3.2 Stress

Figure 3.2: Stress components acting on infinitesimal cube.

Within a body the forces that an enclosed volume imposes on the remainder of the body must be in equilibrium with the forces upon it from the remainder of the body. This principle was first enunciated by Cauchy and provides a path to calculate stress. The stress at a point may be determined by considering a small element of the body enclosed by areas $\Delta A$, on which forces $\Delta {\ensuremath{\mathitbf{F}}}$ act. By making the element infinitesimally small, the stress vector $\sigma$ is defined as the limit

$${\ensuremath{\mathitbf{\sigma}}} = \lim_{\Delta A \to 0} \frac {\... ...}}} {\Delta A} = \frac {\mathrm{d}{\ensuremath{\mathitbf{F}}}} {\mathrm{d}A}\ .$$ (3.8)

From Figure 3.2 one can observe that the force acting on a plane can be decomposed into a force within the plane, the shear components, and one force perpendicular to the plane, the normal component. The shear stress can be further decomposed into two orthogonal force components giving rise to three total stress components acting on each plane.

Thus, the stress on an infinitesimally small element can be characterized by nine stress components in total, namely three normal and six shear components. These components form the stress tensor

$$\ensuremath{{\underaccent{\bar}{\sigma}}} = \begin{pmatrix}\sigma... ...igma_{yy} & \sigma_{yz}\\ \sigma_{xz} & \sigma_{yz} & \sigma_{zz}\end{pmatrix},$$ (3.9)

where the shear stress components across the diagonal are identical as a result of static equilibrium

$$\sigma_{xy} = \sigma_{yx}\ ,\quad \sigma_{yz} = \sigma_{zy}\ ,\quad \sigma_{zx} = \sigma_{xz}.$$

E. Ungersboeck: Advanced Modelling Aspects of Modern Strained CMOS Technology