Stress And Strain


Stress is the ratio of applied force F and cross section A, defined as “force per area”.

Direct Stress or Normal Stress

Stress normal to the plane is usually denoted “normal stress” and can be expressed as

σ = Fn / A (1)


σ = normal stress ((Pa) N/m2, psi)

Fn = normal component force (N, lbf)

A = area (m2, in2)

Shear Stress

Stress parallel to the plane is usually denoted “shear stress” and can be expressedas

τ = Fp / A (2)


τ = shear stress ((Pa) N/m2, psi)

Fp = parallel component force (N, lbf)

A = area (m2, in2)


Strain is defined as “deformation of a solid due to stress” and can be expressed as

ε = dl / lo = σ / E (3)


dl = change of length (m, in)

lo = initial length (m, in)

ε = unitless measure of engineering strain

E = Young’s modulus (Modulus of Elasticity) (Pa, psi)

Hooke’s Law – Modulus of Elasticity (Young’s Modulus or Tensile Modulus)

Most metals have deformations that are proportional with the imposed loads over a range of loads. Stress is proportional to load and strain is proportional to deformation expressed by the Hooke’s law like

E = stress / strain = (Fn / A) / (dl / lo) (4)


E = Young’s modulus (N/m2) (lb/in2, psi)

Modulus of Elasticity or Young’s Modulus are commonly used for metals and metal alloys and expressed in terms 106 lbf/in2,

N/m2 or Pa. Tensile modulus are often used for plastics and expressed in terms 105 lbf/in2 or GPa.

Poisson’s Ratio

υ = – εt / εl


υ = Poisson’s ratio

εt = transverse strain

εl = longitudinal or axial strain

Strain can be expressed as

ε = dl/L


dl = change in length

L = initial length

For most common materials the Poisson’s ratio is in the range 0 – 0.5.


Elasticity is a property of an object or material which will restore it to its original shape after distortion. A spring is an example of an elastic object – when stretched, it exerts a restoring force

which tends to bring it back to its original length. This restoring force is in general proportional to the stretch described by Hooke’s Law.

Hooke’s Law

One of the properties of elasticity is that it takes about twice as much force to stretch a spring twice as far. That linear dependence of displacement upon stretching force is called Hooke’s law which can be expressed as

Fs = -k dL (4)


Fs = force in the spring (N)

k = spring constant (N/m)

dL = elongation of the spring (m)

Yield strength

Yield strength, or the yield point, is defined in engineering as the amount of stress that a material can undergo before moving from elastic deformation into plastic deformation.

Ultimate Tensile Strength

The Ultimate Tensile Strength – UTS – of a material is the limit stress at which the material actually breaks, with sudden release of the stored elastic energy.

Modulus of Rigidity (or Shear Modulus) is the

coefficient of elasticity for a shearing force. It is defined as “the ratio of shear stress to the displacement per unit sample length (shear strain)” . Modulus of Rigidity can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material.

Definition of Modulus of Rigidity:

The ratio of shear stress to the displacement per unit sample length (shear strain)



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