塑性成形理论-stress and strain relationship

The displacements of elements in the plastically deforming
4 Stress-strain relations
4.1 Elastic stress-strain relations 4.2 Plastic stress-strain relations
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.
6d
y
2 z
6 yz2d2
(dy dz )2 ( y z )2d2
6d zx 2 6 zx 2d2
(dz dx )2 ( z x )2 d2
6d zx 2 6 zx 2d2
9 d 2
2
x y
2 y z
2 z x
2
6
2 xy
2 yz
2 zx
2 d2
1
2
( x
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.
ijl j pi
ui su ui0 vi sv vi0
Boundary conditions
x
ux x
y
u y y
z
uz z
xy
1 (ux 2 y
u y x
)
yz
1 (uy 2 z
uz ) y
zx
1 (uz 2 x
ux z
)
4.4 Fundamental equations for plane plastic flow and plane strain
4.2.2 Prandtl-Reuss equations
The stress-strain relations for an elastic-perfectly plastic material were first proposed by Prandtl in 1924 for plane strain deformation and generalized independently by Reuss in 1930.
[
y
( z
x )]
z
1 E
[
z
( x
y )]
xy
1 2G
xy
yz
1 2G
yz
zx
1 2G
zx
Generalized Hooke’s law
(4-1)
E —Young’s modulus
— Poisson’s ratio
G — the modulus of rigidity
G E
2(1 )
Assumed that the plastic strain increment is, at any instant of
loading, proportional to the instantaneous stress deviation and the
shear stresses ,
d
p x
O→A →C →E →F
Subsequent yield
p x
p y
p z
p x
/
2
p xy
p yz
p zx
ቤተ መጻሕፍቲ ባይዱ
0
Initial yield O→B →D →G →F
p xy
p x
p y
p z
p yz
p zx
0
Stress-strain relations
4.2 Plastic stress-strain relations
(4-7)
where,
1
E
1
2
x y z
Stress-strain relations
4.2 Plastic stress-strain relations
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.
Hooke’s law become,
ij
1 2G
ij
3
E
m
ij
(4-3)
Summing the first three equations in (4-1) ,
m
1 2
E
m
(4-4)
Showing that mean strain be proportional to mean stress, if the
with axial symmetry
4.4.1 Equations for plane plastic flow
4.4.2 Equations for plastic strain with axial symmetry
4.4.1 Equations for plane plastic flow
independently reaffirmed by von Mises in 1913.
d 3d
(4-9)
2
These equations are strictly applicable to a rigid-perfectly
plastic material for which the elastic component of the total
1 1 2
ij
ij
m 2G ij
E
m ij
(4-5) (4-6)
4.1 Elastic stress-strain relations
From equation (4-3),(4-4),the stress can be expressed in strain
ij 2Gij ij
volume be not change,
1
2
Stress-strain relations
4.1 Elastic stress-strain relations
x
x
m
1 E
[
x
( y
z
)]
1
2
E
m
1 2G
x
3
E
m
1
2
E
m
1
E
( x
m)
1 2G
x
Therefore,
ij
1 2G
ij
Generalized Hook’s law can be written as,
Stress-strain relations
4.3 Solution of plastic forming
x
x
yx
y
zx
z
px
0
xy
x
y
y
zy
z
py
0
xz
x
yz
y
z
z
pz
0
(
x
y
)2
(
y
z
)2
(
z
x
)2
6(
2 xy
2 yz
2 zx
)
2
2 s
6k 2
d x d y d z d xy d yz d zx d x y z xy yz zx
4.1 Elastic stress-strain relations
Hooke first proposed a linear relation between stress and strain for a uniaxial stress state.
x
1 E
[ x
(
y
z )]
y
1 E
When plastic deformation occurs, the strain state is dependent on stress history and the stress-strain relation is generally nonlinear.
Stress-strain relations
d
p y
d
p z
d
p xy
d
p yz
d
p zx
d
x y z xy yz zx
In tensor notation, d p d
ij
ij
(4-10)
d is a scalar non-negative constant of proportionality which
is not a material constant and may vary throughout the stress history.
strain increment is zero.
d 2 3
d x d y
2
d y d z
2
d z d x
2
6
d
2 xy
d
2 yz
d
2 zx
2
9 d 2
2
d x d y
2
d y d z
2
d z d x
2
6
d
2 xy
d
2 yz
d
2 zx
2
(dx dy)2 ( x y)2d2
dij
dijp
d
e ij
(4-11)
d
e ij
d ij
G
1 2
E
d mij
d
p ij
d ij
(4-12)
d 3d p 2
Stress-strain relations
4.2 Plastic stress-strain relations
Example:
A long, thin-walled tube with caped ends is pressurized internally until it yields. As pressure is increased, explain what happens to the length of the tube because of plastic effects only.
The Levy-von Mises equations The Prandtl-Reuss equations
Stress-strain relations
4.2 Plastic stress-strain relations
4.2.1 Levy-Mises equations
The strain increment is proportional to the instantaneous stress
y)2
( y
z)2
( z
x )2
6( xy2
y
2 z
zx 2 )
2
2
( x
y)2
( y
z )2
( z
x )2
6(
2 xy
y
2 z
zx 2 )
9 d 2 2 2d2
2
d 3 d 2
Stress-strain relations
4.2 Plastic stress-strain relations
(4-2)
Stress-strain relations
4.1 Elastic stress-strain relations
The first equation in (4-1) can be written as
x
1 [(1 E
) x
( x
y
z )]
1 2G
x
3
E
m
Similar expressions can be obtained for y and z, the generalized
However, if a proportional stress path is followed, such that all the stresses increase in the same ratio, then the plastic strain state is independent of the stress history and depends only on the final stress state.
deviation and the shear stress such that
d x d y d z d xy d yz d zx d x y z xy yz zx
or dij d ij
(4-8)
This equation was originally proposed by Levy in 1871 and
Stress-strain relations
4.2 Plastic stress-strain relations
4.2.1 Prandtl-Reuss equations
The total strain increment is the sum of the elastic strain increment and the plastic strain increment,