“women's blue dress shirt” by Tra Nguyen on Unsplash

Mechanical Properties of Solids(Elasticity)

Jyotiraditya
Sep 7, 2018 · 12 min read

When we think about solids, we just imagine about a Wooden Block or a Iron Ingot or simply an Ice. We have studied from childhood about the Properties of 3 states of matter which are :-

a)Solid(Ice)- intermolecular forces of attraction are maximum.

b) Liquid(Water)- intermolecular forces of attraction are intermediate.

c)Gas(Vapour)- intermolecular forces of attraction are minimum.

Now, we can see that a Fluid & a Gas flows easily and can never be of a definite shape by itself. Then what is such present in a solid which provides it it’s rigid shape? Why do Solids Rupture or Break after constant expose to preassure while fluids and vapours don’t?

The reason is due to the Mechanical properties of Solid.

  • Brittleness: Ability of a material to break or shatter without significant deformation when under stress; opposite of plasticity,examples:glass,concrete,cast iron,ceramics etc.
  • Bulk modulus: Ratio of pressure to volumetric compression (GPa) or ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume. brass has highest bulk modulus of 116 GPa.
  • Coefficient of restitution:the ratio of the final to initial relative velocity between two objects after they collide. Range : 0–1, 1 for perfectly elastic collision.
  • Compressive strength: Maximum stress a material can withstand before compressive failure (MPa)
  • Creep: The slow and gradual deformation of an object with respect to time
  • Ductility: Ability of a material to deform under tensile load (% elongation)
  • Durability: Ability to withstand wear, pressure, or damage; hard-wearing.
  • Elasticity: Ability of a body to resist a distorting influence or stress and to return to its original size and shape when the stress is removed
  • Fatigue limit: Maximum stress a material can withstand under repeated loading (MPa)
  • Flexibility: Ability of an object to bend or deform in response to an applied force; pliability; complementary to stiffness
  • Flexural modulus
  • Flexural strength : The stresses in a material just before it yields.
  • Fracture toughness: Ability of a material containing a crack to resist fracture (J/m²)
  • Hardness: Ability to withstand surface indentation and scratching (e.g. Brinnell hardness number)
  • Mass diffusivity: Ability of one substance to diffuse through another
  • Plasticity: Ability of a material to undergo irreversible or permanent deformations without breaking or rupturing; opposite of brittleness
  • Poisson’s ratio: Ratio of lateral strain to axial strain (no units)
  • Resilience: Ability of a material to absorb energy when it is deformed elastically (MPa); combination of strength and elasticity
  • Shear modulus: Ratio of shear stress to shear strain (MPa)
  • Shear strength: Maximum shear stress a material can withstand
  • Slip: A tendency of a material’s particles to undergo plastic deformation due to a dislocation motion within the material. Common in Crystals.
  • Specific modulus: Modulus per unit volume (MPa/m³)
  • Specific strength: Strength per unit density (Nm/kg)
  • Specific weight: Weight per unit volume (N/m³)
  • Stiffness: Ability of an object to resist deformation in response to an applied force; rigidity; complementary to flexibility
  • Surface roughness:the deviations in the direction of the normal vector of a real surface from its ideal form.
  • Tensile strength: Maximum tensile stress of a material can withstand before failure (MPa)
  • Toughness: Ability of a material to absorb energy (or withstand shock) and plastically deform without fracturing (or rupturing); a material’s resistance to fracture when stressed; combination of strength and plasticity
  • Viscosity: A fluid’s resistance to gradual deformation by tensile or shear stress; thickness
  • Yield strength: The stress at which a material starts to yield plastically (MPa)
  • Young’s modulus: Ratio of linear stress to linear strain (MPa)
  • Strength of materials (relation of various strengths)

What is the Elasticity or Elastic Behaviour of a Solid?

Elasticity is the property of a body, by virtue of which it tends to regain its original shape when the appiled force is remove.

People often confuse Plasticity with Elasticity!!!!

Plasticity is the property of a body in which it have no gross tendency to regain the previous shape and they get permanent deformed

Stress and Strain.

Stress:-

When an external force is applied to a body then at each cross section of the body an internal restoring force is developed which tends to restore the body to its original state. The internal restoring force per unit area of cross section of the deformedbody is called Stress. it is usually denoted by “σ”(Sigma)

Strain:-

When the size or shape of a body is changed under an external force, the body is said to be under Strain.The change occured in the unit size of the body is called Strain.

Hooke’ Law and Modulus of Elasticity:-

Hooke’s law, law of elasticity discovered by the English scientist Robert Hooke in 1660, which states that, for relatively small deformations of an object, the displacement or size of the deformation is directly proportional to the deforming force or load. Under these conditions the object returns to its original shape and size upon removal of the load.

