Elasticity, ability of a deformed material body to return to its original shape and size when the forces causing the deformation are removed. A body with this ability is said to behave (or respond) elastically.
To a greater or lesser extent, most solid materials exhibit elastic behaviour, but there is a limit to the magnitude of the force and the accompanying deformation within which elastic recovery is possible for any given material. This limit, called the elastic limit, is the maximum stress or force per unit area within a solid material that can arise before the onset of permanent deformation. Stresses beyond the elastic limit cause a material to yield or flow.
Elasticity is a physical property of a material whereby the material returns to its original shape after having been stretched out or altered by force. Substances that display a high degree of elasticity are termed “elastic.” The SI unit applied to elasticity is the pascal (Pa), which is used to measure the modulus of deformation and elastic limit.
Young’s modulus (E or Y) is a measure of a solid’s stiffness or resistance to elastic deformation under load. It relates stress (force per unit area) to strain (proportional deformation) along an axis or line. The basic principle is that a material undergoes elastic deformation when it is compressed or extended, returning to its original shape when the load is removed. More deformation occurs in a flexible material compared to that of a stiff material. In other words:
- A low Young’s modulus value means a solid is elastic.
- A high Young’s modulus value means a solid is inelastic or stiff.
Equation and Units
The equation for Young’s modulus is:
E = σ / ε = (F/A) / (ΔL/L0) = FL0 / AΔL
Where:
- E is Young’s modulus, usually expressed in Pascal (Pa)
- σ is the uniaxial stress
- ε is the strain
- F is the force of compression or extension
- A is the cross-sectional surface area or the cross-section perpendicular to the applied force
- Δ L is the change in length (negative under compression; positive when stretched)
- L0 is the original length
While the SI unit for Young’s modulus is Pa, values are most often expressed in terms of megapascal (MPa), Newtons per square millimeter (N/mm2), gigapascals (GPa), or kilonewtons per square millimeter (kN/mm2). The usual English unit is pounds per square inch (PSI) or mega PSI (Mpsi).
Modulii of Elasticity
A modulus is literally a “measure.” You may hear Young’s modulus referred to as the elastic modulus, but there are multiple expressions used to measure elasticity:
- Young’s modulus describes tensile elasticity along a line when opposing forces are applied. It is the ratio of tensile stress to tensile strain.
- The bulk modulus (K) is like Young’s modulus, except in three dimensions. It is a measure of volumetric elasticity, calculated as volumetric stress divided by volumetric strain.
- The shear or modulus of rigidity (G) describes shear when an object is acted upon by opposing forces. It is calculated as shear stress over shear strain.
The axial modulus, P-wave modulus, and Lamé’s first parameter are other modulii of elasticity. Poisson’s ratio may be used to compare the transverse contraction strain to the longitudinal extension strain. Together with Hooke’s law, these values describe the elastic properties of a material.
Calculating Stress
Stress is defined as force per unit area as shown in the equation σ = F / A.
Stress is often represented by the Greek letter sigma (σ) and expressed in newtons per square meter, or pascals (Pa). For greater stresses, it is expressed in megapascals (106 or 1 million Pa) or gigapascals (109 or 1 billion Pa).
Force (F) is mass x acceleration, and so 1 newton is the mass required to accelerate a 1-kilogram object at a rate of 1 meter per second squared. And the area (A) in the equation is specifically the cross-sectional area of the metal that undergoes stress.
Let’s say a force of 6 newtons is applied to a bar with a diameter of 6 centimeters. The area of the cross section of the bar is calculated by using the formula A = π r2. The radius is half of the diameter, so the radius is 3 cm or 0.03 m and the area is 2.2826 x 10-3 m2.
A = 3.14 x (0.03 m)2 = 3.14 x 0.0009 m2 = 0.002826 m2 or 2.2826 x 10-3 m2
Now we use the area and the known force in the equation for calculating stress:
σ = 6 newtons / 2.2826 x 10-3 m2 = 2,123 newtons / m2 or 2,123 Pa
Calculating Strain
Strain is the amount of deformation (either stretch or compression) caused by the stress divided by the initial length of the metal as shown in the equation ε = dl / l0. If there is an increase in the length of a piece of metal due to stress, it is referred to as tensile strain. If there’s a reduction in length, it’s called compressive strain.
