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Results achieved by the authors in the course of research activity on continuum mechanics and electrodynamics (CME) during the past twenty years are illustrated, revised and discussed. Adoption of a geometric approach leads to renewal of concepts and methods of classical CME and to formulation in Euclid (3+1)D ambient spacetime wherein innovation, clarity and depth of a geometric treatment naturally emerge. The dissemination of novel concepts and methods in CME is not delayable, with critical revisitation of problematic notions, analyses and results still currently on the scene. Material frame indifference, equilibrium in a reference configuration, extremality principles in Dynamics, finite elasticity, chain decomposition of finite strain in elasto-thermo-plasticity, variance of electro-magnetic induction laws under frame changes, action on electric charges moving in a magnetic field, are under the spotlight of innovation and advancement. Fostering basic knowledge of Differential Geometry and application of geometric notions and methods contribute effective tools in formulating meaningful rules, amending misstatements and dimming debates based on vague affirmations. [ABSTRACT FROM AUTHOR]

The contribution at hand focuses on the introduction of a novel approach to model biological growth. The proposed formulations are chosen to represent plant like structures. Therefore, thermomechanically open systems are considered. The balance laws are presented for such systems. Furthermore, the proposed formulations are coupled with an adaptive meshing framework. Therefore, a so-called structural generator is presented and utilized in this work. Since no growth formulations within the framework of continuum mechanics exist so far for plant like systems, a novel set of constitutive equations is shown. The newly described principles are the phototropism and graviotropism. In the numerical examples, it is shown that the proposed formulation yields physically meaningful results. The combination of different growth principles results in plausible interactions of the aforementioned principles. Furthermore, results of numerical simulations are shown, which represent the growth process of plant like biological structures. [ABSTRACT FROM AUTHOR]

Capturing a shear band in a metallic glass during its propagation experimentally is very challenging. Shear bands are very narrow but extend rapidly over macroscopic distances, therefore, characterization of large areas at high magnification and high speed is required. Here we show how to control the shear bands in a pre-structured thin film metallic glass in order to directly measure local strains during initiation, propagation, or arrest events. Based on the experimental observations, a model describing the shear banding phenomenon purely within the frameworks of continuum mechanics is formulated. We claim that metallic glasses exhibit an elastic limit of about 5% which must be exceeded locally either at a stress concentrator to initiate a shear banding event, or at the tip of a shear band during its propagation. The model can successfully connect micro- and macroscopic plasticity of metallic glasses and suggests an alternative interpretation of controversial experimental observations. The mechanisms of shear band initiation and propagation in amorphous metallic alloys is an open question of materials science. Here, the authors measure local strains around propagating shear bands and propose a model describing the plasticity of metallic glasses at different length scales. [ABSTRACT FROM AUTHOR]

A multidomain macro-hybrid mixed variational subpotential approach of models in continuum mechanics is developed. Variational principles of mass conservation, balance momentum and energy, as well as material constitutive relations, are formulated as subpotential subdifferential evolution systems. Under the basis of variational divergence formulae and subdifferential generalized boundary conditions, variational mixed initial/boundary-value inclusion problems are established in reflexive Banach trajectory real functional frameworks. Multidomain motion velocity, mass conservation density, material constitutive stress, deformation porosity as well as balance internal energy evolution coupled variational mechanical systems are achieved. As a distinctive modeling feature, multidomain synchronizing transmission constraints are implemented in terms of dual Lagrangian internal boundary subpotential subdifferential inclusions. Lastly, evolution internal variational approximations, in general, for nonconforming macro-hybrid mixed finite element spatial discretizations as well as semi-implicit time marching fully discrete schemes are briefly discussed. [ABSTRACT FROM AUTHOR]