The deforming force may be applied to a solid by stretching, compressing, squeezing, bending, or twisting. Thus, a metal wire exhibits elastic behaviour according to Hooke’s law because the small increase in its length when stretched by an applied force doubles each time the force is doubled. Mathematically, Hooke’s law states that the applied force F equals a constant k times the displacement or change in length x, or F = kx. The value of k depends not only on the kind of elastic material under consideration but also on its dimensions and shape.

It als states that Stress=E(strain) where E= Modulus of Elasticity.

Young’s modulus, numerical constant, named for the 18th-century English physician and physicist Thomas Young, that describes the elastic properties of a solid undergoing tension or compression in only one direction, as in the case of a metal rod that after being stretched or compressed lengthwise returns to its original length. Young’s modulus is a measure of the ability of a material to withstand changes in length when under lengthwise tension or compression. Sometimes referred to as the modulus of elasticity, Young’s modulus is equal to the longitudinal stress divided by the strain. Stress and strain may be described as follows in the case of a metal bar under tension.

If a metal bar of cross-sectional area A is pulled by a force F at each end, the bar stretches from its original length L0 to a new length Ln. (Simultaneously the cross section decreases.) The stress is the quotient of the tensile force divided by the cross-sectional area, or F/A. The strain or relative deformation is the change in length, LnL0, divided by the original length, or (LnL0)/L0. (Strain is dimensionless.) Thus Young’s modulus may be expressed mathematically as

Young’s modulus = stress/strain = (FL0)/A(LnL0).

This is a specific form of Hooke’s law of elasticity. The units of Young’s modulus in the English system are pounds per square inch (psi), and in the metric system newtons per square metre (N/m2). The value of Young’s modulus for aluminum is about 1.0 × 107 psi, or 7.0 × 1010 N/m2. The value for steel is about three times greater, which means that it takes three times as much force to stretch a steel bar the same amount as a similarly shaped aluminum bar

Bulk modulus, numerical constant that describes the elastic properties of a solid or fluid when it is under pressure on all surfaces. The applied pressure reduces the volume of a material, which returns to its original volume when the pressure is removed. Sometimes referred to as the incompressibility, the bulk modulus is a measure of the ability of a substance to withstand changes in volume when under compression on all sides. It is equal to the quotient of the applied pressure divided by the relative deformation.

In this case, the relative deformation, commonly called strain, is the change in volume divided by the original volume. Thus, if the original volume Vo of a material is reduced by an applied pressure p to a new volume Vn, the strain may be expressed as the change in volume, VoVn, divided by the original volume, or (VoVn)/Vo. The bulk modulus itself, which, by definition, is the pressure divided by the strain, may be expressed mathematically as

When the bulk modulus is constant (independent of pressure), this is a specific form of Hooke’s law of elasticity.

Because the denominator, strain, is a ratio without dimensions, the dimensions of the bulk modulus are those of pressure, force per unit area. In the English system the bulk modulus may be expressed in units of pounds per square inch (usually abbreviated to psi), and in the metric system, newtons per square metre (N/m2), or pascals.

The value of the bulk modulus for steel is about 2.3 × 107 psi, or 1.6 × 1011 pascals, three times the value for glass. Thus, only one-third the pressure is needed to reduce a glass sphere the same amount as a steel sphere of the same initial size. Under equal pressure, the proportional decrease in volume of glass is three times that of steel. One may also say that glass is three times more compressible than steel. In fact, compressibility is defined as the reciprocal of the bulk modulus. A substance that is difficult to compress has a large bulk modulus but a small compressibility. A substance that is easy to compress has a high compressibility but a low bulk modulus.

Shear modulus, numerical constant that describes the elastic properties of a solid under the application of transverse internal forces such as arise, for example, in torsion, as in twisting a metal pipe about its lengthwise axis. Within such a material any small cubic volume is slightly distorted in such a way that two of its faces slide parallel to each other a small distance and two other faces change from squares to diamond shapes. The shear modulus is a measure of the ability of a material to resist transverse deformations and is a valid index of elastic behaviour only for small deformations, after which the material is able to return to its original configuration. Large shearing forces lead to flow and permanent deformation or fracture. The shear modulus is also known as the rigidity.

Mathematically the shear modulus is equal to the quotient of the shear stress divided by the shear strain. The shear stress, in turn, is equal to the shearing force F divided by the area A parallel to and in which it is applied, or F/A. The shear strain or relative deformation is a measure of the change in geometry and in this case is expressed by the trigonometric function, tangent (tan) of the angle θ(theta), which denotes the amount of change in the 90°, or right, angles of the minute representative cubic volume of the unstrained material. Mathematically, shear strain is expressed as tan θ or its equivalent, by definition, x/y. The shear modulus itself may be expressed mathematically as

shear modulus = (shear stress)/(shear strain) = (F/A)/(x/y) .

