A New Representation Formula for the Logarithmic Corotational Derivative—A Case Study in Application of Commutator Based Functional Calculus

This paper is devoted to a systematic study of local universal relations in continuum mechanics. We show that it is possible to determine the complete set of independent universal relations whose characterization is obtained by linear universal rules. A historical review of the literature on the topic and various significant examples are given.

In this Lecture, we are concerned with the range of applicability of the balance relations in continuum mechanics. These relations have been already introduced in Lecture 1 of the present book as an important foundation of the dynamics and control of structures and machines, Belyaev (2004). In the present Lecture, we start with the general form of the relations of balance, which we afterwards specialise to the equations of balance of mass, momentum and energy (the first law of thermodynamics) and to the balance relation of entropy (the second law of thermodynamics), the latter in the form of the Clausius-Duhem inequality. We also discuss the frequently used equations of balance of kinetic energy (the law of power), and we point out the consequences of inserting the latter into the first and second law of thermodynamics, leading to the balance relations for the internal energy and to the Clausius-Planck inequality, respectively. We lay special emphasis on reviewing the continuity conditions that must be satisfied in order that the global and local forms of the relations of balance do hold. When a surface of discontinuity, a so-called singular surface, is present, across which some entity shows different values when approaching from the two sides of the surface, jump relations are needed in order to connect the local forms of the relations of balance at the two sides. We summarise and extend a recent formulation, which allows to connect the local equations of balance of mass, momentum, energy, kinetic energy and internal energy in a consistent manner, Irschik (2003). In the latter reference, it has been shown that surface growth terms must be introduced for the sake of consistency, and relations between these surface growth terms have been derived. In the present Lecture, these results are extended with respect to the second law of thermodynamics, and with respect to the resolution of a contradictory result in the literature on the jump relations of energy and of internal energy.

In this paper, volume-average relations related to the multilevel modelling process in continuum mechanics are analysed and the concept of average consistency is investigated both analytically and numerically. These volume averages are used in the computational homogenization technique, where a transition of the mechanical properties from the local, microscopic, to the global, macroscopic, length scale is obtained. The representative volume element (RVE) is used as a reference placement and the solution, in terms of volume-averaged stress, will depend on which boundary conditions are chosen for the RVE. Three types of boundary conditions – periodic, affine and anti-periodic – are analysed with respect to the average consistence for the kinematical and stress relations used in continuum mechanics. The inconsistence is quantified by introducing the inconsistence ratio. It is shown analytically that some average stress relations are fulfilled, assuming the periodic boundary condition and anti-periodic traction vector, whereas the average relations connected to the deformation are in general not average consistent. The inconsistence is investigated in a plane model using the finite element technique. The numerical investigation has shown that the inconsistence ratios related to the deformation are also average consistent in the examples considered.

It is widely accepted that numerous cell types respond to mechanical stimuli, yet there is no general agreement as to whether particular cells respond directly to stress, strain, strain-rate, strain-energy, or other mechanical quantities. By recalling the definitions of the mathematical (not physical) concepts of stress and strain, it is suggested herein that cells cannot respond directly to these continuum metrics or to quantities derived from them--mechanistic models will need to be based on more fundamental quantities, as, for example, inter-atomic forces or conformational changes of the appropriate molecules. Nonetheless, the concepts of stress and strain should continue to play an important role in mechanobiology, both in the identification of empirical correlations and in the development of phenomenological constitutive models, each of which can contribute to our basic understanding as well as help in the design of future experiments and some clinical interventions. It is important to remember, therefore, that empirical correlations and most constitutive relations in continuum mechanics do not seek to model the actual physics--rather, their utility is in their predictive capability, which is often achieved via different relations in terms of different metrics for the same material under different conditions. Hence, with regard to quantifying cellular responses to mechanical stimuli, we must delineate between the identification of fundamental mechanisms and the formulation of phenomenological correlations, the latter of which only requires convenient metrics that need not be unique or physical.

A mathematical model was developed to explain the anomalous penetrant diffusion behavior in glassy polymers. The model equations were derived by using the linear irreversible thermodynamics theory and the kinematic relations in continuum mechanics, showing the coupling between the polymer mechanical behavior and penetrant transport. The Maxwell model was used as the stress–strain constitutive equation, from which the polymer relaxation time was defined. An integral sorption Deborah number was proposed as the ratio of the characteristic relaxation time in the glassy region to the characteristic diffusion time in the swollen region. With this definition, an integral sorption process was characterized by a single Deborah number and the controlling mechanism was identified in terms of the value of the Deborah number. The model equations were two coupled nonlinear differential equations. A finite difference method was developed for solving the model equations. Numerical simulation of integral sorption of penetrants in glassy polymers was performed. The simulation results show that (1) the present model can predict Case II transport behavior as well as the transition from Case II to Fickian diffusion and (2) the integral sorption Deborah number is a major parameter affecting the transition. © 1993 John Wiley & Sons, Inc.

A basic scheme of establishing experiments to find three material functions of tensor nonlinear constitutive relations in continuum mechanics is described. These material functions depend on the three invariants of a stress state. It is proposed to use long hollow cylindrical specimens suitable to implement any combination of the following realizable stress states: uniaxial tension, torsion, longitudinal shear, and uniform compression.

2 - Kinematics of deformation

On universal relations in continuum mechanics: A discussion centred on shearing motions

Walter Noll is personally acknowledged to have been an inspirational teacher and an exemplar in his analysis of fundamental concepts and relations in Continuum Mechanics. Attention is drawn to unfounded criticisms of two aspects of his work and how considerations of local spatial and temporal averaging of microscopic behaviour can complement and inform the continuum viewpoint. Remarks on such considerations briefly summarise the results of published works which both utilised Noll’s direct notation and attempted to emulate his precision and clarity.

