9.1 INTRODUCTION [LO1]

Stresses given by relatively simple equations in the strength of materials for structures or machine members are based on the assumed continuity of the elastic medium. However, the presence of discontinuity destroys the assumed regularity of stress distribution in a member and a sudden increase in stresses occurs in the neighborhood of the discontinuity. In developing machines, it is impossible to avoid abrupt changes in cross-sections, holes, notches, shoulders, etc. Abrupt changes in cross-section also occur at the roots of gear teeth and threads of bolts. Some examples are shown in Figure 9.1.

Any such discontinuity acts as a stress raiser. Ideally, discontinuity in materials such as non-metallic inclusions in metals, casting defects, residual stresses from welding may also act as stress raisers. In this chapter, however, we shall consider only the geometric discontinuity that arises from design considerations of structures or machine parts.

Many theoretical methods and experimental techniques have been developed to determine stress concentrations in different structural and mechanical systems. In order to understand the concept, we shall begin with a plate with a centrally located hole. The plate is subjected to uniformly distributed tensile loading at the ends, as shown in Figure 9.2.

Introduction

All metals and alloys exhibit a reduction in electrical resistance as they cool. As the temperature drops, atoms’ thermal vibrations become less intense, and conduction electrons scatter less frequently. The resistivity should decrease toward zero as the temperature approaches zero Kelvin for a perfect pure metal, where the only thing standing in the way of an electron's travel is the thermal vibrations of the lattice. This zero resistance, which a hypothetical perfect specimen would acquire if it could be cooled to absolute zero, is the phenomenon of superconductivity. Any real specimen of metal cannot be perfectly pure and will contain some impurities. As a result, in addition to being scattered by the thermal vibrations of the lattice atoms, the electrons are also dispersed by impurities, and this impurity scattering is largely temperature independent. As a result, at the lowest temperature, there will be some residual resistance. The residual resistivity of a metal increases with the degree of impurity.

The phenomenon of superconductivity was first discovered by Dutch physicist H. Kamerling Onnes of Leiden University in 1911 during the investigation of the variation of electrical resistance of mercury in the newly available range of low temperatures, in the neighborhood of temperature of liquid helium (or 4.2 K). He observed that the resistance of mercury suddenly falls from 0.08 ohm at about 4 K to less than 3 × 10−6 ohm over a very small temperature of 0.01 K.

Introduction

The nonconducting materials such as paper, wood, glass, ceramics, polymers and so on do not have free charge carriers, that is, electrons or holes. Therefore, they prevent the flow of electrical current and heat through them.

When the main function of nonconducting materials is to provide electrical isolation then they are called insulators.

When the main function of nonconducting materials is for charge storage then it is called dielectric.

The dielectrics are polarized under the influence of an external electric field.

Dielectric Constant

Let us consider two parallel plates separated by a distance “d” connected with a dc supply of voltage V, as shown in Figure 6.1(a). Now the circuit is disconnected, and the dielectric is inserted between the plates, as shown in Figure 6.1(b).

Then, the voltage across the capacitor is reduced from V to V′. The change in voltage across the plates can be related by a factor as

Since V < V , the relative permittivity or dielectric constant ɛr 1 >.

The capacitance without dielectric is given as

The capacitance with dielectric is given as

Now, put the value of C and C¢ in equation (6.1), the relative permittivity or dielectric constant is

If “A” is the area of the plates, then

where ɛ is the permittivity of the material.

Introduction

In the early days, an operational amplifier (op-amp) was the only linear integrated circuit (IC) that was used in the design of linear IC circuits and systems. Typical applications of the op-amps were mathematical operations, such as summation, subtraction, integration, small signal amplification, and generating oscillations. Over the years, other devices, such as operational transconductance amplifiers, current conveyors, and so on, have also come into common use; still, it has not reduced the importance and areas of application of op-amps. Rather, it became possible to realize many more advanced functions with linear ICs and many applications coming under the domain of nonlinear applications with advances in the process technology and increased level of integration. Some of the more common nonlinear applications are precision rectifiers, voltage-level detectors, and Schmitt trigger circuits. The Schmitt trigger circuit itself is very popular in generating varieties of pulses and other waveforms like triangular waveforms. Some other nonlinear applications such as log and antilog amplifiers, analog multiplier, charge amplifier, and isolation amplifiers are discussed in brief; phase lock loop and its basic function are also included.

Precision Rectifiers

Conventional rectifiers work well for converting alternating supply to a pulsating one. Filters are normally used to remove ripples in the pulsating voltage to obtain dc. It is observed that these rectifiers have some limitations. One of the main limitations is that when a diode conducts during rectification, it has a voltage drop across its terminals, which is approximately 0.7 V. Hence, the ac voltage available for conversion to dc is reduced by that amount.

