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).

Limited research has been devoted to investigating assumptions about competition dynamics established through a neoliberal lens. Advocates argue that competition fosters innovation and benefits consumers by incentivizing private enterprises to develop better products or services at competitive prices compared to their rivals. Critics argue that competition exacerbates inequality by disproportionately rewarding high achievers. Rewarding high achievers reflects the meritocratic aspect of competition, which has been widely assumed to be rooted in the individualistic culture of Western countries. Contrary to this assumption, the ideology of meritocratic competition thrived in ancient collectivist Asian countries. Moreover, the assumed linear relationship between individualism, competition, and inequality is contradicted by economic literature, which suggests more individualistic nations display lower income inequality. Despite extensive economic and cultural examination of competition, competition’s political dimensions remain understudied. This interdisciplinary book challenges conventional assumptions about competition, synthesizing evidence across economics, culture, and politics.

16.1 Introduction to Artificial Neural Network

Neural networks and deep learning are the buzzwords in modern-day computer science. And, if you think that these are the latest entrants in this field, you probably have a misconception. Neural networks have been around for quite some time, and they have only started picking up now, putting up a huge positive impact on computer science.

Artificial neural network (ANN) was invented in the 1960s and 1970s. It became a part of common tech talks, and people started thinking that this machine learning (ML) technique would solve all the complex problems that were challenging the researchers during that time. But sooner, the hopes and expectations died off over the next decade.

The decline could not be attributed to some loopholes in neural networks, but the major reason for the decline was the “technology” itself. The technology back then was not up to the right standard to facilitate neural networks as they needed a lot of data for training and huge computation resources for building the model. During that time, both data and computing power were scarce. Hence, the resulting neural network remained only on paper rather than taking centerstage of the machine to solve some real-world problems.

Later on, at the beginning of the 21st century, we saw a lot of improvements in storage techniques resulting in reduced cost per gigabyte of storage. Humanity witnessed a huge rise in big data due to the Internet boom and smartphones.

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.

This chapter outlines how the Principle of Multispecies Legality offers solutions to the barriers to legal inclusion facing animals in both criminal and civil law contexts: by enabling animals to take legal action; by ensuring that, in civil suits, harms to animals are taken seriously and benefits are awarded to the animals themselves; and that defences of ‘necessity’ in animal welfare laws only apply when the otherwise harmful action is taken for the ultimate benefit of the animal him- or herself. The chapter then explores four institutional safeguards needed to ensure the PML is effective: that legislation is developed under the principle of anticipatory accommodation; that there is the establishment of independent offices of animal welfare; that there is the establishment of dedicated animal crime units and public prosecutors; and that there is equal access to legal services to ensure that all humans who seek to assist animals in taking legal action can do so, regardless of their financial circumstances. Finally, the chapter considers how we need to learn to recognise more expansive conceptions of (political) communication and learn how to be more receptive to them.

13.1 Implementation of k-means Clustering and Hierarchical Clustering

In the previous chapter, we discussed various clustering algorithms. We learned that clustering algorithms are broadly classified into partitioning methods, hierarchical methods, and density-based methods. The k-means clustering algorithm follows partitioning method; agglomerative and divisive algorithms follow the hierarchical method, while DBSCAN is based on density-based clustering methods.

In this chapter, we will implement each of these algorithms by considering various case studies by following a step-by-step approach. You are advised to perform all these steps on your own on the mentioned databases stated in this chapter.

The k-means algorithm is considered a partitioning method and an unsupervised machine learning (ML) algorithm used to identify clusters of data items in a dataset. It is one of the most prominent ML algorithms, and its implementation in Python is quite straightforward. This chapter will consider three case studies, i.e., customers shopping in the mall dataset, the U.S. arrests dataset, and a popular Iris dataset. We will understand the significance of k-means clustering techniques to implement it in Python through these case studies. Along with the clustering of data items, we will also discuss the ways to find out the optimal number of clusters. To compare the results of the k-means algorithm, we will also implement hierarchical clustering for these problems.

We will kick-start the implementation of the k-means algorithm in Spyder IDE using the following steps.

• Step 1: Importing the libraries and the dataset—The dataset for the respective case study would be downloaded, and then the required libraries would be imported.

• Step 2: Finding the optimal number of clusters—We will find the optimal number of clusters by the elbow method for the given dataset.

• Step 3: Fitting k-means to the dataset—A k-means model will be prepared by training the model over the acquired dataset.

• Step 4: Visualizing the clusters—The clusters formed by the k-means model would then be visualized in the form of scatter plots.

Learning Outcomes

After reading this chapter, the reader will be able to

• Express the meaning of sphere of influence and collision frequency

• Derive the distribution function for the free paths among the molecules and demonstrate the concept of mean free path

• Calculate the expression for mean free path following Clausius and Maxwell

• Derive the expression for pressure exerted by a gas using the survival equation

• Calculate the expressions for viscosity, thermal conductivity, and diffusion coefficient of a gaseous system

• Demonstrate Brownian motion with its characteristics and calculate the mean square displacement of a particle executing Brownian motion

• State the idea of a random walk problem

• Solve numerical problems and multiple choice questions on the mean free path, viscosity, thermal conduction, diffusion, Brownian motion, and random walk

4.1 Introduction

The molecules of an ideal gas are considered as randomly moving point particles. From the concept of kinetic theory of gases (KTG), it is well established that even at room temperature, such point molecules of the ideal gas move at very large speeds. The average value of this speed can be determined assuming that the molecules obey Maxwell's speed distribution law and is given by the following expression

This chapter considers how, with animals recognised as a part of nature, legally enshrined ‘rights of nature’ could provide a basis for animals’ legal subjecthood. The chapter centres on the case of Estrellita, an Ecuadorean woolly monkey who was declared to be a subject of rights under Ecuador’s constitutionally enshrined rights of ‘pachamama’ or ‘Mother Earth’. Yet, while Estrellita’s case highlights the potential for rights of nature to serve as a source of animals’ legal subjectivity, the chapter stresses caution. First, several rights-of-nature provisions have arguably co-opted Indigenous ideas, and served to justify continued resource extraction under the guise of living in balance with nature. Second, rights-of-nature provisions maintain the ontological human/all-other-nature divide that exists in current legal systems. Finally, the rights of nature may operate as a kind of ‘eco-coverture’ by encapsulating the interests of individual animals within the sphere of nature’s interests, thereby limiting the potential scope of animals’ legal protection. The chapter concludes that we can do better than grounding animals’ legal subjecthood in the rights of nature.

