Stress and strain - basic concepts

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

Stress

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

normal stress 

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)  in the SI system is the Pascal* (Pa).  In engineering applications stress values are typically in a range expressed in MPa.

* 1 Pa = 1 N/m2

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

triaxial stress model 

Normal stress (σ) and shear stress (τ) are positive when acting in the positive direction of the references axes. For equilibrium     τxy= τyx     τyz= τzy     τzx= τxz

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.

biaxial stress model 

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 σx σy τxy τyx   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 σxxy and σyyx 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.

Mohr's circle 

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

biaxial stress element 

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

Mohr's circle 

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

Mohr's circle 

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 σ1 and σ2 shown on the diagram below.

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

σ1 and σ2 are respectively the maximum and minimum normal stresses.

biaxial stress element 

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  (σ1 + σ2}/2

Mohr's circle 

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

Strain

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

strain in bar under tension

The diagram above shows a bar of material length L0, 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 εx is the longitudinal strain and εy the lateral strain.

Poisson's ratio 

Provided the material under load remains in the linear-elastic range for stress and strain (see below) the ratio between  εy and εx  is effectively constant.  This constant is designated ν (Greek Nu) and known as Poisson's ratio.

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.

definition of shear strain 

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.

   Relation between stress and strain

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

stress-strain diagram

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.

triaxial stress element with strains

The diagram shows the principal strains produced on each axis for an arbitrary set of tensile principal stresses where  σ1 > σ2 > σ3 .  The following general relations between principal stresses and principal strains are derived from the definition of Poisson's ratio ν .

 

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Alistair's tutorials 2019