The field of Mechanics can be subdivided into three major areas:
Theoretical
Applied
Computational
Theoretical mechanics deals with fundamental laws and principles of mechanics studied for their intrinsic scientific value.
Applied mechanics transfers this theoretical knowledge to scientific and engineering applications, especially as regards the construction of mathematical models of physical phenomena. Computational mechanics solves specific problems by simulation through numericalmethods implemented on digital computers.
Paraphrasing an old joke about mathematicians, one may define a computational mechanicianas a person who searches for solutions to given problems, an applied mechanician as a person who searchesfor problems that fit given solutions, and a theoretical mechanician as a person who can prove the existence ofproblems and solutions.
Computational Mechanics
Several branches of computational mechanics can be distinguished according to the physical scale of the focus of attention:
Computational Mechanics
Nanomechanics and micromechanics
Continuum mechanics
Fluids
Multiphysics
Systems
Solids and Structures
Nanomechanics deals with phenomena at the molecular and atomic levels of matter. As such it isclosely linked to particle physics and chemistry. Micromechanics looks primarily at the crystallographicand granular levels of matter. Its main technological application is the design and fabricationof materials and microdevices.
Continuum mechanics studies bodies at the macroscopic level, using continuum models in which the microstructure is homogenized by phenomenological averages. The two traditional areas of application are solid and fluid mechanics. The former includes structures which, for obvious reasons,are fabricated with solids. Computational solid mechanics takes an applied sciences approach,whereas computational structural mechanics emphasizes technological applications to the analysis and design of structures.
Computational fluid mechanics deals with problems that involve the equilibrium and motion of liquid and gases. Well developed subsidiaries are hydrodynamics, aerodynamics, acoustics, atmosphericphysics, shock, combustion and propulsion.
Multiphysics is a more recent newcomer. This area is meant to include mechanical systems that transcend the classical boundaries of solid and fluid mechanics, as in interacting fluids and structures.Phase change problems such as ice melting and metal solidification fit into this category, as do theinteraction of control, mechanical and electromagnetic systems.
Finally, system identifies mechanical objects, whether natural or artificial, that perform a distinguishablefunction. Examples of man-made systems are airplanes, buildings, bridges, engines, cars,microchips, radio telescopes, robots, roller skates and garden sprinklers. Biological systems, such asa whale, amoeba, inner ear, or pine tree are included if studied from the viewpoint of biomechanics.Ecological, astronomical and cosmological entities also form systems.
In the progression above system is the most general concept. Asystem is studied by decomposition:its behavior is that of its components plus the interaction between components. Components arebroken down into subcomponents and so on. As this hierarchical breakdown process continues,individual components become simple enough to be treated by individual disciplines, but componentinteractions get more complex. Consequently there is a tradeoff art in deciding where to stop.
Statics vs. Dynamics
Continuum mechanics problems may be subdivided according to whether inertial effects are taken into account or not:
Continuum mechanics
Statics
Dynamics
In dynamics actual time dependence must be explicitly considered, because the calculation of inertial(and/or damping) forces requires derivatives respect to actual time to be taken.
Problems in statics may also be time dependent but with inertial forces ignored or neglected. Accordinglystatic problems may be classed into strictly static and quasi-static. For the former time need notbe considered explicitly; any historical time-like response-ordering parameter, if one is needed, willdo. In quasi-static problems such as foundation settlement, metal creep, rate-dependent plasticityor fatigue cycling, a realistic measure of time is required but inertial forces are still neglected.
Linear vs. Nonlinear
A classification of static problems that is relevant is
Statics
Linear
Nonlinear
Linear static analysis deals with static problems in which the response is linear in the cause-and effectsense. For example: if the applied forces are doubled, the displacements and internal stressesalso double. Problems outside this domain are classified as nonlinear.
Discretization methods
A final classification of CSM static analysis is based on the discretization method by which thecontinuum mathematical model is discretized in space, i.e., converted to a discrete model with a finite number of degrees of freedom:
Spatial discretization method
Finite Element (FEM)
Boundary Element (BEM)
Finite Difference (FDM)
Finite Volume (FVM)
Spectral
Meshfree
In CSM linear problems finite element methods currently dominate the scene as regards spacediscretization. Boundary element methods post a strong second choice in specific applicationareas. For nonlinear problems the dominance of finite element methods is overwhelming.
Space finite difference methods in solid and structural mechanics have virtually disappeared frompractical use. This statement is not true, however, for fluid mechanics, where finite difference discretization methods are still important.
Finite-volume methods, which directly address the discretization of conservation laws, are important in difficult problems of fluid mechanics, for examplehigh-Re gas dynamics.
