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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.
This type of stress may be called (simple) normal stress or uniaxial stress; specifically, (uniaxial, simple, etc.) tensile stress. [13] If the load is compression on the bar, rather than stretching it, the analysis is the same except that the force F and the stress σ {\displaystyle \sigma } change sign, and the stress is called compressive ...
We then plot two points A(50,40) and B(-10,-40), representing the state of stress at plane A and B as show in both Figure 8 and Figure 9. These points follow the engineering mechanics sign convention for the Mohr-circle space (Figure 5), which assumes positive normals stresses outward from the material element, and positive shear stresses on ...
The Cauchy's stress quadric, also called the stress surface, is a surface of the second order that traces the variation of the normal stress vector as the orientation of the planes passing through a given point is changed. The complete state of stress in a body at a particular deformed configuration, i.e., at a particular time during the motion ...
Lamé's stress ellipsoid is an alternative to Mohr's circle for the graphical representation of the stress state at a point. The surface of the ellipsoid represents the locus of the endpoints of all stress vectors acting on all planes passing through a given point in the continuum body. In other words, the endpoints of all stress vectors at a ...
The stress is proportional to the strain, that is, obeys the general Hooke's law, and the slope is Young's modulus. In this region, the material undergoes only elastic deformation. The end of the stage is the initiation point of plastic deformation. The stress component of this point is defined as yield strength (or upper yield point, UYP for ...
Figure 7.1 Plane stress state in a continuum. In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analysis is considerably simplified, as the stress state can be represented by a tensor of dimension 2 ...
However, by means of the von Mises yield criterion, which depends solely on the value of the scalar von Mises stress, i.e., one degree of freedom, this comparison is straightforward: A larger von Mises value implies that the material is closer to the yield point. In the case of pure shear stress, =, while all other =, von Mises criterion becomes: