Mechanics of Solids and Fluids

Kinematics is that division of Mechanics which describes the geometry of motion (deformation) of a body, regardless of the forces and stresses, the sources of that motion. Either the position or displacement vector, both the velocity and acceleration vector are key to analyzing the motion of a point. The fields of those vectors determine the kinematics of simple continua (point-continua). The deformation gradients, the spatial derivatives of the displacements, determine the local deformations and, hence, define the strains. Thus, the elongation of a fiber, the extension of a volume element, and the angular change of a configuration of two perpendicular fibers can be calculated. The kinematic model of a rigid (undeformable) body is characterized by a constant distance between any pair of points in motion. It is the principal reference system (eg for the deformations) and the velocity field is represented by means of the angular velocity vector. Polar cones of a spatial pendulum and polar curves (centrodes) associated with a rigid body in plane motion illustrate the velocity field of such an idealized model and introduce the notion of pure rolling contact.

A natural combination of motion (deformation) and force (stress) is given by the notion of mechanical work and power.

“Nonlocal” material laws of solids or fluids are not considered in this textbook. Hence, the constitutive relations in every material point are either finite or time differential equations, which relate the stress tensor to the proper strains (the influence of temperature or other nonmechanical fields are excluded at this stage of consideration). Simple material laws are given for elastic, visco-elastic, elastic-plastic, and elastic-visco-plastic bodies. Especially with respect to the latter, the deformations are restricted to relatively small strains and strain rates, ie problems associated with plastic-forming techniques, like forging, are to be excluded. A loose guide of the subsequent considerations is given by the extensive and organized material testing that is performed in daily routine to render the numerical values of material parameters.

Only linearized problems are considered in this section, ie linearized geometric relations and Hooke’s law are taken into account [see Eqs. (1.21) and (4.15)]. The consequences of these assumptions are illustrated by considering a linear elastic body loaded on its surface by a self-equilibrating system of single forces F ₁, F ₂, …, F _n (reaction forces in the point supports are included). The loading is applied to the body by means of a common load factor λ, which is slowly increased from 0 ≤ λ ≤ 1 to reach the terminal configuration by passing successively through states of equilibrium.

Dynamics is understood in the narrow sense of kinetics, which means that the inertia of accelerated masses is of crucial importance when considering the stresses and deformations in moving bodies. Newton’s law of motion is applicable if the velocities remain small with respect to the speed of light. In vacuum, the speed of light is the natural constant c = 299 792 458 m/s. Since 1983, the definition of the unit of length, 1 meter, is based on the above derivable fraction of the distance traveled by light in vacuum within 1 second. The basic dynamic law within the limits of newtonian mechanics in the formulation of Euler-Cauchy states that proportionality exists in every material point of a continuum between the force density f and the absolute acceleration a.

The stability of the equilibrium configuration of floating bodies has already been considered by simple static means, and proper conditions were found in the form of inequalities of the type (2.115). In this section, the stability of conservative systems at rest is observed by analyzing the motion that follows any perturbation of the equilibrium state (ie by the dynamic method of small perturbations) and, equivalents, by the static Dirichlet stability criterion. Simple applications are given, including the balance problem of rigid heavy bodies in contact, the structural problem of the buckling of slender columns and thin plates (a bifurcation problem), as well as the snapping of shallow arches when the lateral load becomes critical. The extension of the dynamic method of small perturbations to include the stability of a given (main) motion is illustrated through the analyses of a mechanical control device, the centrifugal governor, and the gyroscope without moment. In a third subsection, the stability of elastic-plastic structures is considered statically by limit load analysis and, quasistatically, for alternating loadings by the shake-down theorems of Melan and Koiter. Furthermore, the hydrodynamic stability of incompressible flow in open channels is discussed and the loss of energy in the hydraulic jump where the rapid flow changes “abruptly” to tranquil streaming is given. Finally, flutter instability is described well by the self-excited oscillations of a simplified model of an airfoil that occur above a critical speed of flight.

The principle of virtual work, Eq. (5.3), can be generalized to include the inertia forces of dynamics. In statics, the equilibrium configuration of a system at rest has to be considered; in dynamics, the instant configuration of a moving body at some time t is to be observed. In analogy to the virtual variation of the equilibrium configuration, virtual displacements are applied to the instant configuration, keeping the time t constant.