As economies become more complicated with increasing interdependence tied to exchange and specialization, inequality appears as an outcome of dispersed versus concentrated flows and accumulations of value that affect differences in well-being, power, and institutional formations. We look at the complicated institutional arrangements that favor or limit inequality, perhaps the most important of which is the development of institutional property and how it allowed control over production and distribution. The theoretical and empirical breadth of inequality is vast. For this comparative effort, we formulate an approach that can analyze inequalities in wealth and property from widely different social formations, including the segmentary societies of Pare, Tanzania, and Zuni in the American Southwest, chiefdoms in the Scandinavian Bronze Age (BA), and advanced states and empires such as Rome and the Inca. Within this broad spectrum, differences in the control of wealth, prestige, ranking and/or ascribed rank are intertwined but not necessarily overlapping. Our approach focusses on how access to and control over material wealth is distributed in our sample.

1.1 INTRODUCTION [LO1]

Mechanics is one of the oldest physical sciences, dating back to the times of Aristotle and Archimedes. The subject deals with force, displacement, and motion. The concepts of mechanics have been used to solve many mechanical and structural engineering problems through the ages. Because of its intriguing nature, many great scientists including Sir Isaac Newton and Albert Einstein delved into it for solving intricate problems in their own fields.

Engineering mechanics and mechanics of materials developed over centuries with a few experiment-based postulates and assumptions, particularly to solve engineering problems in designing machines and structural parts. Problems are many and varied. However, in most cases, the requirement is to ensure sufficient strength, stiffness, and stability of the components, and eventually those of the whole machine or structure. In order to do this, we first analyze the forces and stresses at different points in a member, and then select materials of known strength and deformation behavior, to withstand the stress distribution with tolerable deformation and stability limits. The methodology has now developed to the extent of coding that takes into account the whole field stress, strain, deformation behaviors, and material characteristics to predict the probability of failure of a component at the weakest point. Inputs from the theory of elasticity and plasticity, mathematical and computational techniques, material science, and many other branches of science are needed to develop such sophisticated coding.

The theory of elasticity too developed but as an applied mathematics topic, and engineers took very little notice of it until recently, when critical analyses of components in high-speed machinery, vehicles, aerospace technology, and many other applications became necessary. The types of problems considered in both the elementary strength of material and the theory of elasticity are similar, but the approaches are different. The strength of the materials approach is generally simple. Here the emphasis is on finding practical solutions to a problem with simplifying assumptions.

Wave optics is the branch of modern physics in which the nature of light and its propagation are studied.

Interference

When two waves of the same frequency, having a constant phase difference between them, and traveling in the same medium are allowed to superimpose each other, there is a modification in the intensity pattern. This phenomenon is known as interference of light.

When the resultant amplitude at certain points is the sum of the amplitudes of the two waves, this interference is known as constructive interference.

When the resultant amplitude at certain points is the difference of the amplitudes of the two waves, this interference is known as destructive interference, as shown in Figure 11.1.

COHERENT SOURCES

Two sources are said to be coherent if the waves emitted from them have a constant phase difference with time.

THEORY OF INTERFERENCE

Let us consider two coherent sources S1 and S2 that are equidistant from source S. Let a1 and a2 be the amplitudes of the waves originated from source S1 and S2, respectively, as shown in Figure 11.2. Then the displacement y1 from the source S is given by

where δ is the phase difference between the two waves.

Now, according to the law of superposition, the resultant wave is given by

Band Theory ofSolids

The band theory of solids is different from the others because the atoms are arranged very close to each other such that the energy levels of the outermost orbital electrons are affected. But the energy level of the innermost electrons is not affected by the neighboring atoms.

In general, if there is n number of atoms, then there will be n discrete energy levels in each energy band. In such a system of n number of atoms, the molecular orbitals are called energy bands shown in Figure 7.1.

CLASSIFICATION OF SOLIDS ON THE BASIS OF BAND THEORY

The solids can be classified on the basis of band theory. The parameter that differentiates the solids among insulator, conductor, and semiconductor is known as energy band gap and represented by (Eg), as shown in Figure 7.2. When the energy band gap (Eg) between conduction band and valence band is greater than 5 eV (electron-volt) then the solid is classified as insulator. When the energy band gap (E g)between conduction band and valence band is 0 eV (electron-volt), that is, overlapping of bands occurs then the solid is classified as conductor. When the energy band gap (Eg) between conduction band and valence band is approximately equals to 1 eV (electron-volt) then the solid is classified as semiconductors.

4.1 CONSTITUTIVE EQUATIONS [LO1]

So far, we have discussed the strain and stress analysis in detail. In this chapter, we shall link the stress and strain by considering the material properties in order to completely describe the elastic, plastic, elasto-plastic, visco-elastic, or other such deformation characteristics of solids. These are known as constitutive equations, or in simpler terms the stress–strain relations. There are endless varieties of materials and loading conditions, and therefore development of a general form of constitutive equation may be challenging. Here we mainly consider linear elastic solids along with their mechanical properties and deformation behavior.

