configuration is close to its map in the reference configuration. With this assumption all stress measures (first and second Piola-Kirchhoff stress tensors, Cauchy stress tensor) are approximately the same. The same holds for the moment tensors. Thus with the assumption we can write