Elasticity and Fracture

When an object is at rest, we can think that it is not interesting to study because it does not have speed or acceleration and because the forces which are acting on it are zero.

   But this does not imply that there is no force acting on the objects. In fact, it is virtually impossible to find a body over which no force acts. Sometimes the forces can be so great that the object is seriously deformed and can even fracture (break); avoiding these types of problems has made the field of statics very important.

   The objects with which we are in contact every day they have at least one force that acts on them (gravity) and, if they are at rest, there must therefore be other forces acting on them so that the net force is zero.

For example: an object that rests on a table has two forces that act on it: the force of gravity, down, and the force exerted by the table, upwards.

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Due to the net force is zero, the upward force exerted by the table must have the same magnitude as the gravitational force acting downward.

“ We can say that a body is in balance when it is under the action of those two forces ”.

   The determination of these forces, then, allows to determine if the structures can withstand the forces without significantly deforming or fracturing. The techniques for this can be applied in a wide variety of fields. Architects and engineers must be able to calculate the forces acting on the structural components of buildings, bridges, machines, vehicles and other structures, since any material can be broken or deformed if a very large force is applied.

Conditions for balance

For a body which is at rest, the sum of the forces acting on it must be zero. Due to the force is a vector, the components of the net force must be zero. Therefore, if the forces on the object act on a plane, a condition for equilibrium is that:

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If the forces act in three dimensions, then also the third component must be zero. 

   Although the above equations must be valid for an object that is in equilibrium, they are not a sufficient condition. The following figure shows us an object over which the net force is zero.

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Although the two forces, identified in the blocks, add a net force of zero, they give rise to a touch that will turn the object. Therefore mathematically, “for the case of coplanar forces, it must be fulfilled that the arithmetic sum of the moments related to anti-hour rotations must be equal to the arithmetic sum of the moments related to hourly rotations”. That is, it must be fulfilled that:

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Stability and balance

A body in static equilibrium, if it is not disturbed, it does not undergo translation or rotation acceleration, because the sum of all the forces that act on it are zero. However, if the body moves slightly, three results are possible:

  • That it returns to its original position, in which case it is said to be in stable equilibrium such as a ball hanging freely from a thread that we move to the side, will quickly return to its original position.
  • That it moves further away from its position, in which case it is said to be in unstable equilibrium such as a pencil held on its tip.
  • That it remains in its new position, in which case it is said to be in neutral or indifferent equilibrium.

   In most cases, as in the design of structures and work with the human body, we are interested in maintaining a stable balance or balance, as we sometimes say. In general, an object whose center of gravity is below its point of support, like a ball held by a thread, will be in stable equilibrium. The critical point is reached when the center of gravity no longer falls on the base of the support. In general, a body of its center of gravity is above its support base, it will be in stable equilibrium if a vertical line projected from its center of gravity passes through the support base. This is because the upward force on the object, which balances gravity, can only be exerted within the contact area, so that if the force of gravity acts beyond that area there will be a net force that will turn the object.

   In this sense, human beings are much less stable than quadruped mammals, which not only have a greater support base on their four legs, but have a lower center of gravity.

 

Elasticity; strain and unit deformation

Every object changes its shape under the action of the applied forces. If the forces are large enough, the object will break or fracture.

   If a force is exerted on an object, such as the vertically hanging metal bar that we can see below, its length will change. The amount of elongation, or elongation, ΔL, is proportional to the weight or force that is exerted on the object.

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This proportionality can be written in the form of an equation, which is known as:  “Hooke’s law”.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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Where:

  • F represents the force or weight that pulls the object 
  • ∆L is the increase in length
  • k is a constant of proportionality

   If the force is too large, the object will lengthen too much and finally it will break.

   As we apply a force on a solid, it begins to lengthen, finding itself in its zone or elastic region, up to a point called the proportionality limit. If before reaching this point the force ceases, the object will return to its original length, reaching at this point the elastic limit of the solid. If the object is stretched further it enters the plastic region, where it will not return to its original length when removing the applied force, but will remain permanently deformed. The maximum elongation is reached at the point of rupture. The maximum force that can be applied without rupture is determined by the maximum strength of the material.

   The amount of elongation of an object depends not only on the force applied, but also on the material with which it is manufactured and its dimensions.

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If we compare bars made of the same material, but of different lengths and transversal areas, we will find that for the same applied force the amount of elongation will be proportional to the original length and inversely proportional to the area of ​​the cross section. That is, the longer the object is, the longer it will stretch for a certain force; and the thicker it is, the less it will lengthen. 

   These experimental findings can be combined with the Hooke’s Law equation to obtain the expression:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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Where:

  • L0  is the original length of the object 
  • A is the area of its transverse surface 
  • ∆L is the change in length due to the applied force 
  • E is a proportionality constant called “modulus of elasticity or Young’s modulus” and its value depends only on the material

   Since E is a unique property of the material and is independent of the size or shape of the object, it is a very useful constant for practical calculations, as we can see in the following table for different materials.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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As we have already seen, the change in length of an object is directly proportional to the product of its length L0 by the force applied per unit area. It is common to define the force per unit area as an effort whose units are N/m2:

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Also, the unit strain is defined as the ratio between the length change and the original length: 

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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Then, the unit strain is the fractional change in length of an object, and it is a measure of how much the solid has been deformed. Therefore, we can affirm that:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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Or even:

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We see that the unit deformation is directly proportional to the stress.

   External forces applied to an object give rise to internal forces, or efforts, within the material itself. The deformation due to tension is only a type of effort to which the materials can be subjected. 

There are two other common types of effort:

  • the compression effort  
  • the shear stress

   The compression stress is the exact opposite of the tension stress. Instead of stretching, the material is compressed: the forces act towards the interior of the body.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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The shear stress is that in which the object undergoes equal and opposite forces applied on opposite faces, for example, a fixed book on a table on which a force is exerted parallel to its upper surface. The table exerts an equal and opposite force on the lower face. Although the dimensions of the object do not change appreciably, its shape changes.

To calculate the shear stress you can use the equation:

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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Where:

  • A is the area of ​​the surface parallel to the applied force 
  • ∆L is perpendicular to L0
  • G is a constant of proportionality called “shear modulus”, and in general corresponds to half or third of the modulus of elasticity, E.

Fracture

 If the stress on a solid object is too great, the object can fracture or break.

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Below we can see the maximum resistances of various materials to tension, compression and cutting.

Bioprofe | Exercises of Physics, Chemistry and Mathematics | Mechanics | Elasticity and Fracture

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 It is the maximum forces per unit area that can resist an object under each of these three types of effort. However, they are only representative values ​​and the real value for a given solid can be very different.

 

 

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