The bi-linear equation
with the essential boundary condition of u(0) = U0 and natural boundary condition of
at x = L , and the domain 0 < x <L appears in several forms in engineering.
CABLE(ROPE) UNDER A TRANSVERSE LOAD
u = deflection
q(x)= transverse load (eg: weight of cable)
Q = Axial force
BAR IN TENSION
u = deflection
q(x)= Friction(Traction on the bar)
Q = Axial force
u = Temperature
a= Thermal conductivity(k)
q(x)= Heat generated
Q = Heat flux
ONE DIMENSIONAL LAMINAR INCOMPRESSIBLE FLOW (Gradient(p)=const)
u = velocity
a= viscosity (η)
q(x)= Pressure gradient
Q = Axial stress
FLOW IN POROUS MEDIA IN ONE DIMENSION
u = Fluid head
a= Permeability const
Q = Flow
u = Electrostatic potential
a= dielectric constant
q(x)= charge density
Q = Electric Flux
Hot water seems to freeze faster than cold water, known as the Mpemba effect. The effect was first observed by Aristotle in the 4th century BC.
Theories for the Mpemba effect have included faster evaporation of hot water, therefore reducing the volume left to freeze; formation of a frost layer on cold water, insulating it; and different concentrations of solutes such as carbon dioxide, which is driven off when the water is heated. Unfortunately the effect doesn’t always appear - cold water often does actually freeze faster than hot, as you would expect. But this Mpemba effect occurs regularly, and no one has ever been able to definitively answer why.
Some scientists have found evidence that it is the chemical bonds that hold water together that provide the effect. Each water molecule is composed of one oxygen atom bonded covalently to two hydrogen molecules. The separate water molecules are also bound together by weaker forces generated by hydrogen bonds. These forces occur when a hydrogen atom from one molecule of water sits close to an oxygen atom from another.
They suggest that it is these bonds that cause the Mpemba effect. They propose that when the water molecules are brought into close contact, a natural repulsion between the molecules causes the covalent bonds to stretch and store energy. When the liquid warms up, the hydrogen bonds stretch as the water gets less dense and the molecules move further apart.
The stretching in the hydrogen bonds allows the covalent bonds to relax and shrink somewhat, which causes them to give up their energy. The process of covalent bonds giving up their energy is essentially the same as cooling, and so warm water should in theory cool faster than cold.
Stress strain curve is 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. Shown below in the stress strain curve foe mild steel.
From the diagram one can see the different mark points on the curve when a ductile material like mild steel is subjected to tensile test, then it passes various stages before fracture. They are:
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.
Hook’s law is applicable up to this point.
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.
UPPER YIELD 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.
LOWER YIELD POINT
After the yield point, the curve typically decreases slightly because of dislocations escaping from Cottrell atmospheres.
ULTIMATE TENSILE 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.
Breaking point or breaking stress is point where strength of material breaks. The stress associates with this point is known as breaking strength or rupture strength. On the stress strain curve, point B is the breaking stress point. As brittle materials fail due to shear, it will be equal to the ultimate shear stress.
Toughness can be determined by integrating the stress-strain curve. It is the energy of mechanical deformation per unit volume prior to fracture.
The area under the stress-strain curve upto the elastic limit depicts the Modulus of resilience(MR) which signifies the ability of material to store or absorb energy without permanent deformation.
Young's modulus is the ratio of stress to strain(slope of the curve) up to P.