Stress, Strain and Young's Modulus

Stress

Stress is the ratio of applied force F to a cross section area - defined as "force per unit area".

• tensile stress - stress that tends to stretch or lengthen the material - acts normal to the stressed area
• compressive stress - stress that tends to compress or shorten the material - acts normal to the stressed area
• shearing stress - stress that tends to shear the material - acts in plane to the stressed area at right-angles to compressive or tensile stress

Tensile or Compressive Stress - Normal Stress

Tensile or compressive stress normal to the plane is usually denoted "normal stress" or "direct stress" and can be expressed as

σ = F_n/ A                                    (1)

where

σ = normal stress (Pa (N/m²), psi (lb_f/in²))

F_n = normal force acting perpendicular to the area (N, lb_f)

A = area (m², in²)

• a kip is an imperial unit of force - it equals 1000 lb_f (pounds-force)
• 1 kip = 4448.2216 Newtons (N) = 4.4482216 kilo Newtons (kN)

A normal force acts perpendicular to area and is developed whenever external loads tends to push or pull the two segments of a body.

Example - Tensile Force acting on a Rod

A force of 10 kN is acting on a circular rod with diameter 10 mm. The stress in the rod can be calculated as

σ = (10 10³ N) / (π ((10 10^-3 m) / 2)²)

   = 127388535 (N/m²) 

   = 127 (MPa)

Example - Force acting on a Douglas Fir Square Post

A compressive load of 30000 lb is acting on short square 6 x 6 in post of Douglas fir. The dressed size of the post is 5.5 x 5.5 in and the compressive stress can be calculated as

σ = (30000 lb) / ((5.5 in) (5.5 in))

   = 991 (lb/in², psi)

Shear Stress

Stress parallel to a plane is usually denoted as "shear stress" and can be expressed as

τ = F_p/ A                               (2)

where

τ = shear stress (Pa (N/m²), psi (lb_f/in²))

F_p = shear force in the plane of the area (N, lb_f)

A = area (m², in²)

A shear force lies in the plane of an area and is developed when external loads tend to cause the two segments of a body to slide over one another.

Strain (Deformation)

Strain is defined as "deformation of a solid due to stress". 

• Normal strain - elongation or contraction of a line segment
• Shear strain - change in angle between two line segments originally perpendicular

Normal strain and can be expressed as

ε = dl / l_o

   = σ / E                              (3)

where

dl = change of length (m, in)

l_o = initial length (m, in)

ε = strain - unit-less

E = Young's modulus (Modulus of Elasticity) (Pa, (N/m²), psi (lb_f/in²))

• Young's modulus can be used to predict the elongation or compression of an object when exposed to a force

Note that strain is a dimensionless unit since it is the ratio of two lengths. But it also common practice to state it as the ratio of two length units - like m/m or in/in.

• Poisson's ratio is the ratio of relative contraction strain

Example - Stress and Change of Length

The rod in the example above is 2 m long and made of steel with Modulus of Elasticity 200 GPa (200 10⁹ N/m²). The change of length can be calculated by transforming (3) to

 dl = σ l_o / E

     = (127 10⁶ Pa) (2 m) / (200 10⁹ Pa) 

     = 0.00127 m

     = 1.27 mm

• How to calculate radial contraction
• Thermal stress and force

Strain Energy

Stressing an object stores energy in it. For an axial load the energy stored can be expressed as

U = 1/2 F_n dl

where

U = deformation energy (J (N m), ft lb)

Young's Modulus - Modulus of Elasticity (or Tensile Modulus) - Hooke's Law 

Most metals deforms proportional to imposed load over a range of loads. Stress is proportional to load and strain is proportional to deformation as expressed with Hooke's Law.

E = stress / strain

   = σ / ε

   = (F_n / A) / (dl / l_o)                                     (4)

where

E = Young's Modulus (N/m²) (lb/in², psi)

Modulus of Elasticity, or Young's Modulus, is commonly used for metals and metal alloys and expressed in terms 10⁶ lb_f/in², N/m² or Pa. Tensile modulus is often used for plastics and is expressed in terms 10⁵ lb_f/in² or GPa.

Shear Modulus of Elasticity - or Modulus of Rigidity

G = stress / strain

   = τ / γ

   = (F_p / A) / (s / d)                                    (5)

where

G = Shear Modulus of Elasticity - or Modulus of Rigidity (N/m²) (lb/in², psi)

τ  = shear stress ((Pa) N/m², psi)

γ = unit less measure of shear strain

F_p = force parallel to the faces which they act

A = area (m², in²)

s = displacement of the faces (m, in)

d = distance between the faces displaced (m, in)

• Ductile vs. Brittle materials

Bulk Modulus Elasticity

The Bulk Modulus Elasticity - or Volume Modulus - is a measure of the substance's resistance to uniform compression. Bulk Modulus of Elasticity is the ratio of stress to change in volume of a material subjected to axial loading.

Elastic Moduli

Elastic moduli for some common materials:

• 1 GPa = 10⁹ Pa (N/m²)
• 10⁶ psi = 1 Mpsi = 10³ ksi