Strain and Hooke's Law

Strain represents the intensity of deformation experienced by a material under load. While stress describes the internal force intensity, strain quantifies the geometric distortion. The relationship between stress and strain is foundational to solid mechanics and is governed by Hooke's Law.

Types of Strain

Normal Strain (\epsilon)

Normal Strain: Also known as axial strain, it is the deformation per unit length. It is a dimensionless quantity, although it is often expressed in terms of mm/mm\text{mm/mm}, in/in\text{in/in}, or microstrain (μϵ\mu\epsilon).
ϵ=δL0\epsilon = \frac{\delta}{L_0}
where:
  • δ\delta is the total change in length (elongation or contraction).
  • L0L_0 is the original, undeformed length.
Sign Convention:
  • Positive (+): Tensile strain (elongation).
  • Negative (-): Compressive strain (contraction).

Shear Strain (\gamma)

Shear Strain: Unlike normal strain which changes the length of a material, shear strain changes the shape (angle) of a material element. It is defined as the change in the initial right angle between two perpendicular line segments, measured in radians.
γ=δsL≈tan⁡γ\gamma = \frac{\delta_s}{L} \approx \tan \gamma
where δs\delta_s is the shear deformation and LL is the length over which the deformation occurs. For small deformations, tan⁡γ≈γ\tan \gamma \approx \gamma.

Constitutive Relations

Hooke's Law (1D)

Hooke's Law: For linear elastic materials, stress is directly proportional to strain within the elastic limit. This fundamental relationship was discovered by Robert Hooke in 1676.
For Normal Stress and Strain:
σ=Eϵ\sigma = E \epsilon
where EE is the Modulus of Elasticity (or Young's Modulus), which represents the stiffness of the material (e.g., E≈200 GPaE \approx 200 \text{ GPa} for steel).

Shear Hooke's Law

For Shear Stress and Strain:
τ=Gγ\tau = G \gamma
where GG is the Modulus of Rigidity (or Shear Modulus).

Poisson's Ratio (\nu)

Poisson's Ratio: When a material is stretched in one direction (longitudinal), it tends to contract in the transverse directions. The ratio of transverse strain to longitudinal strain is a material property called Poisson's ratio, denoted by ν\nu.
ν=−ϵtransverseϵlongitudinal\nu = -\frac{\epsilon_{\text{transverse}}}{\epsilon_{\text{longitudinal}}}
For most isotropic engineering materials, 0≤ν≤0.50 \le \nu \le 0.5. For example, steel has ν≈0.3\nu \approx 0.3, while rubber is nearly incompressible with ν≈0.5\nu \approx 0.5.

Relationship between Elastic Constants

The Modulus of Elasticity (EE), Modulus of Rigidity (GG), and Poisson's ratio (ν\nu) are related by the following equation for isotropic materials:
G=E2(1+ν)G = \frac{E}{2(1 + \nu)}

Generalized Hooke's Law

Generalized Hooke's Law (Multiaxial Loading)

For a material subjected to stresses in three mutually perpendicular directions (σx\sigma_x, σy\sigma_y, σz\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.
ϵx=σxE−νσyE−νσzE\epsilon_x = \frac{\sigma_x}{E} - \frac{\nu \sigma_y}{E} - \frac{\nu \sigma_z}{E}
ϵy=σyE−νσxE−νσzE\epsilon_y = \frac{\sigma_y}{E} - \frac{\nu \sigma_x}{E} - \frac{\nu \sigma_z}{E}
ϵz=σzE−νσxE−νσyE\epsilon_z = \frac{\sigma_z}{E} - \frac{\nu \sigma_x}{E} - \frac{\nu \sigma_y}{E}
These equations are essential for analyzing stresses in complex 3D states, such as in pressure vessels or near stress concentrations.

Generalized Hooke's Law (Plane Stress)

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., zz-direction) is zero (σz=0,τxz=τyz=0\sigma_z = 0, \tau_{xz} = \tau_{yz} = 0).
The generalized Hooke's Law equations simplify for plane stress:
ϵx=1E(σx−νσy)\epsilon_x = \frac{1}{E} (\sigma_x - \nu \sigma_y)
ϵy=1E(σy−νσx)\epsilon_y = \frac{1}{E} (\sigma_y - \nu \sigma_x)
ϵz=−νE(σx+σy)\epsilon_z = -\frac{\nu}{E} (\sigma_x + \sigma_y)
Notice that even though there is no stress in the zz-direction, there is still a normal strain in the zz-direction due to the Poisson effect from the xx and yy stresses.

