Foundations of Continuum Mechanics

In manufacturing processes anisotropic metals are often exposed to the loading with high strain rates in the range from 102 s-1 to 106 s-1 (e.g. stamping, cold spraying and explosive forming). These types of loading often involve generation and propagation of shock waves within the material. The material behaviour under such a complex loading needs to be accurately modelled, in order to optimise the manufacturing process and achieve appropriate properties of the manufactured component.The presented research is related to development and validation of a thermodynamically consistent physically based constitutive model for metals under high rate loading. The model is capable of modelling damage, failure and formation and propagation of shock waves in anisotropic metals. The model has two main parts: the strength part which defines the material response to shear deformation and an equation of state (EOS) which defines the material response to isotropic volumetric deformation [1]. The constitutive model was implemented into the transient nonlinear finite element code DYNA3D [2] and our in house SPH code. Limited model validation was performed by simulating a number of high velocity material characterisation and validation impact tests.The new damage model was developed in the framework of configurational continuum mechanics and irreversible thermodynamics with internal state variables. The use of the multiplicative decomposition of deformation gradient makes the model applicable to arbitrary plastic and damage deformations.To account for the physical mechanisms of failure, the concept of thermally activated damage initially proposed by Tuller and Bucher [3], Klepaczko [4] was adopted as the basis for the new damage evolution model. This makes the proposed damage/failure model compatible with the Mechanical Threshold Strength (MTS) model Follansbee and Kocks [5], 1988; Chen and Gray [6] which was used to control evolution of flow stress during plastic deformation. In addition the constitutive model is coupled with a vector shock equation of state which allows for modelling of shock wave propagation in orthotropic the material.Parameters for the new constitutive model are typically derived on the basis of the tensile tests (performed over a range of temperatures and strain rates), plate impact tests and Taylor anvil tests.The model was applied to simulate explosively driven fragmentation, blast loading and cold spraying impacts.

1R1. Tensor Analysis and Continuum Mechanics. - YR Talpaert (Dept of Sci and Eng, Algiers Univ, Algeria). Kluwer Acad Publ, Dordrecht, Netherlands. 2002. 591 pp. ISBN 1-4020-1055-9. $161.00.Reviewed by JG Simmonds (Dept of Civil Eng, Univ of Virginia, Thornton Hall, Charlottesville VA 22903).This is a carefully written treatise that begins by (over?) developing the tensor machinery necessary to state and manipulate various global and local forms of the general equations of continuum mechanics (conservation of mass, translational and rotational momentum, first and second laws of thermodynamics, excluding chemical and electromagnetic effects). Each of the book’s six chapters ends with a good number of worked-out exercises. Although considerable attention is given to the kinematics and kinetics of deforming continua, the important topic jumps and shocks are not treated and the only specific constitutive relations considered are those of linear elasticity, mostly for isotropic bodies. However, no matter how well done, one must ask in such a well-plowed field as continuum mechanics if a new book supercedes in any way—novelty of presentation, new insights into well-established principles and equations, range of phenomena, or variety of constitutive relations—some of the older, standard treatments, such as those of Truesdell and Toupin (1960), Truesdell and Noll (1965), Chadwick (1976), Spenser (1980), Gurtin (1981), or Truesdell (1984, 1991). Unfortunately, in my opinion, the answer is “No.” (Incidentally, none of the books just cited appears in the author’s list of references.) By no means am I implying that continuum mechanics is a closed field. But it seems to me that today’s focus is on the establishment, analysis, and implications of ever more comprehensive constitutive relations, an area that the author of the book under review barely touches. In summary, Tensor Analysis and Continuum Mechanics is a sound, well-written book, suitable for a first-year graduate course, but researchers in the various branches of continuum mechanics will find little that is new.

A brief outline of classical and extended irreversible thermodynamics is presented. Classical irreversible thermodynamics (CIT) is known as an active and fast developing field with numerous applications in continuum mechanics, chemistry and statistical mechanics. Its foundations and principal results are shortly commented. Extended irreversible thermodynamics (EIT) has fuelled an increasing interest during the last decade. Its main objective is to extend the domain of validity of classical non-equilibrium thermodynamics. Here, a short analysis of Lebon-Jou-Casas' version of EIT is presented, it is compared with CIT and the Gyarmati wave approach.

