Mechanics of Solid Deformable Body

317 Pages·2023·6.856 MB·English

by  Michael Zhuravkov, Yongtao Lyu, Eduard Starovoitov

Michael Zhuravkov
Yongtao Lyu
Eduard Starovoitov
Mechanics
of Solid
Deformable
Body
Mechanics of Solid Deformable Body
· ·
Michael Zhuravkov Yongtao Lyu
Eduard Starovoitov
Mechanics of Solid
Deformable Body
Michael Zhuravkov Yongtao Lyu
Theoretical and Applied Mechanics Department of Engineering Mechanics
Belarusian State University DUT-BSU Joint Institute
Minsk, Belarus Dalian University of Technology
Dalian, Liaoning, China
Eduard Starovoitov
Construction Mechanics
Belarusian State University of Transport
Gomel, Belarus
ISBN 978-981-19-8409-9 ISBN 978-981-19-8410-5 (eBook)
https://doi.org/10.1007/978-981-19-8410-5
© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature
Singapore Pte Ltd. 2023
This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether
the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse
of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmission or information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or
the editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd.
The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721,
Singapore
Preface
In this textbook, we consider the problems connecting with selecting and constructing
mechanics-mathematical models that adequately describe the stress–strain state of
deformed solids.
We do not describe detailed approaches and methods for solving specific classes
of boundary problems of deformable solids. These issues are set out in the relevant
special publications.
This publication is compiled on the basis of lecture courses for students of the joint
institutes “DTU-BSU” and “BSU-DTU” as well as textbooks and lecture courses
compiled by the authors for students of Belarussian State University, among them:
Zhuravkov M. A. Mathematical modeling of deformation processes in solid
deformable media (on the example of problems of rock mechanics). Minsk. Pub.
by Belarusian State University, 2002. 456 p. (in Russian).
Zhuravkov M. A. Fundamental solutions of the theory of elasticity and some of
their applications in geomechanics, soil and base mechanics. Course of lectures.
Minsk. Pub. by Belarusian State University, 2008. 247 p. (in Russian).
Zhuravkov M. A., Starovoitov E. I. Continuum Mechanics. Theory of Elasticity
and Plasticity. Coursebook. Minsk. Pub. by Belarusian State University, 2011.
543 p. (Classical University Edition). (in Russian).
Zhuravkov M. A., Starovoitov E. I. Mathematical models of solid mechanics.
Coursebook. Minsk. Pub. by Belarusian State University, 2021. 535 p. (Classical
University Edition). (in Russian).
Minsk, Belarus Michael Zhuravkov
Dalian, China Yongtao Lyu
Gomel, Belarus Eduard Starovoitov
v
Introduction
Deformed Solid Body Mechanics or Solid Mechanics (SM) is a collection of
disciplines that study the stress–strain state (SST) of deformed solids under the
various physical laws of material behavior.
The main “classical” sections of SM are the theories of elasticity, plasticity, and
viscoelasticity. This can also include material strength and structural mechanics.
The difference among these sections of mechanics lies in the objects under consid-
eration, assumptions and physical equations defining material behavior, and so on.
In the theories of elasticity, plasticity, and viscoelasticity, various physical laws that
establish the connection between stresses and strains are used. Often these sections
of SM are divided into mathematical and applied theories. Therefore, for example,
the mathematical theory of elasticity does not use any deformation hypotheses, and
the resulting equations are solved either by exact methods or by such approximate
methods that allow us to limitlessly increase the degree of approximation to an exact
solution. Therefore, one can consider the results obtained when solving problems by
the mathematical theory of elasticity can be used as a standard for evaluating the accu-
racy of various approximate theories and methods for solving similar problems. The
applied theory of elasticity differs from the mathematical one in that when solving
problems, in addition to the Hooke’s1 law, some additional hypotheses are used (the
hypothesis of flat sections for rods, straight normal for plates, and shells, etc.). In
solving problems of the applied theory of elasticity, along with accurate methods
for solving the corresponding equations, approximate methods are also used. Note
that there is no “clear boundary” between the applied theory of elasticity and the
resistance of materials.
