Stress, Strain and Effective Stress Frameworks

Introduction and Context

The behaviour of a soil mass is governed by the interaction of stresses, strains, pore fluids, and the soil skeleton. Unlike many structural materials, soil is a particulate medium whose response depends strongly on drainage, loading rate, fabric, saturation, stress history, and changes in effective stress. A foundation may therefore experience very different deformation and strength responses under the same total load, depending on whether pore water can escape and how the applied stress is transferred between the water and the soil skeleton.

This chapter develops a unified framework for analysing these problems. It begins with stress and strain tensors, proceeds to principal stresses and stress invariants, and then introduces Mohr-circle representations and stress paths. The effective-stress principle is developed for saturated soils and extended to partially saturated geomaterials. The final sections connect the theory to foundation engineering, embankments, excavations, slopes, and laboratory interpretation.

Learning Objectives

After studying this chapter, the reader should be able to:

1. Represent three-dimensional stress and strain states using tensors.

2. Determine principal stresses, principal directions, and stress invariants.

3. Construct and interpret Mohr circles for three-dimensional stress states.

4. Distinguish infinitesimal, engineering, and finite strain measures.

5. Explain total stress, pore-water pressure, effective stress, and stress transfer.

6. Construct stress paths for drained, undrained, and coupled loading.

7. Apply effective-stress concepts to saturated and partially saturated soils.

8. Identify assumptions, limitations, and measurement issues in practical analysis.

1. Stress in a Continuum and in a Soil Mass

Stress is the intensity of internal force acting across an imagined plane. At a point in a continuum, the traction vector depends on the orientation of the plane. Cauchy’s stress theorem expresses the traction on a plane with unit normal vector n as

t(n) = σ n

where t(n) is the traction vector and σ is the second-order Cauchy stress tensor. In Cartesian coordinates,

σ = [ σxx τxy τxz ; τyx σyy τyz ; τzx τzy σzz ]

For a classical non-polar continuum with moment equilibrium, the stress tensor is symmetric:

τxy = τyx, τxz = τzx, τyz = τzy.

Thus, six independent stress components define the local stress state: three normal stresses and three shear stresses. In geotechnical engineering, compression is often taken as positive, whereas continuum mechanics texts commonly use tension as positive. The sign convention must be stated before interpreting equations or plotting stress paths.

A soil element is not homogeneous at the particle scale. The continuum stress represents an average force per unit area over a sufficiently large representative volume. This scale must be large compared with individual particles but small compared with the engineering problem. At smaller scales, contact forces form chains and stress is highly heterogeneous.

1.1 Normal and shear stress on an inclined plane

For a plane with unit normal n, the normal stress is

σn = n · t(n) = n · σ n.

The shear traction vector is

ts = t(n) − σn n,

and its magnitude is

τn = |ts|.

These expressions are general and apply to any plane orientation. They show why a single vertical stress is insufficient to describe the state around a foundation, retaining wall, tunnel, or slope.

2. Principal Stresses and Stress Invariants

A principal plane is one on which the shear traction is zero. Its normal vector satisfies

σ n = σp n,

which is an eigenvalue problem. The three eigenvalues are the principal stresses, conventionally ordered as σ1 ≥ σ2 ≥ σ3 under a compression-positive convention. The corresponding eigenvectors are the principal directions.

The characteristic equation is

det(σ − λI) = 0,

where λ is an eigenvalue and I is the identity tensor. Expanding it gives

λ³ − I1 λ² + I2 λ − I3 = 0,

where the stress invariants are

I1 = σ1 + σ2 + σ3 = tr(σ),

I2 = σ1σ2 + σ2σ3 + σ3σ1,

I3 = σ1σ2σ3 = det(σ).

The invariants remain unchanged when the coordinate axes are rotated. They are therefore useful for constitutive models and for comparing stress states independently of orientation.

The mean normal stress is

p = I1/3 = (σ1 + σ2 + σ3)/3.

Define the deviatoric stress tensor as

s = σ − pI.