In recent years, it has been attempted to clarify the damage evolution in solid materials from irreversible thermodynamics. In this study, tensile tests were carried out polyimide and entropy generation amounts were calculated until failure based on irreversible thermodynamics. The parameters of viscoelastic/plastic model were determined by fitting with the result of creep and recovery tests. We predicted fracture of polyimide in the tensile test by using viscoelastic/plastic model which considered entropy generation amounts as criterion of damage.

Development of radial strains in hollow cylinders of rock subjected to radial compression

This chapter deals with how to measure nonlinear viscoelastic functions and various nonlinear viscoelastic constitutive equations. Since there have been developed a number of constitutive equations which cannot be included in a single chapter, this chapter considers only a few popularly used ones, which are classified to four groups: the models based on mathematical approximations; the models obtained from the generalization of linear viscoelastic models; the models based on the simplification of polymer structure; and the Leonov model which is a class of constitutive equations based on irreversible thermodynamics.

In order to enhance currently used beam theories in $$\mathbb {R}^2$$ and $$\mathbb {R}^3$$ to include mechanisms of dissipation and memory, it is necessary to establish if the mathematical models for these theories can be derived using the conservation and the balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the beam mathematical models. Thermodynamic consistency of the currently used beam models will permit use of entropy inequality to establish constitutive theories in the presence of dissipation and memory mechanism in the currently used beam theories. This is the main motivation for the work presented in this paper. The currently used beam theories are derived based on kinematic assumptions related to the axial and transverse displacement fields. These are then used to derive strain measures followed by constitutive relations. For linear beam theories, strain measures are linear functions of displacement gradients and stresses are linear functions of strain measures. Using these stress and strain measures, energy functional is constructed over the volume of the beam consisting of kinetic energy, strain energy and potential energy of loads. The Euler’s equation(s) extracted from the first variation of this energy functional set to zero yields the differential equations describing the evolution of the deforming beam. Alternatively, principle of virtual work can also be used to derive mathematical models for beams. For linear elastic behavior with small deformation and small strain, the two approaches yield same mathematical models. In this paper we examine whether the currently used beam mathematical models with the corresponding kinematic assumption (i) can be derived using the conservation and balance laws of classical continuum mechanics or (ii) are the conservation and balance laws of non-classical continuum mechanics necessary in their derivation. In order to ensure that the mathematical models for various beam theories result in deformation that is in thermodynamic equilibrium we must establish the consistency of the beam theories with regard to the conservation and the balance laws of continuum mechanics, classical or non-classical in conjunction with their corresponding kinematic assumptions. Currently used Euler–Bernoulli and Timoshenko beam mathematical models that are representative of most beam mathematical models are investigated. This is followed by details of general and higher-order thermodynamically consistent beam theory that is free of kinematic assumptions and other approximations and remains valid for slender as well as deep beams. Model problem studies are presented for slender as well as deep beams. The new formulation presented in this paper ensures thermodynamic equilibrium as it is derived using the conservation and the balance laws of continuum mechanics and remains valid for slender as well as non-slender beams.

On countable structures computability is usually introduced via numberings. For uncountable structures whose cardinality does not exceed the cardinality of the continuum the same can be done via representations. Which representations are appropriate for doing real number computations? We show that with respect to computable equivalence there is one and only one equivalence class of representations of the real numbers which make the basic operations and the infinitary normed limit operator computable. This characterizes the real numbers in terms of the theory of effective algebras or computable structures, and is reflected by observations made in real number computer arithmetic. Demanding computability of the normed limit operator turns out to be essential: the basic operations without the normed limit operator can be made computable by more than one class of representations. We also give further evidence for the well‐known non‐appropriateness of the representation to some basebby proving that strictly less functions are computable with respect to these representations than with respect to a standard representation of the real numbers. Furthermore we consider basic constructions of representations and the countable substructure consisting of the computable elements of a represented, possibly uncountable structure. For countable structures we compare effectivity with respect to a numbering and effectivity with respect to a representation. Special attention is paid to the countable structure of the computable real numbers.