This equation is a specific form of Hooke’s law of elasticity. Because the denominator is a ratio and thus dimensionless, the dimensions of the shear modulus are those of force per unit area. In the English system the shear modulus may be expressed in units of pounds per square inch (usually abbreviated to psi); the common SI units are newtons per square metre (N/m2). The value of the shear modulus for aluminum is about 3.5 × 106 psi, or 2.4 × 1010 N/m2. By comparison, steel under shear stress is more than three times as rigid as aluminum.

Poisson’s Ratio, When a metal bar under tension is elongated, its width is slightly diminished. This lateral shrinkage constitutes a transverse strain that is equal to the change in the width divided by the original width. The ratio of the transverse strain to the longitudinal strain is called Poisson’s ratio. The average value of Poisson’s ratio for steels is 0.28, and for aluminum alloys, 0.33. The volume of materials that have Poisson’s ratios less than 0.50 increase under longitudinal tension and decrease under longitudinal compression.

The Stress-Strain Curve:-

Stress strain curve is a behavior of material when it is subjected to load. In this diagram stresses are plotted along the vertical axis and as a result of these stresses, corresponding strains are plotted along the horizontal axis. As shown below in the stress strain curve.

From the diagram one can see the different mark points on the curve. It is because, when a ductile material like mild steel is subjected to tensile test, then it passes various stages before fracture.

These stages are;

PROPORTIONAL LIMIT

Proportional limit is point on the curve up to which the value of stress and strain remains proportional. From the diagram point P is the called the proportional limit point or it can also be known as limit of proportionality. The stress up to this point can be also be known as proportional limit stress.

Hook’s law of proportionality from diagram can be defined between point OP. It is so, because OP is a straight line which shows that Hook’s law of stress strain is followed up to point P.

ELASTIC LIMIT

Elastic limit is the limiting value of stress up to which the material is perfectly elastic. From the curve, point E is the elastic limit point. Material will return back to its original position, If it is unloaded before the crossing of point E. This is so, because material is perfectly elastic up to point E.

YIELD STRESS POINT

Yield stress is defined as the stress after which material extension takes place more quickly with no or little increase in load. Point Y is the yield point on the graph and stress associated with this point is known as yield stress.

ULTIMATE STRESS POINT

Ultimate stress point is the maximum strength that material have to bear stress before breaking. It can also be defined as the ultimate stress corresponding to the peak point on the stress strain graph. On the graph point U is the ultimate stress point. After point Umaterial have very minute or zero strength to face further stress.

BREAKING STRESS (POINT OF RUPTURE)

Breaking point or breaking stress is point where strength of material breaks. The stress associates with this point known as breaking strength or rupture strength. On the stress strain curve, point B is the breaking stress point.

ELASTIC POTENTIAL ENERGY IN A STRETCHED WIRE :

When a wire is stretched, some work is done against the internal restoring forces acting between particles of the wire. This work done appears as elastic potential energy in the wire.

Consider a wire of length l and area of cross section a. Let F be the stretching forces applied on the wire and Δl be the increase in length of the wire.

Initially, the internal restoring force was zero but when length is increased by Δl ,

the internal force for an increase in length Δl of the wire

= (0 + F)/2 = F/2

Hence, work done on the wire, w = average force × increase in length = [F/2 ] × Δl

This is stored as elastic potential energy U in the wire.

∴ U = (1/2) × F × Δl = (1/2) × F/a × Δl/l × al

= (1/2)(stress) × (strain) × volume of the wire

∴ elastic potential energy per unit volume of the wire

u = (1/2)(stress) × (strain) = (1/2) (Young’s modulus × strain) × strain

( Young’s modulus = stress / strain)

∴ u = (1/2) (young’s modulus ) × {(strain)²}

Hope this Have covered quite a fair amount of topics on Mechanical Properties of Solid.

Thank You for Reading..

Jyotiraditya

Jyotiraditya

Written by

Astronaut | Bhavanite | NASA Blogger | Martian| ICTS fellow| Advance Astronaut Trainee - Project Possum | President Mars Society Bharat.

Welcome to a place where words matter. On Medium, smart voices and original ideas take center stage - with no ads in sight. Watch
Follow all the topics you care about, and we’ll deliver the best stories for you to your homepage and inbox. Explore
Get unlimited access to the best stories on Medium — and support writers while you’re at it. Just $5/month. Upgrade