The Schur--Frechet method of evaluating matrix functions consists of putting the matrix in upper triangular form, computing the scalar function values along the main diagonal, and then using the Frechet derivative of the function to evaluate the upper diagonals. This approach requires a reliable method of computing the Frechet derivative. For the logarithm this can be done by using repeated square roots and a hyperbolic tangent form of the logarithmic Frechet derivative. Pade approximations of the hyperbolic tangent lead to a Schur--Frechet algorithm for the logarithm that avoids problems associated with the standard inverse scaling and squaring method. Inverting the order of evaluation in the logarithmic Frechet derivative gives a method of evaluating the derivative of the exponential. The resulting Schur--Frechet algorithm for the exponential gives superior results compared to standard methods on a set of test problems from the literature.

Molecular-dynamic modeling of mechanical properties of free defect metal nanocrystals

The interaction between fluid and structure occurs in a wide range of engineering problems. The solution for such problems is based on the relations of continuum mechanics and is mostly solved with numerical methods. It is a computational challenge to solve such problems because of the complex geometries, intricate physics of fluids, and complicated fluid‐structure interactions. The way in which the interaction between fluid and solid is described gives the largest opportunity for reducing the computational effort. One possibility for reducing the computational effort of fluid‐structure simulations is the use of one‐way coupled simulations. In this paper, different problems are investigated with one‐way and two‐way coupled methods. After an explanation of the solution strategy for both models, a closer look at the differences between these methods will be provided, and it will be shown under what conditions a one‐way coupling solution gives plausible results.

An algorithm for obtaining the stiffness matrix of a mixed finite element in the form of a quadrangular fragment of the middle surface of a thin-walled structure is presented. The following are accepted as nodal unknowns: kinematic quantities (displacement increments and their first derivatives) and deformation quantities (strain increments and curvature increments of the middle surface). To approximate the kinematic quantities of an internal point of a finite element, bicubic shape functions with Hermite polynomials of the third degree were accepted. Strain increments and curvature increments were approximated through nodal unknowns by bilinear shape functions. Physical nonlinearity is implemented based on the theory of plastic flow. The components of the elastic strain increment tensor were determined through the components of the stress increment tensor in a curvilinear coordinate system based on the relations of continuum mechanics, taking into account the Kirchhoff – Love hypothesis. The components of the plastic strain increment tensor are obtained in a curvilinear coordinate system based on the hypothesis of the plastic flow theory about their proportionality to the components of the stress deviator with a straight normal to the middle surface. To obtain the stiffness matrix of the developed finite element, a nonlinear functional was used based on the equality of possible and actual works of specified loads and arising internal forces at the loading step, with the replacement of the actual work of internal forces by the difference between the total and additional work of internal forces at the loading step. After substituting the approximating expressions into the nonlinear mixed functional and performing minimization over the nodal unknowns, two systems of algebraic equations are obtained with respect to the sought nodal unknowns. Based on the obtained systems of algebraic equations, a 36x36 finite element stiffness matrix is obtained. The structure stiffness matrix is formed in the usual way for FEM. The efficiency of the developed finite element is shown in a numerical example.

This work deals with fluid-structure interaction (FSI), one of the emerging areas of numerical simulation and calculation. This research shows a numerical study investigating heat transfer enhancement and fluid-structure interaction in a circular finned tube by using alumina nanofluid as a working fluid with a typical twisted tape that has a twisting ratio of 1.85. The studied nanofluid volumes of fraction are φ=0, 3, 5 % under conditions of laminar and turbulent flow. The solution for such problems is based on the relations of continuum mechanics and is mostly done with numerical methods. FSI occurs when the flow of fluid influences the properties of a structure or vice versa. It is a computational challenge to deal with such problems due to complexity in defining the geometries, nature of the interaction between a fluid and solid, intricate physics of fluids and requirements of computational resources. CFD investigations were made based on the numerical finite volume techniques to solve the governing three-dimensional partial differential equations to get the influence of inserted twisted tape and concentration of nanofluid on heat transfer enhancement, friction loss, average Nusselt number, velocity profile, thermal performance factor characteristics, and two-way interaction in a circular tube at laminar and turbulent flow. The governing continuity, momentum and energy transfer equations are solved using Ansys Fluent and Transient Structural. The simulation results show that the deformations of two-way coupling fluctuate from side to side, with 0.004 mm, as maximum amplitude, located at the typical twisted tape center. Heat transfer dissipation improved by adding fins and as Reynolds numbers increase the heat transfer behavior increases.

A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this extension to logics is strongly helped if we submerge the elementary logical calculus in a matrix-vector formalism that naturally includes a kind of fuzzy-logic. In this way, guided by the laws of matrix algebra, we can construct compact representations for the derivatives and the integrals of logical functions. Inside this semantic-algebraic calculus, we obtain expressions for the derivatives of some of the basic logical operations and show the general way to obtain the derivatives of any well-formed formula of propositional calculus. We show that some of the basic tautologies (Excluded middle, Modus ponens, Hypothetical syllogism) are members of a kind of hierarchical system linked by the differentiation algorithm. In addition using the logical derivatives we show that relatively complex formulas can collapse in simple expressions that reveal clearly their hidden logical meaning. The search for the antiderivatives produces naturally an integral calculus. Within this logical formalism an indefinite integral can always be found for any logical expression. Moreover, particular integrals can be constructed based on detachment properties that lead to logical expressions of growing complexity. We show that these particular integrals have some similarities with the 'generalizing deduction' procedures investigated by Lukasiewicz.