Metal Forming & Machining

MF C314

Yield Conditions and Stress Strain Relations

For analysis of metal forming problem, it isnecessary to know the conditions of yielding,work hardening and stress-strain relations ofmetal under the action of multi-axial stressesmetal under the action of multi-axial stressesand strains.

The elastic stress-strain laws are based onconservation of energy.

On the other hand, the plastic stress-strain lawsare based on dissipation of energy.

Yield Conditions Stress at a point

Stress at a point

For uni-axial stress state For uni-axial stress state

The material is in the plastic state if

Yield condition in multi-axial stress state

Yield condition is a material property, therefore, theyield function for homogeneous and isotropicmaterials will be function of invariants of stresses.According to Bridgeman, the hydrostatic pressure According to Bridgeman, the hydrostatic pressuredoes not make a metal yield. It only produces elasticvolume change.

Conversion of stresses at a point as sum of twofactors.

First factor is called deviator component. Second factor is called hydrostatic pressure. Any stress component of the deviator part may be Any stress component of the deviator part may be

written as below,

is the Kronecker delta and it has following valuewhenwhen

Hydrostatic component produces change in the volume of the body. Its value for the general stress state is given by

In terms of principal stresses it can be written as

The second factor of the stress state (deviator part) is responsible for the change in shape of the body.

The deviator components can be written as:

The yield function is a function of invariants of theabove components of the deviator part of the stresstensor.

Let be the three invariants of thedeviator part of the stress tensor.

It can be written as It can be written as

The value of first invariant is zero.

The yield function can be written as

Expanded form of the function

Different yield criteria may be constructed depending upon the values of constant A,B,C,D etc and number of terms taken in the above expression.

Yield Functions

Von Mises hypothesis of yieldingTrescas hypothesis of yielding

Von Mises hypothesis of yielding First two terms of general equations:

Von Mises hypothesis: A stressed metal body reaches theyield point if the following equation is satisfied.

In terms of principal stresses it will be

For uniaxial loading

Relationship of yield strength of metal in shear to its yield strength in tension.

Trescas Hypothesis of yielding A metal body reaches the yield point when the

maximum shear stress in the body reaches the valueequal to the yield strength of the metal in shear.

Relations among three principal stresses and threemaximum shear stresses.

Calculate greatest value of shear stress:

Trescas hypothesis:

For uniaxial loading

According to the Trescas hypothesis, the yieldingstrength in shear is equal to the half the yieldstrength in tension.

General form of Tresca Hypothesis.

If any of the three factors on the left hand side of theabove equation becomes zero, material reachesplastic state.

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