It is a dimensionless quantity, although it is often expressed in terms of $\text{mm/mm}$, $\text{in/in}$, or microstrain ($\mu\epsilon$).

Sign Convention:

• Positive (+): Tensile strain (elongation).
• Negative (-): Compressive strain (contraction).

Unlike normal strain which changes the length of a material, shear strain changes the shape of a material element. For small deformations, $\tan \gamma \approx \gamma$.

When a material undergoes a temperature change, it expands or contracts. If unconstrained, this results in thermal strain. It is directly proportional to the temperature change.

• Engineering Strain ($e$): Easy to compute, used for most engineering applications where deformations are small. Computed as $e = \delta / L_0$.
• True Strain ($\epsilon$): More accurate for large deformations (like necking in a tensile test). Computed by integrating the incremental strain over the current length: $\epsilon = \int_{L_0}^{L} \frac{dL}{L} = \ln(L/L_0)$.

This fundamental relationship was discovered by Robert Hooke in 1676. It applies to both normal and shear stress.

When a material is stretched in one direction (longitudinal), it tends to contract in the transverse directions. For most isotropic engineering materials, $0 \le \nu \le 0.5$. For example, steel has $\nu \approx 0.3$, while rubber is nearly incompressible with $\nu \approx 0.5$.

The Modulus of Elasticity ($E$), Modulus of Rigidity ($G$), and Poisson's ratio ($\nu$) are related by the following equation for isotropic materials:

For a material subjected to stresses in three mutually perpendicular directions ($\sigma_x$, $\sigma_y$, $\sigma_z$), the strain in any one direction is affected by the Poisson effect from the stresses in the other two directions. This is the Generalized Hooke's Law for triaxial loading.

These equations are essential for analyzing stresses in complex 3D states, such as in pressure vessels or near stress concentrations.

Often, structural members (like thin plates or pressure vessels) are subjected to forces in only two directions. This is a Plane Stress condition, where the stress in the third dimension (e.g., the $z$-direction) is zero ($\sigma_z = 0, \tau_{xz} = \tau_{yz} = 0$).

The generalized Hooke's Law equations simplify for plane stress:

The elastic constants ($E, G, \nu, K$) discussed previously assume the material is isotropic, meaning its mechanical properties are the same in all directions (e.g., steel, aluminum).

• Orthotropic Materials: Have different mechanical properties along three mutually perpendicular axes (e.g., wood, rolled metals). Wood is much stronger along the grain than across it.
• Anisotropic Materials: Have different mechanical properties in all directions (e.g., complex composites).

The volumetric strain can be related to the principal stresses and elastic constants. The Bulk Modulus ($K$) relates the hydrostatic stress to the volumetric strain.

Similar to plane stress transformation, the strains measured at a specific point on a surface can be transformed to determine the strains along different coordinate axes oriented at an angle $\theta$ relative to the original $x$-$y$ axes.

These equations allow us to find the Principal Strains (maximum and minimum normal strains, where shear strain is zero) using a method identical to Mohr's Circle for stress, merely by substituting normal strain ($\epsilon$) for normal stress ($\sigma$) and half-shear strain ($\gamma/2$) for shear stress ($\tau$).

In experimental stress analysis, electrical resistance strain gauges are bonded to the surface of a loaded part. However, a single strain gauge can only measure normal strain along its specific axis; it cannot measure shear strain.

Because the complete state of strain requires three independent values ($\epsilon_x$, $\epsilon_y$, and $\gamma_{xy}$), we must use three strain gauges arranged in a specific pattern, known as a Strain Rosette. By measuring the normal strain in three known directions ($\theta_a$, $\theta_b$, $\theta_c$), we can set up a system of three transformation equations to solve for the unknown 2D strain state.

The most common arrangement is the $45^\circ$ Rectangular Rosette:

• Gauge A aligned at $\theta = 0^\circ$ ($\epsilon_a = \epsilon_x$)
• Gauge B aligned at $\theta = 45^\circ$
• Gauge C aligned at $\theta = 90^\circ$ ($\epsilon_c = \epsilon_y$)

From these three readings, the in-plane shear strain can be derived.