Transcript Title

SM2-10: Yielding & rupture criteria
YIELDING AND RUPTURE CRITERIA
(limit hypothesis)
M.Chrzanowski: Strength of Materials
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SM2-10: Yielding & rupture criteria
The knowledge of stress and strain states and displacements in each point of a
structure allows for design of its members. The dimensions of these members
should assure functional and safe exploitation of a structure.
In the simplest case of uniaxial tension (compression) it can be easily
accomplished as stress matrix is represented by one component 1 only, and
displacement along bar axis can be easily measured to determine axial strain 1
Measurements taken during the
?
expl
tensile test allow also for
determination of material
characteristics: elastic and plastic
expl<<Rm
limits as well as ultimate strength.
expl<RH
With these data one can easily
design tensile member of a structure
to assure its safety.
expl = 1 =RH /s
M.Chrzanowski: Strength of Materials
1
Rm
RH
Ultimate strength
Elastic limit
s-1
arctanE
1
Safety coefficient
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SM2-10: Yielding & rupture criteria
In the more complex states of stress (for example in combined bending and
shear) the evaluation of safe dimensioning (related to elastic limit) becomes
ambiguous.
txz
tzx
z
Do we need to satisfy two independent conditions
x< RH
x
x
tzx
txz
tx< RH
where RHt i RHs denote elastic limits in tension
and shear, respectively?
x
z
x
Transformation to the principal axis of stress matrix
does not help either, as we do not know whether the
modulus of combined stresses is smaller then RH …
Thus, we need to formulate a hypothesis defining
which stress components should be taken as basis
for safe structure design.
M.Chrzanowski: Strength of Materials
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SM2-10: Yielding & rupture criteria
In general case of 3D state of stress we introduce a function in 9-dimensional
space of all stress components (or 3-dimensional in the case of principal axes)
which are called the exertion function:
W  F ( ij )  f (1 , 2 , 3 )
0
Wlim
W  Wlim
In uniaxial sate of stress:
We postulate that exertion function will take the same
value in given 3D state of stress as that in uniaxial case.
The solution of this equation with respect to 0:
W  W0  F ( 0 )
W  F ( ij )  F ( 0 )
 0   ( ij )
is called substitute stress according to the adopted hypothesis defining
function F and thus – function , as well.
M.Chrzanowski: Strength of Materials
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SM2-10: Yielding & rupture criteria
Let the exertion measure be:

p

mW  p
stress vector in main principal
axis
2
RNR
N
RN
3
2
W  p   12   22   32
W0   02
W  W0
The ratio:
f 1, 2 , 2 
12   22   32  0 RN
R  0
0 N
1
RN
gives „the distance” from unsafe
state.
This distance can be dealt with as the
 1 exertion in a given point.
SUCH A HYPOTHESIS DOES NOT EXIST !
A very similar one, which does exist
mW  max1 ,  2 ,  3 
is called Gallieo-Clebsh-Rankine hypothesis
Associated  function appears to bo not-analytical one (derivatives on edges are
indefinable)
M.Chrzanowski: Strength of Materials
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SM2-10: Yielding & rupture criteria
GALIEO-CLEBSH-RANKINE hypothesis
(GCR)
mW  max1 ,  2 ,  3 
2
RN
 RN
1
 RN
1  RN  RN  1  RN
 2  RN  RN   2  RN
 3  RN  RN   3  RN
RN
RN
3
 RN
For plane stress state it
reduces to a square.
M.Chrzanowski: Strength of Materials
It is seen, that materials which obey this
hypothesis are isotopic with respect to
their strength.
They are also isonomic, as their
strength properties are identical for
tension and compression.
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 2 RN
SM2-10: Yielding & rupture criteria
1
Exertion ≤ 100%
Exertion ≤ 80%
1
1 RN
1
Exertion ≤ 60%
Exertion ≤ 40%
1
Exertion 0%
1
 2 RN
1
1
1 RN
1
M.Chrzanowski: Strength of Materials
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SM2-10: Yielding & rupture criteria
GALILEO hypothesis
2
Material isonomic and isotropic
RNr
RNtesion  RNcomrpessio n
RNr
sc
N
R
Material insensitive to compression.
(classical Galileo hypothesis)
mW  max 1 ,  2 ,  3

1
RNsc
Material isotropic but not
isonomic
where
a 
a
when a>0
0
when a<0
M.Chrzanowski: Strength of Materials
RNtension  RNcompressio n
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SM2-10: Yielding & rupture criteria
 2  1  t
For torsion:
COULOMB-TRESCA-GUEST hypothesis
CTG
2
t 0 
T 

0 t 
 2  RNr
RNr
Many materials are sensitive to torsion
This hexagon represents CoulombTresca hypothesis (for plane stress
state); the measure of exertion is
maximum shear stress:
R
mW  maxt1 , t 2 , t 3 
 1   2  2   3  3  1 
mW  max
,
,

2
2 
 2
In uniaxial state of stress:
 
mW0  max o 
 2 
M.Chrzanowski: Strength of Materials
RNr
sc
N
RNsc
1
Uniaxial
tension
1  RNr
 0  max1   2 ,  2   3 ,  3  1 
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SM2-10: Yielding & rupture criteria
HUBER-MISES-HENCKY hypothesis HMH
2
Small but important improvement has been made by
M.T. Huber followed by von Mises and Hencky:
RN
It is distortion energy only which decides on
the material exertion:
1
D D
2
1
1
   f   v  D D  A A
2
2
mW   f 
RN
RN
For elastic materials (Hooke law obeys):



1
1
 1   2 2   2   3 2   3   1 2
 ij ij  3 m2 
4G
6G
In uniaxial state of stress:
f 
 0f 
1
2 02
6G
0 
1
2

1
RN
1   2 2   2   3 2   3  1 2
In 3D space of principal stresses (Haigh space) this hypothesis is represented by a cylinder
with open ends. In 2D plane stress state for  3  0 is an ellipse shown above.
M.Chrzanowski: Strength of Materials
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SM2-10: Yielding & rupture criteria
Hypothesis
GCR
Maximum
normal stress
Exertion
measure
CTG
Maximum
shear stress
2
2
RN
RN
RN
RN
1
Substitute
stress for
beams
 0  max(1 ,  2 )
0 

1
 x   x2  4t xy2
2
M.Chrzanowski: Strength of Materials
RN
RN
RN
RN
Substitute
stress
Circular cylinder with
uniformly inclined
axis
2
RN
2D image
Deformation energy
Hexagonal prism
with uniformly
inclined axis
Cube with sides
equal to 2R
3D image
HMH
 0  1   2

 0   x2  4t xy2
1
RN
RN
1
RN
 0   12   22   1 2
 0   x2  3t xy2
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SM2-10: Yielding & rupture criteria
stop
M.Chrzanowski: Strength of Materials
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