Plane Stress and Plane Strain

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1. Introduction

In solid mechanics, simplifying a three-dimensional stress or strain state to a two-dimensional approximation is often valid and practical, especially in aerospace structural analysis. Two such common idealizations are:

  • Plane Stress – Suitable for thin flat elements like aircraft skins.
  • Plane Strain – Applicable to long structural members where deformation in one direction is restricted.

2. Plane Stress Condition

2.1 Assumptions

Plane stress assumes negligible stress in the thickness direction (say, zz-axis):

\sigma_z = 0,\quad \tau_{xz} = 0,\quad \tau_{yz} = 0

Non-zero stress components:

\sigma_x,\quad \sigma_y,\quad \tau_{xy}

2.2 Strain Relations

The 3D strain–stress relations reduce to:

\varepsilon_x = \frac{1}{E}(\sigma_x - \nu \sigma_y)

\varepsilon_y = \frac{1}{E}(\sigma_y - \nu \sigma_x)

\gamma_{xy} = \frac{\tau_{xy}}{G}

\varepsilon_z = -\frac{\nu}{E}(\sigma_x + \sigma_y)

Where:

  • E = Young’s modulus
  • \nu = Poisson’s ratio
  • G = \frac{E}{2(1+\nu)} = Shear modulus

3. Plane Strain Condition

3.1 Assumptions

Plane strain assumes zero strain in the zz-direction:

\varepsilon_z = 0,\quad \gamma_{xz} = 0,\quad \gamma_{yz} = 0

Non-zero strain components:

\varepsilon_x,\quad \varepsilon_y,\quad \gamma_{xy}

Typical in long bodies where the length dimension is constrained (e.g., long fuselage sections or nozzles).

3.2 Stress–Strain Relations

From Hooke’s law in 3D:

\varepsilon_x = \frac{1}{E}(\sigma_x - \nu \sigma_y - \nu \sigma_z)

\varepsilon_y = \frac{1}{E}(\sigma_y - \nu \sigma_x - \nu \sigma_z)

0 = \varepsilon_z = \frac{1}{E}(\sigma_z - \nu(\sigma_x + \sigma_y))

Solving for \sigma_z:

\sigma_z = \nu(\sigma_x + \sigma_y)

So, out-of-plane stress is non-zero, even though \varepsilon_z = 0.

4. Effective Elastic Moduli (Plane Strain)

In many numerical methods (like FEM), plane strain uses modified elastic constants:

  • Effective Young’s Modulus:

E' = \frac{E}{(1 - \nu^2)}

Effective Poisson’s Ratio:

\nu' = \frac{\nu}{(1 - \nu)}

5. Comparison Table

FeaturePlane StressPlane Strain
\sigma_z0\nu(\sigma_x + \sigma_y)
\varepsilon_z-\frac{\nu}{E}(\sigma_x + \sigma_y)0
Common ScenarioThin plates, aircraft skinsLong, constrained structures
Governing Equations2D Hooke’s LawModified 3D Hooke’s Law

6. Relevance in Aerospace Applications

  • Plane Stress:
    • Used for thin sheets under in-plane loads (e.g., fuselage skins, wing panels).
    • Critical in analyzing riveted and composite skin structures.
  • Plane Strain:
    • Found in long aerospace structures where end constraints dominate (e.g., nozzle flanges, spars under axial compression).
    • Also relevant in structural dynamic simulations where transverse motion is minimal.

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