Chapter 26 Strain Stress and Mohr Diagrams

Year 2 · Structural Geology · G221-L2

Calculate extension and normal and shear stress, distinguish strain from stress, and interpret idealized Mohr diagrams in geology.

Fold geometry and shortening in a schematic layered succession. Original Geology Hub teaching diagram.

This independent lesson follows topics in the supplied University of Peshawar BS Geology curriculum labelled session 2004–05 onward. It is a learning resource, not an official university course or a statement of current degree requirements.

Learning objectives

Calculate simple measures of finite strain.
Resolve traction into normal and shear components.
Explain what a Mohr diagram represents and what it cannot determine alone.

Measuring deformation

Strain describes a change in size or shape. Stress describes force acting across an area. One is an observed response and the other is a mechanical quantity, so a folded rock does not directly reveal the magnitude of the stress that formed it. Temperature, fluids, deformation rate and rock composition influence the relationship. A laboratory stress–strain curve applies to the particular conditions of that experiment.

Engineering extension is e = (L − L₀)/L₀, with L₀ the initial length and L the final length. Stretch is λ = L/L₀. Natural strain is ln λ and is useful when successive coaxial stretches are combined. Shortening gives negative extension under this convention. Always state the strain measure; saying “strain is 0.2” without a definition is ambiguous.

An initially circular marker may become an ellipse under homogeneous two-dimensional deformation. The ellipse axes show principal finite stretches. A sphere similarly becomes an ellipsoid in three dimensions. Rotational deformation includes a component of rigid-body rotation; non-rotational deformation does not. A present-day finite strain ellipse records the cumulative result, while incremental strain describes a small part of the path. Different deformation histories can produce the same final ellipse.

Resolving stress

Traction on a surface has a normal component perpendicular to it and a shear component parallel to it. Principal stress directions are directions for which the shear traction on the perpendicular plane is zero. In the common rock-mechanics convention, compression is positive and σ₁ is the greatest compressive principal stress. Other disciplines use other signs, so mark the convention on calculations and plots.

For a two-dimensional principal-stress state, σₙ = (σ₁ + σ₃)/2 + (σ₁ − σ₃)cos(2θ)/2 and τ = (σ₁ − σ₃)sin(2θ)/2. Here θ is measured from the σ₁ direction to the plane normal. The doubled angle is essential. A Mohr stress circle has centre (σ₁ + σ₃)/2 and radius (σ₁ − σ₃)/2. Its endpoints correspond to the principal stresses; its top and bottom show the maximum shear magnitude. A strain Mohr construction uses strain variables and a chosen shear-strain convention, rather than stress units.

Traction depends on the chosen surface

Imagine cutting a small cube from a stressed rock. Force transmitted across each cut face has a direction as well as a magnitude per area. Rotating the imagined cut changes the normal and shear components even though the surrounding stress state remains unchanged. This is why “the stress on the fault” must specify whether normal stress, shear stress or total traction is meant. Principal planes are special orientations with zero shear traction; their normals define principal directions. The stress ellipsoid summarizes these directional relationships, rather than representing an actual ellipsoidal cavity inside the rock.

Combine successive length changes correctly

If a 10 cm marker stretches by 10% and then stretches by another 10% of its new length, it becomes 12.1 cm, not 12 cm. Stretches multiply: 1.1 × 1.1 = 1.21. Natural strains add for these successive coaxial stretches: ln 1.1 + ln 1.1 = ln 1.21. More general three-dimensional deformation includes rotations and changing principal directions, so scalar addition cannot reconstruct every strain history. A finite strain ellipsoid is the accumulated shape change, while incremental analysis examines the progression that produced it.

Separate mechanical models from geological inference

Fluid pressure can reduce effective normal compression on a fracture. In a simple illustrative model, total normal stress 50 MPa and pore pressure 20 MPa give effective normal stress 30 MPa. A friction coefficient of 0.6 would then give a frictional resistance of 18 MPa if cohesion is assumed zero. These are invented model inputs, not a failure forecast for a real fault. Cohesion, roughness, fluid connectivity, loading history and temperature complicate rock behaviour. Geological fabrics can constrain deformation and relative directions, but converting them into an absolute past stress magnitude requires additional assumptions and independent evidence.

Worked activity

A marker originally 10 cm long is now 12 cm. Its engineering extension is (12 − 10)/10 = 0.20, its stretch is 1.20 and its natural strain is ln(1.20) ≈ 0.182. For σ₁ = 80 MPa and σ₃ = 20 MPa, the Mohr circle centre is 50 MPa and radius 30 MPa. At θ = 45°, normal stress is 50 MPa and shear magnitude is 30 MPa.

These invented stresses describe a model, not a measurement inferred from the marker. Record the distinction between the geometry calculation and the mechanical assumption.

Self-check

What are the units of engineering extension?
Answer
It is dimensionless because it is a ratio of lengths.
Does a finite ellipse uniquely determine the strain path?
Answer
No. More than one sequence of increments can yield the same final shape.
Why label θ carefully?
Answer
The stress equations use the angle to the plane normal, not the plane itself.

Chapter 26 multiple choice questions

Choose one best answer for each question. Open each answer panel after making your choice.

A marker lengthens from 8 cm to 10 cm. What is its engineering extension?
1. 0.20
2. 0.25
3. 1.25
4. 2.00
Answer and explanation
B. 0.25 Engineering extension is (10 − 8)/8 = 0.25. The value 1.25 is the stretch.
Under the compression-positive convention, σ₁ = 100 MPa and σ₃ = 40 MPa. What are the Mohr circle centre and radius?
1. 60 MPa and 40 MPa
2. 30 MPa and 70 MPa
3. 70 MPa and 30 MPa
4. 140 MPa and 60 MPa
Answer and explanation
C. 70 MPa and 30 MPa The centre is (100 + 40)/2 = 70 MPa and the radius is (100 − 40)/2 = 30 MPa.
In the chapter’s stress equations, θ is the angle between which two directions?
1. The plane itself and its map strike
2. The fault strike and geographic north
3. The two principal stress directions
4. The σ₁ direction and the plane normal
Answer and explanation
D. The σ₁ direction and the plane normal The doubled-angle equations use the plane normal measured relative to σ₁. Using the plane itself changes the interpretation.
A rock has the same finite strain ellipse after two different hypothetical deformation sequences. What conclusion follows?
1. The final ellipse alone cannot distinguish the two strain paths
2. The two stress histories must be identical
3. Both histories must have zero rotation
4. The ellipse is an invalid strain measure
Answer and explanation
A. The final ellipse alone cannot distinguish the two strain paths Finite strain records the accumulated result. Different sequences of increments may produce that same result.
Which observation measures strain rather than stress?
1. Force divided by surface area
2. A change in a marker’s length relative to its initial length
3. Normal traction measured in MPa
4. Shear traction resolved on a plane
Answer and explanation
B. A change in a marker’s length relative to its initial length A relative length change describes deformation. Force per area and traction are stress quantities.

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