Introduction to Plasticity:
Plastic Deformation, and yield criteria:
States of stress
When a body is subjected to a stress below the yield strength, it will deform elastically.
The moment the stress is removed, the body comes to initial position.
In contrast, when the body is stressed beyond the yield point, it will undergo permanent
deformation. If it is a ductile material, it will plastically deform continuously with increase in
stress applied.
If a certain object is subjected to uniaxial tensile load, it will start yielding –
deforming plastically – when the stress reaches the uniaxial yield stress Y.
However, when the state of stress is triaxial, a single shear stress can not be used to
predict yielding.
It is the combination of the three stress states which alone can predict yielding.
The relationship among the stresses which predict the yielding of a material is called yield
criterion. The inherent assumptions involved in defining the yielding are: the material is
isotropic & incompressible, Poisson’s ratio equals 0.5 and the hydrostatic or mean stress does
not cause yielding of the material. Porous materials like powder metallurgy alloys can be
assumed compressible. They have Poisson’s ratio less than 0.5.
Commonly, for ductile materials, there are two important yield criteria. They are von Mises
yield criterion – also called distortion energy criterion and Tresca criterion also called Maximum
shear stress theory.
The hydrostatic stress is given by:
Total state of stress at a point can be represented as sum of hydrostatic and deviatoric stresses.
For plane stress, the deviatoric stress is given by: etc .
,Yielding in normal materials is caused by the deviatoric stress
-(σx-σy)/2
σy (σx+σy)/2
τyx
(σx+σy)/2 (σx-σy)/2
σx
= +
Total Stress Hydrostatic stress Deviator stress
Fig.1: States of stress on a plane
From the above figures, we could understand that the given state of biaxial stress can be
replaced by a sum of hydrostatic and deviatoric stresses. Hydrostatic stress, though does not
influence the yielding, it does increase ductility of a material, when it is applied.
Yield criteria:
Commencement of plastic deformation in materials is predicted by yield criteria. Yield criteria
are also called theories of yielding. A number of yield criteria have been developed for ductile
and brittle materials.
Tresca yield criterion:
It states that when the maximum shear stress within an element is equal to or greater than
a critical value, yielding will begin.
τmax k
Where k is shear yield strength.
Or τmax = (σ1 – σ3)/2 = k where σ1 and σ3 are principal stresses
Or σ1 – σ3 = Y
For uniaxial tension, we have k = Y/2
Here Y or k are material properties. The intermediate stress σ2 has no effect on yielding.
Von Mises criterion:
According to this criterion, yielding occurs when
, For plane strain condition, we have: σ2=( σ1 + σ3)/2
Hence, from the distortion energy criterion, we have σ1 – σ3 = Here, is called plane
strain yield strength. Von Mises criterion can also be interpreted as the yield criterion which
states that when octahedral shear stress reaches critical value, yielding commences.
The octahedral shear stress is the shear stresses acting on the faces of an octahedron, given by:
]1/2
According to Tresca criteria we know, (σ1 – σ3)/2 = k.
Therefore, k =
σ3, Tension
Tresca Criterion
Y
Von Mises Criterion
Y
Compression σ1, Tension
Fig. 2: Yield loci for the two yield criteria in plane stress
Von Mises yield criterion is found to be suitable for most of the ductile materials used in
forming operations. More often in metal forming, this criterion is used for the analysis. The
suitability of the yield criteria has been experimentally verified by conducting torsion test on
thin walled tube, as the thin walled tube ensures plane stress. However, the use of Tresca
criterion is found to result in negligible difference between the two criteria. We observe that
, the von Mises criterion is able to predict the yielding independent of the sign of the stresses
because this criterion has square terms of the shear stresses.
Effective stress and effective strain:
Effective stress is defined as that stress which when reaches critical value, yielding can
commence.
For Tresca criterion, effective stress is σeff= σ1 – σ3
For von Mises criterion, the effective stress is
1/2
The factor 1/ is chosen such that the effective stress for uniaxial tensile loading is equal
to uniaxial yield strength Y.
The corresponding effective strain is defined as:
εeff =
From von Mises criterion:
1/2
Effective strain = (√2/3)
For Tresca:
Effective strain = (2/3)
For uniaxial loading, the effective strain is equal to uniaxial tensile strain.
Note: The constants in effective strain expressions, given above are chosen so that for uniaxial
loading, the effective strain reduces to uniaxial strain.
Normal strain versus shear strain:
We know for pure shear: σ1 = - σ3 and σ1 = τ
Therefore from the effective stress equation of Tresca we get: Effective stress = 2σ 1 = 2τ1
Similarly using von Mises effective stress, we have
Plastic Deformation, and yield criteria:
States of stress
When a body is subjected to a stress below the yield strength, it will deform elastically.
