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:

Stress, Strain and Young's Modulus for some common Materials

Material	Young's Modulus - E -	Shear Modulus - G -	Bulk Modulus - K -
	(GPa) (106 psi)	(GPa) (106 psi)	(GPa) (106 psi)
Aluminum	70	24	70
Brass	91	36	61
Copper	110	42	140
Glass	55	23	37
Iron	91	70	100
Lead	16	5.6	7.7
Steel	200	84	160

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