Stress II

Cauchy formulaConsider a small cubic element of rock extracted from the earth, and imagine a plane boundary with an outward normal, n, and an area, A cutting through this element - so it is reduced to a triangular element with sides 1 and 2.

Stress II

The force components acting on sides 1 and 2 are:1.

Note that:2.

Replacing 2 in 1 gives:3.

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f1x = −σ xxδAcosγf1y = −σ xyδAcosγf2x = −σ yxδAsinγf2y = −σ yyδAsinγ .

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cosγ = nxsinγ = ny.

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f1x = −σ xxδAnxf1y = −σ xyδAnxf2x = −σ yxδAnyf2y = −σ yyδAny .

Stress II

Force balance leads to:

Rearranging the above:

This is equivalent to:

where tj is the traction acting on ni.

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fx∑ = txδA −σ xxδAnx −σ yxδAny = 0

fy∑ = tyδA −σ xyδAnx −σ yyδAny = 0.

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tx =σ xxnx +σ yxny

ty =σ xynx +σ yyny .

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t j =σ ijni,

Stress II

Principal stresses

We have learned that the stress tensor is symmetric. A property of symmetric matrices is that they may be diagonaliszd. The transformation from the non-diagonal to the diagonal tensor requires transformation of the coordinate system. The axes of the new coordinate system are the principal axes, and the diagonal elements of the tensor are referred to as the principal stresses.

Note that the shear stresses along the principal axes are equal to zero.

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σ ij =

σ 1 0 0

0 σ 2 0

0 0 σ 3

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟.

Stress II

Using principal stresses to calculate shear and normal stresses on a given plane

Adding vectors in directions parallel and normal to the plane inquestion:

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FN = F1 cosθ + F3 sinθ

FS = F1 sinθ − F3 cosθ .

Stress II

This is equivalent to:

Substituting the following trigonometric identities:

gives:

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σN =σ 1 cos2θ +σ 3 sin2θ

σ S = (σ 1 −σ 3)sinθ cosθ .

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sin2θ = (1− cos2θ) /2

sinθ cosθ = sin2θ /2.

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σN =σ 1 +σ 3

2+σ 1 −σ 3

2cos2θ

σ S =σ 1 −σ 3

2sin2θ .

Stress II

The above equation defines a circle with a center on the horizontal axis at (σ1 + σ 3)/2, and a radius that is equal to (σ 1 - σ 3)/2.

(σ1 + σ 3)/2 is the mean stress.

(σ1 - σ 3)/2 is the deviatoric stress.

is the angle between σ1 and the normal to the plane - positive when measured counter-clockwise from σ1 .

Stress II

Note that for a given stress tensor, the mean stress is independent of the plane in question, that is:

We can thus write the stress tensor as a sum of the mean stress field and the deviatoric stress field:

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σmean =σ 1 +σ 2

2=σ 11 +σ 22

2.

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σ11 σ 12 σ 13

σ 21 σ 22 σ 23

σ 31 σ 32 σ 33

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟=

σ mean 0 0

0 σ mean 0

0 0 σ mean

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟+

σ 11 −σ mean σ 12 σ 13

σ 21 σ 22 −σ mean σ 23

σ 31 σ 32 σ 33 −σ mean

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Stress II

Note that:

• Shear stresses equal to zero at =0 and 90 degrees.

• Maximum shear stress is equal to (σ 1 - σ 3)/2 at =45 degrees.

• The shear stresses along the principal directions are equal to zero.

• The principal axes are orthogonal.

Stress II

Mohr circle in 3D

A single Mohr circle describes the variation of shear and normal stress along a principal plane (a plane that contains 2 principal axes). The representation of a 3D state of stress is obtained by the superposition of three Mohr circles, as follows:

The state of stress on planes that are not perpendicular to a principal plane fall within the shaded area.

Stress II

Mohr circles and the state of stress

Uniaxial stress: Only one non-zero principal stress. For example:

Biaxial stress: One principal stress equals zero, the other two do not. For example:

Stress II

Mohr circles and the state of stress

Triaxial stress: All principal stresses are non-zero. For example:

Axial stress: Two of the three principal stresses are equal. For example:

Stress II

Pressure

This is a special state of stress in which the shear stress is equal to zero, i.e.: σ1= σ2= σ3.

Question: How does this state of stress plot on a Mohr diagram?

It is useful to consider two pressures: Lithostatic and hydrostatic.

Lithostatic pressure:The stress equals the weight of the overlying column of rock.

In the absence of tectonic forces or fluids, the state of stress would be lithostatic.

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Plithostatic = g ρ(z)dzz

∫ .

Stress II

Pressure

Hydrostatic pressure:The stress equals the weight of a column of water.

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Phydrostatic = gρwaterz.

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Role of fluid pressure and effective stress

Pore fluid: Is the fluid within the pores.

Pore pressure: Is the pressure within the pore fluid.

Usually the fluid is water, but it can also be oil or gas.

In a granular medium, the pore pressure acts to reduce the contact between the grains.

Stress II

Effective stress:

The effective stress tensor is:

Question: Is pressure a vector or a scalar?

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effective stress = normal stress - pore pressure

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σ11 − P σ 12 σ 13

σ 21 σ 22 − P σ 23

σ 31 σ 32 σ 33 − P

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟=

σ 11 σ 12 σ 13

σ 21 σ 22 σ 23

σ 31 σ 32 σ 33

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟−

P 0 0

0 P 0

0 0 P

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Stress II

Effective stress:

The effect of pore pressure increase (for example, due to water pumping) is to lower the effective stress. Graphically, this may be illustrated as follows:

P