The moment the stress is removed, the body comes to initial position.
In contrast, when the body is stressed beyond the yield point, it will undergo permanent
deformation. If it is a ductile material, it will plastically deform continuously with increase in
stress applied.
If a certain object is subjected to uniaxial tensile load, it will start yielding –
deforming plastically – when the stress reaches the uniaxial yield stress Y.
However, when the state of stress is triaxial, a single shear stress can not be used to
predict yielding.
It is the combination of the three stress states which alone can predict yielding.
The relationship among the stresses which predict the yielding of a material is called yield
criterion. The inherent assumptions involved in defining the yielding are: the material is
isotropic & incompressible, Poisson’s ratio equals 0.5 and the hydrostatic or mean stress does
not cause yielding of the material. Porous materials like powder metallurgy alloys can be
assumed compressible. They have Poisson’s ratio less than 0.5.
Commonly, for ductile materials, there are two important yield criteria. They are von Mises
yield criterion – also called distortion energy criterion and Tresca criterion also called Maximum
shear stress theory.
The hydrostatic stress is given by:
Total state of stress at a point can be represented as sum of hydrostatic and deviatoric stresses.
For plane stress, the deviatoric stress is given by: etc .
,Yielding in normal materials is caused by the deviatoric stress
-(σx-σy)/2
σy (σx+σy)/2
τyx
(σx+σy)/2 (σx-σy)/2
σx
= +
Total Stress Hydrostatic stress Deviator stress
Fig.1: States of stress on a plane
From the above figures, we could understand that the given state of biaxial stress can be
replaced by a sum of hydrostatic and deviatoric stresses. Hydrostatic stress, though does not
influence the yielding, it does increase ductility of a material, when it is applied.
Yield criteria:
Commencement of plastic deformation in materials is predicted by yield criteria. Yield criteria
are also called theories of yielding. A number of yield criteria have been developed for ductile
and brittle materials.
Tresca yield criterion:
It states that when the maximum shear stress within an element is equal to or greater than
a critical value, yielding will begin.
τmax k
Where k is shear yield strength.
Or τmax = (σ1 – σ3)/2 = k where σ1 and σ3 are principal stresses
Or σ1 – σ3 = Y
For uniaxial tension, we have k = Y/2
Here Y or k are material properties. The intermediate stress σ2 has no effect on yielding.
Von Mises criterion:
According to this criterion, yielding occurs when
, For plane strain condition, we have: σ2=( σ1 + σ3)/2
Hence, from the distortion energy criterion, we have σ1 – σ3 = Here, is called plane
strain yield strength. Von Mises criterion can also be interpreted as the yield criterion which
states that when octahedral shear stress reaches critical value, yielding commences.
The octahedral shear stress is the shear stresses acting on the faces of an octahedron, given by:
]1/2
According to Tresca criteria we know, (σ1 – σ3)/2 = k.
Therefore, k =
σ3, Tension
Tresca Criterion
Y
Von Mises Criterion
Y
Compression σ1, Tension
Fig. 2: Yield loci for the two yield criteria in plane stress
Von Mises yield criterion is found to be suitable for most of the ductile materials used in
forming operations. More often in metal forming, this criterion is used for the analysis. The
suitability of the yield criteria has been experimentally verified by conducting torsion test on
thin walled tube, as the thin walled tube ensures plane stress. However, the use of Tresca
criterion is found to result in negligible difference between the two criteria. We observe that
, the von Mises criterion is able to predict the yielding independent of the sign of the stresses
because this criterion has square terms of the shear stresses.
Effective stress and effective strain:
Effective stress is defined as that stress which when reaches critical value, yielding can
commence.
For Tresca criterion, effective stress is σeff= σ1 – σ3
For von Mises criterion, the effective stress is
1/2
The factor 1/ is chosen such that the effective stress for uniaxial tensile loading is equal
to uniaxial yield strength Y.
The corresponding effective strain is defined as:
εeff =
From von Mises criterion:
1/2
Effective strain = (√2/3)
For Tresca:
Effective strain = (2/3)
For uniaxial loading, the effective strain is equal to uniaxial tensile strain.
Note: The constants in effective strain expressions, given above are chosen so that for uniaxial
loading, the effective strain reduces to uniaxial strain.
Normal strain versus shear strain:
We know for pure shear: σ1 = - σ3 and σ1 = τ
Therefore from the effective stress equation of Tresca we get: Effective stress = 2σ 1 = 2τ1
Similarly using von Mises effective stress, we have
