Computational Contact Mechanics - The Coolest Thing in the Field of Mechanics
Can we walk without the aid of friction between our feet and the ground? Why is contact between two interacting surfaces necessary to transfer the actions? What are types of contacts exist in mechanics? Why is it difficult to solve the mechanics’ problems involving contact analytically? What are the main objectives for solving the problems computationally? How did this discipline emerge with time?
Read this article to find out the answers to these questions.
This article will walk you through the following topics.
What is computational contact mechanics (CCM)
Virtually all movements on this planet involve contact and friction. Frictional contact is an essential part of our life, without which we cannot walk or run; no vehicle could move. If we closely see any physical system, it involves the mechanical interactions across the interfaces. They are significant in the mechanical, civil, environmental engineering industries and medical applications. From the design of prosthetics in biomedical engineering to the crash analysis of cars in automotive engineering, relative slip between concrete and reinforcing steel in structural engineering involves contact interactions. At the micro and nano-scale level, contact exists between adhesion systems, like the adhesion mechanism of geckos and insects. Sometimes, contact analysis includes the consideration of other fields like thermal or electromagnetic in the contact area. Applications include the analysis of cooling of electronic devices, heat removal with nuclear power plant vessels.
Before the advent of computing in science, computational contact mechanics was considered an engineering sub-discipline within applied mechanics. Now, it is well adapted as a sub-discipline within computational science, where numerical computing methods are used to study the phenomena governed by the principles of mechanics. In general, the discipline itself says about its definition. To put it simply, The CCM is the study of contact mechanics with the aid of computers. Hence, it is based on the foundation of mechanics, mathematics, and computer science and physics. Before knowing about the topic, first, we have to understand the classical contact mechanics.
From the mechanical point of view, contact is the notion for all interactions between separate bodies coming in touch or two portions of the same body. The former case is frequently referred to as “direct contact”, and the latter with “self-contact”. Direct contact between two solid bodies can be utilized for many purposes like transferring loads, heat, and an electric charge. Both the cases are shown in the figures below. These figures, tangent and normal to the contact surface, play a significant role in the contact analysis. If we see the contact interaction in terms of frictional behaviour at the contact surface, it can distinguish between frictionless and frictional contact. We can classify them into small and large deformation problems regarding deformation behaviour at the contact surface. Furthermore, according to the space dimension, where the contact occurs, we classify them into one, two, and three-dimensional problems.
Contact between two different objects (DIRECT CONTACT).
Contact between two portions of the same object (SELF CONTACT).
So, the problems can be classified in the following way.
In reality, the physics of contact interaction is multiscale, making the problem highly non-linear and non-smooth in computing science. This non-linearity stems from the fact that the actual contact surface on which bodies comes into contact and the stresses developed on it is unknown before applying the load to a body. The multiscale nature of the problems comes from friction, deformations, and geometry complexity of the objects in contact. Sometimes, the rigorous description of the interacting surfaces creates mathematical difficulties. The situations can be the penetration of contacting bodies, discretizations of a domain into a finite number of elements. Because of this, problems are often approximated by assumptions. One of the main assumptions is that the contact interface has no thickness and can sustain only the compressive stress perpendicular to the contact surface. Today, we can simulate these problems with numerical computation tools due to the rapid improvement of modern computer technology.
Generally, contact at the interface is accompanied by two types of stresses. One is of compressive or adhesive nature, acting perpendicular to the contact plane. Another is of frictional or shearing nature, acting tangentially between the contact surfaces. The compressive nature of stress perpendicular to the contact plane assures the contact between the two objects. If the stress is tensile, bodies will be separated, which has no meaning in the contact mechanism. The stress in the tangential direction deals with the situations of frictional contact. According to this contact mechanism, two surfaces of the contact objects slide relative to each other without tangential efforts. In other words, we can say the frictionless contact interface allows two surfaces to slide. However, in frictional contact, it does not allow any sliding until critical shear stress is reached. Two bodies stick to each other until the critical stress is reached. After the critical stress is reached, bodies start to slip and create the energy of the system.
So, the contact mechanism involves the stresses acting in the following directions.
A real example
Have you ever thought about why the stability of the building structure is important? The stability of the structure depends upon soil–foundation interaction. The eccentric loads acting on a building are the leading cause of instability of the structure. These loads transmit from the structure to the soil through the footing. Soil exerts an upward pressure, called contact pressure, on the footing. So, it is essential to know about the contact pressure. You can see the lifting of the foundation from the soil due to eccentric loads acting on the building in the figure shown below. Even though the physical nature of soil and foundation is different, the contact between soil and foundation extensively deals with stability.