The development of the History of Rational Mechanics covers about 150 years only. However, historical studies at the end of the nineteenth and the beginning of the twentieth century were usually restricted to the principles of the Mechanics of Discrete Systems. The first professional review of the history of elasticiry theory was given by Adhémar Barré de Saint-Venant in his Historique abrégé des recherches sur la résistance et sur l'élasticite des corps solides (1864). Another exception is the History of the theory of Elasticity and of the Strength of Materials from Galilei to Lord Kelvin (1886-1893) by Isaac Todhunter and Karl Pearson, which presents on 2200 pages summaries of most of the works on selected topics. During the nineteenth century, Mechanics was mainly considered to be a part of Applied Mathematics. A significant change occurred at the beginning of the twentieth century when Continuum Mechanics acquired special significance. In the middle of that century, Stephen Timoshenko published the History of Strength of Materials with a brief Account of the History of Theory of Elasticity and Theory of Structures (1953). Although this historical essay was extremely informative, Clifford Truesdell was the first to begin in the 1950s broad and profound investigations into the History of Continuum Mechanics. First of all, his work is distinguished by a thorough knowledge of the sources. He was perhaps the only famous, creatively working scientist in Rational Mechanics who was professionally engaged in the History of Mechanics as well, and he can be rightly considered the founder of its modern trends. The history of structural mechanics was treated in the spirit of Truesdellian fundamentals later by Edoardo Benvenuto in his numerous studies and, particularly, in An Introduction to the History of Struccural Mechanics (1991).KeywordsTwentieth CenturyNineteenth CenturyElasticity TheoryRational MechanicIntroductory ArticleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Recently we have modified the theory of Nobel Prize winners Modigliani and Miller (MM theory), which is a perpetuity limit case of the general theory of capital cost and capital structure—Brusov–Filatova–Orekhova theory (BFO theory), into two ways: we apply it for rating methodologies needs and later we generalized it for the case of advance payments of tax on profit, which is widely used in practice (MMM theory) (Brusov et al. (Mathematics 9(11):1286 (WoS Q1), 2021b). In current chapter, we use the Modified Modigliani–Miller theory (MMM theory) and apply it for rating methodologies needs. The financial “ratios” (main rating parameters) were introduced into MMM theory. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate in financial flows discounting in rating methodologies, on coverage and leverage ratios, is analyzed. Obtained results will help improve the existing rating methodologies. During a couple of years, we have suggested a new approach to rating methodology of nonfinancial issuers, as well for project rating (for long-term projects as well as for projects of arbitrary duration) [Brusov et al. (Modern corporate finance, investments, taxation and ratings. Springer Nature, 2018a; Modern corporate finance and investments. Knorus Publishing House, 2018b; J Rev Glob Econ 7:37–62, 2018c; J Rev Glob Econ 7:88–103, 2018d; J Rev Glob Econ 7:63–87, 2018e; J Rev Glob Econ 7:104–122, 2018f; J Rev Glob Econ 7:360–376, 2018g; J Rev Glob Econ 8:437–448, 2019; J Rev Glob Econ 9:257–267, 2020c); Brusov (J Rev Glob Econ 7:i–vi, 2018a; J Rev Glob Econ 7:i–v, 2018b); Filatova et al. (J Rev Glob Econ 7:645–661, 2018). The key factors of a new approach are: (1) The adequate use of discounting of financial flows virtually not used in existing rating methodologies, (2) The incorporation of rating parameters (financial “ratios”) into the modern theory of capital structure (Brusov–Filatova–Orekhova (BFO) theory) [Brusov et al. 2011, 2014, (Modern corporate finance, investments and taxation. Springer International, 2015; Modern corporate finance, investments, taxation and ratings. Springer Nature, 2018a; Modern corporate finance and investments. Knorus Publishing House, 2018b; J Rev Glob Econ 7:37–62, 2018c; J Rev Glob Econ 7:88–103, 2018d; J Rev Glob Econ 7:63–87, 2018e; J Rev Glob Econ 7:104–122, 2018f; J Rev Glob Econ 7:360–376, 2018g; J Rev Glob Econ 8:437–448, 2019); Brusov (J Rev Glob Econ 7:i–vi, 2018a; J Rev Glob Econ 7:i–v, 2018b); Filatova et al. (J Rev Glob Econ 7:645–661, 2018)] and into its perpetuity limit.Recently we have generalized the Modigliani and Miller theory for a more realistic method of payments of tax on profit payments: for the case of advance payments of tax on profit, which is widely used in practice [Brusov et al. (J Rev Glob Econ 9:282–292, 2020b)]. Modigliani–Miller theory accounts these tax payments as annuity-immediate, while in practice, these payments are making in advance and thus should be accounting as annuity-due. We have shown that this generalization leads to some important consequences, which change seriously all the main statements by Modigliani and Miller.In current chapter, we use the modified Modigliani–Miller theory (MMM theory) and apply it for rating methodologies needs. A serious modification of MMM theory in order to use it in rating procedure has been required. The financial “ratios” (main rating parameters) were introduced into MMM theory. The necessity of an appropriate use of financial flows discounting in rating methodologies is discussed. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate, on coverage and leverage ratios is analyzed.Obtained results make it possible to use the power of this theory in the rating and create a new base for rating methodologies, by other words, this allows to develop a new approach to methodology of rating, requiring a serious modification of existing rating methodologies.