Fundamental relation between stress and strain was first given by Robert Hooke in 1676 in the most simplified manner as, “Force varies as the stretch”. This implies a load–deflection relation that was later interpreted as a stress–strain relation. Following this, we can write P = kδ, where P is the force, δ is the stretch or elongation, and k is the spring constant. This can also be written for linear elastic materials as σ = E∈, where σ is the stress, ∈ is the strain, and E is the modulus of elasticity. For nonlinear elasticity, we may write in a simplistic manner σ = E∈n, where n ≠ 1.

Hooke's Law based on this fundamental relation is given as the stress–strain relation, and in its most general form, stresses are functions of all the strain components as shown in equation (4.1.1).

In various applications of computer vision and imageprocessing, it is required to detect points in animage, which characterize the visual content of thescene in its neighborhood and are distinguishableeven in other imaging instances of the same scene.These points are called key points of an image andthey are characterized by the functionaldistributions, such as distribution of brightnessvalues or color values, around its neighborhood foran image. For example, in the monocular and stereocamera geometries, various analyses involvecomputations of transformation matrices such as,homography between two scenes, fundamental matrixbetween two images of the same scene in a stereoimaging setup, etc. These transformation matricesare computed using key points of the same scenepoint of a pair of images. The image points of thesame scene point in different images of the sceneare called points ofcorrespondence or corresponding points. Key points ofimages are good candidates to form such pairs ofcorresponding points between two images of the samescene. Hence detection and matching of key points ina pair of images are fundamental tasks for suchgeometric analysis.

Consider Fig. 4.1, where images of the same scene arecaptured from two different views. Though theregions of structures in the images visuallycorrespond to each other, it is difficult toprecisely define points of correspondences betweenthem. Even an image of a two-dimensional (2-D)scene, such as 2-D objects on a plane, may gothrough various kinds of transformations, likerotation, scale, shear, etc. It may be required tocompute this transformation among such a pair ofimages. This is also a common problem of imageregistration.

learning Outcomes

After reading this chapter, the reader will be able to

• Understand the meaning of three processes of heat flow: conduction, convection, and radiation

• Know about thermal conductivity, diffusivity, and steady-state condition of a thermal conductor

• Derive Fourier's one-dimensional heat flow equation and solve it in the steady state

• Derive the mathematical expression for the temperature distribution in a lagged bar

• Derive the amount of heat flow in a cylindrical and a spherical thermal conductor

• Solve numerical problems and multiple choice questions on the process of conduction of heat

6.1 Introduction

Heat is the thermal energy transferred between different substances that are maintained at different temperatures. This energy is always transferred from the hotter object (which is maintained at a higher temperature) to the colder one (which is maintained at a lower temperature). Heat is the energy arising due to the movement of atoms and molecules that are continuously moving around, hitting each other and other objects. This motion is faster for the molecules with a largeramount of energy than the molecules with a smaller amount of energy that causes the former to have more heat. Transfer of heat continues until both objects attain the same temperature or the same speed. This transfer of heat depends upon the nature of the material property determined by a parameter known as thermal conductivity or coefficient of thermal conduction. This parameter helps us to understand the concept of transfer of thermal energy from a hotter to a colder body, to differentiate various objects in terms of the thermal property, and to determine the amount of heat conducted from the hotter to the colder region of an object. The transfer of thermal energy occurs in several situations:

• When there exists a difference in temperature between an object and its surroundings,

• When there exists a difference in temperature between two objects in contact with each other, and

• When there exists a temperature gradient within the same object.

Introduction

Statistical mechanics bridges the gaps between the laws of thermodynamics and the internal structure of the matter. Some examples are as follows:

• 1. Assembly of atoms in gaseous or liquid helium.

• 2. Assembly of water molecules in solid, liquid, or vapor state.

• 3. Assembly of free electrons in metal.

The behavior of all these abovementioned assemblies is totally different in different phases. Therefore, it is most significant to relate the macroscopic behavior of the system to its microscopic structure.

In this mechanics, most probable behavior of assembly are studied instead of individual particle interactions or behavior.

The behavior of assembly that is repeated a maximum time is known as most probable behavior.

hase Space

Six coordinates can fully characterize the state of any system:

• 1. Three for describing the position x, y, z and three for momentum Px, Py, Pz.

• 2. The combined position and momentum space (x, y, z, Px, Py, Pz) is called phase space.

• 3. The momentum space represents the energy of state,

For a system of N particles, there exists 3N position coordinates and 3N momentum coordinates. A single particle in phase space is known as a phase point, and the space occupied by it is known as µ-space.

olume Element ofµ-Space

4. Consider a particle having the position and momentum coordinates in the range.