Isotropic vs. Anisotropic Materials

The elastic constants (E,G,ν,KE, 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).

Volumetric Strain and Bulk Modulus

The change in volume per unit original volume is the volumetric strain (or dilatation).
ϵv=ϵx+ϵy+ϵz=1−2νE(σx+σy+σz)\epsilon_v = \epsilon_x + \epsilon_y + \epsilon_z = \frac{1 - 2\nu}{E} (\sigma_x + \sigma_y + \sigma_z)
The Bulk Modulus (KK) relates the hydrostatic stress to the volumetric strain:
K=E3(1−2ν)K = \frac{E}{3(1 - 2\nu)}

Strain Transformation and Strain Rosettes

Plane Strain Transformation

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 θ\theta relative to the original xx-yy axes. The plane strain transformation equations are mathematically analogous to stress transformation equations:
ϵx′=ϵx+ϵy2+ϵx−ϵy2cos⁡(2θ)+γxy2sin⁡(2θ)\epsilon_{x'} = \frac{\epsilon_x + \epsilon_y}{2} + \frac{\epsilon_x - \epsilon_y}{2}\cos(2\theta) + \frac{\gamma_{xy}}{2}\sin(2\theta)
ϵy′=ϵx+ϵy2−ϵx−ϵy2cos⁡(2θ)−γxy2sin⁡(2θ)\epsilon_{y'} = \frac{\epsilon_x + \epsilon_y}{2} - \frac{\epsilon_x - \epsilon_y}{2}\cos(2\theta) - \frac{\gamma_{xy}}{2}\sin(2\theta)
γx′y′2=−ϵx−ϵy2sin⁡(2θ)+γxy2cos⁡(2θ)\frac{\gamma_{x'y'}}{2} = -\frac{\epsilon_x - \epsilon_y}{2}\sin(2\theta) + \frac{\gamma_{xy}}{2}\cos(2\theta)
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 (ϵ\epsilon) for normal stress (σ\sigma) and half-shear strain (γ/2\gamma/2) for shear stress (τ\tau).

Strain Rosettes

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 (ϵx\epsilon_x, ϵy\epsilon_y, and γxy\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 (θa\theta_a, θb\theta_b, θc\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∘45^\circ Rectangular Rosette:
  • Gauge A aligned at θ=0∘\theta = 0^\circ (ϵa=ϵx\epsilon_a = \epsilon_x)
  • Gauge B aligned at θ=45∘\theta = 45^\circ
  • Gauge C aligned at θ=90∘\theta = 90^\circ (ϵc=ϵy\epsilon_c = \epsilon_y)
From these three readings, the in-plane shear strain can be derived:
γxy=2ϵb−(ϵa+ϵc)\gamma_{xy} = 2\epsilon_b - (\epsilon_a + \epsilon_c)
Key Takeaways
  • Normal strain (ϵ\epsilon) changes length, while shear strain (γ\gamma) changes shape (angle).
  • Hooke's Law: Relates stress and strain linearly via the Modulus of Elasticity (EE) for normal stress and Modulus of Rigidity (GG) for shear stress.
  • Poisson's Ratio (ν\nu): Quantifies the transverse contraction (or expansion) that occurs alongside longitudinal stretching (or compression).
  • Elastic Constants: The relationship G=E/[2(1+ν)]G = E / [2(1+\nu)] links the three fundamental isotropic material properties.
  • Generalized Hooke's Law: Extends Hooke's Law to 3D states of stress, accounting for the Poisson effect across all principal axes.
  • Volumetric Strain is the sum of the normal strains in three orthogonal directions.
  • Strain Transformation: Determines principal strains and maximum shear strains on arbitrary planes, fully analogous to Mohr's Circle for stress.
  • Strain Rosettes: A configuration of three strain gauges used in experimental mechanics to completely define the in-plane strain state (ϵx,ϵy,γxy\epsilon_x, \epsilon_y, \gamma_{xy}) on the surface of a structure.
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