Its first invariant is zero. The second deviatoric invariant is commonly written

J2 = 1/2 s:s,

where “:” denotes the double contraction of two tensors. In principal stress form,

J2 = [(σ1 − σ2)² + (σ2 − σ3)² + (σ3 − σ1)²]/6.

The octahedral shear stress is

τoct = √(2J2/3).

The third deviatoric invariant, J3 = det(s), influences the dependence of yielding on the intermediate principal stress. This is important because the Mohr–Coulomb criterion is less sensitive to σ2 than many three-dimensional constitutive models.

2.1 Hydrostatic and deviatoric stress

The total stress can be decomposed as

σ = pI + s.

The hydrostatic part pI changes volume but not shape. The deviatoric part s changes shape and is primarily associated with distortion, shear strain, and frictional yielding. In real soils the two effects are coupled: shear stress may cause dilation or contraction, while volumetric changes alter contact forces and effective stress.

3. Mohr Circles

For a three-dimensional principal stress state, the three-dimensional Mohr diagram contains circles associated with the pairs (σ1, σ2), (σ2, σ3), and (σ1, σ3). The circle between σ1 and σ3 is the largest and has

centre C13 = (σ1 + σ3)/2,

radius R13 = (σ1 − σ3)/2.

The maximum shear stress at the point is therefore

τmax = (σ1 − σ3)/2.

For plane stress, the normal and shear stresses on planes of varying orientation are represented by a single circle. If σx, σy, and τxy are known, the centre and radius are

C = (σx + σy)/2,

R = √[((σx − σy)/2)² + τxy²].

The principal stresses in the plane are C + R and C − R. The principal direction θp satisfies

tan(2θp) = 2τxy/(σx − σy),

with quadrant selection determined by the signs of numerator and denominator.

Mohr circles are valuable for visualising stress rotation, maximum shear stress, and failure envelopes. However, they do not by themselves define a soil constitutive law. A point lying below a failure envelope may still deform substantially, and a peak friction angle may be inappropriate for a loose or contractive soil.

3.1 Mohr–Coulomb failure criterion

For effective stresses, the Mohr–Coulomb criterion is commonly written

τf = c′ + σ′n tan φ′,

where τf is shear strength at failure, c′ is effective cohesion intercept, σ′n is effective normal stress, and φ′ is the effective friction angle. For a cohesionless soil, c′ is approximately zero in the idealised drained model.

In triaxial compression, using effective principal stresses,

σ′1 = σ′3 (1 + sin φ′)/(1 − sin φ′) + 2c′ cos φ′/(1 − sin φ′).

This equation assumes a Mohr–Coulomb material, conventional triaxial compression, and a consistent compression-positive convention. Natural soils may exhibit anisotropy, bonding, destructuration, curvature of the failure envelope, and dependence on stress path, so the equation should be treated as an idealisation.

4. Strain Measures

Strain describes deformation relative to an initial configuration. For small displacement gradients, the infinitesimal strain tensor is

ε = 1/2(∇u + ∇uᵀ),

where u is the displacement vector. In Cartesian coordinates,

εxx = ∂u/∂x,

εyy = ∂v/∂y,

εzz = ∂w/∂z,

and

γxy = 2εxy = ∂u/∂y + ∂v/∂x.

Engineering shear strain γxy is twice the tensorial shear-strain component εxy. Confusing these two definitions is a frequent source of error.

The volumetric strain for small deformation is

εv = tr(ε) = εxx + εyy + εzz.

With compression taken as positive for geotechnical strain notation, some texts define compressive volumetric strain as positive; others use the continuum convention in which extension is positive. The sign convention must remain consistent with the stress convention and constitutive equations.

The strain tensor can also be decomposed into volumetric and deviatoric parts:

ε = (εv/3)I + e,

where e is the deviatoric strain tensor. A useful distortion measure is the deviatoric strain invariant

εq = √(2/3 e:e).

For triaxial testing, an often-used strain measure is the axial deviatoric strain or a q–εq representation. The selected definition should always be stated because different research and laboratory standards use different conventions.