This chapter presents an alternative to legal personhood and the rights of nature as the means to better include animals within the scope of legal justice. It offers the Principle of Multispecies Legality as not merely an account of animals’ legal subjectivity but of the legal subjectivity of all those beings and entities that have – or that we might, as a democratic society, choose to recognise as having – interests. The PML holds that interests-bearing entitles one to recognition as a subject of the law, with the capacity to take legal action and have one’s interests considered impartially. In rejecting sentience as the grounds of animals’ politico-legal inclusion, the PML’s account of legal subjectivity provides for animals alongside existing sentient and non-sentient legal subjects, like humans and corporations. It also leaves the door open for other valuable entities that currently lack legal subjecthood, such as plants, fungi, bodies of water, and ecosystems. The chapter argues that the inclusivity of the PML is beneficial not only for animals and other non-human entities but also for those humans whose legal subjectivity remains tenuous under existing personhood paradigms.

Learning Outcomes

After reading this chapter, the reader will be able to

• State the assumptions of kinetic theory of gases (KTG)

• Explain the concept of pressure and calculate the expression for it

• Demonstrate mathematically the gas laws using the expression for pressure derived from KTG

• Present the kinetic interpretation of temperature

• Derive the expression for specific heat at constant volume 𝐶𝑉 and constant pressure 𝐶𝑃

• Explain the concept of degree of freedom

• Solve numerical problems and multiple choice questions on KTG

2.1 Introduction

The kinetic theory of gases (KTG) is a theoretical model that describes the physical properties of a gaseous system in terms of a large number of submicroscopic particles, such as atoms, molecules, and small particles. These constituent elements are in random motion and collide constantly with each other and also with the walls of the container. Considering the molecular composition and characteristic features of such random motion of the molecules, various macroscopic properties of the gaseous system, such as pressure, temperature, viscosity, thermal conductivity, and mass diffusivity can be explained with the help of KTG. In this theory, it is postulated that the pressure exerted by a gas is due to the collision of atoms or molecules moving at different velocities on the walls of a container. It basically attempts to explain the macroscopic properties that are related to the microscopic phenomenon. The physical properties of solids and liquids, in general, are described by their shape, size, mass, volume, etc. Gases, however, have no definite shape, and size. Furthermore, their mass and volume are not directly measurable. In such cases, the KTG can be successfully applied to extract the physical properties of the gaseous system.

7.1 INTRODUCTION [LO1]

The problems of stresses and deformations in disks rotating at high speeds are important in the design of both gas and steam turbines, generators and many such rotating machinery in industry. As discussed in earlier chapters, this is another example of axisymmetric problems in polar coordinates. Although the theoretical treatment of a flat disk is simpler, in many industrial applications, disks are tapered. They are usually thicker near the hub, and their theoretical analysis is slightly more involved. We shall first take up the analysis for flat disks.

In the case of rotating disks with centrifugal force as body force, the equation of equilibrium reduces to as in equation (6.1.3).

Combining this with displacement equations, we have, as in equation (6.1.5), a general equation for determining the stress distribution in axisymmetric problems. This is given as

This is a nonhomogeneous differential equation. The associated homogeneous equation (complementary equation) is

The solution of this equation is Lame's equation as discussed in Chapter 6, equation (6.2.3), and taking into consideration the particular solution, the solution to equation (7.1.2) turns out to be

We may also determine the radial displacement from equation (6.2.11), and this is given as

We may therefore write the stresses and displacement for the rotating disk under one bracket as

With these introductory basic equations, we shall now set out to discuss the stress distribution and displacement in rotating disks.

14.1 Introduction to Association Rule Mining

Association rule mining is a rule-based technique to discover the relation between the attributes of a dataset. It is used to find the relation between the sales of item X and item Y. It is often called a “market basket” analysis, as shown in Figure 14.1. Here, the market analyst examines the items that consumers often purchase together to find the relation between the sale of item X and item Y.

In other words, when customers visit a store, they may buy a certain type of items together during a shopping trip. For example, as shown in Figure 14.1, a database of customer’s transactions (e.g., shopping baskets) is shown where each transaction consists of a set of items (e.g., products) purchased during a visit, machine learning (ML) engineers can use association mining for finding out a group of items which are frequently purchased together (customers purchasing behavior). This is also referred to as an analysis of customer purchasing behavior. For example, “IF one buys bread, THEN there is a high probability of buying butter with it”, as it is common that people who buy bread often buy butter with it. The store manager can use this information and arrange the items accordingly to increase sales and the overall efficiency of the store.

Let us consider a situation where the store manager feels that there is a lot of rush and customers always complain about the slow working of his store. He is exploring different ways to improve the efficiency of his store. He performed an association analysis and prepared a list of associated items like bread and butter. He may decide to put all these associated items together on the same shelf or near each other so that customers can find them quickly, reducing their shopping time. It will also improve the overall efficiency of the store and the sale of the products. To further improve the shopping experience of his customers, he can create different combos and put sales over these combos.