This tutorial provides an overview of stress and strain in an engineering context.

At a basic level we define:

Normal stress** σ** in a material as
**σ** = **F/A** where **F**
is the normal force applied on area **A**

Shear stress **τ** in a material as
**τ = V/A** where **V** is the tangential shearing
force applied on area **A**

At a macro level these definitions of stress can be applied in situations where stresses are uniform over a large area, for example a steel bar in pure tension or compression. However, in many situations consideration of stress on a micro level is necessary and definitions are based on limits over a small element of area δA :

The unit for stress (**σ ** and** τ**)**
i**n the SI system is the *Pascal** (Pa). In
engineering applications stress values are typically in a range
expressed in MPa.

* 1 Pa = 1 N/m^{2}

The model for normal and shear stresses on a three-dimensional
(triaxial) stress element is shown below.

Normal stress (σ) and shear stress (τ) are positive when acting in
the positive direction of the references axes. For equilibrium
τ_{xy}= τ_{yx} τ_{yz}=
τ_{zy } τ_{zx}= τ_{xz}

Note the convention for designating shear stresses – the first subscript is the co-ordinate normal to the face; the second subscript is the axis parallel to the face.

Corresponding stresses exist on the three hidden faces of the element. Directions of these stresses are such as to maintain static equilibrium of the element.

For some situations a two-dimensional (biaxial or plane) stress analysis suffices.

Directions of σ shown are positive and represent tension; reverse directions represent compression.

Directions of τ are designated clockwise (cw) or counter-clockwise (ccw).

We will now show a graphical method of determining normal and shear stresses
based on known values of σ_{x} σ_{y} τ_{xy} τ_{yx} where the stress element is
oriented at any angle θ to the xy axes using a diagram known as **Mohr's circle of stress**.

To construct Mohr's circle set out orthogonal axes of σ
and τ. Mark off the known values of σ_{x},τ_{xy} and
σ_{y},τ_{yx} The line joining these points forms the
diameter of the circle. The orientation of this diameter
represents the reference axes xy in the diagram above.

Now consider stresses σ^{/} , σ^{//} and τ^{/}
on planes rotated clockwise by θ degrees.

These stresses are resolved on Mohr's circle by rotating the base
diameter by **2**θ degrees and projecting from the
circumference on to the σ and τ axes. Note shear
stresses are of equal magnitude on all four faces.

Consider below what occurs on planes oriented at angle φ degrees anti-clockwise from the reference xy axes.

The circle diameter rotates 2φ degrees anti-clockwise, this being
the angle that brings the base diameter coincident with the σ axis.
Hence there are no shear stresses on these planes. We call
these normal stresses **principal stresses**,
designated σ_{1} and σ_{2 }shown on the diagram
below.

The principal stresses act on the **principal planes**
which are 90 degrees apart (180 degrees on the Mohr circle).

σ_{1} and σ_{2} are respectively the maximum and
minimum normal stresses.

It is clear from Mohr's circle below that the maximum shear
stress occurs on planes oriented 45 degrees (90 degrees on the
circle) from the principal planes. It is easily shown that
normal stresses are equal on the planes of maximum shear stress and
have the value (σ_{1} + σ_{2}}/2

Mathematical expressions showing the derivation of
Mohr's circle can be found
in text books.

Materials subject to normal stress **σ** experience
deformation which in general terms we call ** strain**.

The diagram above shows a bar of material length L_{0}, with
applied axial force F. The elongation of the bar is ΔL and the
strain **ε** is defined as:

Note that this definition of strain has no dimensions. The diagram shows the tensile strain resulting from tensile stress in the bar. A compressive force produces compressive strain.

Sometimes the term *total strain* is used for ΔL, i.e. the
actual elongation or compression of the material. The units of
ΔL are length.

When material such as a round bar in the diagram below is subject to
longitudinal tensile strain there is a corresponding contraction in
the lateral direction. Conversely there is lateral expansion
in a bar subject to compressive strain. The respective
elongations and contractions are shown where ε_{x} is the
longitudinal strain and ε_{y} the lateral strain.

Provided the material under load remains in the linear-elastic range
for stress and strain (see below) the ratio between ε_{y}
and ε_{x} is effectively constant. This constant
is designated ** ν** (Greek Nu) and known as

For most engineering materials the value of * ν *is
approximately 0.3

We have considered strain resulting from normal stresses σ.
Strain arising from shear stress **τ** is defined in
the diagram below.

The biaxial stress condition shows a rectangular block of material
deformed under shear stress **τ** into the form of a
parallelogram. The angle **γ** defines the shear
strain. As with ε, shear strain has no dimensions and is
measured in radians.

The diagram below illustrates the relation between stress and strain in the linear-elastic range for a material.

Stress-strain diagrams such as this are generally obtained from tensile load tests using a test piece in the form of a round bar. As the load on the test piece increases the stress increases and the resulting strain is measured.

The diagram shows a linear relationship between stress and strain up
to a point called the *proportional limit,* sometimes called
the *elastic limit*. In this
region the material is said to be *elastic*, meaning that
when the load is removed it regains its original dimensions without
deformation.

In the linear-elastic region the relationship between stress and strain is expressed as:

**σ = E. ε **
where the constant** E** is called **Young's
modulus **or the **modulus of elasticity**. Note
that E has the same units (Pa) as stress. E is a measure of
the *stiffness* of a material, the greater the value of E the
stiffer the material.

For most engineering materials the strain in the linear-elastic
region is very small, typically up to 0.001. Behaviour beyond
the elastic limit is outside the scope of this tutorial but in broad
terms materials reach a *yield point *beyond which increasing
stress produces permanent deformation and eventually fracture.
There are significant differences in behaviour between ductile and
brittle materials.

Values of E vary considerably for different materials. Typical values are:

carbon steel E = 207 GPa

brass E = 106 GPa

cast iron E = 100 GPa

aluminium alloy E = 70 Gpa

Material subjected to pure shear stress also exhibits linear-elastic behaviour. In this case the relation is:

**τ = G.γ **
where the constant **G** is called the *modulus of
rigidity *or *shear modulus*. Again, G has the same
units (Pa) as stress. Values of G are generally about 1/3
those of E.

To complete this overview oft he basic concepts of stress and strain
we will look again at the condition of triaxial stress (see above) where the element is aligned on the axes of the principal
stresses and thus shear stresses are zero on all faces of the
element. The strains associated with the principal stresses
are called *principal strains*.

The diagram shows the principal strains produced on each axis for an
arbitrary set of tensile principal stresses where σ_{1} > σ_{2}
> σ_{3} . The following general relations between
principal stresses and principal strains are derived from the
definition of Poisson's ratio **ν .**

I welcome feedback at:

alistair@alistairstutorials.co.uk