Learning topic
Generalized Hooke's Law
Learn generalized Hooke’s law for isotropic linear elasticity: 3D normal and shear stress-strain relations using Young’s modulus, Poisson’s ratio, and shear modulus.
Generalized Hooke's law relates stress and strain components in a linearly elastic isotropic material. Unlike the uniaxial relation \(\sigma=E\varepsilon\), it accounts for simultaneous normal stresses in three directions and for shear deformation.
Poisson effect
A normal stress acting in one direction produces longitudinal strain and transverse strains. This coupling is governed by Poisson's ratio \(\nu\). For example, a bar stretched in the \(x\)-direction becomes thinner in the transverse directions.
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.
Three-dimensional normal-strain relations
When axes are aligned with the principal stress directions, shear stresses are zero and
\[\varepsilon_x=\frac{1}{E}[\sigma_x-\nu(\sigma_y+\sigma_z)],\]
\[\varepsilon_y=\frac{1}{E}[\sigma_y-\nu(\sigma_x+\sigma_z)],\qquad\varepsilon_z=\frac{1}{E}[\sigma_z-\nu(\sigma_x+\sigma_y)].\]
Here \(E\) is Young's modulus. The equations show that each normal strain depends on all three normal stress components, not only on the stress with the same subscript.
Shear relations
For isotropic linear elasticity, shear stresses and engineering shear strains are related by
\[\tau_{xy}=G\gamma_{xy},\qquad \tau_{yz}=G\gamma_{yz},\qquad \tau_{zx}=G\gamma_{zx},\]
where \(G\) is the shear modulus. Only two elastic constants are independent:
\[G=\frac{E}{2(1+\nu)}.\]
Volumetric strain and bulk modulus
For small strains, the relative change in volume is the sum of the normal strains:
$$\varepsilon_v=\varepsilon_x+\varepsilon_y+\varepsilon_z.$$
For an isotropic linearly elastic material:
$$\varepsilon_v=\frac{1-2\nu}{E}(\sigma_x+\sigma_y+\sigma_z).$$
The bulk modulus is:
$$K=\frac{E}{3(1-2\nu)}.$$
Under uniform hydrostatic pressure $p$, if tension is positive, $\sigma_x=\sigma_y=\sigma_z=-p$ and $\varepsilon_v=-p/K$.
- $\varepsilon_v$ — volumetric strain;
- $\varepsilon_x$, $\varepsilon_y$, $\varepsilon_z$ — normal strains;
- $\sigma_x$, $\sigma_y$, $\sigma_z$ — normal stresses;
- $E$ — Young's modulus;
- $\nu$ — Poisson's ratio;
- $K$ — bulk modulus;
- $p$ — hydrostatic pressure.
The volumetric strain is \(\varepsilon_v=\varepsilon_x+\varepsilon_y+\varepsilon_z\). Under hydrostatic stress \(\sigma_x=\sigma_y=\sigma_z=\sigma_m\),
\[\varepsilon_v=\frac{3(1-2\nu)}{E}\sigma_m=\frac{\sigma_m}{K},\qquad K=\frac{E}{3(1-2\nu)},\]
where \(K\) is the bulk modulus. Hydrostatic stress changes volume; deviatoric stress changes shape.
Plane stress and plane strain
For a thin plate, \(\sigma_z\approx0\), but \(\varepsilon_z\) is generally not zero:
\[\varepsilon_z=-\frac{\nu}{E}(\sigma_x+\sigma_y).\]
In plane strain, used for a long constrained body, one strain component is approximately zero instead. Plane stress and plane strain therefore have different stress–strain relations and must not be interchanged.
Example
Let \(E=200\) GPa, \(\nu=0.30\), \(\sigma_x=100\) MPa, \(\sigma_y=40\) MPa, and \(\sigma_z=0\). Then \(\varepsilon_x=0.00044\), \(\varepsilon_y=0.00005\), and \(\varepsilon_z=-0.00021\). Although no stress acts through the thickness, the plate contracts through the thickness owing to the Poisson effect.
Limits of applicability
These relations assume small strains, homogeneous isotropic material, and linear-elastic behaviour. Anisotropic composites, plastic deformation, cracking concrete, viscoelasticity, and large strains require other constitutive models.
About this topic
Generalized Hooke's law defines linear relationships between components of the strain tensor and stress tensor for isotropic elastic bodies under triaxial stress states. This section covers linear strains along principal axes, shear strains, volumetric deformation, and bulk modulus. Practical engineering applications include stress analysis of thin-walled pressure vessels and pipes.