Computational multiscale methods are highly sophisticated numerical approaches to predict the constitutive response of heterogeneous materials from their underlying microstructures. However, the quality of the prediction intrinsically relies on an accurate representation of the microscale morphology and its individual constituents, which makes these formulations computationally demanding. Against this background, the applicability of an adaptive wavelet-based collocation approach is studied in this contribution. It is shown that the Hill–Mandel energy equivalence condition can naturally be accounted for in the wavelet basis, (discrete) wavelet-based scale-bridging relations are derived, and a wavelet-based mapping algorithm for internal variables is proposed. The characteristic properties of the formulation are then discussed by an in-depth analysis of elementary one-dimensional problems in multiscale mechanics. In particular, the microscale fields and their macroscopic analogues are studied for microstructures that feature material interfaces and material interphases. Analytical solutions are provided to assess the accuracy of the simulation results. [ABSTRACT FROM AUTHOR]

This paper presents a system of cyclic tensor algebra for operations involving fourth-order tensors. The advantages are that the system is objectively and consistently defined in three ways that each fall into one of three universal classes. Operators within a given class are called conjugate operators such that many familiar and fundamental identities of scalars and second-order tensors are maintained in fourth order; this provides greater insight along with anthropological and pedagogical advantages over current systems, while also revealing new identities and solutions. The relationship between operators of a different class is such that a property of cyclic symmetry arises whereby mixed-class product operators can be cycled around without invalidating an equation. In defining this system, we have considered the following: preservation from identities in zeroth- (scalar) and second-order tensor identities to fourth-order tensor identities; possible permutations of definitions and subsequent logical restrictions; the visual notational consistency throughout the system; and maintenance to legacy definitions and operator symbols. Additionally, we present many new and useful algebraic identities and provide a comparison to some selected contemporary systems used in the literature. We also provide, to complete at least a basic exposition of our proposed system, a set of identities for matrix-equivalent operations, which facilitate programming for numerical computing. This article is designed to be used as a reference work for anyone choosing to adopt this system of tensor operations in continuum mechanics theory involving fourth-order tensors. [ABSTRACT FROM AUTHOR]

The main aim of the present work is to investigate the role of the Maxwell stress tensor in the study of active materials. Despite the importance of this tensor in modeling mechatronic devices used in sophisticated applications, its non–symmetry still generates controversies in the literature, probably because classical continuum mechanics assumes a symmetric Cauchy stress, although the sum of Cauchy and Maxwell stresses is non–symmetric. In the framework of generalised continuum mechanics–a more advanced formalism than the classical one–, each material point has an associated microstructure so that the micro–rotations of the electric/magnetic dipoles present in real active materials may be simulated. To this end, a modified total stress formulation, including an angular momentum balance, is developed and implemented into a finite element research code using a complex–step formulation. It is concluded that generalised mechanics allows for incorporating both symmetric and non–symmetric contributions of the Maxwell tensor. Consequently, the rotations generated by the electromagnetic field may be analysed. The influence of the complete Maxwell tensor in a magnetostrictive actuator is studied by several magneto–mechanical numerical experiments of a Terfenol–D rod encircled by air, and several conclusions are highlighted. [ABSTRACT FROM AUTHOR]

Based on numerical simulation within the framework of a two-dimensional axisymmetric problem of continuum mechanics, an analysis has been made of penetration into a semi-infinite barrier by dummy strikers with diameters of 14 and 18 mm and an initial velocity of 1200 m/s with and without account for the effects of the detachable sabot used for accelerating the dummies in the barrel of a ballistic installation. In simulating the penetration by dummy strikers in a sabot, a reduction was registered in the depth of penetration by a little over 10% with increase in the maximum overload compared to the case of penetration in the absence of a sabot. A cause has been identified for the sabot′s negative effects on dummy strikers′ penetration, lying in an increase of tangential stresses acting on the surface of the dummies′ contact with the adjacent layer of loosened concrete due to precompression of this layer by the sabot until the sabot′s complete detachment from the dummy striker. [ABSTRACT FROM AUTHOR]

This brief note is a personal recollection of the human and scientific traits of a great figure of the twentieth century's continuum mechanics. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the development of the method for searching an approximate solution of non-stationary problems on a variable interval. This method was first proposed by L.I. Slepyan. Here, the field of its applicability has been expanded to systems of equations, including equations of both hyperbolic and parabolic type. We describe the procedure of such approach in detail on a number of classical partial differential equations and compare the obtained results with exact analytical solution. Also, we consider a dynamic non-coupled thermoelastic problem. The method of expansion on a variable interval allows us to estimate the material response to a thermal disturbance at a large distance from the source. [ABSTRACT FROM AUTHOR]

