Indian Institute of Technology, Kanpur

Department of Earth Sciences

ES303: STRUCTURAL GEOLOGY

Lecture 8. Stress (Part – I)

Santanu Misra
Department of Earth Sciences
Indian Institute of Technology, Kanpur
smisra@iitk.ac.in http://home.iitk.ac.in/~smisra/
Aims of this lecture

Concept of Mechanics, Force and Stress, dimensions and units

Stress on a Surface and at a Point; Stress Tensor

Stress Ellipse and Ellipsoid

Compressive, Tensile and Shear stresses

Structural Geology and Mechanics
We have learnt now that the rock-masses get strained (deformed) and to achieve the
strain the rock-mass must have experienced some natural forces/pressures.

If we consider the rocks are “materials”, the study of the deformation of rocks under
forces falls under the subject “Mechanics”, which deals the science related to the
behaviour of physical materials subjected to force and displacements. Think of the
term we use “Rock Mechanics”.

Therefore, the study of the deformation of rocks can be included and explained under
the broader subject: MECHANICS.
Concept of Continuum
A body is and remains CONTINUOUS under the action of external forces
o Consisting of continuous material points
o Neighboring points remain neighbors CONTINUUM
o Neglecting its atomistic structure MECHANICS

A continuum, or continuous medium, is represented as a continuous aggregates of idealized

material particles (elemental volumes). They are small enough that their position can be given
in terms of points in some co-ordinate systems, yet large enough that local value of any
variable does not depend on fluctuations at the atomic scale in the immediate neighbourhood
of the point.

In order to deal with the properties and mechanics of continuum, it is therefore necessary to
refer the body to a system of co-ordinates.
Force
An object, in motion or in equilibrium is a function of the object’s mechanical
interaction with the other objects. FORCE is the quantitative measure and description
of the mechanical interaction.

A force is a vector quantity [first-order tensor] and has magnitude, direction and point
of application

SI unit of Force is Newton [1Newton is required to accelerate a 1kg mass at 1

meter/second2] ……[mlt-2]
Dyne [1Dyne is required to accelerate a 1gm mass at 1 cm/second2]
Force

Being a vector, the components of the Force (F:

magnitude F, and , ,  are the angles it makes
with the axes of a Cartesian coordinate system) in
3D:

𝐹𝑋1 = 𝐹 cos 𝛼 𝐹𝑋2 = 𝐹 cos 𝛽 𝐹𝑋3 = 𝐹 cos 𝛾

and

𝐹 2 = 𝐹𝑋21 + 𝐹𝑋22 + 𝐹𝑋23

Force

The forces acting on a body can be divided into two groups:

Internal forces: Internal forces represent the interaction between the particles in the body.

External forces: refer to the action of other bodies on the particles of a given body.

Body forces act on unit mass or unit volume of the body (e.g., gravity and magnetic
forces).

Surface forces act on the surface of a body when it comes in contact with another body.
The surface forces are mostly responsible for the deformation of rocks at various scales.
The surface forces acting on an area are often referred as traction.
Stress on a surface - TRACTION

The stress on a surface (traction) can be idealised in geological context in many different ways:
on a fault plane, on the contact areas between adjacent grains, meteoritic impacts etc.

In mechanics, the stress on a surface (traction, T) is defined as the ratio 𝐹റ

𝑇=
between the Reactive Force (F) and the Surface area (S), on which the 𝑆
force is acting.

As, Force is a vector, the Stress on a surface (traction) is also a vector.

SI unit of Stress is Pascal (Pa) = Newton/(Meter)2 = 1 kg/m.s2 ……[ml-1t-2]

1 Pa = 10-5 bar = 0.000145 psi 1 MPa = 10 bar = 145 psi

Pressures in normal bicycle and car tyres are 0.6 and 0.24 MPa, respectively. Lithostatic Pressures at the
lower-upper mantle boundary (670 km) ~28 GPa; at core-mantle boundary 330 GPa and at the center of
the earth ~400 GPa.
Stress on a surface element
Let a continuous medium occupy a volume V in Cartesian space and let ΔV be an element of volume inside V.

Two forces can act over the volume ΔV

Body Forces, per unit volume or unit mass – acceleration of Gravity (Xi)
Surface Forces, on the surface ΔS bounding ΔV

S be a small surface element of ΔS and ni is the outer

normal unit vector of S

Fi be the surface force acting on that the part of the body
outside ΔS exerts on the part through S.

The surface S will exert an equivalent (and

opposite) force and produce a couple on S
[Newton’s 3rd Law].
Stress on a surface element
Assuming S tends to be zero (very small area), the
couple vanishes and the ratio Fi /S tends to a finite
limit and can be defined by a vector - TRACTION
𝛿𝐹𝑖
𝑇 𝑛 𝑖 = lim
𝛿𝑆→0 𝛿𝑆

The stress vector T(n)i can be resolved into two

components (a) NORNAL STRESS VECTOR () – acting
normal to the surface and (b) SHEAR STRESS VECTOR () –
acting along (or parallel) to the surface.

