Abstract
For a new engineering design, we have to perform various analyses. Many of these analyses belong to mechanics. As a consequence of a static or dynamic loading, deformation and stress occur in the continuum body. If the stress lies below the yield limit, the deformation is recoverable upon unloading. This behavior is called an elastic response. This elastic response is instantaneous, i.e., rate of loading does not matter. In order to bring in the effect of the loading rate, we need a viscoelastic response. This behavior is modeled by changing the constitutive (material) equation. The deformation is still recovered upon unloading. In the case of remaining deformation after unloading, we need a constitutive equation modeling a plastic behavior. In all of the aforementioned phenomena, we ignore any change in temperature, thus the process is isothermal. We will discuss mechanical systems and compute the motion of particles belonging to a continuum body. We start with linear elasticity, then solve a problem with geometric nonlinearities, and finally incorporate material nonlinearities. All of these computations belong to elastostatics. By including time rate in the equations, we start off with dynamics; examples of linear and fractional rheology are presented. Moreover, the plastic deformation is addressed, where the material starts flowing beyond the yield stress. We change the understanding of motion from a solid body to fluid. We present the computation of flows of linear and nonlinear fluids. We finalize the chapter with an application of a deformable solid in a viscous fluid, known as a fluid-structure interaction. All applications are discussed theoretically and implemented by using open-source codes.
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Notes
- 1.
It is named after Joseph-Louis Lagrange.
- 2.
The frame is named after Leonhard Euler.
- 3.
The stress is named after Augustin-Louis Cauchy.
- 4.
See [12] for materials data.
- 5.
Kronecker delta, \(\delta _{ij}\), is simply the identity matrix having 1 on its diagonal and zero as the non-diagonal components. It is named after Leopold Kronecker.
- 6.
The summation convention is named after Albert Einstein.
- 7.
Rate means a change in time, given by the derivative in time.
- 8.
The law is named after Robert Hooke .
- 9.
It is named after Woldemar Voigt .
- 10.
The law is named after Carl Friedrich Gauß.
- 11.
For a clear treatment of all the classes you can refer to the lessons of Bernhard J. Wuensch in MIT freely available under: http://www.academicearth.org/courses/symmetry-structure-and-tensor-properties-of-materials
- 12.
They are named after Gabriel Lamé.
- 13.
It is named after Thomas Young.
- 14.
It is named for Siméon Denis Poisson.
- 15.
It is named after Peter Gustav Lejeune Dirichlet.
- 16.
It is named for Carl Gottfried Neumann.
- 17.
Mathematicians use the notation \({\mathrm d}x={\mathrm d}x_1 {\mathrm d}x_2 {\mathrm d}x_3\) for a three-dimensional volume element in space \(\varvec{x}=(x_1,x_2,x_3)\), in order to generalize this to \({\mathrm d}x={\mathrm d}x_1 {\mathrm d}x_2 \dots {\mathrm d}x_n\) for an n-dimensional volume element in space \(\varvec{x}=(x_1,x_2,\dots x_n)\). Also for the area element they use \({\mathrm d}s\) by referring to surface element. This notation is also used in the code that we will employ.
- 18.
The theorem is named after Carl Friedrich Gauß and Mikhail Vesilyevic Ostrogradskiy .
- 19.
It is named after George Green .
- 20.
The method is named after Boris Grigoryevich Galerkin.
- 21.
It is named after Carl Friedrich Gauß.
- 22.
See [17].
- 23.
In analytical mechanics the choice of the coordinate system is of importance. In a particular coordinate system the analytical solution might be much easier than in another coordinate system. In numerical mechanics this is very rarely the case, so we simply select Cartesian coordinates.
- 24.
There is a small difference between a coordinate system and a frame. A frame is only the observer in any coordinate system. Of course, in order to define physical quantities, the observer uses a coordinate system in its frame. We use them together in a sense that we immediately append a coordinate system to the frame.
- 25.
This mapping is according to the tensor transformation rules, as we will see it in Sect. 1.4 on p. 34.
- 26.
It is named after Augustin Louis Cauchy, George Green, Josef Finger, and Heinrich Hencky.
- 27.
