What is Strain Hardening and Plasticity?

Materials when loaded with a specific load, deform both elastically and plastically. Elastic deformations are recoverable and hence temporary, whereas plastic deformations are permanent. In elastic deformation, stretching of atomic bonds occurs, while on the other hand in plastic deformation, the atomic bonds break.

 

 

Most of the metals are made up of unit cell crystal structure as shown in figure 1.1. The metal structure is made up of an individual crystalline area called grain (figure 1.2). Every crystal has a defect which is mainly classified as point defect, line defect and planar defect. Plastic deformations in the material occur due to movement of linear defects (also called dislocations). 

The linear defect occurs when the group of atoms in the structure is in an irregular position. The movement of these dislocations involves breaking of atomic bonds in the crystal. This movement makes atoms in the crystal planes slip over one another. The slip occurs along the parallel plane within the grain. Any defects in the regular crystal affect the motion of dislocation which makes the movement of dislocation difficult. So the strength of the metal can be raised by increasing the number of dislocations. 

In plastic deformation, the dislocation movement produces additional dislocations, the sliding of which often hinders the movement. This increases the force needed to move the dislocation and strengthen the metal. 

 

 

Figure 1.4 depicts the uniaxial stress-strain behaviour of a typical metal. When metal is loaded (standard tensile test) it deforms elastically till the yield point (A) and then hardens after that point. In hardening, the stress value rises with an increase in strain. Perfectly plastic behaviour is also shown in the figure. Here, after the yield point, the stress value remains constant and the material only strains under constant stress without any hardening. If the load is removed at point B after some hardening, there is elastic unloading. The elastic strain (εe) is recovered and there is an only permanent plastic strain (εp) left in the metal. This phenomenon is described by elasto-plastic material models.

Material nonlinearity comes into picture when there is a non-linear stress-strain behaviour. Further rate-independent plasticity is identified by an irreversible deformation (strain) which occurs after some stress value. Elasto-plastic behaviour can be characterized by plasticity theory. The important aspects of this theory are yield criteria and hardening rule.

 

Yield criteria define the elasticity limit and the onset of the plastic deformation in the material under load. Widely used and the most famous yield criterion is the Von Mises yield criteria. It states yielding occurs when the distortional energy density reaches a value equal to the distortional energy density at yielding in a uniaxial case. Von-Mises stress is used to check the yielding in material under complex loading from the stress results from a uniaxial tensile test. Any stress condition can be plotted using the principal stress axis `(σ_1,σ_2,σ_3)`which are orthogonal to each other. For multi-axial loading, this criterion is represented as,

 

   `(σ_1-σ_2)^2+ (σ_2-σ_3)^2+(σ_3-σ_1)^2- 2σ_y^2=0`                                                     (1)

                                                                   

 

 

This yield criterion can be described using principle stress as shown above and defines the yield surface. If plotted in 3D with all principal stresses as axes then the surface is a cylinder. The cylinder is aligned with the `σ_1=σ_2=σ_3=σ_(Hydrostatic stress)`. There is no yielding if the point lies on the hydrostatic stress line vector. So, hydrostatic stress does not induce any plastic deformation. 

Figure 1.6 represents the yield surface in 2D which is an ellipse in the principal stress domain. When the stress state is inside the yield surface then it has elastic deformations. If the stress state is equal to the yield surface then the material develops plastic deformation. In a perfectly plastic case, the yield surface remains unchanged but in general cases, it can change shape, size and position. The equation describing the change in the yield surface with plastic deformation is called a hardening rule. Material behaviour can be modelled by two basic hardening rules namely, isotropic hardening and kinematic hardening rule. 

 

 

ISOTROPIC HARDENING

In isotropic hardening, the yield surface remains at the centre and of the same shape but only expands with an increase in plastic deformation. If you plastically deform a solid and unload it, then try to load it in compression you will find that its yield stress would have increased compared to what it was in the first cycle. Again if the solid is unloaded and reloaded, the elastic limit (yield stress) further increases until some limit.

Basically it means if the yield stress in tension increases due to hardening the yield stress in compression grows by the same amount. Figure 1.7 represents the linear isotropic behaviour where the yield surface expands uniformly. The concept will be explained below for one tension and compression cycle:

 

• If the material is loaded in tension from points 1 to 3, it deforms plastically after point 2. The corresponding yield surface expands uniformly after point 2 as well. 
• The new yield strength after the isotropic hardening (points 2 to 3) now is `σ_A`. The new yield surface is denoted by ‘B’ and shown on the right side in figure 1.7.
• If the material is unloaded and further loaded in compression (points  3 to 4) then it will yield at (negative `σ_A`, at point 4). 
• This is because it gains the strength in tension and compression uniformly. The increase in strength is shown by ‘S’ in the figure below.
•  If the material is further loaded in compression it will plastically deform (point 4 to 5) and attain a new elastic limit (yield stress), the yield surface expands further uniformly and is denoted by ‘C’.

 

Isotropic hardening is generally used for large strain problems and is not a good choice for cyclic load problems.

 

 

KINEMATIC HARDENING:

In kinematic hardening, the yield surface does not change its size and shape but translates with plastic deformation. If you plastically deform a solid and unload it, and then try to load it in compression, you will find that its yield stress would have decreased compared to what it was in the first cycle (tension).

In simple language, the amount of strength we gain in tension through plastic deformation, we lose the same strength in compression. Generally, metals show kinematic hardening behaviour for small cyclic strain problems. Figure 1.8 depicts the linear kinematic hardening behaviour where the difference of  (`2σ_y`) in yielding exists.

 

• If the material is loaded in tension from points 1 to 3, it deforms plastically after point 2. The corresponding yield surface translates after point 2 but remains the same in size. 
• The new yield strength after the kinematic hardening (points 2 to 3) is `ᓂ_A`. The new yield surface is denoted by ‘B’ and shown on the right side in figure 1.8
• If the material is unloaded and further loaded in compression (points 3 to 4) then it will yield before the previous yield stress `ᓂ_Y`, (at point 4).
• This is because it loses the same strength in compression as much as it gains in tension, shown by ‘S’ in the figure below. 
• If the material is further loaded in compression it will plastically deform (point 5) and attain a new elastic limit (yield stress), the yield surface shifts accordingly.

 

This effect shown in the kinematic hardening is known as the Bauschinger effect. For large strain problems, kinematic hardening model is not a good choice because of the Bauschinger effect. For higher strain, the reverse yielding can happen in the material when it is still in tension. This phenomenon is depicted in figure 1.9. When the material undergoes large plastic deformation (point 3 in figure 1.9) it gains the strength ‘S’ in tension but loses the same strength ‘S’ in compression. When it is unloaded, this causes reverse yielding (point 4) in the tension region. Kinematic hardening is generally used for cyclic load or small strain problems.