2. failure mechanics

FAILURE MECHANICS

James A. Craig Omega 2011

Concepts of Failure

Tensile Failure

Shear FailureFailure CriteriaMohr-Coulomb Failure Criterion

Failure occurs to any solid material when:Sufficiently large stress is applied.The material does not return to its original state

after stress relief.Mode of failure depends on:

Stress stateType and geometry of material

Fatigue makes failure to occur below the stress level.

CONCEPTS OF FAILURE

Uniaxial TestStress is applied to the

end faces of the specimen.

No radial (confining stress)

Also called Unconfined Compression Test.

Elastic regionSpecimen returns to its original state after stress relief.

Yield PointPermanent changes beyond this point. Specimen does not return to its original state after removal of stress.

Uniaxial compressive strengthThe peak stress.

Ductile regionPermanent deformation, but can still support load.

Brittle regionAbility to withstand stress decreases rapidly as deformation increases.

Triaxial TestIn addition to axial

stress, confining pressure of different magnitude is applied to the circumference of the cylinder (by a confining oil bath).

Two of the principal stresses are equal.

Process:Axial & confining loads are increased

simultaneously until a prescribed hydrostatic stress level is reached.

Confining pressure is kept constant while axial load increases until failure occurs.

Difference in principal stresses is plotted against axial deformation.

Specimen can still support load after failure due to high confining pressure. It is called Work Hardening or Strain Hardening.

Uniaxial test

X → abrupt brittle failure

Tensile failure occurs when the effective tensile stress across some plane is the sample exceeds a critical limit called Tensile Strength.

TENSILE FAILURE

Tensile failure is caused by the stress concentrations at the edges of thin cracks oriented normal to the direction of the least compressive principal stress.

For isotropic rocks, conditions for failure will always be fulfilled first for the lowest principal stress.

To = tensile strength (in Pa, atm, psi or bar).

3 3 oP T

Most sedimentary rocks have a rather low tensile strength, typically only a few MPa or less.

Standard approximation for several applications is that the tensile strength is zero

It occurs when the shear stress along some plane in the sample is too large.

SHEAR FAILURE

Mohr–Coulomb

Hoek–Brown

Drucker–Prager

Griffith (tensile)

Failure criteria

So = cohesion or inherent shear strength of material (in Pa, atm, psi or bar).

µ = coefficient of internal friction.

Shear stress must overcome the cohesion plus the internal friction in order to produce a macroscopic shear failure.

Mohr-Coulomb Criterion

f

oS

If the Mohr’s circle lies below the failure line, the rock does not fail and remains intact.

Failure Line

Mohr Circle

tan Slope =

cottan

oo

SA S

φ = angle of internal friction. It varies from 0 to 90o (approx. 30o)

A = attraction (in Pa, atm, psi or bar).β = angle that fulfils the failure criterion. It gives

orientation of the failure plane. Varies between 45o and 90o.

At point P:Angle 2β gives the position of coincidence of Mohr’s

circle and the failure line.Coordinates are given as:

OR

1 3

1sin 2

2 1 3 1 3

1 1cos 2

2 2

2 90o 4 2

Co = uniaxial compressive strength (in Pa, atm, psi or bar).

2 cos

2 tan1 sino

o o

Sa C S

1 sin

tan1 sin

b

tan 1

sintan 1

21 3 tanoC

1 3a b

© Haimson and Song (1995)

Principle of effective stress is introduced, i.e. subtract fluid pressure from the total stress.

Previously:

And

Then:

Mohr-Coulomb Criterion on Saturated Rocks

1 3

1 sin2 cos

1 sin 1 sino

f f

SP P

1 3a b

1 1 fP 3 3 fP

1 3a b

Pore fluid can affect the failure of the rock in 2 ways:Mechanical effect of pore pressure.Chemical interactions between the rock and the

fluid.

Effect of pore pressure on failure:Shear stress is unaffected by the pore pressureMinimum & maximum principal stresses are

decreased by the same amount.Radius of the Mohr circle in unchanged.Center of the circle has shifted to the left.Circle moves towards the failure line when the

fluid pressure is increased for a material obeying the criterion.