The outbreak of COVID-19 in 2020 has led to a surge in interest in the research of the mathematical modeling of epidemics. Many of the introduced models are so-called compartmental models, in which the total quantities characterizing a certain system may be decomposed into two (or more) species that are distributed into two (or more) homogeneous units called compartments. We propose herein a formulation of compartmental models based on partial differential equations (PDEs) based on concepts familiar to continuum mechanics, interpreting such models in terms of fundamental equations of balance and compatibility, joined by a constitutive relation. We believe that such an interpretation may be useful to aid understanding and interdisciplinary collaboration. We then proceed to focus on a compartmental PDE model of COVID-19 within the newly-introduced framework, beginning with a detailed derivation and explanation. We then analyze the model mathematically, presenting several results concerning its stability and sensitivity to different parameters. We conclude with a series of numerical simulations to support our findings. [ABSTRACT FROM AUTHOR]

Constitutive equations of continuum mechanics of the solid phase of anisotropic material is focused in the paper. First, a synoptic one-dimensional Maxwell model is explored, subjected to arbitrary deformation load. The explicit form is derived for stress on strain dependence. Further, the analogous explicit constitutive equation is taken in three spatial dimensions and treated mathematically. Later on, a simply supported straight concrete beam reinforced by the steel fibres is taken as an investigated domain. The reinforcement is considered and dealt as scattered within the beam. Material characteristics are determined in line with the theory of the reinforcement. Sinusoidal load is taken as the action, stress reaction function is observed. By exploitation of the Fourier transform within the stress-strain relation analysis, both time and frequency interpretation of the constitutive relation can be performed. [ABSTRACT FROM AUTHOR]

We revisit the question of finding the most general representation of vector-valued functions such that f (Q A Q T , Q u) = Q f (A , u) for all rotation tensors Q , all symmetric tensors A , and all vectors u . Seven scalar invariants and three vectors are identified that form the basis for the functional dependence in its simplest form. Six of the seven scalar invariants are required in the case of isotropy with respect to the full orthogonal group. The results are relevant to hemitropic vector-valued functions in continuum mechanics, such as the thermoelastic heat flux. [ABSTRACT FROM AUTHOR]

A general structure for space-time that preserves the classical notion of absolute time, and to which we refer as Proto-Galilean space-time, is presented. In this setting, space-time is modeled as a general fiber bundle over the time axis. The motion of a particle is a section of the fiber bundle. A recent framework for continuum mechanics, where configurations are represented by sections of a fiber bundle over the body manifold, is used to formulate the dynamics of a particle and the dynamics of a continuous body in a Proto-Galilean space-time. [ABSTRACT FROM AUTHOR]

Tests of rocket and space equipment objects in terrestrial conditions for shock resistance to the impact of meteoroids and space debris can be carried out using shaped charges. To substantiate the design parameters of shaped charges ensuring the formation of aluminum particles in a broad velocity range (from 2.5 to 16 km/s), numerical simulation has been conducted for the process of the said formation within the framework of a two-dimensional axisymmetric problem of continuum mechanics using ANSYS/AUTODYN, ERUDIT, and KOLDUN computer systems. In calculations, consideration has been given to shaped charges with a diameter from 20 to 100 mm with aluminum liners of various shape. It has been shown that the formation of particles with velocities close to the lower boundary of the considered range is ensured by gently sloping segmental degressive-thickness liners. For the formation of higher-velocity particles with velocities above 5 km/s, it is suggested that use be made of combined liners whose jet-forming portion has the shape of a hemisphere of constant thickness or the shape of a semi-ellipsoid or semisuperellipsoid of rotation. [ABSTRACT FROM AUTHOR]