Strain is often represented by the Greek letter epsilon (ε), and in the equation, dl is the change in length and l0 is the initial length.
Strain has no unit of measurement because it’s a length divided by a length and so is expressed only as a number. For example, a wire that’s initially 10 centimeters long is stretched to 11.5 centimeters; its strain is 0.15.
ε = 1.5 cm (the change in length or amount of stretch) / 10 cm (initial length) = 0.15
The shear modulus is defined as the ratio of shear stress to shear strain. It is also known as the modulus of rigidity and may be denoted by G or less commonly by S or μ. The SI unit of shear modulus is the Pascal (Pa), but values are usually expressed in gigapascals (GPa). In English units, shear modulus is given in terms of pounds per square inch (PSI) or kilo (thousands) pounds per square in (ksi).
- A large shear modulus value indicates a solid is highly rigid. In other words, a large force is required to produce deformation.
- A small shear modulus value indicates a solid is soft or flexible. Little force is needed to deform it.
- One definition of a fluid is a substance with a shear modulus of zero. Any force deforms its surface.
Shear Modulus Equation
The shear modulus is determined by measuring the deformation of a solid from applying a force parallel to one surface of a solid, while an opposing force acts on its opposite surface and holds the solid in place. Think of shear as pushing against one side of a block, with friction as the opposing force. Another example would be attempting to cut wire or hair with dull scissors.
The equation for the shear modulus is:
G = τxy / γxy = F/A / Δx/l = Fl / AΔx
Where:
- G is the shear modulus or modulus of rigidity
- τxy is the shear stress
- γxy is the shear strain
- A is the area over which the force acts
- Δx is the transverse displacement
- l is the initial length
Shear strain is Δx/l = tan θ or sometimes = θ, where θ is the angle formed by the deformation produced by the applied force.
Example Calculation
For example, find the shear modulus of a sample under a stress of 4×104 N/m2 experiencing a strain of 5×10-2.
G = τ / γ = (4×104 N/m2) / (5×10-2) = 8×105 N/m2 or 8×105 Pa = 800 KPa
Isotropic and Anisotropic Materials
Some materials are isotropic with respect to shear, meaning the deformation in response to a force is the same regardless of orientation. Other materials are anisotropic and respond differently to stress or strain depending on orientation. Anisotropic materials are much more susceptible to shear along one axis than another. For example, consider the behavior of a block of wood and how it might respond to a force applied parallel to the wood grain compared to its response to a force applied perpendicular to the grain. Consider the way a diamond responds to an applied force. How readily the crystal shears depends on the orientation of the force with respect to the crystal lattice.
Effect of Temperature and Pressure
As you might expect, a material’s response to an applied force changes with temperature and pressure. In metals, shear modulus typically decreases with increasing temperature. Rigidity decreases with increasing pressure. Three models used to predict the effects of temperature and pressure on shear modulus are the Mechanical Threshold Stress (MTS) plastic flow stress model, the Nadal and LePoac (NP) shear modulus model, and the Steinberg-Cochran-Guinan (SCG) shear modulus model. For metals, there tends to be a region of temperature and pressures over which change in shear modulus is linear. Outside of this range, modeling behavior is trickier.
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 = Fn / A (1)
where
σ = normal stress (Pa (N/m2), psi (lbf/in2))
Fn = normal force acting perpendicular to the area (N, lbf)
A = area (m2, in2)
- a kip is an imperial unit of force – it equals 1000 lbf (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 = Fp / A (2)
where
τ = shear stress (Pa (N/m2), psi (lbf/in2))
Fp = shear force in the plane of the area (N, lbf)
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 / lo
= σ / E (3)
where
dl = change of length (m, in)
lo = initial length (m, in)
ε = strain – unit-less
E = Young’s modulus (Modulus of Elasticity) (Pa , (N/m2), psi (lbf/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 lo / E
= (127 106 Pa) (2 m) / (200 109 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 Fn 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
= (Fn / A) / (dl / lo) (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 lbf/in2, N/m2 or Pa. Tensile modulus is often used for plastics and is expressed in terms 105 lbf/in2 or GPa.
Shear Modulus of Elasticity – or Modulus of Rigidity
G = stress / strain
= τ / γ
= (Fp / 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
Fp = 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)
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) (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 ks