4.1 Finite strain and large deformation

Infinitesimal strain is adequate when displacement gradients and rotations are small. In large embankment deformation, landslides, tunnelling, and progressive failure, finite-strain measures are more appropriate. The deformation gradient is

F = ∂x/∂X,

where X is the reference position and x is the current position. The right Cauchy–Green tensor is C = FᵀF, and the Green–Lagrange strain tensor is

E = 1/2(C − I).

For finite deformation, the choice between material and spatial measures affects interpretation, numerical implementation, and calibration. Many routine geotechnical calculations remain within the small-strain range, but the assumption should not be accepted automatically.

5. Total Stress, Pore Pressure and Effective Stress

Terzaghi’s effective-stress principle for a saturated soil is

σ′ = σ − uw I,

or in principal directions,

σ′i = σi − uw,

where σ is total stress, σ′ is effective stress carried by the soil skeleton, and uw is pore-water pressure. The equation assumes that the pore fluid contributes an isotropic stress and that the soil skeleton is the phase responsible for frictional resistance and most deformation.

For normal stress on a plane,

σ′n = σn − uw.

Because water pressure is isotropic, it does not directly contribute to the deviatoric stress:

s′ = s.

However, changes in pore pressure alter mean effective stress p′ and therefore contact forces, stiffness, compressibility, and shear strength. This explains why undrained loading can produce large strength changes even when total stresses change in a similar way.

5.1 Saturated soil under undrained loading

In a saturated, low-permeability soil, rapid loading may occur before drainage. The initial increment of applied stress is shared between the soil skeleton and pore water. For an ideal incompressible soil–water system under perfectly undrained isotropic loading, the increment in pore pressure may approach the increment in total mean stress, so the increment in mean effective stress is small.

In real soils, pore pressure response depends on compressibility, stress history, anisotropy, fabric, and loading mode. A contractive soil tends to generate positive excess pore pressure during undrained shearing; a dilative soil may generate negative excess pore pressure after shearing begins. The sign and magnitude of pore pressure must therefore be measured or predicted using an appropriate constitutive model.

The undrained shear strength of a normally consolidated clay is often represented in a total-stress analysis by su. This is a convenient short-term parameter, not a replacement for effective-stress mechanics. Its value depends on water content, stress history, anisotropy, strain rate, and test procedure.

5.2 Drained loading and consolidation

Under drained loading, pore pressure generated by loading dissipates as water flows through the soil. Effective stress increases during consolidation under sustained total stress. Settlement then develops through changes in void ratio and soil fabric. The time scale depends on permeability, compressibility, drainage length, and boundary conditions.

The distinction between immediate, consolidation, and secondary compression settlement is a practical expression of the stress-transfer framework. Immediate deformation may occur without substantial drainage, primary consolidation is associated with dissipation of excess pore pressure, and secondary compression reflects time-dependent structural adjustment after excess pore pressure has largely dissipated.

6. Stress Paths

A stress path is the trajectory followed by a stress state in stress space. Common coordinates include p′–q space, where p′ is mean effective stress and q is a deviatoric stress measure. In triaxial compression, a common definition is

q = σ′1 − σ′3.

For a conventional triaxial test with constant cell pressure σ3, q increases as axial stress increases. The effective stress path depends on pore-pressure generation:

p′ = (σ′1 + σ′2 + σ′3)/3.

For axisymmetric triaxial conditions, σ2 = σ3, so

p′ = (σ′1 + 2σ′3)/3.

A drained triaxial test with constant σ′3 generally moves toward higher p′ and q. An undrained test may move toward higher q but lower p′ when positive excess pore pressure develops. A dilative specimen may move toward higher p′ during shearing because negative excess pore pressure increases effective stress.

Stress paths are useful because soil response is path-dependent. Two final stress states can produce different strains and structures if reached through different loading histories. Stress-path analysis is central to embankment construction, excavation unloading, cyclic loading, and staged foundation loading.

7. Effective Stress in Partially Saturated Soils

When air and water coexist in the voids, a single Terzaghi effective-stress equation is generally insufficient. The relevant pore pressures are air pressure ua and water pressure uw. Matric suction is

s = ua − uw.

