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.

Chapter 19 looks at two landmark decisions that were issued after the submission of the full manuscript of this book for publication: the Advisory Opinion of the International Tribunal for the Law of the Sea on climate change and marine protection, and the judgment of the European Court of Human Rights in Verein KlimaSeniorinnen Schweiz and Others v Switzerland. The authors situate these decisions within the broader context of climate litigation, examining their implications for future cases and drawing connections to the themes explored in other chapters. They demonstrate how these decisions both reflect and advance emerging best practice in climate jurisprudence, potentially inspiring further innovation based on science and rigorous legal reasoning.

This chapter surveys the transformation of the North African countryside in Late Antiquity, analysing shifts in settlement patterns, agricultural production and economic structures from the late Roman period through the Arab conquest in 689. It highlights regional variations across ancient provinces such as Mauretania, Numidia, Africa Proconsularis, Byzacena and Tripolitana, emphasising that the rural landscape did not experience uniform decline but rather underwent adaptation and reorganisation. Discussing North Africa’s role as a key supplier of olive oil, grain, wine and fine ceramics, notably African Red Slip (ARS) pottery, the chapter challenges earlier assumptions that economic collapse followed the Vandal conquest in 439, showing instead that new industrialised production methods led to increased ARS exports during the late fifth century. However, the Byzantine reconquest in 539 did not reintegrate the region into the larger imperial economy as expected, instead fostering greater regionalisation and isolation. This contribution also addresses the Christianisation of the countryside, analysing the spread of monastic estates and bishoprics and their impact on land use. By examining archaeological surveys and pottery distribution, the chapter concludes that rural settlement patterns in North Africa were shaped by a combination of environmental, economic and political factors.

This chapter centres on the rural landscapes of the late Roman Eastern Empire, focusing on settlement patterns, economic structures and the integration of rural communities into broader imperial networks. It challenges older historiographical models that depict the countryside as declining in tandem with the urban world, arguing instead that many rural regions remained vibrant and adapted to shifting political and economic conditions. This investigation draws on archaeological surveys, architectural remains and epigraphic evidence to analyse how different rural settlements evolved. It contrasts the village-based economies of the Eastern Mediterranean with the villa-centric model of the western provinces, emphasising that villages in the East continued to thrive well into the Byzantine period. The chapter also highlights the role of religious transformation, documenting the conversion of pagan sanctuaries into Christian sites and the spread of monastic settlements in rural landscapes. Another key theme is connectivity, particularly how infrastructure such as roads and maritime trade routes sustained rural economies. The discussion of transhumance and agricultural production further illustrates the dynamism of the eastern countryside. While certain regions experienced decline due to warfare or shifting imperial priorities, the chapter presents the eastern rural world as an adaptive, resilient environment, rather than one in simple decay.

In a small, rectangular dimly lit room, Khatun Shaikh, a female qazi (Islamic judge) in a women's sharia court, lent a patient ear to women who approached her with complaints of marital discord and violence. The sharia court is an alternative dispute resolution forum run by members of the Bharatiya Muslim Mahila Andolan (Indian Muslim Women's Movement, henceforth BMMA), a social movement led by Muslim women aimed at achieving equality and justice in the adjudication of Muslim family law in India. These alternative forums were frequented by women from poor neighbourhoods in Mumbai who did not have the wherewithal to access the formal justice system. As cases of marriage, divorce, maintenance and domestic violence were discussed and debated in these forums, quarrels broke out between the spouses and their relatives. Allegations of abuse and counter-allegations flew thick and fast. In the midst of these heated exchanges between spouses, Shaikh often emphasised the importance of raham (compassion) as an everyday, lived ethical ideal that both the spouses ought to practice. While the disputes revolved around women claiming specific rights during and after the breakdown of their marriage, Shaikh insisted on how both men and women needed to be compassionate. According to Shaikh, one could display compassion in moments of crisis in the marriage by avoiding the use of harsh words, refraining from overt displays of anger and addressing each other respectfully. This practice of compassion thus entailed using the body in specific ways while claiming one's rights. Shaikh construed compassion as a lived ideal that resonated with the teachings of the Quran and the life of the Prophet. The pursuit of this ideal was closely tethered to the realisation of equality (barabari) and justice (insaf) in the domain of the family.

The sharia court emerged as a space of self-making for both the activists of the BMMA and the women visiting the court. Women spoke their mind. They spoke about the violence and injustice in the family. Interactions between activists, lawyers and the women who visited these forums helped in creating a supportive community space for women who faced injustice in their marital homes. On some days, the court room also doubled as a space where activists of the BMMA conducted training sessions on Muslim family law, the Quran and the Constitution for women of the neighbourhood.

This chapter explores the urban and architectural evolution of Ravenna in Late Antiquity, focusing on its transformation from a relatively modest Roman settlement into a major political and religious centre. It examines how Ravenna served as the capital of the Western Roman Empire, the Ostrogothic Kingdom and the Byzantine Exarchate of Italy between 400 and 725 CE. The discussion highlights how Ravenna’s geographical setting – a marshy coastal landscape – shaped its urban development and archaeological record. The study assesses the city’s infrastructure, including its walls, aqueducts, palaces and religious buildings, demonstrating how imperial patronage played a decisive role in its growth. The construction of monumental churches such as the Basilica of San Vitale and Sant’Apollinare in Classe is explored as evidence of Ravenna’s rising Christian prominence. The chapter also details the impact of Theoderic’s rule, particularly in the adaptation of Roman architectural traditions for Ostrogothic purposes. Under Byzantine rule, Ravenna became a centre of ecclesiastical power, reflected in continued church-building projects. Despite political upheavals and economic fluctuations, Ravenna remained a dynamic and influential city. The chapter concludes by pointing out the challenges of archaeological research in the area, emphasising how ongoing excavations continue to refine our understanding of Ravenna’s late antique legacy.

This chapter examines the evolution of monumental civic architecture in Late Antiquity, exploring how urban spaces and public buildings adapted to shifting political, social and economic conditions. Challenging the traditional ‘decline and fall’ narrative, it argues instead for a process of gradual transformation. While some cities in the Western Empire experienced a decline in public building activity, urban centres in the east continued to flourish, with new constructions and adaptations extending well into the Islamic period. In these cities, public buildings such as basilicas, baths and colonnaded streets were frequently repurposed rather than abandoned. Christianity played a crucial role in this shift, as bishops assumed civic responsibilities and redirected resources towards maintaining and adapting urban infrastructure. As a result, late antique cities remained vibrant, with economic and social life persisting despite structural changes. The chapter also highlights the significance of imperial patronage and contributions from the local elite in sustaining urban architecture. By reassessing civic architecture through archaeological and textual evidence, the study presents a more nuanced perspective – one that emphasises urban resilience and transformation rather than outright decline.

This chapter investigates the material culture of saint veneration in Late Antiquity, detailing how objects, architecture and inscriptions influenced devotional practices. It draws on a range of sources to investigate how saints were honoured and how their presence was made tangible through physical artefacts. The chapter argues that material culture played a crucial role in facilitating interactions between the living and the divine. Saints’ memorials, inscriptions and relics served as focal points for devotion, strengthening communal identity and religious experience. Key examples include the engraved mensa (table) from Khirbet Oum el-Ahdam, which was used for communal feasting in honour of martyrs, and wall paintings from Wadi Sarga, which depict saints in orans postures, underscoring their role as intercessors. The evidence suggests that material artefacts did more than simply commemorate saints – they actively shaped devotional practices and evolved over time to meet the needs of different generations.