It is named after Leonhard Euler, Emilio Almansi, and Georg Karl Wilhelm Hamel.
- 28.
The simplest choice is to refer to the particles at the initial time, since we know their initial positions. The Lagrangean frame has no restrictions of using the initial frame as the reference frame. Positions of particles at any moment can be chosen as the reference frame. In this work we will use the initial frame as the reference frame.
- 29.
An identity is just an equation being true for any chosen arguments, here the left side is equal to the right side in every coordinates.
- 30.
Plane normals have no units, they just show the directions. Hence, it is also useful to write \(n_i {\mathrm d}a = {\mathrm d}a_i\) or analogously \(N_i {\mathrm d}A = {\mathrm d}A_i\).
- 31.
It is named after Carl Gustav Jacob Jacobi.
- 32.
There are many names for this stress in the literature: Piola’s stress named after Gabrio Piola, the first Piola–Kirchhoff stress, named after Gabrio Piola and Gustav Robert Kirchhoff, nominal stress, and engineering stress are the most prominent names. Some textbooks distinguish between nominal and Piola’s stresses and use Piola’s stress as the transpose of the nominal stress as introduced as in this book.
- 33.
Although we skip a discussion about the objectivity, the transformation properties of the strain and nominal stress are different. It is easier to construct constitutive equations between terms with equal transformation properties.
- 34.
It is named after Adhémar Jean Claude Barré de Saint-Venant and Gustav Robert Kirchhoff.
- 35.
It is named for Isaac Newton and Joseph Raphson.
- 36.
W(att) is the unit of power named after James Watt. Power is the rate of energy. The energy used from the wall is measured in kWh (kilo-Watt-hours).
- 37.
Alberto Castigliano never wrote the equation in this way but he found out the variational method on a one-dimensional beam by using the relation: force is the energy differentiation with respect to the displacement. We can extend this theorem and motivate the relation that stress is the energy differentiation with respect to the strain.
- 38.
John William Strutt, 3rd Baron Rayleigh.
- 39.
It is named after Carl Friedrich Gauß.
- 40.
A cylindrical and a spherical coordinate system are curvilinear orthogonal coordinate systems, where the base vectors are orthogonal to each other. If the base vectors are not orthogonal then we use the term oblique. An oblique and curvilinear coordinate system is called an arbitrary coordinate system. We use coordinate systems fixed in time (they do not move according to the observer).
- 41.
- 42.
The symbol is named after Elwin Bruno Christoffel.
- 43.
It is named after Clarence Melvin Zener.
- 44.
This phenomenon is called creep in the nanoindentation test, however, creep is a plastic deformation under constant stress such that we omit to use this term in this section.
- 45.
See [9] for such experiments.
- 46.
This historical note is taken from [11].
- 47.
It is named after Georg Friedrich Bernhard Riemann and Joseph Liouville.
- 48.
It is named for Anton Karl Grünwald and Alexey Vasilievich Letnikov.
- 49.
See [18] for these definitions and their equivalence.
- 50.
Makrolon M2200 is a commercial polycarbonate manufactured by Bayer, for the material parameters with 5Â \(\%\) multi-wall carbon nanotubes from Baytubes, see [10].
- 51.
Mathematically, we should expand in time by using a Taylor expansion up to nth integer rate. Then the history is involved in for a limited interval of time given by nth derivative.
- 52.
We lower the rank by contracting indices. In order to contract two indices we multiply by the metric tensor. Kronecker delta is also the metric tensor in Cartesian coordinates.
- 53.
- 54.
There are three invariants in three-dimensional space of the stress tensor. The first invariant of stress is the bulk quantity \(s=\sigma _{ii}\), the second invariant is \(\sigma _{ij}\sigma _{ij}\) and the third invariant is, \(\sigma _{ij}\sigma _{jk}\sigma _{ki}\).
- 55.
See [16].
- 56.
This approach gives the so-called Karush–Kuhn–Tucker conditions:
since in the elastic regime \(f<0\) and
whereas in the plastic regime \(f=0\) and 
. We will not make much use of these relations, they are mostly used in conditional optimization. - 57.
They are named for William Herrick Macaulay.
- 58.
