8.1 INTRODUCTION [LO1]

In simple words, the application of a torque on a prismatic member causes twisting or torsion. This causes shear stress if a torque alone is applied. However, this is rarely true in practical cases. A circular bar, used to transmit torque between a prime mover and a machine, is a typical example of a torsion member. However, in many applications, a torque along with a bending moment and axial loading are applied, and there we need to combine these effects and find the principal stresses. A typical example of such combined stresses is a propeller shaft. Torsional problems are important in many applications both in industry and in our daily life. Therefore, we consider torsion alone in this chapter in some detail.

Torsional problems for circular members are generally solved assuming that plane sections normal to the axis of the bar remain plane even after twisting. This assumption was first made by Coulomb intuitively in 1784, and he came up with a correct usable equation for members with circular sections. However, this assumption does not apply to bars with a noncircular cross-section. Navier attempted to solve torsional problems with noncircular sections using Coulomb's assumption and came up with an erroneous solution. The correct solution was provided by St. Venant in 1853 using a warping function. Much later, in 1903, Prandtl came up with a membrane analogy method that could solve problems with any complicated cross-section. First, we shall consider torsional problems with circular cross-sections.

8.2 TORSION OF MEMBERS WITH CIRCULAR CROSS-SECTION [LO2]

The torsion analysis of members with a circular cross-section starts with simplified assumptions made by Coulomb. In order to establish a relation between the applied torque and shear stress developed and the angle of twist in such cases, the following assumptions are made:

• 1. Material is homogeneous and isotropic.

• 2. Plane sections perpendicular to the axis of a circular member remain plane after twisting. No warping or distortion of the parallel planes occurs.

11.1 INTRODUCTION [LO1]

Stresses developed at the contact between two loaded elastic bodies are generally localized and most machine parts or structures are designed based on the stresses in the main body. However, there are many important machine members where the localized stresses developed at the contact between curved surfaces with initially limited contact area play an important role in their design. Ball or roller bearings, gears, cams, and valve tappets of internal combustion engines are some of the examples of machine parts where contact stresses must be taken into account in order to predict their failure probability.

The localized contact stresses that develop between two curved bodies as they are loaded with small deformations are often referred to as Hertzian stresses, following the work of H. Hertz (1881), who first solved these contact problems elegantly more than a century ago. Since then the topic has received a good deal of attention by the researchers due to its importance in engineering practice and science. Much work has been done on the stress distribution at the Hertzian contact surfaces and sub-surfaces. Ball bearings and gear teeth often fail by pitting. Hertzian stress analysis can precisely locate the depth at which maximum shear stress occurs where cracks may initiate and propagate leading to failure. Thus, a remedy to such failures may be prescribed in terms of limiting stresses. In many rolling contact problems, failure occurs with the initiation of a tiny crack that eventually grows due to repeated contacts. Analysis of crack initiation and growth is often based on Hertzian stress analysis. In this chapter, we shall consider the basics and application of contact stress analysis, beginning with some basic elasticity theory necessary for such analyses.

2.1 MATHEMATICAL PRELIMINARIES [LO1]

In any scientific or engineering field of study, knowledge of some mathematical techniques and methods are essential. Solid mechanics is no exception. To develop proper formulation methods and solution techniques for elasticity problems, it is necessary to have an appropriate mathematical background. In this chapter, we shall discuss Cartesian tensors, which have a special significance in the discussion of stress, strain, and displacement fields, and their manipulation. Other mathematical details will be discussed as and when they are required in solving different problems.

Tensors may be defined in a number of ways. One simple definition is that a tensor is a physical quantity that is governed by certain transformation laws when the coordinate system is changed. A tensor is invariant under any change of coordinate system, but its components along the coordinate axes change with the changed coordinate system. Tensors of order zero are called scalars. Common examples of scalars are temperature, density, Young's modulus, or Poisson's ratio. They have a single magnitude at each point in space, and they are invariant with coordinate transformations. A typical example of scalars is often taken as temperature T at a point in space with coordinates (x, y, z) represented as T(x, y, z). Temperature at the same point does not change if we choose a different coordinate system (x′, y′, z′) represented as T′(x′, y′, z′) and we may say

T=T′. (2.1.1)

Tensors of first order are vectors, and we know that a vector has a magnitude and a direction. A typical example of a vector is a velocity vector V. It is sometimes taken as a convention to represent a vector by a bold letter. Consider the velocity vector V in (x, y, z) coordinate system.

Ideologue, reformer, feminist, firebrand secessionist: these are some of the many things E. V. Ramasamy Periyar has been called. If there is one strand of Periyar's thought that runs through all these titles and the politics that informed them, it would be his clarion call for self-respect. Not one to stop at dismantling the hegemonic power structures that he saw around him in India, Periyar was committed to the cause of reform in and for the Tamil diaspora as well. His views on nationhood thus ‘constantly violated the certitude about boundaries, identities, agents of change, and went beyond the territoriality of the nation’ (Pandian, 1993, p. 2282). Periyar emphasized that foreign settlement could enable the regeneration of Tamil society abroad, unfettered by India's oppressive traditions. Moreover, he saw the diaspora as an important source of financial support for the Dravidian movement. To this end, Self-Respect literature often asserted that Tamil people everywhere were bound by obligations of mutual assistance and reciprocity (Alagirisamy, 2016). Periyar visited British Malaya and Singapore twice in his lifetime: once in 1929–1930 and again in 1954–1955. Both visits were pivotal in aiding the development of a settled Tamil political consciousness in Singapore.

