Theories of failure

Over view

Stress- strain curve

Hardening types

Combined stresses

Principal stresses, Hydrostatic, Deviatoric and octahedral stresses.

Failure theories

Yield function

Yield criteria

Flow rule

Various material models

Finite element example

Uniaxial loading

Simple tension

For some materials yield point is so poorly defined that it is taked as 0.2 percent of the permanent strain.

A few materials such as annealed mild steel, exhibit a sharp drop in yield after the upper yield point B is reached, this is because of the luder bands.

True stress can be obtained from the nominal stresses by considering no volumetric change.

Uniaxial loading

True stress

True strain

Hardening Types

Kinematic Hardening: Elastic range remains constant

Isotropic Hardening: yield in tension is same as yield in compression

Hardening types

Mixed hardening: Yield in tension and compression are independent

Actual Hardening is above the mixed hardening yield in compression

Stress strain idealized curves

Perfectly linear elasticElastic Perfectly plastic

Rigid perfectly plasticBilinear hardening material

M

M

M

Emperical equations for stress strain curves

Ludwick equation

Ramberg Osgood equation

Tangent modulus

Multi directional loads

Unit stress:

The unit stress is not normal to the plane. The value of unit stress is referred to a particular plane.

Stress at a point

Representation of stress at a point

Stresses on an arbitrary plane

l, m and n be the direction cosines of the normal acting on plane ABC, then

Cauchy's stress

Stresses on an arbitrary plane

Principal stresses

Consider a plane on which the resultant stress is perpendicular to the plane

Substituting the above values in the Cauchy's stress equation we get

In indical notation

Prinicpal stresses and Invariants

For a solution to be non- trivial, determinant of the three equations is zero

Where

at a point on any plane the values of these invariants doesn't change.

Please write the values of invariants in other formats alsoPrincipal stresses and Invariants

The cubic equation has three real roots and consequently three principal stresses and .

From the pricipal stress values we can get the eigen vectors l,m and n, if in addition

If Hydrostatic stress (any three perpendicular directions are principal)

If all principal directions are unique and orthogonal.

If one principal direction will be unique, but the other two directions can be any two directions orthogonal to first.

If and are the co-ordinate axes then

Octahedral shear stresses

Hydrostatic and deviatoric stress tensor

Stress tensor can be divided in to hydrostatic part and deviatoric part.

The value of hydrostatic tensor is same for any streess state at a point

principal deviatoric stresses can be found out similarly like the previous one

Pricipal deviatoric stresses

Where

in terms of pricipal deviatoric stresses

Haigh-Westergaard Stress Space

How to geomertrically represent a stress state?a) Considering six indepenedent stresses as six components of positional coordinates.b) Use the pricipal stresses.

Substituting the above values in the Cauchy's stress equation we get