Theories of (static) Failure
Consider biaxial (plane stress) case first:
Basic idea is that if some combination of the
principal stresses gets too large, the material will
fail. We can think of the "safe" stresses as defining
some region in terms of principal stress
components:
σ p2
failure
failure
safe region
σ p1
failure
failure
Maximum Normal Stress Theory
Failure occurs if either σ p1 or σ p 2 = σ f where σ f
fracture stress as determined in a uniaxial tension test
σf
σ p2
σf
σf
σ p1
is the
σ p2
σ p1
cast iron
Maximum Normal Stress Theory
1. Used to describe fracture of brittle materials such as cast iron
2. Limitations
doesn't distinguish between tension or compression
doesn't depend on orientation of principal planes so only
applicable to isotropic materials
3. Generalization to 3-D stress case is easy:
σ p2
σ p1
σ p3
Maximum Shearing Stress (Tresca) Theory
Failure by slip (yielding) occurs when the maximum shearing stress, τ max
exceeds the yield stress τ f as determined in a uniaxial tension test
τf
σf
0
For biaxial (plane stress) case
τ max
⎧ σ p1 − σ p 2 / 2
⎪
⎪
= τ f = σ f / 2 = max ⎨ σ p1 / 2
⎪
⎪⎩ σ p 2 / 2
σf
τ max
⎧ σ p1 − σ p 2 / 2
⎪
⎪
= τ f = σ f / 2 = max ⎨ σ p1 / 2
⎪
⎪⎩ σ p 2 / 2
defines a hexagon in terms of the
principal stresses
σ p2
σ p2 = σ f
σ p 2 − σ p1 = σ f
σ p1 = σ f
45 D
σ p1 = −σ f
σ p1
σ p1 − σ p 2 = σ f
σ p 2 = −σ f
1. Limitation – doesn't depend on orientations of planes of
extreme shear so only strictly applicable to isotropic materials
Maximum Distortional Strain Energy (von Mises) Theory
elongation
δ
For 1-D state of stress
axial strain
σf
P
load
L
e=
δ
L
A
Work done by P = strain energy (potential energy) of the bar
Wk
=
U
Wk = ∫ Pd δ
= ∫ σ Ad [ eL ]
= AL ∫ σ de
∫
so that U = AL σ de
AL =V = volume of bar
u=
and
U
= σ de = strain energy/unit volume
V ∫
(strain energy density)
Maximum Distortional Strain Energy (von Mises) Theory
u = ∫ σ de
But σ = Ee so
u = E ∫ e de
Ee 2 σ e
=
=
2
2
For a general 3-D state of stress with three principal stresses
1
u = (σ p1e p1 + σ p 2 e p 2 + σ p 3e p 3 )
2
Maximum Distortional Strain Energy (von Mises) Theory
u=
1
σ p1e p1 + σ p 2 e p 2 + σ p 3e p 3 )
(
2
By generalized Hooke's law
1
⎡σ p1 −ν (σ p 2 + σ p 3 ) ⎤
⎦
E⎣
1
e p 2 = ⎡⎣σ p 2 −ν (σ p1 + σ p 3 ) ⎤⎦
E
1
e p 3 = ⎡⎣σ p 3 −ν (σ p1 + σ p 2 ) ⎤⎦
E
e p1 =
so
u=
1
⎡σ p21 + σ p2 2 + σ p2 3 − 2ν (σ p1σ p 2 + σ p1σ p 3 + σ p 2σ p 3 ) ⎤
⎦
2E ⎣
Maximum Distortional Strain Energy (von Mises) Theory
u=
1
⎡σ p21 + σ p2 2 + σ p2 3 − 2ν (σ p1σ p 2 + σ p1σ p 3 + σ p 2σ p 3 ) ⎤
⎦
2E ⎣
Experiments have shown that a pure hydrostatic pressure will not
cause yielding even under extremely large stresses so that failure
by slip must be independent of the hydrostatic part of u.
Let
σ p1 = σ p 2 = σ p 3 = − p
Then
up =
and if we let
we have
3 (1 − 2ν ) 2
1
⎡⎣3 p 2 − 6ν p 2 ⎤⎦ = −
p
2E
2E
p=
up =
− (σ p1 + σ p 2 + σ p 3 )
3
(1 − 2ν ) σ + σ + σ 2
( p1 p 2 p3 )
6E
Maximum Distortional Strain Energy (von Mises) Theory
Thus if we define the distortional strain energy
density , ud , as
ud = u − u p
it can be shown that
ud =
(1 +ν ) ⎡ σ
(
6 E ⎢⎣
2
2
2
σ
σ
σ
σ
σ
−
+
−
+
−
( p1 p3 ) ( p 2 p3 ) ⎤⎥⎦
p1
p2 )
The von Mises failure theory predicts failure with respect to slip
(yielding) whenever ud equals uf as determined by a uniaxial tension
test
1 +ν )
(
⎡⎣ 2σ 2f ⎤⎦
uf =
6E
Maximum Distortional Strain Energy (von Mises) Theory
ud = u f
gives
(σ p1 − σ p 2 ) + (σ p1 − σ p3 ) + (σ p 2 − σ p3 ) = 2σ 2f
2
2
2
For the biaxial (plane stress) case this reduces to
σ p21 − σ p1σ p 2 + σ p2 2 = σ 2f
rotated ellipse
σ p2
σf
45 D
σf
σ p1
Maximum Distortional Strain Energy (von Mises) Theory
Recall that the total shear stress on the octahedral plane is given by
τ oct =
1
3
(σ p1 − σ p 2 ) + (σ p1 − σ p3 ) + (σ p 2 − σ p3 )
2
2
2
so that if we say that failure occurs when τ oct equals the value of the
octahedral stress at failure in a uniaxial tension test given by
(τ oct ) f =
1
2σ 2f
3
we get again
(σ p1 − σ p 2 ) + (σ p1 − σ p3 ) + (σ p 2 − σ p3 ) = 2σ 2f
2
2
2
Maximum Distortional Strain Energy (von Mises) Theory
1. In formulating this failure theory we used generalized Hooke's law for
an isotropic material so the theory given is only applicable to those
materials but it can be generalized to anisotropic materials.
2. The von Mises theory is a little less conservative than the Tresca
theory but in most cases there is little difference in their predictions of
failure. Most experimental results tend to fall on or between these two
theories.
σ p2
45 D
σ p1
steel
copper
aluminum
Tresca and von Mises Theories for 3-D stresses
Tresca
von Mises
τ max
(σ
⎧ σ p1 − σ p 2 / 2
⎪
⎪
= τ f = σ f / 2 = max ⎨ σ p1 − σ p 3 / 2
⎪
⎪⎩ σ p 2 − σ p 3 / 2
2
−
σ
+
σ
−
σ
+
σ
−
σ
=
2
σ
)
(
)
(
)
p1
p2
p1
p3
p2
p3
f
2
2
2
Both of these equations remain unchanged if we add equal principal
stresses to all components, i.e.
σ ′p1 = σ p1 + σ
σ ′p 2 = σ p 2 + σ
σ ′p 3 = σ p 3 + σ
since both theories are independent of adding or subtracting a
hydrostatic component
This means that in 3-D the yield surfaces are cylinders whose sides are
parallel to a line that makes equal angles with all three principal stress
directions
σ p2
n=
1
1
1
e p1 +
e p2 +
e p3
3
3
3
σ p1
σ p3
hexagon for Tresca, ellipse for von Mises