Stress (mechanics)

 ( Solid Mechanics ) 

As in the case of an axially loaded bar, in practice the shear stress may not be uniformly distributed over the layer; so, as before, the ratio F/ A will only be an average ("nominal", "engineering") stress. That average is often sufficient for practical purposes. Shear stress is observed also when a cylindrical bar such as a axle is subjected to opposite torques at its ends. In that case, the shear stress on each cross-section is parallel to the cross-section, but oriented tangentially relative to the axis, and increases with distance from the axis. Significant shear stress occurs in the middle plate (the "web") of under bending loads, due to the web constraining the end plates ("flanges").

In these situations, the stress across any imaginary internal surface turns out to be equal in magnitude and always directed perpendicularly to the surface independently of the surface's orientation. This type of stress may be called isotropic normal or just isotropic; if it is compressive, it is called hydrostatic pressure or just pressure. Gases by definition cannot withstand tensile stresses, but some liquids may withstand very large amounts of isotropic tensile stress under some circumstances. see Z-tube.

Cauchy observed that the stress vector $T$ across a surface will always be a linear function of the surface's surface normal $n$, the unit-length vector that is perpendicular to it. That is, $T\; =\; \backslash boldsymbol\{\backslash sigma\}(n)$, where the function $\backslash boldsymbol\{\backslash sigma\}$ satisfies $$\backslash boldsymbol\{\backslash sigma\}(\backslash alpha\; u\; +\; \backslash beta\; v)\; =\; \backslash alpha\backslash boldsymbol\{\backslash sigma\}(u)\; +\; \backslash beta\backslash boldsymbol\{\backslash sigma\}(v)$$ for any vectors $u,v$ and any real numbers $\backslash alpha,\backslash beta$. The function $\backslash boldsymbol\{\backslash sigma\}$, now called the (Cauchy) stress tensor, completely describes the stress state of a uniformly stressed body. (Today, any linear connection between two physical vector quantities is called a tensor, reflecting Cauchy's original use to describe the "tensions" (stresses) in a material.) In tensor calculus, $\backslash boldsymbol\{\backslash sigma\}$ is classified as a second-order tensor of type (0,2) or (1,1) depending on convention.

Like any linear map between vectors, the stress tensor can be represented in any chosen Cartesian coordinate system by a 3×3 matrix of real numbers. Depending on whether the coordinates are numbered $x\_1,x\_2,x\_3$ or named $x,y,z$, the matrix may be written as $$\backslash begin\{bmatrix\}$$

 \sigma _{11} & \sigma _{12} & \sigma _{13} \\
 \sigma _{21} & \sigma _{22} & \sigma _{23} \\
 \sigma _{31} & \sigma _{32} & \sigma _{33}
     

 \sigma _{xx} & \sigma _{xy} & \sigma _{xz} \\
 \sigma _{yx} & \sigma _{yy} & \sigma _{yz} \\
 \sigma _{zx} & \sigma _{zy} & \sigma _{zz} \\
     

 \begin{bmatrix} T_1 & T_2 & T_3 \end{bmatrix} =
 \begin{bmatrix} n_1 & n_2 & n_3 \end{bmatrix}
 \cdot
 \begin{bmatrix}
   \sigma_{11} & \sigma_{21} & \sigma_{31} \\
   \sigma_{12} & \sigma_{22} & \sigma_{32} \\
   \sigma_{13} & \sigma_{23} & \sigma_{33}
 \end{bmatrix}
     

The linear relation between $T$ and $n$ follows from the fundamental laws of conservation of linear momentum and static equilibrium of forces, and is therefore mathematically exact, for any material and any stress situation. The components of the Cauchy stress tensor at every point in a material satisfy the equilibrium equations (Cauchy's equations of motion for zero acceleration). Moreover, the principle of conservation of angular momentum implies that the stress tensor is symmetric matrix, that is $\backslash sigma\_\{12\}\; =\; \backslash sigma\_\{21\}$, $\backslash sigma\_\{13\}\; =\; \backslash sigma\_\{31\}$, and $\backslash sigma\_\{23\}\; =\; \backslash sigma\_\{32\}$. Therefore, the stress state of the medium at any point and instant can be specified by only six independent parameters, rather than nine. These may be written $$$$

 \begin{bmatrix}
   \sigma_x & \tau_{xy} & \tau_{xz} \\
   \tau_{xy} & \sigma_y & \tau_{yz} \\
   \tau_{xz} & \tau_{yz} & \sigma_z
 \end{bmatrix}
     

As a symmetric 3×3 real matrix, the stress tensor $\backslash boldsymbol\{\backslash sigma\}$ has three mutually orthogonal unit-length eigenvectors $e\_1,e\_2,e\_3$ and three real eigenvalues $\backslash lambda\_1,\backslash lambda\_2,\backslash lambda\_3$, such that $\backslash boldsymbol\{\backslash sigma\}\; e\_i\; =\; \backslash lambda\_i\; e\_i$. Therefore, in a coordinate system with axes $e\_1,e\_2,e\_3$, the stress tensor is a diagonal matrix, and has only the three normal components $\backslash lambda\_1,\backslash lambda\_2,\backslash lambda\_3$ the principal stresses. If the three eigenvalues are equal, the stress is an isotropic compression or tension, always perpendicular to any surface, there is no shear stress, and the tensor is a diagonal matrix in any coordinate frame.

In that view, one redefines a "particle" as being an infinitesimal patch of the plate's surface, so that the boundary between adjacent particles becomes an infinitesimal line element; both are implicitly extended in the third dimension, normal to (straight through) the plate. "Stress" is then redefined as being a measure of the internal forces between two adjacent "particles" across their common line element, divided by the length of that line. Some components of the stress tensor can be ignored, but since particles are not infinitesimal in the third dimension one can no longer ignore the torque that a particle applies on its neighbors. That torque is modeled as a bending stress that tends to change the curvature of the plate. These simplifications may not hold at welds, at sharp bends and creases (where the radius of curvature is comparable to the thickness of the plate).