Physics
Elasticity
Volume Stress
11
⚡ Quick Summary
Volume stress is the force acting normally on a surface area of a body immersed in a fluid, equal to the pressure at that location.
Volume Stress = F/A = Pressure (P)
When a body is acted upon by forces acting everywhere on the surface such that:
(a) the force at any point is normal to the surface, and
(b) the magnitude of the force on any small surface area is proportional to the area.
This is the case when a small solid body is immersed in a fluid. If the pressure at the location of the solid is P, the force on any area ΔS is PΔS directed perpendicularly to the area. The force per unit area is then called volume stress.
Volume Strain
11
⚡ Quick Summary
Volume strain is the fractional change in volume when a body is subjected to volume stress.
Volume strain = ΔV/V
When a body is subjected to a volume stress, its volume changes. The volume strain is defined as the fractional change in volume. If V is the volume of unstressed body and V + ΔV is the volume when the volume stress exists, the volume strain is defined.
Longitudinal Strain
11
⚡ Quick Summary
Longitudinal strain is the fractional change in length of a rod being pulled by equal and opposite forces.
Longitudinal strain = ΔL/L
Consider a rod of length l being pulled by equal and opposite forces. The length of the rod increases from its natural value L to L + ΔL. The fractional change ΔL/L is called the longitudinal strain.
If the length increases from its natural length, the longitudinal strain is called tensile strain. If the length decreases from its natural length, the longitudinal strain is called compressive strain.
Shearing Strain
11
⚡ Quick Summary
Shearing strain is the displacement of a layer divided by its distance from the fixed layer when a shearing stress is present.
Shearing strain = x/h (where x is the displacement of the layer and h is the distance from the fixed layer)
This type of strain is produced when a shearing stress is present over a section. Consider a body with square cross section and suppose forces parallel to the surfaces are applied.
We define the shearing strain as the displacement of a layer divided by its distance from the fixed layer.
Hooke's Law
11
⚡ Quick Summary
Hooke's Law states that stress is proportional to strain for small deformations.
Tensile stress = Y * Tensile strain (where Y is Young's modulus)
If the deformation is small, the stress in a body is proportional to the corresponding strain. This fact is known as Hooke’s law. Thus, if a rod is stretched by equal and opposite forces F each, a tensile stress F/A is produced in the rod where A is the area of cross section. The length of the rod increases from its natural value L to L + ΔL. Tensile strain is ΔL/L.
Stress-Strain Relation
11
⚡ Quick Summary
The relationship between stress and strain in a material as it's deformed. Different materials behave differently. Important concepts include proportional limit, elastic limit, plastic behavior, ductile vs brittle materials, and elastic hysteresis. Rubber and steel are compared to illustrate elasticity concepts.
Hooke's Law: Stress = Young's Modulus * Strain (valid only up to the proportional limit)
- Stress-Strain Curve: A graph showing the relationship between stress (force per unit area) and strain (deformation) as a material is subjected to increasing force.
- Proportional Limit (a): The point on the stress-strain curve up to which stress is directly proportional to strain (Hooke's Law applies).
- Elastic Limit/Yield Point (b): The point beyond which the material will not return to its original length when the force is removed. Beyond this point, permanent deformation occurs.
- Plastic Behavior: The behavior of a material when it is deformed beyond its elastic limit, resulting in permanent deformation. Represented by a dashed line from point 'c' in Figure 14.7.
- Fracture Point (d): The point at which the material breaks. The stress at this point is the breaking stress.
- Ductile Material: A material that undergoes large deformation between the elastic limit and the fracture point.
- Brittle Material: A material that breaks soon after the elastic limit is crossed, with little deformation.
- Elasticity Comparison (Rubber vs. Steel): Rubber can be stretched much more and still return to its original shape (more 'elastic' in this sense). However, steel requires much larger forces for a given deformation (larger Young's modulus), making it 'more elastic' in terms of resistance to deformation.
- Elastic Hysteresis: The phenomenon where the stress-strain curve during loading is different from the curve during unloading. Energy is absorbed during the cycle and released as heat.
