文档介绍：4 Stress-strain relations
Elastic stress-strain relations
Plastic stress-strain relations
Elastic strain state is a unique function of the elastic stress state pertaining at a given instant and is independent of how that stress state was attained.
When plastic deformation occurs, the strain state is dependent on stress history and the stress-strain relation is generally nonlinear.
Stress-strain relations
Generalized Hooke’s law
E —Young’s modulus
— Poisson’s ratio
G — the modulus of rigidity
Elastic stress-strain relations
Hooke first proposed a linear relation between stress and strain for a uniaxial stress state.
(4-1)
(4-2)
Stress-strain relations
The first equation in (5-1) can be written as
Summing the first three equations in (5-1) ,
Elastic stress-strain relations
Similar expressions can be obtained for y and z, the generalized Hooke’s law e,
(4-3)
Showing that mean strain be proportional to mean stress, if the volume be not change,
(4-4)
Stress-strain relations
Elastic stress-strain relations
Therefore,
(4-5)
Generalized Hook’s law can be written as,
(4-6)
Stress-strain relations
Elastic stress-strain relations
From equation (5-3),(5-4),the stress can be expressed in strain
where,
(4-7)
Elastic stress-strain relations
For plastic deform,
then
In plastic range,the stress-strain relations are generally nonlinear. The strains are not uniquely determined by the stress state but depend on the history of how the stress state was reached.
Initial yield
Subsequent yield
O→A →C →E →F
O→B →D →G →F
Plastic stress-strain relations
Stress-strain relations
Because of the dependence of the plastic strains on the stress path it is usually necessary to consider incremental plastic strains throughout the stress history and determine the total plastic strain by integration.
However, if a proportional stress path is followed, such that all the stresses increase in the same ratio, then the plastic strai