Modeling and Simulation in Science, Engineering and Technology Antonio Romano Addolorata Marasco Continuum Mechanics using ® Mathematica Fundamentals, Methods, and Applications Second Edition Modeling and Simulation in Science, Engineering and Technology Series Editor Nicola Bellomo Politecnico di Torino Torino, Italy Editorial Advisory Board K.J Bathe Department of Mechanical Engineering Massachusetts Institute of Technology Cambridge, MA, USA M Chaplain Division of Mathematics University of Dundee Dundee, Scotland, UK P Degond Department of Mathematics, Imperial College London, London, United Kingdom A Deutsch Center for Information Services and High-Performance Computing Technische Universität Dresden Dresden, Germany M.A Herrero Departamento de Matematica Aplicada Universidad Complutense de Madrid Madrid, Spain P Koumoutsakos Computational Science & Engineering Laboratory ETH Zürich Zürich, Switzerland H.G Othmer Department of Mathematics University of Minnesota Minneapolis, MN, USA K.R Rajagopal Department of Mechanical Engineering Texas A&M University College Station, TX, USA T.E Tezduyar Department of Mechanical Engineering & Materials Science Rice University Houston, TX, USA A Tosin Istituto per le Applicazioni del Calcolo “M Picone” Consiglio Nazionale delle Ricerche Roma, Italy More information about this series at http://www.springer.com/series/4960 Antonio Romano • Addolorata Marasco Continuum Mechanics using Mathematica R Fundamentals, Methods, and Applications Second Edition Antonio Romano Department of Mathematics and Applications “R Caccioppoli” University of Naples Federico II Naples, Italy Addolorata Marasco Department of Mathematics and Applications “R Caccioppoli” University of Naples Federico II Naples, Italy Additional material to this book can be downloaded from http://extras.springer.com ISSN 2164-3679 ISSN 2164-3725 (electronic) ISBN 978-1-4939-1603-0 ISBN 978-1-4939-1604-7 (eBook) DOI 10.1007/978-1-4939-1604-7 Springer New York Heidelberg 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Science+Business Media (www.birkhauser-science.com) Preface The motion of any body depends both on its characteristics and on the forces acting on it Although taking into account all possible properties makes the equations too complex to solve, sometimes it is possible to consider only the properties that have the greatest influence on the motion Models of ideals bodies, which contain only the most relevant properties, can be studied using the tools of mathematical physics Adding more properties into a model makes it more realistic, but it also makes the motion problem harder to solve In order to highlight the above statements, let us first suppose that a system S of N unconstrained bodies Ci , i D 1; : : : ; N , is sufficiently described by the model of N material points whenever the bodies have negligible dimensions with respect to the dimensions of the region containing the trajectories This means that all the physical properties of Ci that influence the motion are expressed by a positive number, the mass mi , whereas the position of Ci with respect to a frame I is given by the position vector ri t / versus time To determine the functions ri t /, one has to integrate the following system of Newtonian equations: mi rR i D Fi Á fi r1 ; : : : ; rN ; rP ; : : : ; rP N ; t /; i D 1; : : : ; N , where the forces Fi , due both to the external world and to the other points of S , are assigned functions fi of the positions and velocities of all the points of S , as well as of time Under suitable regularity assumptions about the functions fi , the previous (vector) system of second-order ordinary differential equations in the unknowns ri t / has one and only one solution satisfying the given initial conditions ri t0 / D r0i ; rP i t0 / D rP 0i ; i D 1; : : : ; N: A second model that more closely matches physical reality is represented by a system S of constrained rigid bodies Ci , i D 1; : : : ; N In this scheme, the v vi Preface extension of Ci and the presence of constraints are taken into account The position of Ci is represented by the three-dimensional region occupied by Ci in the frame I Owing to the supposed rigidity of both bodies Ci and constraints, the configurations of S are described by n Ä 6N parameters q1 ; : : : ; qn , which are called Lagrangian coordinates Moreover, the mass mi of Ci is no longer sufficient for describing the physical properties of Ci since we have to know both its density and geometry To determine