Elements of Theory of Plasticity

Plasticity describes the behavior of a material that remains permanently deformed after an external force is removed. Many substances, such as clay or ductile metals, become plastic once stress passes a certain threshold. This differs from brittle materials, which generally fracture rather than stretch when forces reach a limit. Notably, while fracture is usually induced by normal stresses pulling material apart, plastic deformation is primarily driven by shear stresses sliding material layers past one another.

Historically, engineers studied plasticity mainly to ensure structures remained elastic and returned to their original shape. While predicting the start of plasticity is simple if a body is being pulled in only one direction, real-world structures face complex stresses in many directions at once. We need plasticity theory to determine exactly when and where parts of a material under complex loading will begin to yield and become plastic.

Furthermore, looking at a typical ductile metal, the elastic zone is very small compared to how much the material can stretch before rupturing. Engineers realized that striving to keep material strictly in the elastic zone wastes a great deal of its load-carrying and energy-absorbing potential. Therefore, modern design utilizes plasticity theory to safely access this potential.

Why Engineers Need Plasticity Theory

Engineers need plasticity theory for three main reasons: first, to calculate precisely when materials will yield under complex, multi-directional forces; second, to design safer structures that can absorb massive energy during extreme events like car crashes or earthquakes without collapsing; and third, to simulate manufacturing processes where permanently reshaping metal is the goal.

Why Modeling Plastic Behavior Is Much More Complicated Than Elastic Behavior

Modeling and understanding plastic deformation is significantly more complex than linear elasticity for several reasons:

1. Nonlinearity: Plastic deformation is inherently nonlinear. In mathematics and physics, nonlinear systems are generally more difficult to solve than linear ones, much like how nonlinear differential equations are far harder to solve than linear differential equations. While there are established methods for solving systems of linear equations, no such universal method exists for nonlinear equations.
2. Path Dependence: In linear elasticity, stress is uniquely determined by the current strain. However, in plasticity, a single strain state can be associated with various stress values depending on the loading history. Because the material behavior is path-dependent, we cannot simply relate total stress to total strain; instead, we must relate increments of stress to increments of strain.
3. Constitutive Variability: Plastic behavior varies fundamentally from one material to another. Also various parameters, such as temperature, loading rate, and grain size, can fundamentally alter a material's plastic behavior. In contrast, linear elasticity is uniform across materials, differing only by their compliance constants (the slope of the stress-strain curve).
4. Irreversibility and Energy Dissipation: Elastic deformation is a conservative process; the energy stored in the material is essentially a potential energy that is fully recoverable upon unloading. Plastic deformation, however, is dissipative. Energy is permanently lost (mostly converted to heat) and cannot be recovered. This means plastic modeling must satisfy strict thermodynamic laws to ensure that the model is physically valid, adding a layer of thermodynamic complexity that is not present in elasticity.
5. Geometric Nonlinearity and Large Deformations: In linear elasticity, deformations are assumed to be infinitesimal. This allows for major simplifications: equilibrium is calculated on the original shape, and all stress/strain definitions essentially coincide. In plasticity, these assumptions may break down due to large stretches and rotations, introducing two layers of complexity:
   • Distinction of Configurations: Because the geometry changes significantly, we cannot assume a fixed shape. We must strictly distinguish between the reference (undeformed) configuration and the current (deformed) configuration, and enforce equilibrium on a continuously moving and deforming body.
   • Multiplicity of Stress and Strain Measures: While a single definition suffices for small strains, large deformations require specific measures to ensure physical accuracy and energy balance (work conjugacy):
     • Cauchy Stress: Defined on the current configuration (force per current area).
     • Piola-Kirchhoff Stresses: Mapped back to the reference configuration. The 1st Piola-Kirchhoff relates current force to original area, while the 2nd Piola-Kirchhoff transforms the force vector itself to account for material rotation.
     • Conjugacy: Each stress measure must be paired with a mathematically corresponding strain rate to ensure the calculation of internal work is correct.