Stress,Strain and Young's Modulus

Stress

Stress is "force per unit area" - the ratio of applied force F to cross section area - defined as "force per 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)
F_n = normal component force (N, lb_f (alt. kips))
A = area (m², in²)

• a kip is a non-SI unit of force - it equals 1,000 pounds-force
• 1 kip = 4448.2216 Newtons (N) = 4.4482216 kilonewtons (kN)

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 the plane is usually denoted "shear stress" and can be expressed as

τ = F_p / A         (2)
where
τ = shear stress ((Pa) N/m², psi)
F_p = parallel component force (N, lb_f)
A = area (m², in²)

Strain

Strain is defined as "deformation of a solid due to stress" 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 = unit less measure of engineering strain
E = Young's modulus (Modulus of Elasticity) (N/m² (Pa), lb/in² (psi))

• Young's modulus can be used to predict the elongation or compression of an object.

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. The change of length can be calculated by transforming (3) as

 dl = σ l_o / E

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

     = 0.00127 (m)

     = 1.27 (mm)

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

S = stress / strain

   = τ / γ

   = (F_p / A) / (s / d)         (5)

where

S = shear modulus (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)

Elastic Moduli

Elastic moduli for some common materials:

Material	Young's Modulus		Shear Modulus		Bulk Modulus
	1010 N/m2	106 lb/in2	1010 N/m2	106 lb/in2	1010 N/m2	106 lb/in2
Aluminum	7.0	10	2.4	3.4	7.0	10
Brass	9.1	13	3.6	5.1	6.1	8.5
Copper	11	16	4.2	6.0	14	20
Glass	5.5	7.8	2.3	3.3	3.7	5.2
Iron	9.1	13	7.0	10	10	14
Lead	1.6	2.3	0.56	0.8	0.77	1.1
Steel	20	29	8.4	12	16	23

Source: http://www.engineeringtoolbox.com