the motion of S , that is, the functions q1 t /; : : : ; qn t /, the Lagrangian expressions of the kinetic energy T q; q/ P and active forces Qh q; q/ P are necessary Then a possible motion of S is a solution of the Lagrange equations d @T dt @qP h @T D Qh q; q/; P @qh h D 1; : : : ; n; satisfying the given initial conditions qh t0 / D qh0 ; qP h t0 / D qP h0 ; h D 1; : : : ; n; which once again fix the initial configuration and the velocity field of S We face a completely different situation when, to improve the description, we adopt the model of continuum mechanics In fact, in this model the bodies are deformable and, at the same time, the matter is supposed to be continuously distributed over the volume they occupy, so that their molecular structure is completely erased In this book we will show that the substitution of rigidity with the deformability leads us to determine three scalar functions of three spatial variables and time, in order to find the motion of S Consequently, the fundamental evolution laws become partial differential equations This consequence of deformability is the root of the mathematical difficulties of continuum mechanics This model must include other characteristics which allow us to describe the different macroscopic material behaviors In fact, bodies undergo different deformations under the influence of the same applied loads The mathematical description of different materials is the object of the constitutive equations These equations, although they have to describe a wide variety of real bodies, must in any case satisfy some general principles These principles are called constitutive axioms and they reflect general rules of behavior These rules, although they imply severe restrictions on the form of the constitutive equations, permit us to describe different materials The constitutive equations can be divided into classes describing the behavior of material categories: elastic bodies, fluids, etc The choice of a particular constitutive relation cannot be done a priori but instead relies on experiments, due to the fact that the macroscopic behavior of a body is strictly related to its molecular structure Since the continuum model erases this structure, the constitutive equation of a particular material has to be determined by experimental procedures However, the introduction of deformability into the model does not permit us to describe all the phenomena accompanying the motion In fact, the viscosity of S as well as the friction between S and any external bodies produce heating, which in turn causes heat exchanges among parts of S or between S and Preface vii its surroundings Mechanics is not able to describe these phenomena, and we must resort to the thermomechanics of continuous systems This theory combines the laws of mechanics and thermodynamics, provided that they are suitably generalized to a deformable continuum at a nonuniform temperature The situation is much more complex when the continuum carries charges and currents In such a case, we must take into account Maxwell’s equations, which describe the electromagnetic fields accompanying the motion, together with the thermomechanic equations The coexistence of all these equations gives rise to a compatibility problem: in fact, Maxwell’s equations are covariant under Lorentz transformations, whereas thermomechanics laws are covariant under Galilean transformations This book is devoted to those readers interested in understanding the basis of continuum mechanics and its fundamental applications: balance laws, constitutive axioms, linear elasticity, fluid dynamics, waves, etc It is self-contained, as it illustrates all the required mathematical tools, starting from an elementary knowledge of algebra and analysis It is divided into 13 chapters In the first two chapters the elements of linear algebra are presented (vectors, tensors, eigenvalues, eigenvectors, etc.), together with the foundations of vector analysis (curvilinear coordinates, covariant derivative, Gauss and Stokes theorems) In the remaining ten chapters the foundations of continuum mechanics and some fundamental applications of simple continuous media are introduced More precisely, the finite deformation theory is discussed in Chap 3, and the kinetic principles, the singular surfaces, and the general differential formulae for surfaces and volumes are presented in Chap Chapter contains the general integral balance laws of mechanics, as well as their local Eulerian or Lagrangian forms In Chaps