Integral Equation Methods in Solving Problems of Elasticity and Potential Theory: A Unified Approach

Classical problems in elasticity and potential theory often involve complex boundary conditions and domain geometries. Traditional domain-based approaches, such as finite difference and finite element methods, though versatile, can become computationally expensive, particularly in higher dimensions or multiply connected domains. An alternative lies in integral equation methods, which reformulate the original partial differential equations into equivalent boundary integral representations. This transformation not only reduces the dimensionality of the problem but also facilitates the handling of infinite or semi-infinite domains.

The response or reaction of the material against the internal relative motion or simple displacement is what we mean when we talk about elasticity. Whenever the medium is subjected to a system of forces that is equilibrating, the mathematical theory of elasticity reduces the computation of the state of strain to an equation of equilibrium, in addition to stress-strain equations and compatibility relations. Hooke proposed that the extension of spring-like entities or elastic bodies, which are formed by tensile force, were inextricably linked to the pressures exerted on them. Dry and scaly started with the many assumptions that resulted in the development of the linear theory of elasticity, which is still applicable today. Making use of a variety of assumptions resulted in the construction of the linear theory of elasticity, which continues to be consistent with the current state of affairs. As part of our ongoing research endeavours, we will be conducting investigations on thermo-elasticity and thermal stress. During the crack analysis process, the phenomena of fracture mechanics is related with the crack. A significant impact on human existence was brought about by the phenomena of brittle fracture. The principle of brittle fracture was used by prehistoric man in the process of crafting the stone implements [1].

It seems that potential theory originated in quest of knowledge about fundamental forces of nature. Lagrange (1736-1813) named gravitational function as a potential function and then Gauss in 1840 called it Potential. Gauss along with his team found out how potentials are implemented in various issues of mathematical physics. Certain type of potentials became important for boundary value issues like Dirichlet problem, Neumann problem, the electrostatic problems, Rabin problem. In this line a list of types of potentials is: i. Double-layer potential. ii. Logarithmic potential. iii. Green potential. iv. It is the Newtonian potential. The modern theory of potentials exists in close connection with the theory of analytic functions, harmonic functions, and subharmonic functions, as well as with the theory of probability. Two dimensional potential theory is particularly important for its connections to analytic functions theory. Thus we reached to the point to say that the function which satisfies Laplace equation is called Potential Function. The solution of physical problem is obtained as solution of Laplace’s equation. This solution will be reduced to n-tuple integral equations [2].

The most of the work dealt within this thesis owes its genesis to the most beneficent manifestations of brittle fracture i.e., sudden or catastrophic failure of structures. One of the earliest on record was falling of a mill at Oldham; England in 1844. In the first quarter of 29th century many disasters were recorded involving bridges, naphtha conduits, gas tanks, water mains and tanks of steam boilers, large and small guns, rail road tracks and other railway equipment, molasses tanks under both active and quiescent loads. A.A. Griffith laid a theoretical foundation from continuum mechanics point of view in 1920. The first T2 tanker built by Kaisen company at Portland, Oregon had just completed successful sea trials and had returned to shipyard on 16 Jan. 1943. At 10:30 p.m. in calm, cool weather (air temp. 26°F and water temp. 48°F) the ship suddenly broken into two parts. These types of incidents stirred the ship building industries. Griffith had already attached the importance to normal stress (crack opening mode) at crack tips he defined stress-intensity factors at crack tips [3].

Earlier research in the subject of thermoelasticity was preceded by significant research in the so-called theory of thermal stresses. This is something that should be emphasised a lot. This later theory is a theory of the state of strain and stress in an elastic body, which is caused by heating. Due to its oversimplification, it allows us to disregard the effect of deformation on the temperature field. Thermoelasticity encompasses a wide variety of phenomena. It encompasses not only the generalised theories of thermal stress and heat conduction, but also the phenomena of thermo elastic dissipation and the temperature distribution produced by deformation. both of these theories are included. Heat conduction is a possible variant that is considered in the thesis that is now being presented. One day, we are going to broaden the scope of the study to include this aspect. In the current thesis, we have explored the issues of potentials and elasticity across multiply-connected domains, which reduce the physical problem to n-tuple integral equations or n-tuple series equations. These equations are used to solve the physical problem [4].

In an integral equation, an unknown function is represented by one or more integral signs. This kind of equation is known as an integral equation. One example of the most common kind of linear integral equation has the form, which may be presented as

Whereas, depending on the circumstances, the top limit of integration can be either fixed or adjustable. Finding the unknown function g(s) is more challenging than finding the known functions h(s), f(s), and K(s,t). The function denoted as K(s,t) is known as the kernel. These functions may have complicated values since s and t are real variables [5].

Types of integral equations

There are primarily three groups of integral equations: (i) Fredholm Integral Equations (also known as FIE)

The letter b, which represents the top limit of integration, is constant in Fredholm integral equations.

To begin, the Fredholm integral equation of the first sort is as follows:

h(s) = 0 in equation (1.1.1), Thus

(b) Equation (1.1.1) of the second sort of the Fredholm integral equation has h(s) = 1.

Thus

(c) A Fredholm integral equation of second kind is called the homogeneous Fredholm integral equation if f(s) is equal to zero in equation (1.1.3). Thus

(ii). The integral equations of Volterra

The variable upper limit of integration, denoted by b = s, is the only difference between the definitions given above for first, second, and homogeneous type Volterra integral equations.

(iii) Integrals with a singular value

When either the limits of integration or the kernel becomes infinite at some point within the range of integration, or when both of these things happen, the integral equation is called the singular integral equation. Take, as an example, the integral equations.

and

are singular integral equations.