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Soil – foundation interaction.
Applications of contact mechanics
Phenomena of contact mechanics exist in various fields of engineering and sciences. The most common applications that can be seen in the mechanical and civil engineering and medical industry are summarized below.
A few rules to solve a mechanical problem
A contact problem can be formulated as a boundary value problem (BVP), where the governing differential equation is fulfilled in the domain, and boundary conditions are imposed on the domain’s boundary. Here, we are talking about the solution procedure in the context of the finite element method (FEM).
As the differential equations (strong form) of real BVP are difficult to solve in computers, they are converted into equivalent algebraic equations, easily solved computationally. To get the algebraic system of equations, we need to convert the strong form into an integral form (weak form). So, the weak form (integral form) of the BVP is written by balancing the virtual work, which will be considered a basis on which FEM is constructed. The principle upon which the process is based is called the variational principle.
Then, the contact constraints are formulated as a set of inequalities, which results in variational inequality. These are the conditions that make the BVP solvable. As these steps are slightly different from the classical variational equality, which is generally used in the finite element analysis of the structures, a new solution technique is required to solve the problem.
Next, discretization of the contact interface is done to obtain the algebraic set of solvable equations. There are multiple methods available to discretize. For the conforming meshes, where each node of the contacting surface has a corresponding node on the other surface, a Node-to-Node method is used. This method has a limitation of solving the problems involving small deformations and infinitely small relative sliding. The second method is the Node-to-Segment discretization, where contact pairs consist of a node of one surface and a corresponding segment of the other surface. This discretization has a limitation of instability. A new method called Segment-to-Segment discretization was introduced to overcome the drawbacks arising from the previous methods. Here we have two different options available to choose from. One is Nitsche, and the other is Mortar methods. However, the computer implementation of these methods for a general case presents a real challenge.
So, we have the following choices in the discretization.
Lastly, we follow the detection process to determine the contact pairs on discretized surfaces. This step saves time for complex problems and depends on the contact interface's discretisation type (above three) and type of contact (direct or self contact). In this step, care should be taken to efficiently treat contact problems, both for rapidity and robustness.
Different methods are applied to solve different classes of contact problems.
The design of robust algorithms to treat contact problems efficiently within the finite element method needs input from different sources. Theoretical aspects of continuum mechanics, contact kinematics, and the contact interface’s constitutive behaviour are required to understand the contact mechanism.
Historical remarks
The actual application of frictional contact is dated back to the era of ancient Egyptian people. Large stone blocks were moved to build the pyramids, overcoming the associated frictional forces, which clearly shows they knew the process of lubrication. In the 15th century, a famous researcher Da Vinci investigated the influence of contact area on the frictional force using blocks of different contact areas and the same weight. The experiment showed that the frictional force depends proportionally on the weight of the blocks but is independent of the contact area.
Euler carried out the analysis to represent surface roughness mathematically and showed that the kinetic coefficient of friction is smaller than the static coefficient of friction. He also introduced the symbol?μ?for the friction coefficient. A comprehensive experimental study on frictional phenomena was later performed by Coulomb in 1785. He put these findings into a mathematical expression,?f = μN, also known as Coulomb’s friction law. In this equation,??f is the friction force,??N is the normal force, and?μ?is the coefficient of friction.
In the year 1882, Hertz gave the theory of elasticity in the framework of contact mechanics. He investigated the elastic contact of two spheres and derived the pressure distribution in the contact area. However, very few problems involving contact were solved analytically.
Researchers worked on developing a computational method to solve the problems of mechanics. The finite element method (FEM) can be traced back to the early 1940s. Attempts were made to solve the first structural problem using FEM in the late 1950s. However, FEM obtained its real impetus in the 1960s with the growing power of computers. After this, literature grew because industrial applications had problems that could not be solved analytically. It took another ten years for the first paper, in which FEM solved contact problems. Wilson and Parsons (1970), Chan and Tuba (1971) were the first contributors who treated contact using geometrically linear theory. Following this, the explicit codes DYNA2D and DYNA3D and the implicit codes NIKE2D and NIKE3D were developed by Hallquist, beginning in the mid-seventies. For the first time, these codes provided the possibility to solve the general contact problems efficiently.
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