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