Plane strain problem
Theory and Physics
What is Plane Strain?
During the discussion on plane stress, the term "plane strain" came up. Could you please explain it in detail?
Plane strain (plane strain) is the assumption that the strain in the depth direction (the $z$ direction) of a structure is zero:
No deformation in the depth direction... Under what circumstances does that happen?
It's for structures that are sufficiently long in the depth direction compared to their cross-sectional dimensions. Near the center of the structure, the influence of the ends is negligible, and deformation in the depth direction is constrained.
Typical examples:
- Dam cross-section — Very long in the depth (river flow) direction
- Tunnel cross-section — Uniform cross-section along the axis
- Long embankment — Uniform along the longitudinal direction of the embankment
- Rolling mill roll — Uniform deformation in the width direction
- Soil slip surface — Assumed uniform in depth
So plane stress is for "thin plates," and plane strain is for "cross-sections of long prismatic bodies."
That understanding is perfect. They are used differently based on "which direction is special." Zero stress in the thin direction (plate thickness) → plane stress. Zero strain in the long direction (depth) → plane strain.
Constitutive Law for Plane Strain
How does Hooke's law for plane strain differ from that for plane stress?
The constitutive law (matrix form) for plane strain:
There's a $(1-2\nu)$ in the denominator! The stiffness becomes infinite as $\nu \to 0.5$.
This is the most important characteristic of plane strain. For incompressible materials ($\nu = 0.5$), volume change is zero, yet $\varepsilon_{zz} = 0$ is also required, severely restricting the degrees of freedom for in-plane deformation. This is the cause of volumetric locking.
So plane strain analysis is difficult for rubber or nearly incompressible materials?
Exactly. For $\nu > 0.49$ or so, standard elements become unusable. Hybrid elements (making pressure an independent variable) or reduced integration elements are essential.
Stress in Plane Strain
Even though $\varepsilon_{zz} = 0$, $\sigma_{zz} \neq 0$, right?
Yes. Even though the strain in the $z$ direction is zero, stress in the $z$ direction occurs due to the Poisson effect:
If $\sigma_x + \sigma_y$ is tensile, then $\sigma_{zz}$ is also tensile... A tensile stress develops in the depth direction.
This $\sigma_{zz}$ is a constraint stress that automatically arises as long as the plane strain assumption holds. If you solve the same problem with 3D analysis, you can confirm this $\sigma_{zz}$ in the central part of the structure, and see that $\sigma_{zz} \to 0$ (approaching plane stress) near the ends.
Plane Strain in Soil Mechanics
I've heard that plane strain is standard in geotechnical engineering.
For geotechnical problems with cross-sections uniform along the longitudinal direction, such as excavations, embankments, retaining walls, and tunnels, plane strain is effectively the standard.
However, there are points to note:
- Soil constitutive laws — Mohr-Coulomb, Cam-Clay, etc., are defined for a 3D stress state, but in plane strain, the intermediate principal stress $\sigma_2 = \sigma_{zz} = \nu(\sigma_1 + \sigma_3)$ is automatically determined. This $\sigma_2$ affects failure criteria.
- Anisotropy — Sedimentary soils have different stiffness in the horizontal and vertical directions (transverse isotropy). This anisotropy should be considered even in plane strain.
Since the Mohr-Coulomb failure criterion ignores the intermediate principal stress, does that mean plane strain gives a conservative (safer) prediction?
Because the Mohr-Coulomb criterion ignores the influence of the intermediate principal stress, plane strain often yields conservative (safer side) predictions. Using criteria that consider the intermediate principal stress, like Drucker-Prager or Lade, leads to more realistic strength assessments.
Summary
Let me organize the theory of plane strain.
Key points:
- Assumption of $\varepsilon_{zz} = 0$ — Applied to cross-sectional analysis of long prismatic structures
- $\sigma_{zz} = \nu(\sigma_x + \sigma_y)$ — Constraint stress develops in the depth direction
- $(1-2\nu)$ in the denominator — Stiffness diverges as $\nu \to 0.5$ (volumetric locking)
- Standard assumption in soil mechanics — Excavations, tunnels, embankments
- Confusion with plane stress is strictly forbidden — The assumptions are completely different
Plane stress and plane strain look similar, but their underlying physics is fundamentally different.
Yes. Both reduce to 2D, but "what is zero" differs. Zero stress (plane stress) or zero strain (plane strain). This difference in the starting point creates all the other differences.
Background of Plane Strain Theory
The plane strain assumption (εz=γyz=γxz=0) was established by Barré de Saint-Venant in 1856 as part of his theory on shear stress distribution. It is applicable to structures like tunnels and long dams, where "the depth is sufficiently large compared to the cross-sectional dimensions," allowing 3D problems to be reduced to 2D. In geotechnical engineering, it remains the mainstay design standard for retaining wall and embankment analysis.
