MMU-307 DESIGN OF MACHINE ELEMENTS

FAILURES RESULTING FROM STATIC LOADING• Modification of the Mohr Theory for Brittle Materials

• Selection of Failure Theory

Asst. Prof. Özgür ÜNVER November 19th, 2019

Brittle materials, εf < 0.05, do not exhibit an identifiable yield strength,

Typically classified by ultimate tensile and compressive strengths, Sut and Suc, respectively (where Suc is given as a positive quantity).

Yield strength and Ultimate tensile strengths are very close to each other, in general they cannot be seperated.

Brittle Materials

•Brittle-Coulomb-Mohr (BCM) theory

•Modified Mohr (MM) theory

Two modifications of the Mohr theory for

brittle materials

Modifications of the Mohr Theory for Brittle Materials

Failure Theories for Brittle Materials

Brittle-Coulomb-Mohr Theory For Brittle Materials

Stress

Region

Mohr’s

Circle

Failure Factor of

Safety

sA,B >0

sA,B≤ 0

sA > 0

-Sut < sB

sB < 0

sA > 0

sB < -Sut

See

Equation

A

See

Equation

B

utA Ss

ucB Ss

A

utS

s

B

ucS

s

|sB| = Sut

Sut

Sut

utA SsA

utS

s

ButAutuc

utuc

SSS

SS

ss

1

ButAutuc

utuc

SSS

SS

ssA: B:

Which to use? (C-M or Mod-M)

In general, Mod-M is more accurate

Modified Mohr Theory For Brittle Materials

Modified Mohr Theory For Brittle Materials

Do not use these equations, instead;• Draw the mohr circle,• Draw the lines using line equations,• Use line intersection equations to find out the factor of safety.

• In the first quadrant the data appear on both sides and along the failure curves of Coulomb-Mohr and modified Mohr. All failure curves are the same, and data fit well.

• In the fourth quadrant the modified Mohr theory represents the data best.

• In the third quadrant the points A, B, C, and D are too few to make any suggestion concerning a fracture locus.

Failure of Brittle Materials Summary

Cast iron (ASTM grade 30)

‘F’ required to fracture the part using;

• Coulomb-Mohr Failure Theory

• Modified Mohr Failure Theory

Assume DC is strong enough!

Example

Given: Shaft of ASTM G25 cast iron subject to loading

Find: For a factor of safety of = 2.8, what should the diameter of the shaft be? (Sut = 26 kpsi, Suc = 97 kpsi)

Example

Ductile and Brittle Failure

• Tensile Test:

Ductile Material (Limited by Shear Strength)

Brittle Material (Limited by Tensile Strength )

• Compression Tests:

• Bending Tests:

14

Ductile Material Brittle Material

Ductile Material Brittle Material

• Torsion Tests:

Ductile Material Brittle Material

Brittle failure or ductile failure? Key: is the fracture surface on a plane of max shear or max normal stress.

TORQUE:

DUCTILE BRITTLE

Brittle Ductile

AXIAL

Fracture Mechanics• The idea that cracks exist in parts even before service begins,

and that cracks can grow during service, has led to the descriptive phrase “damage-tolerant design.”

• Crack growth increases until it becomes critical, and the part is removed from service.

• The analysis tool is linear elastic fracture mechanics (LEFM).

Fracture Mechanics

• When there exists a crack, flaw, inclusion, or defect of unknown small radius in a part, the elastic stress-concentration factor approaches infinity as the root radius approaches zero.

• Brittle fracture happens so rapidly that we think of it as instantaneous.

Note that when a = b, the ellipse becomes a circle and Eq. (5–33) gives a stressconcentration factor of 3.

• For a fine crack, b/a →0

• However, on a microscopic level, an infinitely sharp crack is ahypothetical abstraction that is physically impossible, and when plastic deformation occurs, the stress will be finite at the crack tip.

• Crack growth can be stable or unstable.

Fracture Mechanics

Summary

Static failure

• Ductile

• Brittle

• Stress concentration