Recently we have modified the theory of Nobel Prize winners Modigliani and Miller (MM theory), which is a perpetuity limit case of the general theory of capital cost and capital structure—Brusov–Filatova–Orekhova theory (BFO theory), into two ways: we apply it for rating methodology needs and later we generalized it for the case of advance payments of tax on profit, which is widely used in practice (MMM theory). In the current chapter, we use the modified Modigliani–Miller theory (MMM theory) and apply it for rating methodology needs. The financial “ratios” (main rating parameters) were introduced into MMM theory. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate in financial flows discounting in rating methodologies, on coverage and leverage ratios is analyzed. Obtained results will help improve the existing rating methodologies. A couple of years ago, we have suggested a new approach to rating methodology of non-financial issuers, as well as for project rating (for long-term projects as well as for projects of arbitrary duration) (Brusov et al. Modern corporate finance, investments, taxation and ratings. Springer, 2018a; J Rev Glob Econ 7:37–62, 2018b; J Rev Glob Econ 7:88–103, 2018c; J Rev Glob Econ 7:63–829, 2018d; J Rev Glob Econ 7:104–122, 2018e; J Rev Glob Econ 7:360–376, 2018f; Modern corporate finance and investments, monograph. Knorus Publishing House, 2018g; J Rev Glob Econ 8:437–448, 2019; J Rev Glob Econ 9:257–268, 2020b). The key factors of a new approach are: (1) The adequate use of discounting of financial flows virtually not used in existing rating methodologies, (2) The incorporation of rating parameters (financial “ratios”) into the modern theory of capital structure (Brusov–Filatova–Orekhova (BFO) theory) (Brusov et al. 2011, 2014, 2015; Modern corporate finance, investments, taxation and ratings. Springer, 2018a; J Rev Glob Econ 7:37–62, 2018b; J Rev Glob Econ 7:88–103, 2018c; J Rev Glob Econ 7:63–829, 2018d; J Rev Glob Econ 7:104–122, 2018e; J Rev Glob Econ 7:360–376, 2018f; Modern corporate finance and investments, monograph. Knorus Publishing House, 2018g; J Rev Glob Econ 8:437–448, 2019) and into its perpetuity limit.

