1

Stresses (1)

ForceForce is a kind of mechanical action between different objects, it tends to change the shape, volume or movement state of the object with a force upon it.

Force = mass × acceleration (kg m s-2) [Newton][N]

Force is a vector quantity, and thus possesses both magnitude and direction; it can be represented by an arrow whose length specifies the magnitude and whose orientation specifies the orientation of the force.

F

Unit: Newton 1 Newton = 1 kilogram meter per second squared

vectorscalar (only magnitude)

Resolution and resultant of forcesA.A force F be resolved into two components F1 and

F2. B. Two forces F1 and F2 be represented by the resultant F

F1

F2

F F1

F2

F

A B

Surface Forces and Body Forces

Surface forces: the forces acting on the contact surface between adjacent parts of rock system, between adjacent blocks or adjacent lithosphere plates. The contact surface may be or may be not a visible material boundary. It can be a imaginary surface inside the object considered.

Body forces: the forces can work at a distance and depend on the amount of material affected, so, we can call body forces distant forces. Gravitational force is an example of body forces. The gravitational force on a rock body of mass m is

F = mgwhere g is the acceleration of gravity. g varies with depth in the earth and with position on the earth’s surface, but for the purpose of structural geology, it is a constant 9.8m/sec2.

Body forces

Uniform forces

Nonuniform forces

External forces

Imaginary plane

Uniform Internal Forces

a b

a

F F

F

σ

N=F

σ=N/A=F/A

Internal forces and stresses

Stress on a plane: internal forces acting on unit area of thegiven plane within the considering body.

x

2

pdFdp

FP

F==

ΔΔ

→Δlim

0

m

Interna l fo rc e area

F

P

F—P—

Externalforces

Internal forces area stress

Stress acting at a point m on a plane n is a vector, it can be resolved into two components σ and τ, σ is normal to the plane, called normal stress, τ is tangential to the plane, called shear stress.

Normal stress and shear stress

Magnitude of stress

Stress = Force / Area, limit Area approaching zero

Units of stress

[ Newton / m2 or a ‘Pascal’], or simply say ‘Pa’

That is 1 pascal = 1 newton per square meter.

1 newton = 1 kilogram meter per second squared (1 kg m s-2)

A more commonly used unit is the bar or the kilobar,

Where:

1 bar = 105 pascals = 0.1 Mpa (105: 10 to the power 5)

Magnitude and Units of Stress

Normal and shear stresses at a fault plane (A) and a bedding plane during flexural slip folding (B)

(Park,2007)

p p p

x

yz τx

τy

Resolution of stress in two dimensions (A) and in tree dimensions (B)

(Park,2007)

Stress components in three dimensions

Infinitesimal cube

τxy=τyxτyz=τzyτzx=τxz

3

Nine components of stress at a point in matrix form

Since τxy=τyx τyz=τzy τzx=τxz , only six stresscomponents left: three normal stresses σx, σy ,σzand three shear stresses τxy , τy z, τzx

For an arbitrarily chosen set of orthogonal axes x, y, z, six independent quantities are necessary to specify completelythe state of stress at a point.

(5-2)