The first mathematician who was engaged in systematic studies of the resistance of
solid bodies to destruction was Galileo.2 Although he considered solids to be inelastic
and did not know the law linking displacements and forces, his work indicated the
path that other researchers followed.
1 Robert Hooke (1635–1703), English scientist, architect, and polymath.
2 Galileo di Vincenzo Bonaiuti de’ Galilei (1564–1642), Italian astronomer, physicist, and engineer,
polymath.
vii
viii Introduction
The discovery of Hooke’s law in 1660 and the establishment of Navier’s3 general
equations in 1821 represent two important milestones in the further development of
the theory that began with Galileo. Hooke’s law provided the necessary experimental
justification for the theory. Finding common equations made it possible to reduce all
problems related tosmalldeformations of elastic bodies to mathematical calculations.
In the period between the discovery of the Hooke’s law and the establishment of
general differential equations of the theory of elasticity, the interest of researchers
was focused on the problems of oscillations of rods and plates, as well as on the
stability of columns. These were, first of all, the fundamental work of J. Bernoulli,4
devoted to the definition of the elastic curve shape, and the work of Euler,5 which laid
the foundation for the research in the field of stability of elastic systems. Lagrange6
applied Euler’s theory to determine the most reliable shape of columns.
The mathematical theory of elasticity as a science has been developed in the
first half of the nineteenth century, mainly thanks to the work of French engineers
and scientists. Therefore, for the first time, equilibrium and oscillation equations
of elastic solids, assuming the body discrete molecular structure, were obtained by
Navier. He deduced not only differential equations, but also boundary conditions
that must be satisfied on the surface of the body. Lamé7 and Clapeyron8 developed
Navier’s theory in relation to engineering. They wrote a special and scientific work
on the internal equilibrium of solids, which solved the problem of stresses and strains
of a thick-walled pipe under axisymmetric loading (Lamé’s problem).
By the fall of 1822, Cauchy9 discovered most of the main elements of the pure
theory of elasticity. He introduced the concept of stress and strain at a point. He
showed that they (stress and strain) can be defined by six relevant components. Based
on the hypothesis of a continuous and homogeneous solid structure, Cauchy obtained
equations of motion (or equilibrium). He first introduced two elastic constants into the
3 Claude-Louis Navier (1785–1836), French mechanical engineer, a physicist whose work was
specialized in continuum mechanics. First introduced the concept of stress.
4 Jacob Bernoulli (1655–1705), one of the many prominent mathematicians in the Bernoulli family.
He was an early proponent of Leibnizian calculus and sided with Gottfried Wilhelm Leibniz during
the Leibniz–Newton calculus controversy.
5 Leonhard Euler (1707–1783), Swiss mathematician, physicist, astronomer, geographer, logician,
and engineer who made important and influential discoveries in many branches of mathematics. He
is also known for his work in mechanics, fluid dynamics, optics, and astronomy.
6 Joseph-Louis Lagrange (1736–1813), Italian mathematician and astronomer, later naturalized
French. He made significant contributions to the fields of analysis, number theory, and both classical
and celestial mechanics.
7 Gabriel Lamé (1795–1870), French mathematician who contributed to the theory of partial differ-
ential equations by the use of curvilinear coordinates and the mathematical theory of elasticity (for
which linear elasticity and finite strain theory elaborate the mathematical abstractions).
8 Benoît Paul Émile Clapeyron (1799–1864), French engineer and physicist, one of the founders of
thermodynamics.
9 Baron Augustin-Louis Cauchy (1789–1857), French mathematician, engineer, and physicist who
made pioneering contributions to several branches of mathematics, including mathematical analysis
and continuum mechanics.
Introduction ix
equations of elastic theory, while Navier’s equations contained only one. Relations
that link small strains and displacements are named after him.
A significant contribution to the development of the theory of elasticity belongs
to Saint-Venan.10 He proposed a new approach for solving a number of applications
(the semi-reverse method of Saint-Venan). Using this method, important problems
were solved regarding the bending and torsion of a bar with a non-circular cross
section. He also owns research on oscillation, shock, and plasticity theory.
In the second half of the nineteenth century, Kirchhof11 formulated the main equa-
tions of thin rod theory, which laid the foundation for the development of methods
for calculating elastic springs. In addition, he developed a consecutive theory of thin
plates. The first attempt in this direction was made by Lagrange and Sophie Germain12
in 1814, but they were not able to correctly formulate the boundary conditions.