Spectral methods are based on transforms that map space and/or timedimensions to spaces (for example, the frequency domain) where the problem is easier to solve.
A recent newcomer to the scene are the meshfree methods. These combine techniques and tools of finite element methods such as variational formulation and interpolation, with finite difference features such as non-local support.
What Does a Finite Element Look Like?
The term admits of two interpretations,For now the underlying concept will be partly illustrated through a truly ancient problem: find the perimeter L of a circle of diameter d. Since L = π d, this is equivalent toobtaining a numerical value for π.Draw a circle of radius r and diameter d = 2r as in Figure (a). Inscribe a regular polygon ofn sides, where n = 8 in Figure (b). Rename polygon sides as elements and vertices as nodes.Label nodes with integers 1, . . . 8. Extract a typical element, say that joining nodes 4–5, as shown inFigure (c). This is an instance of the generic element i– j pictured in Figure (d). The elementlength is Li j = 2r sin(π/n). Since all elements have the same length, the polygon perimeter isLn = nLi j , whence the approximation to π is πn = Ln/d = n sin(π/n).
Values of πn obtained for n = 1, 2, 4, . . . 256 are listed in the second column of Table 1.1. As canbe seen the convergence to π is fairly slow. However, the sequence can be transformed by Wynn’salgorithm5 into that shown in the third column. The last value displays 15-place accuracy.Some key ideas behind the FEM can be identified in this example. The circle, viewed as a sourcemathematical object, is replaced by polygons. These are discrete approximations to the circle.The sides, renamed as elements, are specified by their end nodes. Elements can be separated bydisconnecting nodes, a process called disassembly in the FEM. Upon disassembly a generic elementcan be defined, independently of the original circle, by the segment that connects two nodes i and j .The relevant element property: side length Li j , can be computed in the generic element independentlyof the others, a property called local support in the FEM. The target property: the polygon perimeter,is obtained by reconnecting n elements and adding up their length; the corresponding steps in theFEM being assembly and solution, respectively. There is of course nothing magic about the circle;the same technique can be be used to rectify any smooth plane curve.
THE SOLIDS & STRUCTURES
Here I have introduced these basic concepts and classical theories ina brief and easy to understand manner. Solids and structures are stressed when they aresubjected to loads or forces. The stresses are, in general, not uniform, and lead to strains,which can be observed as either deformation or displacement. Solid mechanics and structural mechanics deal with the relationships between stresses and strains, displacements andforces, stresses (strains) and forces for given boundary conditions of solids and structures.These relationships are vitally important in modelling, simulating and designing engineeredstructural systems.
FORCES
Forces can be static and/or dynamic. Statics deals with the mechanics of solids and structures subjected to static loads such as the deadweight on the floor of buildings. Solidsand structures will experience vibration under the action of dynamic forces varying withtime, such as excitation forces generated by a running machine on the floor. In this case,the stress, strain and displacement will be functions of time, and the principles and theoriesof dynamics must apply. As statics can be treated as a special case of dynamics, the staticequations can be derived by simply dropping out the dynamic terms in the general, dynamicequations. My approach would be to derive the dynamic equation first, andobtaining the static equations directly from the dynamic equations derived.
MATERIAL PROPERTY
1. Depending on the property of the material, solids can be elastic, meaning that thedeformation in the solids disappears fully if it is unloaded. There are also solids that areconsidered plastic, meaning that the deformation in the solids cannot be fully recoveredwhen it is unloaded. Elasticity deals with solids and structures of elastic materials, andplasticity deals with those of plastic materials. We shall see the problems ofsolids and structures of elastic materials. In addition, with problems ofvery small deformation, where the deformation and load has a linear relationship. Therefore,our problems will mostly be linear elastic.
2. Materials can be anisotropic, meaning that the material property varies with direction.Deformation in anisotropic material caused by a force applied in a particular directionmay be different from that caused by the same force applied in another direction. Compositematerials are often anisotropic. Many material constants have to be used to definethe material property of anisotropic materials. Many engineering materials are, however,isotropic, where the material property is not direction-dependent. Isotropic materials area special case of anisotropic material. There are only two independent material constantsfor isotropic material. Usually, the two most commonly used material constants are theYoung's modulus and the Poisson's ratio. We mostly deal with isotropic materials.Nevertheless, most of the formulations are also applicable to anisotropic materials.
BOUNDRY CONDITIONS
Boundary conditions are another important consideration in mechanics. There aredisplacement and force boundary conditions for solids and structures. For heat transfer problemsthere are temperature and convection boundary conditions.