Further Kinds of Jntegral Equations

In the past half-century, mixed boundary value problems in mathematical physics have been mostly solved using specific types of integral equations, such as dual integral equations, triple integral equations, etc. What follows is a brief outline of the literature's evolution of integral equations of this kind.

Methods of Solutions

Solving dual integral equations is now possible using a variety of techniques. What follows is a discussion of some of the methods:

 (i) Use of Mellin Transform

The use of the Mellin transform method has been shown to play a key role in the development of several novel concepts within the theory of dual integral equations. Titchmarsh [6] was the first to find the formal solution of dual integral equations of the type using the Mellin transform.

for g(x) = 0. His method of solution is closely related with the Wiener-Hopf technique [7] and solution is valid for

This section has been divided into two parts

·                    Crack problems in dissimilar media,

·                    Thermal problems of elasticity.

In this section, we will discuss some of the issues that arise from diverse elastic medium and how integral equation techniques may be used to solve them. Bonding together two or more materials that have differing mechanical characteristics is the starting point for the construction of many engineering structures. In order for the dissimilar material system to function as a single unit, it is necessary for the loads to be transferred from one material to the next across the interfaces. It is possible that the existence of faults or cracks in one of the materials or at the interface might result in a significant increase in the local stresses, which could ultimately lead to failure if the fracture reach a critical magnitude. Because of this, it is essential to have a solid understanding of the stress state that is connected to these fissures in the dissimilar material system. The analytical stress solution exhibits a peculiar behaviour near the tip of an interface crack where the stresses undergo a rapid reversal of sign. This highly oscillatory character takes the form of the argument  in where an is the size of the crack, r 1 is the radial distance from the fracture border, and E is a bimaterial constant that depends on the mechanical characteristics of the materials next to it. The investigation of two-dimensional fracture issues was where this phenomenon was first uncovered [8].

The fundamental problem of two semi-infinite elastic media with different properties that are connected along a plane is considered by Erdogan [9]. Furthermore, these mediums have a series of flat holes fashioned like concentric rings. Using Green's functions for semi-infinite planes, the problem may be expressed as a system of simultaneous singular integral equations. The dominant system's closed-form solution corresponding to Cauchy kernels is shown here. The method of converting the problem to a system of linear algebraic equations is described here in order to get the whole answer, instead of solving two Fredholm integral equations. He illustrated the contact stresses with examples such as the penny-shaped fracture and bonded media with an external notch that is axially symmetric and subject to homogeneous temperature fluctuations or axial load.

Bassani and Erdogan considered the antiplane shear problem for two bonded different half planes with a semi-infinite fracture or two randomly placed collinear fractures in their work [10]. The problem is resolved for a concentrated wedge force in the semi-infinite fracture scenario, and the stress intensity factor and angular distribution of stresses are calculated. If the number of fractures is unlimited, the problem may be reduced to two integral equations. Numerical results for fractures completely embedded in homogeneous media, one crack tip touching the interface, and a crack crossing the interface are found for different fracture angles [11].

Through the use of fractional integral operators, [12] have been able to acquire the solution to a dual integral equation that involves the G-function of two variables. For the purpose of this chapter, we will attempt to find a solution to simultaneous dual integral equations that include G-functions of two variables. According to [13-15], G-functions are defined as

 ……..(1.1.7)

 ……(1.1.8)

The sequence of parameters  do not correspond with any of the integer's poles. If needed, the integration pathways are indented so that all of the  lie to the right and those of   lie to the left of the imaginary axis.

The integral (2. 1.1) converges if

 ………..(1.1.9)

The simultaneous dua l integral equations to be discussed here are as follows:

 ……(1.1.10)

 ……(1.1.11)

The historical development of integral formulations began with the recognition of structural similarities between Laplace’s equation in potential theory and the Navier-Cauchy equations in elasticity. Pioneers like Betti and Somigliana extended potential-theoretic techniques to elastostatics. Green’s identities, particularly the third identity, play a critical role in expressing the solution inside a domain in terms of boundary data, laying the groundwork for the boundary integral equations (BIEs).

In potential theory, the use of single-layer and double-layer potentials allows the transformation of Laplace and Poisson equations into boundary integrals. These formulations are well-suited to treat Dirichlet, Neumann, and Robin boundary conditions. The simplicity of the Laplace kernel ensures smooth behavior, and the methods enjoy excellent analytical properties and convergence behavior when implemented numerically.

In elasticity, the governing equations are vector-valued and more complex due to the tensorial nature of stress and strain. Nevertheless, the use of Somigliana’s identity, derived from Betti’s reciprocal theorem, enables a similar reduction to boundary integrals. Rizzo (1967) advanced this approach by presenting a vector boundary formula connecting boundary displacements and tractions, applicable to both simply and multiply connected domains. This was a significant conceptual advancement, especially for three-dimensional problems.

Numerical approaches like the Boundary Element Method (BEM) discretize the boundary into elements, transforming the integral equations into a solvable linear system. Methods for treating singular kernels (e.g., Cauchy Principal Value) are crucial in ensuring accuracy. Modern developments also include fast multipole methods and adaptive mesh refinement to handle large-scale and complex geometries effectively.

Integral equation methods offer a powerful alternative to classical domain methods in addressing elasticity and potential theory problems. By leveraging the mathematical analogy and converting volume integrals into boundary formulations, these methods facilitate both analytical insight and numerical efficiency. With ongoing advancements in computational techniques and broader applications in engineering and physics, integral methods continue to gain prominence as a robust tool in applied mathematics and mechanics.