Learning topic
Stress and Strain State
Understand plane and 3D stress states, stress-tensor components, principal planes and stresses, strain components, and the link to failure criteria.
The stress and strain state at a point describes the set of stresses and strains acting on planes of different orientations passing through the same material point. This framework is required whenever a simple uniaxial stress model is insufficient.
Stress components
On an arbitrary plane, the traction can be resolved into normal and shear components. In Cartesian coordinates, a three-dimensional stress state is described by normal components $\sigma_x$, $\sigma_y$, $\sigma_z$ and shear components $\tau_{xy}$, $\tau_{yz}$, $\tau_{zx}$. In classical continuum mechanics with moment equilibrium, the stress tensor is symmetric, so paired shear components are equal.
Plane and three-dimensional stress
For thin plates loaded in their own plane, a plane stress model is often appropriate, with $\sigma_z$, $\tau_{xz}$, and $\tau_{yz}$ taken as approximately zero. A general triaxial state requires all three principal stresses.
Principal stresses
There are mutually perpendicular principal planes on which shear stresses vanish. The normal stresses acting on these planes are the principal stresses. They provide coordinate-independent characteristics that are convenient for strength assessment.
For a plane stress state defined by $\sigma_x$, $\sigma_y$, and $\tau_{xy}$, the principal stresses are:
$$\sigma_{1,2}=\frac{\sigma_x+\sigma_y}{2}\pm\sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2+\tau_{xy}^2}.$$
Shear stress is zero on the principal planes. Their orientation can be determined from:
$$\tan 2\theta_p=\frac{2\tau_{xy}}{\sigma_x-\sigma_y},$$
with the quadrant and sign convention handled consistently.
- $\sigma_1$, $\sigma_2$ — principal normal stresses;
- $\theta_p$ — orientation angle of a principal plane;
- $\sigma_x$, $\sigma_y$, $\tau_{xy}$ — components of the original plane stress state.
Strain state
Strain at a point is described by normal strains $\varepsilon_x$, $\varepsilon_y$, $\varepsilon_z$ and engineering shear strains $\gamma_{xy}$, $\gamma_{yz}$, $\gamma_{zx}$. For a linearly elastic isotropic material, stresses and strains are related by generalized Hooke's law.
For a homogeneous, linearly elastic, isotropic material under a three-dimensional stress state, the normal strains are:
$$\varepsilon_x=\frac{1}{E}[\sigma_x-\nu(\sigma_y+\sigma_z)],$$
$$\varepsilon_y=\frac{1}{E}[\sigma_y-\nu(\sigma_z+\sigma_x)],$$
$$\varepsilon_z=\frac{1}{E}[\sigma_z-\nu(\sigma_x+\sigma_y)].$$
The engineering shear strains are:
$$\gamma_{xy}=\frac{\tau_{xy}}{G},\qquad \gamma_{yz}=\frac{\tau_{yz}}{G},\qquad \gamma_{zx}=\frac{\tau_{zx}}{G},$$
where $G=E/[2(1+\nu)]$.
- $E$ — Young's modulus;
- $\nu$ — Poisson's ratio;
- $G$ — shear modulus.
Transition to strength assessment
Under a multiaxial stress state, a single relation such as $\sigma=N/A$ is not sufficient. For ductile materials, an equivalent stress based on the Tresca or von Mises criterion is commonly compared with the strength characteristic specified by the adopted design method.
Section structure
The child topics develop stress transformation and Mohr's circle, principal stresses, failure criteria, and generalized Hooke's law. Together they form the basis for analyzing combined loading.
About this topic
The stress state at a point in a deformable body is fully characterized by the stress tensor across all possible planes passing through that point. We distinguish between uniaxial, biaxial (plane), and triaxial (3D) stress states. This page provides a theoretical analysis of stress components, coordinate transformation rules under rotation, and methods for identifying principal planes where shear stresses equal zero.