Physical Meaning of Each Term
- Inertia term (mass term): $\rho \ddot{u}$, meaning "mass × acceleration." Have you ever experienced being thrown forward when slamming on the brakes? That "feeling of being carried forward" is precisely the inertial force. Heavier objects are harder to set in motion and harder to stop once moving. Buildings shake during earthquakes because the ground moves suddenly while the building's mass "gets left behind." In static analysis, this term is set to zero, which assumes "forces are applied slowly enough that acceleration can be ignored." It absolutely cannot be omitted for impact loads or vibration problems.
- Stiffness term (elastic restoring force): $Ku$ or $\nabla \cdot \sigma$. When you stretch a spring, you feel a "force trying to return it," right? That's Hooke's law $F=kx$, the essence of the stiffness term. So, a question—if you pull an iron rod and a rubber band with the same force, which stretches more? Obviously, the rubber band. This "resistance to stretching" is Young's modulus $E$, which determines stiffness. A common misconception: "High stiffness ≠ strong." Stiffness is "resistance to deformation," strength is "resistance to failure"—they are different concepts.
- External force term (load term): Body forces $f_b$ (e.g., gravity) and surface forces $f_s$ (e.g., pressure, contact forces). Think of it this way—the weight of a truck on a bridge is a "force acting on the entire volume" (body force), while the force of the tires pushing on the road surface is a "force acting only on the surface" (surface force). Wind pressure, water pressure, bolt tightening force... all are external forces. A common mistake here: getting the load direction wrong. Intending "tension" but ending up with "compression"—it sounds like a joke, but it actually happens when coordinate systems are rotated in 3D space.
- Damping term: Rayleigh damping $C\dot{u} = (\alpha M + \beta K)\dot{u}$. Try plucking a guitar string. Does the sound continue forever? No, it gradually fades. That's because vibrational energy is converted to heat by air resistance and internal friction in the string. Car shock absorbers work on the same principle—they intentionally absorb vibrational energy to improve ride comfort. What if damping were zero? Buildings would continue swaying forever after an earthquake. Since that doesn't happen in reality, setting appropriate damping is crucial.
Assumptions and Applicability Limits
- Continuum assumption: Treats material as a continuous medium, ignoring microscopic heterogeneity.
- Small deformation assumption (for linear analysis): Deformation is sufficiently small compared to initial dimensions, and the stress-strain relationship is linear.
- Isotropic material (unless otherwise specified): Material properties are independent of direction (anisotropic materials require separate tensor definitions).
- Quasi-static assumption (for static analysis): Ignores inertial and damping forces, considering only the balance between external and internal forces.
- Non-applicable cases: For large deformation/large rotation problems, geometric nonlinearity is required. For plasticity, creep, and other nonlinear material behaviors, constitutive law extensions are needed.
Dimensional Analysis and Unit Systems
| Variable | SI Unit | Notes / Conversion Memo |
|---|---|---|
| Displacement $u$ | m (meter) | When inputting in mm, unify load and elastic modulus to MPa/N system. |
| Stress $\sigma$ | Pa (Pascal) = N/m² | MPa = 10⁶ Pa. Be careful of unit system inconsistency when comparing with yield stress. |
| Strain $\varepsilon$ | Dimensionless (m/m) | Note the distinction between engineering strain and logarithmic strain (for large deformations). |
| Elastic modulus $E$ | Pa | Steel: ~210 GPa, Aluminum: ~70 GPa. Note temperature dependence. |
| Density $\rho$ | kg/m³ | In mm system: tonne/mm³ (= 10⁻⁹ tonne/mm³ for steel). |
| Force $F$ | N (Newton) | In mm system: N, in m system: N (unified). |
Numerical Methods and Implementation
Plane Strain Analysis by FEM
How do FEM elements for plane strain differ from those for plane stress?
The element shapes and meshes are the same. What differs is the content of the constitutive matrix $[D]$. You just swap the $[D]$ for plane stress with the $[D]$ for plane strain.
If the element shapes are the same, then setting errors (mistaking plane stress/plane strain selection) are hard to notice.
That's precisely the biggest pitfall. Even with the same mesh, loads, and boundary conditions, the results change based on just one element type setting.
Element Names by Solver
| Element | Nastran | Abaqus | Ansys |
|---|---|---|---|
| 4-node quadrilateral (plane strain) | CQUAD4 + PLPLANE | CPE4, CPE4R, CPE4H | PLANE182 (KEYOPT3=2) |
| 8-node quadrilateral (plane strain) | CQUAD8 + PLPLANE | CPE8, CPE8R, CPE8RH | PLANE183 (KEYOPT3= |
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