nb: uniaxial strain is a type a non-rotational transformation uniaxial strain

NB: Uniaxial strain is a type a non-rotational transformation

Uniaxial strain

Pure Shear

NB: Pure shear in is a type a non-rotational transformation

Simple Shear

NB: Simple shear is rotational

Progressive pure shear

Progressive pure shear is a type of coaxial strain

Progressive simple Shear

Progressive simple shear is non coaxial

III. Strain and Stress

• Strain

• Stress

• RheologyReadingSuppe, Chapter 3Twiss&Moores, chapter 15

Additional References :Jean Salençon, Handbook of continuum mechanics: general concepts,

thermoelasticity, Springer, 2001Chandrasekharaiah D.S., Debnath L. (1994) Continuum Mechanics

Publisher: Academic press, Inc.

Stress

Stress is force per unit area

– Spreading out the weight reduces the stress with the same force.

F=mg

Normal Stress is skier’s weight distributed over skis surface area.

Thought experiments on stress…

The “flat-jack” experiment…

What if we rotate the slot???

Two stress components…

n

n

1

2

The stress (red vector) acting on a plane at M is the force exterted by one side over the other side divided by plane area…

Stress acting on a plane at point M…Let n be the unit vector defining an oriented surface with elementary area da at point M. (n points from side A to side B)

Let dT be the force exerted on the plane by the medium on side B. It can be decomposed into a normal and shear component parallel to the surface. The stress vector is:

2 2

n n n

n n n

n

Normal stress

Shear stress

,( ) i j j

dTn n n

da

Side B

Side A

Convention: positive in compression

The state of stress at a point can be characterizes from the stress tensor defined as …

i, j 11 12 13 21 22 23

31 32 33

The stress tensor

i, j 11 12 13 21 22 23

31 32 33

Symmetry…

i, j j ,i

Principal stresses

11 1

, 22 2

33 3

0 0 0 0

0 0 0 0

0 0 0 0i j

1 2 3

Engineering sign convention tension is positive,Geology sign convention compression is positive…

Plane perpendicular toprincipal direction has no shear stress…

Because the matrix is symmetric, there is coordinate frame such that….

The deviatoric stress tensor…

Stress tensor = mean stress + deviatoric stress tensor

i, j m 0 0

0 m 0

0 0 m

1 m 0 0

0 2 m 0

0 0 3 m

i, j m i, j

11 22 33 1 2 3

3 3m

mean stress:

Expressed in a reference frame defined by the principal directions:

Do not confuse the deviatoric stress tensor with the ‘differential stress’, often noted , defined as

1 3

Sum of forces in 1- and 2-directions…

2-D stress on all possible internal planes…

The Mohr diagram

Sum of forces in 1- and 2-directions…

2-D stress on all possible internal planes…

Rearrange equations yet again…

Get more useful relationship betweenprincipal stresses andstress on any plane….

Rearrange equations…

The Mohr diagram

[1] What does a point on the circle mean?

[2] What does the center of the circle tell you?

[3] Where are the principle stresses?

[4] What does the diameter or radius mean?

[6] Where is the maximum shear stress?

Any point on the circle gives coordinates acting on the plane at an angle to

Maximum shear stress max occurs for =45°; then max = (

(is the mean or hydrostatic stress= that which produces change in volume

(is the maximum possible shear stress= that which produces change in shape

In direction of and = 0; hence and are on the abscissa axis of Mohr graph

Pole of the Mohr circle

n

A

B

2

P

Poles of the Mohr circle

n

A

P

Poles of the Mohr circle

n

A

P

A represent the state of stress on a facet with known orientationThe geometric construction, based on the pole of the facet (P), allows to infer the state of stress on any orientation

Representation of the stress state in 3-D using the Mohr cirles.

n

The state of stress of a plane with any orientation plots in this domain

This circle represent the state of stress on planes parallel to

This circle represent the state of stress on planes parallel to

This circle represent the state of stress on planes parallel to

Classification of stress state

– General tension

– General compression

– Uniaxial Compression

– Uniaxial tension

– Biaxial stress

Pure Shear(as a state of stress)

The exression ‘Pure shear’ is also used to characterize the a particular case of biaxial stress

Do not confuse with pure shear as a state of strain

n

III. Strain and Stress

• Strain

• Stress

• RheologyReadingSuppe, Chapter 3Twiss&Moores, chapter 15

Additional References :Jean Salençon, Handbook of continuum mechanics: general concepts,

thermoelasticity, Springer, 2001Chandrasekharaiah D.S., Debnath L. (1994) Continuum Mechanics

Publisher: Academic press, Inc.

• A rheological law relates strain to stress and time.

A review of simple rheological models…

(f) Elasto-plastic

F

(e) Perfect plastic

F

Elastic rheology

Elastic deformation is recoverable

In linear elasticity stress is ‘proportional’ to strain

In linear elasticity stress is ‘proportional’ to stress (‘Hooke’s law’)

, , , , ,i j i j k l k l

C

C

C is the elasticity tensor. It is a symmetric tensor (21 elasticity coefficients)

Elastic rheology

For an isotropic linear elastic medium there are only 2 elasticity coefficients, e.g.: the Young modulus and Poisson ratio:

r

l

1 lE Shear modulus

Bulk modulus

Young Modulus

Poisson ratio

Viscous rheology

For Newtonian rheology the strain rate is ‘proportional’ to stress.

Viscous deformation is non recoverable

Plastic rheology

For Newtonian viscous rheology the strain rate is ‘proportional’ to stress.

Plastic deformation is non recoverable

(e) Plastic

F

t1 t2

t1 t2

NB: The strain evolution depends on the experimental conditions (ex: stiffness of the apparatus)

s

s

Coupled rheological models…

Recoverable, with hysteresis loop

Non-recoverable

F

t1 t2

…Coupled rheological models

(f) Elasto-plastic

Non-recoverable