Theories of filure

2. Theories of Failure

3. Theories of Failure
• Why do parts fail?
• You may say “Parts fail because their stresses exceed their
strength”
• Then what kind of stresses cause the failure: Tensile?
Compressive? Shear?
• Answer may be: It depends.
• It depends on the material and its relative strength in
compressive, tension, and shear.
• It also depends on the type loading (Static, Fatigue, Impact) and
• presence of the cracks in the material

4. Theories of Failure
• The failure may be elastic or fracture
• Elastic failure results in excessive deformation, which makes the
machine component unfit to perform its function satisfactorily
• Fracture results in breaking the component into two parts

5. Theories of Failure
• Question: How do one compare stresses induced to the material
properties?
–Generally machine parts are subjected to combined loading and to
find material properties under real loading condition is practically
not economical
–Thus, material properties are obtained from simple
tension/torsion test
–These data like 𝑆𝑦𝑡, 𝑆𝑢𝑡 etc. are available in form of table (Design
Data Book)

6. Theories of Failure
• Theories of failure provide a relationship between the strength of
machine component subjected to complex state of stress with the
material properties obtained from simple test (Tensile)
Strength of machine
component subjected
to complex state of
stress
Strength of standard
component subjected
to uniaxial state of
stress

7. Theories of Failure
• Loads are assumed to not vary over time
• Failure theories that apply to:
– Ductile materials
– Brittle materials
• Why do we need different theories ??
Stress-strain curve of a ductile material Stress-strain curve of a brittle material

8. Theories of Failure – Tension Test
Failure along principal
shear stress plane
Failure along principal
normal stress plane
Cast iron has C between
2.1% to 4% and Si between
1% and 3% C contents less
than 2.1% are steels.

9. Theories of Failure – Compression Test
Why doesn’t it fail ?? Why does it fail ??
Why nearly 45o ??
Shear failure

10. Theories of Failure
• In general, ductile, isotropic materials are limited by their shear
strengths.
• Brittle materials are limited by their tensile strengths.
• If cracks are present in a ductile material, it can suddenly fracture at nominal
stress levels well below its yield strength, even under static loads.
• Static loads are slowly applied and remain constant with time.
• Dynamic loads are suddenly applied (impact), or repeatedly varied with time
(fatigue), or both.
• Ductile materials often fail like brittle materials in dynamic loading.

11. Theories of Failure

12. Theories of Failure
⚫ There is no universal theory of failure for the general case of material properties
and stress state. Instead, over the years several hypotheses have been formulated
and tested, leading to today’s accepted practices most designers do.
⚫ The generally accepted theories are:
Ductile materials (yield criteria)
⚫ Maximum shear stress (MSS)
⚫ Distortion energy (DE)
Brittle materials (fracture criteria)
⚫ Maximum normal stress (MNS)
⚫ Brittle Coulomb-Mohr (BCM)
⚫ Modified Mohr (MM)

13. Maximum Shear Stress Theory for Ductile materials
The most common type of yielding of a ductile material, such
as steel, is caused by slipping, which occurs between the
contact planes of randomly ordered crystals that make up the
material. If a specimen is made into a highly polished thin strip
and subjected to a simple tension test, it then becomes possible
to see how this slipping causes the material to yield, as shown
in Fig. The edges of the planes of slipping as they appear on
the surface of the strip are referred to as Lüder’s lines. These
lines clearly indicate the slip planes in the strip, which occur at
approximately 45° as shown.

14. Maximum Shear Stress Theory for Ductile materials
The slipping that occurs is caused by shear stress. To show
this, consider an element of the material taken from a
tension specimen, Fig. a, when the specimen is subjected to
the yield stress 𝝈𝒀 . The maximum shear stress can be
determined from Mohr’s circle, Fig. b. The results indicate
that

15. Maximum Shear Stress Theory for Ductile materials
Furthermore, this shear stress acts on planes that are 45°
from the planes of principal stress, Fig. c, and since these
planes coincide with the direction of the Lüder lines shown
on the specimen, this indeed indicates that failure occurs by
shear.
Realizing that ductile materials fail by shear, in 1868 Henri Tresca proposed the
maximum shear stress theory or Tresca yield criterion.

