Plane Stress

Plane stress is a simplification in elasticity used to model bodies where

1. one dimension (thickness) is much smaller than the other two, such as thin plates or shells
2. forces acting only in that plane.

Let the plane of the structure be the -plane, and the thickness direction be the -axis.

1. Primary Assumptions

The plane stress formulation consists of the following core assumptions regarding the stress state:

1. The traction forces on the surfaces (at ) are zero.
2. Because the plate is thin, it is assumed that there is no room to develop significant internal stresses in the -direction, nor shear stresses associated with the -face.

Mathematically, this dictates: Consequently, the non-zero stress components () are assumed to be functions of and only, and are uniform through the thickness.

2. The Mathematical Inconsistency

While the plane stress simplification is highly useful and accurate for thin engineering components, it contains a theoretical inconsistency when strictly analyzed through the full three-dimensional theory of elasticity. This inconsistency arises from the relationship between stress, strain, and displacement compatibility.

The Poisson Effect and Out-of-Plane Strain

Even though the out-of-plane stress is assumed to be zero, the out-of-plane strain is not zero due to the Poisson effect. Using generalized Hooke’s Law:

Since , this reduces to:

Because and vary with and , also varies across the plane of the plate. This means the plate changes thickness non-uniformly.

The Conflict with Compatibility

From strain-displacement relations, (where is displacement in the -direction). Integrating with respect to (assuming symmetry about the midplane ) gives: Since is a function of and , then is a function of and . Consequently, the derivatives and are generally non-zero.

Now look at the transverse shear strains, which must be zero based on the stress assumption ():

If and are non-zero, then for these shear strains to remain zero, the in-plane displacements and must depend on .

Conclusion on Inconsistency

The assumption that in-plane stresses are independent of contradicts the requirement of strain compatibility. A stress state with everywhere is generally not possible unless is constant or linear in and . Therefore, plane stress is regarded as an approximate solution. In reality, the assumed stresses are treated as average values across the thickness of the plate.

3. Equations and Unknowns

Despite the theoretical inconsistency regarding the -direction, the 2D plane stress problem is mathematically determinate. We focus only on the variables in the -plane.

The Unknowns (Total: 8)

To fully solve the 2D field problem, we need to determine 8 field variables (all functions of and ):

• Displacements (2):
• Strains (3):
• Stresses (3):

The Governing Equations (Total: 8)

To solve for these 8 unknowns, we utilize 8 fundamental equations of elasticity (neglecting body forces for simplicity):

1. Equilibrium Equations (2): Derived from Newton’s second law (static).
2. Kinematic (Strain-Displacement) Equations (3): Based on geometry of deformation.
3. Constitutive Equations (Hooke’s Law for Plane Stress) (3): Relating stress to strain. Note the modification of due to the condition.

Since there are 8 unknowns and 8 independent equations, the system is closed and solvable, provided appropriate boundary conditions are applied.