One widely used effective-stress expression is

σ′ = (σ − ua I) + χ(ua − uw)I,

or for normal stress,

σ′n = (σn − ua) + χs.

Here χ is an effective-stress parameter, usually between 0 and 1 for many practical conditions, but its value depends on degree of saturation, hydraulic history, fabric, and the adopted model. The expression reduces to the saturated form when ua = uw and becomes a net-stress form when χ = 0.

Alternative formulations use net stress σ − ua and suction as separate independent stress variables. This is often preferable for constitutive modelling because χ is not a universal material constant. It may vary during wetting, drying, scanning, hysteresis, and changes in pore-water connectivity.

7.1 Hydraulic and mechanical coupling

Partially saturated behaviour is coupled: mechanical deformation changes pore geometry and water retention, while suction changes stiffness, strength, and volume. Soil-water characteristic curves relate suction to water content or degree of saturation. They exhibit hysteresis because the wetting path differs from the drying path.

Apparent cohesion arising from suction should not automatically be treated as permanent cohesion. Infiltration, rainfall, irrigation, or a rising water table may reduce suction and trigger large reductions in shear strength. This is particularly important for residual soils, compacted fills, shallow slopes, and unsaturated foundations.

8. Worked Example: Effective Stress and Stress Invariants

A saturated soil element is subjected to total principal stresses σ1 = 300 kPa, σ2 = 180 kPa, and σ3 = 100 kPa. The pore-water pressure is uw = 60 kPa. Compression is taken as positive.

The effective principal stresses are

σ′1 = 300 − 60 = 240 kPa,

σ′2 = 180 − 60 = 120 kPa,

σ′3 = 100 − 60 = 40 kPa.

The total mean stress is

p = (300 + 180 + 100)/3 = 193.33 kPa.

The effective mean stress is

p′ = (240 + 120 + 40)/3 = 133.33 kPa.

As expected for isotropic pore pressure,

p′ = p − uw = 193.33 − 60 = 133.33 kPa.

The maximum shear stress is unchanged by pore pressure:

τmax = (σ1 − σ3)/2 = (300 − 100)/2 = 100 kPa,

and equivalently

τ′max = (σ′1 − σ′3)/2 = (240 − 40)/2 = 100 kPa.

The deviatoric stress measure for the effective state is

J2 = [(240−120)² + (120−40)² + (40−240)²]/6

= (14,400 + 6,400 + 40,000)/6

= 10,133.33 kPa².

Therefore,

τoct = √(2J2/3) = √(6,755.56) = 82.19 kPa approximately.

The example demonstrates that isotropic pore pressure shifts the normal-stress coordinates and mean stress but does not change stress differences or deviatoric stress.

9. Applied Case: Foundation Loading on a Soft Saturated Clay

Background

Consider an illustrative case involving a 4 m thick normally consolidated clay layer beneath a proposed warehouse foundation. The groundwater table is near the ground surface. Construction is planned over six months, and the foundation load will produce a substantial increase in vertical total stress.

Problem

If the load is applied rapidly relative to the clay’s drainage time, the immediate response may be largely undrained. The increase in total stress will initially generate excess pore pressure. If the analysis incorrectly assumes that the entire load immediately becomes effective stress, predicted short-term strength and settlement behaviour may be unsafe.

Analysis

The design team first estimates drainage conditions using permeability, compressibility, layer thickness, and drainage boundaries. Piezometers are installed to measure pore pressure during staged filling. Consolidation parameters are obtained from oedometer tests, while undrained strength is measured using appropriate triaxial or field tests. The stress history is established from sampling, geological interpretation, and in situ testing.

During construction, the total vertical stress rises. Initially, excess pore pressure carries a significant portion of the increment. As drainage proceeds, excess pore pressure dissipates and vertical effective stress increases. The soil compresses, void ratio decreases, and settlement develops. If the soil is contractive under the imposed stress path, lateral deformation and pore-pressure generation may also reduce undrained stability.

