Stress, Strain and Young's Modulus
Stress is force per unit area  strain is the deformation of a solid due to stress.
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 rightangles 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/m^{2}), psi (lb_{f}/in^{2}))
F_{n} = normal force acting perpendicular to the area (N, lb_{f})
A = area (m^{2}, in^{2})
 a kip is an imperial unit of force  it equals 1000 lb_{f} (poundsforce)
 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 10^{3} N) / (π ((10 10^{3} m) / 2)^{2})
= 127388535 (N/m^{2})
= 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/in^{2}, 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/m^{2}), psi (lb_{f}/in^{2}))
F_{p} = shear force in the plane of the area (N, lb_{f})
A = area (m^{2}, in^{2})
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  unitless
E = Young's modulus (Modulus of Elasticity) (Pa , (N/m^{2}), psi (lb_{f}/in^{2}))
 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 10^{9} N/m^{2}). The change of length can be calculated by transforming (3) to
dl = σ l_{o }/ E
= (127 10^{6} Pa) (2 m) / (200 10^{9} Pa)
= 0.00127 m
= 1.27 mm
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/m^{2}) (lb/in^{2}, psi)
Modulus of Elasticity, or Young's Modulus, is commonly used for metals and metal alloys and expressed in terms 10^{6} lb_{f}/in^{2}, N/m^{2} or Pa. Tensile modulus is often used for plastics and is expressed in terms 10^{5} lb_{f}/in^{2} 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/m^{2}) (lb/in^{2}, psi)
τ = shear stress ((Pa) N/m^{2}, psi)
γ = unit less measure of shear strain
F_{p} = force parallel to the faces which they act
A = area (m^{2}, in^{2})
s = displacement of the faces (m, in)
d = distance between the faces displaced (m, in)
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:
Material  Young's Modulus  E   Shear Modulus  G   Bulk Modulus  K  

(GPa) (10^{6} psi)  (GPa) (10^{6} psi)  (GPa) (10^{6} 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 = 10^{9} Pa (N/m^{2})
 10^{6} psi = 1 Mpsi = 10^{3} ksi
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