16. Maximum Shear Stress Theory for Ductile materials
This theory states that
“Regardless of the loading, yielding of the material begins when the absolute maximum
shear stress in the material reaches the shear stress that causes the same material to yield
when it is subjected only to axial tension”.
Therefore, to avoid failure, it is required that 𝝉𝒂𝒃𝒔𝒎𝒂𝒙
in the material must be ≤ Τ
𝝈𝒀
𝟐,
where 𝛔𝐘 is determined from a simple tension test.
Absolute maximum shear stress is represented in terms of the principal stresses 𝛔𝟏 and 𝛔𝟐. If
these two principal stresses have the same sign, i.e., they are both tensile or both compressive,
then failure will occur out of the plane i.e.

17. Maximum Shear Stress Theory for Ductile materials
If instead the principal stresses are of opposite signs,
then failure occurs in the plane i.e.
With these two equations and Eq. of 𝝉𝒎𝒂𝒙 , the
maximum shear stress theory for plane stress can
therefore be expressed by the following criteria:
A graph of these equations is shown in Fig. Therefore, if
any point of the material is subjected to plane stress
represented by the coordinates (𝛔𝟏, 𝛔𝟐) that fall on the
boundary or outside the shaded hexagonal area, the
material will yield at the point and failure is said to
occur.

18. Maximum Distortion Energy Theory for Ductile materials
An external load will deform a material, causing it to store energy
internally throughout its volume. The energy per unit volume of
material is called the strain energy density, and if the material is
subjected to a uniaxial stress the strain-energy density, defined by
If the material is subjected to triaxial stress, Fig. a, then each principal stress
contributes a portion of the total strain-energy density, so that

19. Maximum Distortion Energy Theory for Ductile materials
This strain-energy density can be considered as the sum of two parts.
One part is the energy needed to cause a volume change (hydrostatic
stress) of the element with no change in shape, and the other part is the
energy needed to distort the element. Specifically, the energy stored in
the element as a result of its volume being changed is caused by
application of the average principal stress,

20. Maximum Distortion Energy Theory for Ductile materials
Experimental evidence has shown that materials do not yield when they are
subjected to a uniform (hydrostatic) stress, such as 𝜎𝑎𝑣𝑔. As a result, in
1904, M. Huber proposed that
“Yielding in a ductile material occurs when the distortion energy per unit volume of
the material equals or exceeds the distortion energy per unit volume of the same
material when it is subjected to yielding in a simple tension test”.
This theory is called the maximum distortion energy theory, also known as R. von
Mises theory.

21. Maximum Distortion Energy Theory for Ductile materials

22. Maximum Distortion Energy Theory for Ductile materials

23. Maximum Distortion Energy Theory for Ductile materials

24. Maximum Normal Stress Theory for Brittle materials
Maximum normal stress theory states that
“when a brittle material is subjected to a multiaxial state of stress, the
material will fail when a principal tensile stress in the material reaches a
value that is equal to the ultimate normal stress the material can sustain
when it is subjected to simple tension”.
Therefore, if the material is subjected to plane stress, we require that

25. Maximum Normal Stress Theory for Brittle materials
These equations are shown graphically in Fig. Here the stress coordinates
(𝛔𝟏, 𝛔𝟐) at a point in the material must not fall on the boundary or outside
the shaded area, otherwise the material is said to fracture. This theory is
generally credited to W. Rankine, who proposed it in the mid-1800s.
Experimentally it has been found to be in close agreement with the behavior
of brittle materials that have stress–strain diagrams that are similar in both
tension and compression.

26. Mohr’s Failure Criterion for Brittle materials
In some brittle materials the tension and compression properties are
different. When this occurs, a criterion based on the use of Mohr’s circle
may be used to predict failure. This method was developed by Otto Mohr
and is sometimes referred to as Mohr’s failure criterion. To apply it, one first
performs three tests on the material.
A uniaxial tensile test and uniaxial compressive test are
used to determine the ultimate tensile and compressive
stresses 𝜎𝑢𝑙𝑡 𝑡 and 𝜎𝑢𝑙𝑡 𝑐 , respectively. Also, a
torsion test is performed to determine the material’s
ultimate shear stress 𝜏𝑢𝑙𝑡. Mohr’s circle for each of
these stress conditions is then plotted as shown in Fig.

27. Mohr’s Failure Criterion for Brittle materials