All About Plain Stress and Plain Strain

In Finite Element Analysis (FEA) and solid mechanics, we hear a lot about plane stress and plane strain notions. Plane stress is an approximation, as opposed to plane strain, which is an exact solution. Plane strain is a particular solution of the whole three-dimensional elasticity equations, whereas plane stress is only attained in the limit when the thickness of the loaded body approaches zero. Stress, or the force applied to a material, is defined as the  force divided by the cross-sectional area of the substance. Strain is the deformation or displacement of a material caused by applied stress.

Why Do Things Have to Be Modelled in Two Dimensions?

Computational costs were significant issues with pre-processing and post-processing. It prompted mathematicians to find novel approaches to reducing complex three-dimensional problems to something manageable. Although computers have evolved to a significant extent, 2D formulas are often helpful.

Plane Strain 

In plain strain, the thickness of the object is comparatively much more than the other two dimensions of the object. Plane strain refers to the deformation of a dimension of the object in the perpendicular directions of the plane. Plane strain assumptions are helpful when it comes to modelling real-world components. In-plane strains are generated as they are in a full 3D model, but with out-of-plane or z-direction strains set to zero. It occurs in an object whose strong walls limit it in the z-direction. The formulation allows stress resolution only ‘in (the) plane’, thus the name ‘Plane strain: zz = xz = yz = 0’.

Take the distortion components (εzz, εxz,εyz) from the constitutional equations derived from Hooke’s Law.

Plane strain assumptions provide a reasonable estimate of the behaviour inside a coarse component loaded in only one direction. Because there are many materials in the thickness, the stress through the thickness is essentially irrelevant (or at least negligible).

Plane Stress

Plane stress is a condition where the normal stress remains zero and the shear stress in the perpendicular direction of the applied load is assumed to be zero. In plane stress, one of the dimensions of the object is negligible compared to the other two dimensions.

In simple words, a uniform load is applied on the thickness of the object which is comparatively much smaller than the other two dimensions of the object. Moreover, the shear stress generated by the load can be negligible as it brings negligible deformations to the thickness of the object. There is insufficient limit material to keep the thickness stress constant:

σz = σxz = σyz = 0

This approach is best suited for examining fragile plates loaded in only one direction but might even be used to inspect the surface of more significant elements. The only region where actual plane stress circumstances can exist is on the surface of the plate. It accurately represents the boundary conditions.

In the plane stress constitutive model, we use σz = 0 to make εz = 0 and simply calculate εzz from -v(εxx + εyy), where v is Poisson’s Ratio. However, some FEA codes deal with this differently for various reasons including accounting for thermal expansion. 

Fracture Mechanics

The transition between plane strain and plane stress conditions significantly affects the confinement, so understanding plane stress and plane strain is essential in fracture mechanics research (on the surface). The distance from the front centre affects the stress power aspect.

The constraint is severe and conditions in the middle of the sample, where planar stress conditions predominate, resemble small-scale yields. In contrast, the absence of out-of-plane tension on the surface results in a significantly lower stress power aspect.

This article shows that 2D notions can be beneficial for modelling some idealised geometry. Here are two simple pointers to help you remember:

• Plane strain is used to model the interior of a component that is excessively thick and loaded in the same plane.
• Select a plane tension to simulate a thin element only when loaded in a plane.

Conclusion

In this article, we explored why we use 2D assumptions, plane strain and plane stress assumptions, and how to choose between them depending on the geometry of interest. Though the development of these mathematical techniques is to facilitate problems, they have applicability to modern engineering concerns, especially in fracture mechanics.