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In continuum mechanics, the Cauchy stress tensor (symbol , named after Augustin-Louis Cauchy), also called true stress tensor[1] or simply stress tensor, completely defines the state of stress at a point inside a material in the deformed state, placement, or configuration. The second order tensor consists of nine components and relates a unit ...
Alternative stress measures. In continuum mechanics, the most commonly used measure of stress is the Cauchy stress tensor, often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined: [1][2][3] The Kirchhoff stress (. τ {\displaystyle {\boldsymbol {\tau }}} ).
The effect of stress in the continuum flow is represented by the ∇p and ∇ ⋅ τ terms; these are gradients of surface forces, analogous to stresses in a solid. Here ∇p is the pressure gradient and arises from the isotropic part of the Cauchy stress tensor. This part is given by the normal stresses that occur in almost all situations.
Continuum mechanics. In continuum mechanics, the finite strain theory —also called large strain theory, or large deformation theory —deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strain theory. In this case, the undeformed and deformed configurations of the ...
Thus the stress state of the material must be described by a tensor, called the (Cauchy) stress tensor; which is a linear function that relates the normal vector n of a surface S to the traction vector T across S. With respect to any chosen coordinate system, the Cauchy stress tensor can be represented as a symmetric matrix of 3×3
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
Mohr's circles for a three-dimensional state of stress. Mohr's circle is a two-dimensional graphical representation of the transformation law for the Cauchy stress tensor. Mohr's circle is often used in calculations relating to mechanical engineering for materials' strength, geotechnical engineering for strength of soils, and structural ...
Stress–strain analysis (or stress analysis) is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other ...