and the constitutive axioms, the thermo-viscoelastic materials, and their symmetries are discussed In Chap 8, starting from the characteristic surfaces, the classification of a quasi-linear partial differential system is discussed, together with ordinary waves and shock waves The following two chapters cover the application of the general principles presented in the previous chapters to perfect or viscous fluids (Chap 9) and to linearly elastic systems (Chap 10) In Chap 11, a comparison of some proposed thermodynamic theories is presented The great importance of fluid dynamics in meteorology is showed in Chap 12 In particular, in this chapter the arduous path from the equation of fluid dynamics to the chaos is sketched In the last chapter, we present a brief introduction to the navigation since the analysis of this problem is an interesting example of the interaction between the equations of fluid dynamics and the dynamics of rigid bodies Finally, in Appendix A the concept of a weak solution is introduced This volume has many programs written with Mathematicar [69] These programs, whose files will be available at Extras.Springer.com., apply to topics discussed in the book such as the equivalence of applied vector systems, differential operators in curvilinear coordinates, kinematic fields, deformation theory, classification of systems of partial differential equations, motion representation of perfect viii Preface fluids by complex functions, waves in solids, and so on This approach has already been adopted by two of the authors in other books (see [1, 33]).1 Many other important topics of continuum mechanics are not considered in this volume, which is essentially devoted to the foundations of the theory In a second volume [56], already edited, continua with directors, nonlinear elasticity, mixtures, phase changes, electrodynamics in matter, ferromagnetic bodies, and relativistic continua are discussed Naples, Italy Antonio Romano Addolorata Marasco The reader interested in other fundamental books in continuum mechanics can consult, for example, the references [26, 30, 36, 66] Contents Elements of Linear Algebra 1.1 Motivation to Study Linear Algebra 1.2 Vector Spaces and Bases Examples 1.3 Euclidean Vector Space 1.4 Base Changes 1.5 Vector Product 1.6 Mixed Product 1.7 Elements of Tensor Algebra 1.8 Eigenvalues and Eigenvectors of a Euclidean Second-Order Tensor 1.9 Orthogonal Tensors 1.10 Cauchy’s Polar Decomposition Theorem 1.11 Higher Order Tensors 1.12 Euclidean Point Space 1.13 Exercises 1.14 The Program VectorSys Aim of the Program VectorSys Description of the Problem and Relative Algorithm Command Line of the Program VectorSys Parameter List Worked Examples Exercises 1.15 The Program EigenSystemAG Aim of the Program EigenSystemAG Description of the Algorithm Command Line of the Program EigenSystemAG Parameter List 1 10 12 13 14 20 25 28 29 31 32 38 38 38 39 39 40 42 42 42 43 43 43 ix A A Brief Introduction to Weak Solutions 465 does not have a weak derivative In fact, we have Z Z df d' dx D dx D f dx dx 1 Œ'10 D '.0/; and no function can satisfy the previous relation Definition A.2 The vector space W21 / D f W f L2 /; @f L2 /; i D 1; : : : ; n @xi (A.5) of the functions which belong to L2 /, together with their first weak derivatives, is called a Sobolev space It becomes a normed space if the following norm is introduced: !1=2 à Z n Z  X @f 2 kf k1;2 D f d C d : @xi iD1 (A.6) Recalling that in L2 / we usually adopt the norm ÂZ Ã1=2 kf kL2 / D f d ; we see that the relation (A.6) can also be written as kf k21;2 D kf k2L2 / n X @f C @x i iD1 : (A.7) L2 / The following theorem is given without a proof Theorem A.1 The space W21 /, equipped with the Sobolev norm (A.6), is a Banach space (i.e., it is complete) More precisely, it coincides with the completion H21 / of the space ff C /; kf k1;2 < 1g: This theorem states that for all f W21 /, a sequence ffk g of functions fk C / with a finite norm (A.7) exists such that lim kf k!1 fk k1;2 D 0: In turn, in view of (A.6), this condition can be written as ) (Z à n Z  X @f @fk 2 f fk / d C d D 0; lim k!1 @xi @xi iD1 (A.8) 466 A A Brief Introduction to Weak Solutions or equivalently, Z k!1 Z  lim k!1 @f @xi D 0; fk /2 d f lim @fk @xi Ã2 D 0; d i D 1; : : : ; n: In other words, each element f W21 / is the limit in L2 / of a sequence ffk g of C /-functions and its weak derivatives are also the limits in L2 / of the sequences of ordinary derivatives f@fk =@xi g The previous theorem makes it possible