Scholars of the Indian Ocean world continue to trace the comings and goings of sojourners and settlers, privileging the ocean itself as a key agent of change (Moorthy and Jamal, 2010; Amrith, 2013; Menon et al., 2022). Yet, settlement also brought with it a sea-change in the lived presents and anticipated futures of migrant communities that aspired to citizenship.

The popular focus on Periyar and Dravidian—as a person leading his loyal people—may invite placing nationalism's assertion, rather than critique, at the heart of political thought in twentieth-century Tamil-speaking South India. ‘Nationalism’ names the intuition that ‘France [is] for the French, England for the English … and so forth’ (Shaw, 2013) or, more generally, ‘nationalism is a theory of political legitimacy … requir[ing] that ethnic boundaries should not cut across political ones’, insisting on ‘congruence of state and nation’, and refusing ‘ethnic divergence between rulers and ruled’ (Gellner, 1983, pp. 1, 134). Much turns on ‘ethnic’. Considerations include whether ‘ethnic’ is ‘racial’ or ‘historically constituted’ (Lenin and Stalin, 1970, pp. 66–68) and nationalism's ‘inherent contradictoriness’, both because its ‘rational and progressive’ promises of modernity are often premised on ‘traditional and conservative’ gestures to the past and because its anti-colonial articulation usually adopts the very imperial ‘representational structure … nationalist thought seeks to repudiate’ (Chatterjee, 1986, pp. 22, 38). So, when the August 1944 creation of the Dravidar Kazhagam (DK) was heralded with ‘Long live Periyar, Dravida Nadu for the Dravidian, let the Dravidar Kazhagam flourish’1—entrenching an ‘ethnic’ idiom for Tamil-speaking South India's politics—an invitation for ‘a chapter on Periyar and nationalism’ encourages the interrogation of a twinned presumption of coherence: not simply of ‘Dravidian’ as a people loyal, but of ‘Periyar’ as a person leading.

Solid mechanics, compared to mechanics of materials or strength of materials, is generally considered to be a higher level course. It is usually offered in higher semester to senior students. There are many textbooks available on solid mechanics, but they generally include a large part of theory of elasticity with in depth mathematical formulations. The usual prerequisites are one or two semester course on elementary strength of materials and a thorough mathematical background, including scalar, vector, and tensor field theory and cartesian and curvilinear index notation. The difference in levels between these books and elementary texts on strength of materials is generally formidable. However, in our experience of teaching this course for many years at premier institutes like IIT Kharagpur and Jadavpur University, despite its complexity, senior students generally cope well with the course using the readily available textbooks.

However, there is a vast student population pursuing mechanical, civil, or allied engineering disciplines across the country in colleges where AICTE curriculum is followed. Through several years of interaction with this group of students, we have found that there is no suitable textbook that suits their requirements. The book is primarily aimed at this group of students, attempting to bridge the gap between complex formulations in the theory of elasticity and elementary strength of materials in a simplified manner for better understanding. Index notations have been avoided, and the mathematical derivations are restricted to second-order differential equations, their solution methodologies, and only a few special functions, such as stress function and Laplacian operators.

The text follows more or less the AICTE guidelines and consists of twelve chapters. The first five chapters introduce the engineering aspects of solid mechanics and establish the basic theorems of elasticity, governing equations, and their solution methodologies. The next four chapters discuss thick cylinders, rotating disks, torsion of members with both circular and noncircular cross-sections, and stress concentration in some depth using the elasticity approaches. Thermoelasticity is an important issue in the design of high-speed machinery and many other engineering applications. This is dealt with in some detail in the tenth chapter. Problems on contact between curved bodies in two-dimensional and three-dimensional situations can be challenging, and they have wide applications in mechanical engineering such as in bearing and gear technology.

3.1 STATE OF STRESS AT A POINT [LO1]

When a body is subjected to external forces, its behavior depends on the magnitude and distribution of forces and properties of the body material. Depending on these factors, the body may deform elastically or plastically, or it may fracture. The body may also fail by fatigue when subjected to repetitive loading. Here we are primarily interested in elastic deformation of materials.

In order to establish the concept of stress and stress at a point, let us consider a straight bar of uniform cross-section of area A and subjected to uniaxial force F as shown in Figure 3.1. Stress at a typical section A - A′ is normally given as σ = F/A. This is true only if the force is uniformly distributed over the area A, but this is rarely true. Therefore, definition of stress must be considered by progressively reducing the area until it is small enough such that the force may be considered to be uniformly distributed.

To understand this, consider a body subjected to external forces P1, P2, P3, and P4 as shown in Figure 3.2. If we now cut the body in two pieces,

Internal forces f1, f2, f3, etc. are developed to keep the pieces in equilibrium. Now consider an infinitesimal element of area ΔA Dat the cut section and let the resultant force on the element be Δf.

It is a principle of international law not only that workers should be free to join trade unions and take part in their activities, but also that trade union autonomy should be respected by the State. Trade unions in the United Kingdom in contrast are subject to detailed regulation by legislation, which undermines their right to promote political objects without restraint, decide their own procedures for the selecting and electing senior officials, and determine when disciplinary powers may be used against those who break the rules. This chapter considers these and other questions, as well as the controversial regulatory role of the Certification Officer.

There is a statutory right for employees not to be unfairly dismissed. The right usually requires a qualifying period of continuous employment, and claim has to be made to an employment tribunal within three months of the effective date of termination. The employee has to prove dismissal has occurred, though resignation in response to a fundamental breach of contract by the employer counts as constructive dismissal. The courts have interpreted the statutory test of fairness to require proof that the employer acted outside of the range of reasonable responses to the fault of the employee. Some reasons for dismissal are automatically unfair. The normal remedy for unfair dismissal in practice is not reinstatement but a modest award of compensation for which there is an upper limit.