The stress vectors, in general, act obliquely to the

planes. A structural geologist decomposes the stress
vector to Normal and Shear Stress components with
respect to the orientation of the concerned surface.
We shall learn this technique later.
Stress at a point - concept
In structural geology, we often consider stress at a point, for example a point inside a
crystal of a mineral, or a very tiny inclusion within a grain.

Consider a point P now is inscribed in volume ΔV. You can imagine

infinite number of very small planes around the point and resolve
the TRACTIONS on these small planes to define the STRESS AT A
POINT.

On each pair or opposite planes around the point, one can resolve
two perpendicular and oppositely directed with equal magnitude
(i.e., of equal length) component of the stress vectors.

Different pairs of planes would have different magnitude of stress vectors – when resolved
around the point, it would produce an ellipse (2D) or ellipsoid (3D).

We shall come back to these ellipses and ellipsoids later..

Stress at a point – STRESS TENSOR
For easier mathematical operations, imagine a very small
parallelepiped around the P with dimensions ΔX1, ΔX2 and ΔX3.

The Tractions of the three positive faces of the parallelepiped can be

resolved into their Cartesian components: one normal, and two
tangential to the face on which the tractions acts.

𝑇 1 𝑖 = 𝜎11 , 𝜏12 , 𝜏13 Face normal to X1

𝑇 2 𝑖 = 𝜏21 , 𝜎22 , 𝜏23 Face normal to X2

𝑇 3 𝑖 = 𝜏31 , 𝜏32 , 𝜎33 Face normal to X3

The nine components of the tractions acting at a point can be

expressed in index notation:

𝜎11 𝜏12 𝜏13

STRESS
𝜎𝑖𝑗 = 𝜏21 𝜎22 𝜏23
𝜏31 𝜏32 𝜎33 TENSOR
Stress at a point - Equilibrium

The Conditions of equilibrium for body and surface forces

𝜕𝜎11 𝜕𝜎21 𝜕𝜎31
+ + + 𝜌𝑿1 = 0
𝜕𝑥1 𝜕𝑥2 𝜕𝑥3
𝜕𝜎12 𝜕𝜎22 𝜕𝜎32 𝜕𝜎𝑗𝑖
+ + + 𝜌𝑿2 = 0 + 𝜌𝑿𝑖 = 0
𝜕𝑥1 𝜕𝑥2 𝜕𝑥3 𝜕𝑥𝑗
𝜕𝜎13 𝜕𝜎23 𝜕𝜎33
+ + + 𝜌𝑿3 = 0
𝜕𝑥1 𝜕𝑥2 𝜕𝑥3

The Conditions of equilibrium for moments

𝜕𝜎𝑗𝑖 𝜕𝜎𝑖𝑗
𝜎𝑖𝑗 = 𝜎𝑗𝑖 + 𝜌𝑿𝑖 = 0; + 𝜌𝑿𝑖 = 0
𝜕𝑥𝑗 𝜕𝑥𝑗
Check the derivations in
Ghosh’s book (Chapter 5)
Stress Ellipse and Stress Ellipsoid
𝜎11 𝜏13 Please recall, “Different pairs of
𝜎𝑖𝑗 = 𝜏 𝜎33
31 planes would have different
magnitude of stress vectors – when
𝜎11 𝜏12 𝜏13 resolved around the point, it would
𝜎𝑖𝑗 = 𝜏21 𝜎22 𝜏23 produce an ellipse (2D) or ellipsoid
𝜏31 𝜏32 𝜎33 (3D)”


1  1



2D 3D
Stress Ellipse and Stress Ellipsoid
The geometric disposition of the stress ellipsoid (shape and orientation) reveals the state of
stress at a given point in a rock-mass deforming or even in static-state.

The largest, smallest and intermediate axes (1 3 and

 respectively) of the stress ellipsoid are known as
Principal Stress or Principle Axes of Stress of the stress
ellipsoid. [we will learn more about them later]

Please remember, the stress and strain ellipsoids (and ellipses in 2D) are very similar physically
and mathematically. However, they are different. (A) A stress ellipsoid may not lead to a strain
ellipsoid (i.e., rocks are nor deforming); (B) The shape and orientation of the strain ellipsoid
may be very different to those of a stress ellipsoid responsible for the strain.
Sign convention of Stress Axes
Compressive Stress: The stress on rock-
mass which tends to shrink/shorten the
material along the direction of stress
(e.g., folding, thrust-faults).

Tensile Stress: The stress on rock-mass

which tends to extend the material along
the direction of stress (e.g., boudinage,
normal-faults).

Shear Stress: The stress on rock-mass

which acts along / parallel to the surface
(e.g., stress along fault-planes).
Next Lecture

In this lecture, we have learnt the basics of

force, stress on surface, stress at a point and
the principal axes of stresses.

In the next lecture we shall continue this topic

and focus upon the characteristics of the
principal axes of stresses, their components
and shear stress.