At the end of the nineteenth century, Aron and Love13 gave the first version of
the equations of shell theory of the model, based on the application of the hypothesis
of a non-deformability of the normal rectilinear element. Bussinesk14 investigated
the problem of determining stresses in an elastic body under the influence of a
concentrated force. These studies allowed Hertz15 to formulate the problem about
the interaction of two elastic bodies in contact.
An important role in the development of the theory of elasticity was played by
the work of Russian scientists. The fundamental results of the development of the
principle of possible displacements, the theory of impact, as well as the integration
of dynamics equations belong to Ostrogradsky.16 Gadolin17 studied the stresses in
multilayer cylinders, thereby building the basis for the design of artillery barrels.
Zhuravsky18 formulated the modern theory of bending beams. Significant progress
in the construction of methods for solving 2D problems of the theory of elasticity is
10 Adhémar Jean Claude Barré de Saint-Venant (1797–1886), a mechanician and mathematician
who contributed to early stress analysis and also developed the unsteady open channel flow shallow
water equations, also known as the Saint-Venant equations.
11 Gustav Robert Kirchhoff (1824–1887), German physicist who contributed to the mechanics,
mathematical physics, electricity, and spectral analysis.
12 Marie-Sophie Germain (1776–1831), French mathematician, physicist, and philosopher. One of
the pioneers of elasticity theory.
13 Augustus Edward Hough Love (1863–1940), a mathematician famous for his work on the
mathematical theory of elasticity. He also worked on wave propagation.
14 Joseph Valentin Boussinesq (1842–1929), French mathematician and physicist who made
significant contributions to the theory of hydrodynamics, vibration, light, heat, and theory of
elasticity.
15 Heinrich Rudolf Hertz (1857–1894), a German physicist who first conclusively proved the
existence of the electromagnetic waves predicted by James Clerk Maxwell’s equations of
electromagnetism.
16 Ostrogradsky Mikhail Vasilyevich (1801–1862), Russian mathematician, mechanician, and
physicist. Performed important studies on integral calculus.
17 Gadolin Aksel Vilgelmovich (1828–1892), Russian scientist, developed the theory of fastening
the barrels of artillery guns; works on physics, metal processing.
18 Zhuravsky Dmitriy Ivanovich (1821–1891), Russian scientist and engineer, founder of a school
in the field of construction mechanics and bridge construction.
x Introduction
associated with the names of Kolosov19 and Muskhelishvili,20 who first applied the
method based on the use of complex variable functions. Bubnov21 solved a number
of problems about a plate bending. Fundamental studies on the theory of plates and
shells, vibrations of rods taking into account the influence of shear deformations
were carried out by Tymoshenko.22 Subsequently, many problems were solved by
the energy method proposed by him. Galerkin23 completed a series of studies on
the theory of bending thin plates, thick plates, and the theory of shells. To derive
the equations of shell theory, he apparently first applied the equations of the three-
dimensional theory of elasticity.
Papkovich first proposed to build a solution to the problems of the theory of
elasticity of displacements in the form of harmonic functions. He also performed
studies of general stability theorems of elastic systems. In addition, he solved a large
number of problems about bending plates under various boundary conditions.
Vlasov,24 Novozhilov25 and Rabotnov26 made a great contribution to the devel-
opment of the general theory of shells and other sections of the theory of elasticity.
Therefore, Vlasov is the founder of a new scientific discipline—the construction
mechanics of shells.
The theory of plasticity, which studies irreversible deformations of solid bodies,
as an independent section of mechanics, began “its history” about a hundred and
fifty years ago. The foundations of the theory of plastic flow whose creation was
aimed at describing metal forming processes were formulated in the first publication
of Saint-Venan (1868–1871).
19 Kolosov Guriy Vasikyevich (1867–1936), Russian, Soviet mechanic; works on mathematics,
theory of elasticity, machine theory, and mechanisms.
20 Muskhelishvili Nikolai Ivanovich (1891–1976), a renowned Soviet Georgian mathematician,
physicist, and engineer. Muskhelisvili conducted fundamental research on the theories of phys-
ical elasticity, integral equations, boundary value problems, and others.