1R40. Microcontinuum Field Theories II: Fluent Media. - AC Eringen (15 Red Tail Dr, Littleton CO 80126-5001). Springer-Verlag, New York. 2001. 340 pp. ISBN 0-387-98969-2. $149.00.Reviewed by ME Ryan (Dept of Chem Eng, SUNY, 505 Furnas Hall, Buffalo NY 14260).The book, Microcontinuum Field Theories II. Fluent Media, is a mathematically-based exposition of the continuum mechanics of structured fluid media, such as anisotropic fluids, liquid crystals, suspensions, etc. Volume II builds upon and extends the theoretical foundation of the subject developed in the first volume, Microcontinuum Field Theories I. Foundations and Solids. The development in Volume II extends the classical field theories of elasticity, fluid dynamics, and electromagnetism to continua constituted of material points or internal structures having directors that may be deformable as well as orientable. The book is largely based on the author’s contribution to the subject over the past few decades. The book is a reference source that assumes the reader has sufficient background in the subject to follow the derivation and application of the field equations to a variety of different situations. The book is intended for research scientists and graduate students in the disciplines of mathematics, physics, engineering, and related fields. This volume comprises nine chapters (Chs 9-17) which can be divided into four parts: I) Theory of micropolar fluids (Chs 9-11), II) Liquid crystals (Chs 12-14), III) Microstretch fluids (Chs 15-16), and IV) Micromorphic fluids (Ch 17). Electromag-netic interactions are also incorporated into the first three parts. Chapter 9 introduces basic measures of strain and rotation, constitutive equations, and the basic field equations. Analyses of pipe flow, flow in an orthogonal rheometer, lubrication in a journal bearing, Stokes flow around a sphere, stagnation flow, thermal instability, and boundary layer flow are provided. Mixed convection in vertical flow and a discussion of turbulence are also provided within the context of micropolar fluid dynamics. Chapter 10 develops the constitutive and field equations of magneto-hydrodynamics. Analyses of channel flow, shear flow, and thermal stability are presented. Chapter 11 addresses anisotropic fluids and suspensions. The basic theory is developed and applied to channel and couette flows.Chapter 12 gives a detailed presentation of the kinematics, balance laws, constitutive equations, and field equations of liquid crystals. The electrodynamics of liquid crystals is presented in Chapter 13. A variety of problems involving liquid crystals are analyzed in Chapter 14 including shear flow and heat conduction in a channel. The kinematics, constitutive equations, and field equations relating to microstretch fluids are provided in Chapter 15. Applications to the theory of acoustic waves in bubbly fluids and blood flow in small arteries are presented. Chapter 16 presents a continuum theory of the electrodynamics of polymeric liquid crystals. The last chapter addresses the subject of micromorphic fluids with applications of the theory to channel flow, the rheological behavior of blood in arteries, and the behavior of deformable suspensions. A few problems are briefly stated at the end of each chapter. A list of references, a subject index, and a list of errata for Volume I are provided at the end of the book. The treatment provided in this monograph will serve as a useful reference for the academician or research scientist interested in the fundamental theoretical development of this subject.

Geometric Continuum Mechanics ( GCM) is a new formulation of Continuum Mechanics ( CM) based on the requirement of Geometric Naturality ( GN). According to GN, in introducing basic notions, governing principles and constitutive relations, the sole geometric entities of space-time to be involved are the metric field and the motion along the trajectory. The additional requirement that the theory should be applicable to bodies of any dimensionality, leads to the formulation of the Geometric Paradigm ( GP) stating that push-pull transformations are the natural comparison tools for material fields. This basic rule implies that rates of material tensors are Lie-derivatives and not derivatives by parallel transport. The impact of the GP on the present state of affairs in CM is decisive in resolving questions still debated in literature and in clarifying theoretical and computational issues. As a consequence, the notion of Material Frame Indifference ( MFI) is corrected to the new Constitutive Frame Invariance ( CFI) and reasons are adduced for the rejection of chain decompositions of finite elasto-plastic strains. Geometrically consistent notions of Rate Elasticity ( RE) and Rate Elasto-Visco-Plasticity ( REVP) are formulated and consistent relevant computational methods are designed.

Using geologic structures to constrain constitutive laws not accessible in the laboratory

The starting point of every theory of continuum materials is the choice of the set of independent variables defining the configuration of a material. The aim is that of determining the configuration of the material when it is subjected to given external actions. Both general principles and constitutive equations are exploited. I t is assumed not only that the constitutive equations of the material can, at least in principle, be experimentally determined, but also that the constitutive quantities arc sufficient to characterize the material for the purposes of the theory. Often, no relation between the constitutive quantities is introduced at the outset: they are, therefore, to be considered independent of each other. Consider the case of homogeneous isotropic elastic materials. When thermal effects are taken into account, the set of independent variables is composed of the deformation gradient F, the absolute temperature 0 and the temperature gradient g = grad 0. The constitutive quantities are the stress tensor T, the specific internal energy E, the specific entropy V and the heating-flux vector q. The relevant constitutive equations