Stress: force exerted per unit area of surface on an extended object. STRESS LEAD TO STRAIN
Three types of deformation due to strain:
- Stretching (change in length)
- Twisting (change in angle)
- Compression or Expansion (change in volume)
***Each of these strains is caused by a different form of stress
- Tensile stress= Stretching
- Shearing stress= Twisting
- Hydraulic stress= Expansion or Compression
Tensile Stress and Stretching
Stretching: stressing in vertical direction, caused by two opposite force, F1 & F2, DENSITY is Constant(this only works for solids and liquids)
Strain is the relative change in length, express by ∆l/l, there strain= DIMENSIONLESS QUANTITY
Tensile Stress= F/A F1=-F2- units for pressure (Pa) hence related by Newton’s first law but not Newton’s third law…forces act on the same object…
Elastic: strain and stress are proportional of each other (i.e when length change of object (strain) increases linearly with the stress)
(F/A)= Y (∆l/l) [Ơ= Y x Ɛ]
- Y= Young’s modulus (proportionality factor) units: Pa
- Ơ= Stress (F/A)
- Ɛ= ∆l/l
Large Young’s moduli= STRONG MATERIAL! Even large force acting on a piece of material leads to a small length increase
Small Young’s moduli= SOFT MATERIAL! Even small force acting on piece of material leads to large length increase
EXTENSIVE stretching doesn’t obey (F/A)= Y (∆l/l)….elastic limit: after this strain threshold which the equation is no longer applicable!
LARGE strain below strain threshold…..permanent plastic deformations
PLASTIC DEFORMATIONS
Permanent Plastic deformation- once you pass the elastic point of a material and thought is ultimate strength point
Hooke’s Law
Applies in the elastic regime of a material and states that the stress of the material are linearly proportional.
Fext= k(x-xeq)
k= spring constant in units N/m (the Fext is linearly proportional to the displacement…takes twice as much force to stretch a spring twice as far)
Felast= -k(x-xeq) <- (RESTORING FORCE)
Cases:
1) When position x> xeq the Felast is negative, force pulls the object to back to a smaller value
2) When position x<xeq the Felast is positive, the force pushes the object toward a larger value
This works for ideal spring because REAL springs don’t follow this law.
OBJECT ATTACHED TO AN IDEAL SPRING
Elastic (potential) energy: dependent on object’s relative position to its equilibrium position
Eelast= ½ k (x-xeq)^2
A system with a linear restoring force (Hooke’s law) has elastic potential energy that is proportional to the square of the displacement from the equilibrium position of the system.
Etotal= CONSTANT= Ekin + Eelastic
Eelastic= HIGHEST at release point & collapse (two points)
Amplitude (A): the maximum displacement of a vibration object (when object has moved to the amplitude, its kinetic energy becomes ZERO, Ekin=0 and the Etotal= Eelastic
Etotal= (1/2)k x A^2…………………………A= [(2 x Etotal)/k]^1/2
HENCE: vmax= (k/m)^1/2 x A
Simple Harmonic Motion: an object oscillates about a point of mechanical equilibrium
Frequency (angular): Ѡ= (k/m)^1/2 unit: 1/s
Frequency: (Hertz) f= 1/T