It is named after Oliver Heaviside.
- 59.
Formally, a sinusoidal displacement, \(u=a\sin (bt)\), and thus the velocity, \(v=a\, b\, \cos (bt)\), are independent, \(\int u\, v {\mathrm d}t = 0\), since sinus and cosinus are orthogonal. This independence means that a variation in one does not change the other. A velocity in the current time causes a displacement in a future time, therefore, they are affecting each other in the subsequent times, however, independent at the current time.
- 60.
This effect has been discussed in [4] for the first time and therefore it is named after Johann Bauschinger.
- 61.
See [13].
- 62.
See [19].
- 63.
See [23].
- 64.
See [24, Chap. 3].
- 65.
We simply ignore the surface stress, which holds a drop together.
- 66.
The constitutive equation is named for Claude Louis Marie Henri Navier and George Gabriel Stokes .
- 67.
Positive stress attains a lengthening, whereas pressure shortens the structure. Hence, the mechanical pressure is a minus trace of stress tensor.
- 68.
It is named after Osborne Reynolds.
- 69.
A function of the class \(C^k\) has a continuous kth derivative. Since we use linear finite elements (polynomial order one) the first derivative exists, only \(C^1\) can be represented in one finite element.
- 70.
See [29, p. 32] at 20 \(^\circ \)C.
- 71.
They are named after Eugene Cook Bingham and Alexey Antonovich Il’yushin.
- 72.
A conserved quantity lacks a production term, it cannot vanish or be produced out of nothing.
- 73.
It is formulated by Augustin-Louis Cauchy. For a detailed explanation see [28, Sect. 203].
- 74.
For a detailed derivation see [25 ] .
- 75.
The theorem is named after Arthur Cayley and William Rowan Hamilton.
- 76.
- 77.
It is named after Pierre Carreau.
- 78.
See [21, Sect. 6.2.3] for equations of these models.
- 79.
We follow the ideas in [6, Part I, Sect. 2.2.3].
- 80.
It is named for Brook Taylor.
- 81.
It is named after René Eugène Gâteaux.
- 82.
It is named after Carl Gustav Jakob Jacobi.
- 83.
An assign operator is not a mathematical equality. The value of the object \(\varvec{P}\) is updated as \(\varvec{P} + \varDelta \varvec{P}\). The assign operator in computational algebra is also denoted by \(\leftarrow \) so we may write \(\varvec{P} \leftarrow \varvec{P}+\varDelta \varvec{P}\). In mathematics, the assign operator \(:=\) is used in the meaning of a definition, i.e., an equality introduced for the first time. We use \(=\) for every mathematical equation and \(:=\) for a computational assignment of values.
- 84.
- 85.
- 86.
LBB conditions are named after Olga Aleksandrovna Ladyzhenskaya, Ivo Babuška, and Franco Brezzi.
- 87.
It is named for Cedric Taylor and Paul Hood.
- 88.
There are also other approaches to obtain a stable solution method for fluids in finite element method, see for example [1].
- 89.
In many textbooks the word configuration is used instead of frame.
- 90.
The balance of mass in the initial frame is not a differential equation, see Sect. 1.4 for its derivation.
- 91.
See Sect. 1.2 for the derivation this transformation .
- 92.
Especially in materials science, the nominal stress is called the engineering stress and Cauchy’s stress is called the true stress .
- 93.
We skip a lengthy discussion for this fact. Actually, the balance of linear momentum on singular surfaces suggests this fact. In case of neglecting the surface tension, the traction on the interface—it is a singular surface without mass—is continuous. In order words, the traction vectors experiencing solid and fluid are identical.
- 94.
In some textbooks this term is called a diffusive term, however, a diffusion is a motion of mass. Since a traction on the boundary implies a momentum flow without mass diffusion, we refrain from naming traction as a diffusive term.
- 95.
A vortex means a circulation in the fluid.
whereas in the plastic regime 
. We will not make much use of these relations, they are mostly used in conditional optimization.
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Abali, B.E. (2017). Mechanics. In: Computational Reality. Advanced Structured Materials, vol 55. Springer, Singapore. https://doi.org/10.1007/978-981-10-2444-3_1
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