Mechanics of Solids & Fluids Assignment Help






Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions (e.g., external forces, temperature changes, applied displacements, etc.). It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation. Solid mechanics extensively uses tensors to describe stresses, strains, and the relationship between them

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Response models:

A material has a rest shape and its shape departs away from the rest shape due to stress. The amount of departure from rest shape is called deformation, the proportion of deformation to original size is called strain. If the applied stress is sufficiently low (or the imposed strain is small enough), almost all solid materials behave in such a way that the strain is directly proportional to the stress; the coefficient of the proportion is called the modulus of elasticity or Young's modulus. This region of deformation is known as the linearly elastic region. It is most common for analysts in solid mechanics to use linear material models, due to ease of computation. However, real materials often exhibit non-linear behavior. As new materials are used and old ones are pushed to their limits, non-linear material models are becoming more common.

There are three models that describe how a solid responds to an applied stress:

  • Elastically – When an applied stress is removed, the material returns to its undeformed state. Linearly elastic materials, those that deform proportionally to the applied load, can be described by the linear elasticity equations such as Hooke's law.
  • Viscoelastically – These are materials that behave elastically, but also have damping: when the stress is applied and removed, work has to be done against the damping effects and is converted in heat within the material resulting in a hysteresis loop in the stress–strain curve. This implies that the material response has time-dependence.
  • Plastically – Materials that behave elastically generally do so when the applied stress is less than a yield value. When the stress is greater than the yield stress, the material behaves plastically and does not return to its previous state. That is, deformation that occurs after yield is permanent.

Solids and Fluids:

Intermolecular Forces and Interatomic Forces:-

When two wet glass plates are pressed together, they cannot be separated easily. This is because of the force of attraction existing between the atoms and molecules, and are electrical in origin.

interatomic forces

All mechanical systems have to acquire a state of minimum potential energy, to attain stability. This is achieved by the force of attraction and repulsion acting between the atoms. This can be understood by the following graph showing the variation of potential energy with interatomic distances.

interatomic forces graph

As the two isolated Hydrogen atoms approach each other, the nucleus of one atom attracts the valence electrons of the other and vice versa. As the hydrogen atoms come closer, the attractive force tends to decrease the potential energy of the system. When the potential energy reaches a minimum value, sharing of electrons take place between the two, resulting in a covalent bond. This stage is represented by the distance 'ro' in the graph. It is interesting to note that when the distance between the two atoms becomes even lesser, their nuclei repel each other, thus preventing them from collapsing. In other words, the force of repulsion increases the potential energy of the system.

At distances equal to ro, the two forces balance each other, resulting in a molecule. Similarly, forces of attraction exist between molecules, binding them together. These are intermolecular forces. The intermolecular forces are weaker than interatomic forces, as the former forces are the Van der waals forces.

Crystalline and Amorphous Solids:-

Solids have definite shape and volume because the average distance between the molecules or atoms remain constant and do not change with time. The arrangement of molecules inside a solid differ from one to another. This results in two types of solids

  • Crystalline solids
  • Amorphous solids

In crystalline solids, the atoms or molecules are arranged in an order, extending over a large volume of the crystal. All the bonds have the same bond strength. Therefore, such solids have a precise melting point. They also have a uniform chemical composition. Examples of crystalline solids are quartz, calcite, rocksalt, sugar, mica and diamonds.

Amorphous solids on the other hand, do not have a regular and periodic arrangement of atoms. All the bonds are not equally strong. These solids do not have a precise melting point.

Examples of amorphous solids are rubbers, glass, plastic, cement and paraffin.

amorphous solid

crystalline solid

Elasticity and Deforming Forces

External forces acting on a body, bring about a change in its state or configuration. The latter is possible when the body is not free to move, but the molecules are compelled to change their positions. Such forces are called deforming forces. These forces bring about a change in the length, volume or shape. What happens to the body when these forces are removed? Obviously one expects the body to regain its shape. How does one account for this?

On applying the forces, the interatomic distance becomes more than ro, thus increasing their potential energy (leading to instability). On removing the forces, the system tends to regain a minimum P.E. and as a result, attractive forces develop, restoring them to their original shape. The same applies when a body is subjected to a compressional force, where repulsive forces develop and restore the system to equilibrium.

The property of the material of a body by virtue of which, the body regains its original length, volume and shape after the deforming forces have been removed, is called elasticity.

Do all bodies possess this property of elasticity to the same extent? Substances like putty (clay), kneaded flour and paraffin wax undergo a permanent deformation. This property where bodies do not show a tendency to recover their original form after deforming forces are removed, is called plasticity.

Hooke's law

Experimental study by Hooke revealed that elastic bodies regain their original configuration completely, only upto a limit. He termed this limit as the elastic limit. He found that within the elastic limit, the extension produced in the wire was directly proportional to the load applied.

i.e. Stress a strain

Stress = E strain

where E is constant and is called modulus of elasticity of the material of the body.

Experimental verification of Hooke's law

hookes law

The apparatus is set as shown above. The weights are loaded one by one and unloaded one by one to bring the spring to its elastic mode. Weights are then added in the pan and reading of pointer on the scale is noted. Some more weights are added and the readings are noted once again. The difference between the two gives the extension in the spring due to the weights added in the pan. The procedure is repeated for other weights. On plotting a graph between the load and extension, one gets a straight line as shown below. Thus, the graph verifies the Hooke's law.

hookes law graph


Archimedes' Principle:-

  • A floating ship displaces water equal to its own weight, including that of cargo.
  • Submarines

    A submarine sinks by taking water into its buoyancy tanks. Once submerged, the upthrust is unchanged, but the weight of the submarine increases with the inflow of water and it sinks faster. Compressed air is used to blow water out of the tanks when it has to resurface.

    submarine

    (An atomic submarine. This remains underwater for weeks without surfacing).

    • Balloons filled with hot air or hydrogen, weigh less than weight of cold air that it displaces. Therefore, the upthrust is greater than its weight and the resultant upward force on the balloon, causes it to rise.

    hydrometer

    • Hydrometer is used to measure relative density of a liquid. It consists of a weighted sealed glass tube and a scale and when placed in the liquid, the scale reads the level of the liquid surface. In a denser liquid, the scale floats since less liquid is displaced. The weight of the displaced liquid is equal to its own weight, the number on the scale therefore, increases downwards. This method is use to check the state of a car battery. In a fully charged medium, the R.D of the acid should be 1.25 and recharging is required when reading is less than 1.8.

    Know more

    • Pressure is a scalar quantity, because its direction is unique (normal to the area) and not to be specified.
    • Barometer uses mercury. This is because, if water were used, then a longer glass tube would have to be used (10m, since10m of water column is required to balance the atmospheric pressure.). Also water sticks to the glass tube.
    • Blood pressure is measured in mm of Hg. The systolic pressure is 120mm of Hg and the diastolic pressure is 80mm of Hg.
    • The thrust on the total area of the body of a man of a medium built (1.5m2) is 1.5 ton weight. How is he able to withstand this thrust? This is because there is a large number of opening and pores in the skin, through which air enters the system and presence of air inside, counterbalances the pressure outside.
    • Pressure outside the human body changes at high altitudes or undersea. Therefore, precautions are to be taken to compensate for these differences.

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