Approach and solution

A staged coupled analysis is adopted. Construction is divided into increments, each followed by a period of monitoring and consolidation. The model uses effective-stress constitutive parameters and includes the actual drainage boundaries. Trigger levels are defined for pore pressure, lateral displacement, and settlement. If measurements exceed predicted ranges, construction is paused and the model is updated.

Possible mitigation includes staged construction, prefabricated vertical drains, a temporary surcharge, basal reinforcement, or reduction of foundation contact pressure. The selected solution must consider not only final settlement but also short-term undrained stability and differential settlement.

Outcome

The staged approach allows excess pore pressures to dissipate between construction stages and increases the available effective stress before the next load increment. Monitoring confirms that the observed response remains within the design envelope. The project achieves the required bearing performance with a longer construction period but lower geotechnical risk.

Learning points

1. Total stress and effective stress answer different engineering questions.

2. Construction rate is a mechanical design variable because it controls drainage.

3. Pore-pressure monitoring provides direct evidence of stress transfer.

4. A final-state analysis alone may miss temporary instability.

5. Numerical predictions must be checked against field observations.

10. Limitations, Professional Practice and Contemporary Tools

Effective-stress analysis is powerful but not universal in a simplistic form. The assumptions of isotropic pore pressure, continuum averaging, small strain, homogeneous material, and a unique friction angle may be inappropriate for fissured clay, structured soil, crushable sand, fibrous peat, chemically active soil, or partially saturated ground.

Laboratory specimens may not reproduce field fabric, stress anisotropy, boundary drainage, or strain rate. Sampling disturbance can alter pore-pressure response and compressibility. In situ stress measurements are also uncertain, particularly in stiff or cemented geomaterials. Parameter selection should therefore use multiple sources of evidence rather than a single test.

Finite-element and finite-difference programs increasingly incorporate elastoplastic, critical-state, hypoplastic, bounding-surface, and unsaturated constitutive models. These tools can represent coupled deformation and pore-fluid flow, but numerical sophistication does not remove the need for correct boundary conditions, mesh assessment, parameter calibration, and sensitivity analysis. Artificially high permeability, inappropriate drainage assumptions, or unverified effective-stress parameters can produce persuasive but incorrect results.

AI and data-driven methods are becoming useful for identifying trends in monitoring data, classifying soil fabric, estimating parameters, and detecting anomalous pore-pressure behaviour. They should support—not replace—mechanistic interpretation. Training-data bias, extrapolation beyond the calibration domain, explainability, data governance, and professional accountability remain important. Safety-critical decisions require traceable calculations and engineering judgement.

Best practice includes stating sign conventions, defining stress and strain measures, identifying drainage conditions, distinguishing total- and effective-stress parameters, documenting parameter sources, checking units, and comparing predictions with field observations. In design reports, the assumed stress path and pore-pressure response should be explicit rather than hidden inside software settings.

Chapter Summary

Stress at a point is represented by a tensor, and traction on any plane follows from the tensor and the plane normal. Principal stresses are tensor eigenvalues, while stress invariants provide coordinate-independent descriptions of the stress state. The stress tensor separates naturally into hydrostatic and deviatoric components, which are associated broadly with volumetric and distortional response.

Mohr circles provide a graphical interpretation of normal and shear stresses, principal stresses, and maximum shear stress, but they do not replace a constitutive model. Strain may be represented using infinitesimal measures for small deformation or finite-strain measures for large deformation. Engineering shear strain is twice the tensorial shear component.

For saturated soils, effective stress is the total stress minus isotropic pore-water pressure. Isotropic pore pressure changes mean effective stress but not deviatoric stress. Drainage conditions control the evolution of effective stress, pore pressure, settlement, and strength. Stress paths are therefore essential for understanding staged construction, undrained loading, consolidation, unloading, and cyclic behaviour.

Partially saturated soils require consideration of air pressure, water pressure, and matric suction. Effective-stress parameters may vary with saturation and hydraulic history, so apparent suction-induced strength should not be treated as permanent. Reliable practice combines mechanics, laboratory and field data, monitoring, appropriate numerical modelling, and professional judgement.