21 Bubnov Ivan Grigorievich (1872–1919), Russian engineer, developed the basics of the ship’s
construction mechanics, the designer of two submarines.
22 Tymoshenko Stepan Prokofyevich (1878–1972), Ukrainian, Russian, and later, an American
engineer and academician. He is considered to be the father of modern engineering mechanics.
23 Galerkin Boris Grigoryevich (1871–1945), Soviet mathematician and an engineer. One of the
creators of the theory of bending of plates, his works contributed to the introduction of mathematical
methods in engineering research.
24 Vlasov Vasiliy Zaharovich (1906–1958), Russian, Soviet scientist in the field of mechanics, works
on the resistance of materials, construction mechanics, the theory of elasticity.
25 Novozhilov Valentin Valentinovich (1910–1987), Russian, Soviet mechanic; works on the theory
of elasticity, plasticity, and calculation of shells and ship structures.
26 Rabotnov Uriy Nikolaevich (1914–1985), Russian and Soviet mechanical scientist; works on the
theory of shells, the theory of plasticity, creep, and destruction of materials.
Introduction xi
At the beginning of the twentieth century, Karman,27 Huber,28 Mises,29 Nadai,30
Henki,31 and others put forward new concepts and theories that, although they did not
solve problems, expanded the range of ideas. In the 1920s and 1930s, in addition to
propose different versions of plasticity theory, important fundamental experimental
studies were carried out. It was an important stage in the development of plasticity
theory. However, by the end of the 1930s–first years of the 1940s, a situation devel-
oped when experiments could confirm one theory and at the same time refute other
theories. And the opposite, results from other experiments indicated otherwise.
Clarity was introduced by Ilyushin32 (1943–1945), which pointed out the need for
a clear distinction between the nature of deformation processes (simple and complex
deformation). As a result of these researches, Ilyushin developed the theory of small
elastoplastic deformations.
In the early 1950s, various theories of plasticity were proposed under arbitrary
complex loading. These approaches took the form of three theories: modern flow
theory, sliding theory, and the general theory of elastoplastic deformations.
The construction of the general mathematical deformation theory of plasticity
was based on the isotropy postulate formulated by Ilyushin. The basis for the further
development of the theory of the flow of elastoplastic bodies was the Drucker33
hardening postulate on the non-negativity of the external forces work in a closed
cycle of plastic loading.
Models of viscoelastic behavior of materials began to actively develop in the
second half of the last century. In many materials under operating conditions, the
law of connection between force and displacement depends significantly on time.
This dependence is exerted in the fact that, for example, under constant loading, the
deformations do not remain constant, but increase. On the other hand, if the body is
subject to deformation and some bonds keep the deformation unchanged, the reaction
of the bonds decreases with time, “relaxed”.
27 Theodore von Kármán (1881–1963), Hungarian-American mathematician, aerospace engineer,
and physicist who was active primarily in the fields of aeronautics and astronautics.
28 Hyber Maximilian Tytus (1872–1950), scientist in the field of mechanics; works in the field of
material resistance, the theory of elasticity, the theory of plasticity, proposed a criterion for the
plasticity of the potential energy of shape change.
29 Richard Edler von Mises (1883–1953), an Austrian scientist and mathematician who worked on
solid mechanics, fluid mechanics, aerodynamics, aeronautics, statistics, and probability theory.
30 Nadai Arpad Ludwig (1883–1963), scientist mechanic, engineer; works in the field of the theory
of elasticity, plasticity, together with Lode introduced the Nadai-Lode parameter.
31 Henki Heinrich (1885–1951), scientist mechanic, engineer; the most famous works in the field
of sliding theory, plasticity, and rheology.
32 Ilyushin Alexey Antonovich (1911–1998), Russian and Soviet scientist mechanic, one of the
founders of the theories of plasticity, viscoelasticity, and other sections of mechanics.
33 Daniel Charles Drucker (1918–2001) was an American civil and mechanical engineer. Drucker
was known as an authority on the theory of plasticity in the field of applied mechanics. His key
contributions to the field of plasticity include the concept of material stability described by the
Drucker stability postulates and the Drucker–Prager yield criterion.