- Application of Elastic Hysteresis: Shock absorbers use materials (like vulcanized rubber) with high hysteresis to absorb vibrational energy and dampen oscillations.
Elastic Potential Energy
11
⚡ Quick Summary
When a material is deformed, it stores potential energy due to the internal forces. This is the elastic potential energy. It's the energy stored when a wire is stretched.
No formulas are explicitly provided in this section, but it sets the stage for deriving the formula for elastic potential energy in the following text.
- Definition: Elastic potential energy is the potential energy stored in a deformed body due to the internal forces resisting the deformation.
- Origin: When a body is in its natural shape, its potential energy is minimum (taken as zero). Deformation requires work against internal forces, increasing the potential energy.
Stress and Strain
11
⚡ Quick Summary
Stress is the force per unit area within a material that arises from externally applied forces. Strain is the deformation of the material caused by stress.
Stress = Force / Area, Strain = Change in Length / Original Length
- Stress is defined as the force acting per unit area. It is a measure of the internal forces that molecules within a continuous material exert on each other.
- Strain is the measure of the deformation of the material. It is dimensionless and can be tensile, compressive, or shear strain.
- Young's modulus (Y) relates stress and strain in tensile or compressive deformation: Y = Stress / Strain.
Tension
11
⚡ Quick Summary
Tension is the pulling force exerted by a string, cable, chain, or similar object on another object.
Tension (T) is a force, and its units are Newtons (N).
- Tension is a pulling force transmitted axially through a string, cable, chain, or similar one-dimensional continuous object, or by each end of a rod or similar three-dimensional object.
- Tension is directed along the length of the wire or string.
- The tension in a wire can be calculated by analyzing the forces acting on the objects connected to the wire, considering equilibrium or Newton's laws of motion.
Bulk Modulus and Compressibility
11
⚡ Quick Summary
Bulk modulus measures a substance's resistance to uniform compression. Compressibility is the reciprocal of the bulk modulus and indicates how easily a substance's volume changes under pressure.
Bulk Modulus (B) = - (Change in Pressure) / (Change in Volume / Original Volume), Compressibility (K) = 1 / B
- Bulk Modulus (B) is a measure of a substance's resistance to uniform compression. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.
- Compressibility (K) is the reciprocal of the bulk modulus. A high compressibility means a small change in pressure results in a large change in volume.
Stress
11
⚡ Quick Summary
Stress is the force acting per unit area within a solid material.
Stress = Force / Area
Stress is a measure of the internal forces that molecules within a continuous material exert on each other. It is defined as the force acting per unit area. There are different types of stress: Tensile stress (pulling), compressive stress (pushing), and shear stress (tangential force). Stress = Force / Area
Strain
11
⚡ Quick Summary
Strain is the deformation of a solid material due to stress.
Strain = Change in Length / Original Length
Strain is a measure of the deformation of a material caused by stress. It's a dimensionless quantity, often expressed as a ratio of change in dimension to the original dimension. Strain = Change in Length / Original Length
Young's Modulus
11
⚡ Quick Summary
Young's modulus (Y) is a measure of a solid's stiffness or resistance to deformation under tensile or compressive stress.
Y = Stress / Strain
Young's modulus is defined as the ratio of tensile stress to tensile strain. It represents the stiffness of a material. Y = Stress / Strain
Elasticity
11
⚡ Quick Summary
The ability of a solid material to return to its original shape after external forces are removed.
Stress = Force/Area; Strain = Change in Length/Original Length; Young's Modulus = Stress/Strain; Bulk Modulus = - (Change in Pressure)/(Change in Volume/Original Volume)
Deals with the behavior of solid materials under stress. Key concepts include elastic and inelastic behavior, stress, strain, and elastic moduli.
Poisson's Ratio
XI
⚡ Quick Summary
Ratio of transverse strain to axial strain.
ν = - (transverse strain) / (axial strain)
Ratio of transverse strain to axial strain.
Rigidity Modulus
XI
⚡ Quick Summary
A measure of a solid's resistance to deformation by shear stress.
G = (shear stress) / (shear strain)
A measure of a solid's resistance to deformation by shear stress.