Key Terms

Stress tensor: Second-order tensor describing normal and shear stresses at a point.

Traction: Force per unit area acting on a plane through a point.

Principal stress: Normal stress acting on a plane where shear traction is zero.

Principal direction: Direction normal to a principal plane.

Stress invariant: Scalar quantity unchanged by rotation of coordinate axes.

Mean stress: Average of the three principal normal stresses.

Deviatoric stress: Stress component remaining after subtraction of mean hydrostatic stress.

Mohr circle: Graphical representation of normal and shear stresses on differently oriented planes.

Volumetric strain: Change in volume per unit initial volume, within the selected strain convention.

Deviatoric strain: Strain component associated primarily with distortion.

Effective stress: Stress carried by the soil skeleton after accounting for pore-fluid pressure.

Pore-water pressure: Pressure exerted by water within soil voids.

Excess pore pressure: Pore pressure above the equilibrium hydrostatic value caused by loading or other processes.

Drainage condition: Hydraulic condition governing whether fluid can escape during loading.

Stress path: Trajectory of a stress state in a selected stress-coordinate space.

Matric suction: Difference between pore-air pressure and pore-water pressure.

Degree of saturation: Volume of water divided by volume of voids.

Consolidation: Time-dependent increase in effective stress and deformation caused by drainage.

Undrained strength: Short-term shear strength measured when drainage is prevented or negligible.

Mohr–Coulomb criterion: Idealised frictional failure relation involving cohesion and friction angle.

Multiple Choice Questions with Answers

Question 1

Which quantity is unchanged when an isotropic pore-water pressure is subtracted from the total stress tensor?

A. Mean normal stress

B. Effective vertical stress

C. Deviatoric stress tensor

D. Total stress invariant I1

Correct Answer: C. Deviatoric stress tensor

Question 2

For principal stresses 240 kPa, 120 kPa, and 40 kPa, what is the maximum shear stress?

A. 80 kPa

B. 100 kPa

C. 120 kPa

D. 200 kPa

Correct Answer: B. 100 kPa

Question 3

The effective stress in a saturated soil is commonly expressed as:

A. σ′ = σ + uw I

B. σ′ = σ − uw I

C. σ′ = uw I − σ

D. σ′ = σ − ua I

Correct Answer: B. σ′ = σ − uw I

Question 4

What is the mean effective stress for effective principal stresses of 240, 120, and 40 kPa?

A. 100 kPa

B. 120 kPa

C. 133.3 kPa

D. 160 kPa

Correct Answer: C. 133.3 kPa

Question 5

In a conventional triaxial compression test, q is commonly defined as:

A. σ′1 + σ′3

B. σ′1 − σ′3

C. (σ′1 + σ′3)/2

D. σ′1σ′3

Correct Answer: B. σ′1 − σ′3

Question 6

Matric suction in an unsaturated soil is defined as:

A. uw − ua

B. ua − uw

C. σ − uw

D. σ − ua

Correct Answer: B. ua − uw

Question 7

Which strain component is twice the tensorial shear-strain component εxy?

A. Volumetric strain

B. Principal strain

C. Engineering shear strain γxy

D. Deviatoric stress

Correct Answer: C. Engineering shear strain γxy

Question 8

During rapid undrained loading of a saturated contractive soil, the likely initial response is:

A. Dissipation of all pore pressure

B. Generation of positive excess pore pressure

C. Immediate increase in permeability to infinity

D. Zero change in total stress

Correct Answer: B. Generation of positive excess pore pressure

Question 9

The radius of the largest three-dimensional Mohr circle is:

A. (σ1 + σ3)/2

B. (σ1 − σ3)/2

C. σ2/2

D. σ1 − σ2

Correct Answer: B. (σ1 − σ3)/2

Question 10

In the unsaturated effective-stress expression σ′ = (σ − uaI) + χ(ua − uw)I, χ represents:

A. The void ratio

B. The coefficient of permeability

C. An effective-stress parameter related to saturation and hydraulic history

D. The total unit weight

Correct Answer: C. An effective-stress parameter related to saturation and hydraulic history