to define the weak derivatives of a function as the limits in L2 / of sequences of derivatives of functions belonging to C / as well as to introduce the Sobolev space as the completion H21 / of ff C /; jjf jj1;2 < 1g Let C01 / denote the space of all the C -functions having a compact support contained in and having a finite norm (A.6) Then another important functional space is HO 21 , which is the completion of C01 / In such a space, if the boundary @ is regular in a suitable way, the following Poincaré inequality holds: Z f d à n Z  X @f Äc d @xi iD1 8f HO 21 /; where c denotes a positive constant depending on the domain other norm jjf jjHO / !1=2 à n Z  X @f D d ; @xi iD1 (A.9) If we consider the (A.10) then it is easy to verify the existence of a constant c such that jjf jjHO / Ä jjf jj1;2 Ä c jjf jjHO / ; so that the norms (A.6) and (A.9) are equivalent We conclude this section by introducing the concept of trace If f C /, then it is possible to consider the restriction of f over @ Conversely, if f H21 /, it is not possible to consider the restriction of f over @ , since the measure of @ is zero and f is defined almost everywhere, i.e., up to a set having a vanishing measure In order to attribute a meaning to the trace of f , it is sufficient to remember that, if f H21 , then a sequence of C /-functions fk exists such that lim jjf k!1 fk jj1;2 D 0: (A.11) A A Brief Introduction to Weak Solutions 467 Next, consider the sequence ff k g of restrictions on @ of the functions fk and suppose that, as a consequence of (A.12), it converges to a function f L2 / In this case, it could be quite natural to call f the trace of f on @ More precisely, the following theorem could be proved: Theorem A.2 A unique linear and continuous mapping W H21 / ! L2 / exists such that f / coincides with the restriction on @ Moreover, (A.12) of any function f / HO 21 / D ff H21 /; f / D 0g: (A.13) A.2 A Weak Solution of a PDE Consider the following classical boundary value problems relative to Poisson’s equation in the bounded domain < having a regular boundary @ : n X @2 u Df @xi2 iD1 in uD0 n X @2 u Df @xi2 iD1 ; on @ I (A.14) ; in n X du @u D D on @ I dn @xi iD1 (A.15) where f is a given C /-function The previous boundary problems are called the Dirichlet boundary value problem and the Neumann boundary value problem, respectively Both these problems admit a solution, unique for the first problem and defined T up to an arbitrary constant for the second one, in the set C / C / By multiplying (A.14)1 for any v H21 / and integrating over , we obtain Z n X @2 u v d @xi2 iD1 Z D f vd : (A.16) 468 A A Brief Introduction to Weak Solutions If we recall the identity @2 u @ v D @x @xi i  @u v @xi à @v @u @xi @xi and use Gauss’s theorem, then (A.16) can be written as n Z X iD1 @ @u v ni d @xi n Z X @v @u d @xi @xi iD1 Z D fvd ; (A.17) where ni / is the unit vector normal to @ In conclusion, we have: • If u is a smooth solution of Dirichlet’s boundary value problem (A.14) and v is any function in HO /12 , then the previous integral relation becomes n Z X iD1 @v @u d @xi @xi Z D fvd 8v HO 21 /: (A.18) • If u is a smooth solution of Neumann’s boundary value problem (A.15) and v is any function in H /12 , then from (A.17) we derive n Z X iD1 @v @u d @xi @xi Z D f vd 8v H21 /: (A.19) Conversely, it is easy to verify that if u is a smooth function, then the integral relations (A.18) and (A.19) imply that u is a regular solution of the boundary value problems (A.14) and (A.15), respectively All the previous considerations suggest the following definitions: • A function u HO 21 / is a weak solution of the boundary value problem (A.14) if it satisfies the integral relation (A.18) • A function u H21 / is a weak solution of the boundary value problem (A.15) if it satisfies the integral relation (A.19) Of course, a weak solution is not necessarily a smooth (or strong) solution of the above boundary value problems, but it is possible to prove its existence under very general hypotheses Moreover, by resorting to regularization procedures, which can be applied when the boundary data are suitably regular, a weak solution can be proved to be smooth as well More generally, instead of (A.14) and (A.15), let us consider the following mixed boundary value problem: n X @ ALi x; u; ru/ D fL @x i iD1 uD0 ALi x; u; ru/ D gL x/ 8x 8x @ ; 8x @ @ ; @ ; (A.20) A A Brief Introduction to Weak Solutions 469 where u.x/ is a p-dimensional vector field depending on x D x1 ; : : : ; xn /, L D 1; : : : ; p,