The following will be covered in this Chapter:
– Background Elasticity/Plasticity
– Theory
Yield Criteria
Hardening Rule
– Procedure
– Workshop
The capabilities described in this section are generally applicable to ANSYS Structural licenses and above.
– Exceptions will be noted accordingly
A. Metal Plasticity Overview
What is plasticity?
When a ductile material experiences stresses beyond the elastic limit, it will yield, acquiring large permanent deformations.
– Plasticity refers to the material response beyond yield.
– Plastic response is important for metal forming operations.
– Plasticity is also important as an energy-absorbing mechanism for structures in service.
Materials that fail with little plastic deformation are said to be brittle.
Ductile response is safer in many respects than is brittle response.
This Chapter will review some basics of plasticity by defining certain terminology.
Review of Elasticity:
Before proceeding to a discussion on plasticity, it may be useful to review elasticity of metals.
– In elastic response, if the induced stresses are below the material’s yield strength, the material can fully recover its original shape upon unloading.
– From a standpoint of metals, this behavior is due to the stretching but not breaking of chemical bonds between atoms. Because elasticity is due to this stretching of atomic bonds, it is fully recoverable. Moreover, these elastic strains tend to be small.
– Elastic behavior of metals is most commonly described by the stress-strain relationship of Hooke’s Law:
Review of Plasticity:
Plastic deformation results from slip between planes of atoms due to shear stresses (deviatoric stresses). This dislocation motion is essentially atoms in the crystal structure rearranging themselves to have new neighbors
– results in unrecoverable strains or permanent deformation after load is removed.
– slipping does not generally result in any volumetric strains (condition of incompressibility), unlike elasticity
Rate-Independent Plasticity:
If the material response is not dependent on the rate of loading or deformation, the material is said to be rate-independent.
– Most metals exhibit rate-independent behavior at low temperatures (< 1/4 or 1/3 melting temperature) and low strain rates.
Engineering vs. True Stress-Strain:
While engineering stress-strain can be used for small-strain analyses, true stress-strain must be used for plasticity, as they are more representative measures of the state of the material.
Engineering vs. True Stress-Strain (cont’d):
If presented with engineering stress-strain data, one can convert these values to true stress-strain with the following approximations:
– Up until twice the strain at which yielding occurs:
– Up until the point at which necking occurs: 
– Note that, only for stress conversion, the following is assumed:
Material is incompressible (acceptable approximation for large strains)
Stress distribution across cross-section of specimen is assumed to be uniform.
– Beyond necking:
There is no conversion equation relating engineering to true stress-strain at necking. The instantaneous cross-section must be measured.
B. Yield Criterion
The yield criteria is used to relate multiaxial stress state with the uniaxial case.
– Tensile testing on specimens provide uniaxial data, which can easily be plotted on one-dimensional stress-strain curves, such as those presented earlier in this section.
– The actual structure usually exhibits multiaxial stress state. The yield criterion provides a scalar invariant measure of the stress state of the material which can be compared with the uniaxial case.
In general, a stress state can be separated into two components.
– Hydrostatic stress - generates volume change.
– Deviatoric stress - generates angular distortion.
The von Mises yield criterion predicts that yielding will occur whenever the distortion energy in a unit volume equals the distortion energy in the same volume when uniaxially stressed to the yield strength.
– From this theory, a scalar invariant (von Mises equivalent stress) is derived as:
When von Mises equivalent stress exceeds the uniaxial material yield strength, general yielding will occur.
If plotted in 3D principal stress space, the von Mises yield surface is a cylinder.
At the edge of the cylinder (circle), yielding will occur.
No stress state can exist outside of the cylinder.
Instead, hardening rules will describe how the cylinder changes with respect to yielding.
C. Hardening Rules
The hardening rule describes how the yield surface changes (size, center,shape) as the result of plastic deformation.
The hardening rule determines when the material will yield again if the loading is continued or reversed.
– This is in contrast to elastic-perfectly-plastic materials which exhibit no hardening -- i.e., the yield surface remains fixed.
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