The deformed configuration of a body, in general, is determined by the field of displacement vectors u = u(x, y, z, t; X, Y, Z), ie a continuum possesses an infinite number of degrees of freedom. The basic partial differential equations of such distributed parameter systems, with associated boundary and initial conditions, even in the case of linear elastic solids, but with non-simple geometry, can hardly be solved in an exact manner. Two classes of approximation techniques are commonly used to overcome these difficulties: (1) The essential boundary conditions are built into the approximation that is not a solution of the basic differential equations. The Rayleigh-Ritz-Galerkin method based on such a set of admissible functions is discussed below, and examples of discretization by the finite-element method (FEM) are given. (2) A solution of the basic equations is known that takes on the prescribed boundary values at a number of discrete points only. Such an approximation is the output of the class of collocation methods. The boundary element method (BEM) should be mentioned here. Collocation methods are not discussed in this text any further.

Impact is a process of momentum exchange between two colliding bodies within a short time of contact. With respect to a single impacted body or structure, the loading in such a process acts with high intensity during this short period of time. As a result, the initial velocity distribution is rapidly changed (even pressure wave loadings, eg following an explosion, are events of that category). Such rapid loading in the contacting area is a source where waves are emitted that propagate with finite speeds through the body. In the case of sufficiently small amplitudes (in the far field), linear elastic body waves propagate with the speed of sound waves, the distinction between the fast longitudinal P-wave (primary wave) and the slower transverse S-wave (secondary wave) complicates the pattern (see Sec. 7.6 and Exercise A 7.12), and further refractions and diffractions are encountered. An illustration is given in Sec. 12.7 in which an impacted finite rod is considered. A further complication arises in the near field through large deformations and nonlinear material behavior. Visco-plasticity must be considered, and high deformation rates may even render fracturing in statically ductile materials. This short account of the complex processes involved already indicates that a complete analysis of geometrically nonsimple impacted bodies requires numerical methods of high resolution, ie say, in terms of the finite-element method, large systems must be considered when marching in time with extremely fine steps. Such expensive models are restricted to special problems, like the impact following an airplane crash on an atomic power plant and the like. Many “everyday” impact problems of engineering interest, however, can be drastically simplified by idealization. The most important idealizing assumption is that of a sudden jump in the velocity distribution and, thus, of the momentum in the limit of vanishing contacting time. Taking such a limit implies the infinite speed of the stress signal propagation in the body. Furthermore, the impact forces, accelerations, and stresses take on infinite values such that their time integrals remain finite and assume actual values. Idealized impact becomes a sudden mechanical process at some instant of time, and the colliding bodies keep their positions with unchanged configurations. The motion following impact is initiated with the new velocity distributions after completing the sudden exchange of momentum.

Hydrodynamic forces, resultants of pressure distributions, have been discussed in connection with the conservation of momentum of control volume. One ingredient of the theory is still missing: the “circulation.” It is basic for the understanding of the most important of these forces, the lift. Its connection to the vortex vector, Eq. (1.50), ie to the rotation, is derived below. The effects of viscosity on Newtonian fluids (for the constitutive relations, see Sec. 4.2.1) are illustrated and the Navier-Stokes equations derived. The essential parameters of similarity solutions and, hence, of model testing in wind tunnels as well as in water channels are discussed. The boundary layer that develops in the flow along a plate is calculated. The irrotational motion of ideal fluids, which enters the solution, eg of the outer flow, is studied by means of potential theory, and a brief account of the singularity method is given. One application is the von Karman vortex trail in the wake behind a blunt body moving through a viscous fluid, with the notion of drag as a byproduct. Moving walls in contact with a fluid pose a nonstationary boundary value problem of interest in earthquake engineering and, eg they function as wave makers in a swimming pool. The effects of compressibility and Mach number are discussed in connection with stationary efflux from a pressure vessel into open air, a problem often encountered in engineering. It is, eg a strong source of noise in connection with the operation of steam power plants. For shock waves formed in supersonic flows, the reader is referred to the special literature on gas dynamics.

In addition to the illustrative examples within the text and the exercises added there in the form of an appendix to each of the thirteen previous chapters, all given with full solutions, there is a need for further selected problems. The following sample is put into the order of the book’s sections. Figures are not numbered consecutively but carry the number of the exercise.