In this paper, we investigate stability and optimization of an axially loaded nonlocal beam that is simply supported at one end and elastically restrained against rotation on the other. Nonlocal continuum mechanics is a theoretical framework that extends classical continuum mechanics to account for the influence of small length scales, especially in nanostructures. We analyze elastically buckling nanobeam based on Eringen's nonlocal elasticity theory. The Euler method of adjacent equilibrium configuration is used to derive the nonlinear governing equations. The critical axial force and postbuckling shape are obtained for the beam with the unit cross-sectional area. The shape of the nonlocal beam stable against buckling that has minimal volume is determined by using Pontryagin's maximum principle. The optimality conditions are derived. The cross-sectional area for optimally designed beam is found from the solution of a nonlinear boundary value problem. New numerical results are obtained. The first integral (Hamiltonian) is used to monitor the accuracy of the integration. The numerical analysis includes the influence of the characteristic parameter of the small length scale on the critical load, the postbuckling shape and the optimal shape of the analyzed beams. It is shown that there are the saving materials for optimally designed nanobeam in all numerical examples. [ABSTRACT FROM AUTHOR]

The study uses FEM to analyze the creep behavior of a rotating FGM disc made of Al-SiCp. The disc thickness and SiCp reinforcement in the disc are assumed to vary according to the exponential functions of the disc radius. The yielding and creep behavior of the disc material are described by Tresca's criterion and threshold stress-based law, respectively. The disc is assumed to rotate at constant angular velocity under free-free boundary conditions. The disc domain is discretized radially using 3-noded one-dimensional elements with quadratic shape functions. Galerkin's approach has been used to derive the expressions for stresses and strain rates in the disc. For numerical computation of the creep results, a MATLAB code has been developed corresponding to the FE formulation. The creep results estimated using the FE approach are noticed to be in excellent agreement with those obtained using the analytical technique (continuum mechanics approach), experiment results, and commercial FE software. The study reveals that with the increase in SiCp gradient in the disc, the elastic as well as creep components of the tangential stress (near the inner radius) and the radial stress increase significantly, though the tangential stress exhibits a sizable decrease toward the outer radius. The increase in SiCp gradient results in a significant reduction in the steady-state radial and tangential strain rates in the disc, besides resulting in a relatively more uniform distribution of the strain rates, which is desirable for reducing distortion in the disc. The study indicates that the rupture time of the FGM disc could be enhanced significantly by employing a steeper SiCp gradient in the rotating disc. [ABSTRACT FROM AUTHOR]

Background: In a previous work, the problem of identifying residual stresses through relaxation methods was demonstrated to be mathematically ill-posed. In practice, it means that the solution process is affected by a bias-variance tradeoff, where some theoretically uncomputable bias has to be introduced in order to obtain a solution with a manageable signal-to-noise ratio. Objective: As a consequence, an important question arises: how can the solution uncertainty be quantified if a part of it is inaccessible? Additional physical knowledge could—in theory—provide a characterization of bias, but this process is practically impossible with presently available techniques. Methods: A brief review of biases in established methods is provided, showing that ruling them out would require a piece of knowledge that is never available in practice. Then, the concept of average stresses over a distance is introduced, and it is shown that finding them generates a well-posed problem. A numerical example illustrates the theoretical discussion Results: Since finding average stresses is a well-posed problem, the bias-variance tradeoff disappears. The uncertainties of the results can be estimated with the usual methods, and exact confidence intervals can be obtained. Conclusions: On a broader scope, we argue that residual stresses and relaxation methods expose the limits of the concept of point-wise stress values, which instead works almost flawlessly when a natural unstressed state can be assumed, as in classical continuum mechanics (for instance, in the theory of elasticity). As a consequence, we are forced to focus on the effects of stress rather than on its point-wise evaluation. [ABSTRACT FROM AUTHOR]

A nonlinear phenomenological model of flexoelectricity in thermoelastic solids is presented within the frame of continuum mechanics and extended thermodynamics, incorporating the quasi-electrostatic approximation where the time derivative of the electric displacement vector can be neglected in Maxwell–Ampère's law. An expression for the energy flux is proposed, including many internal variables. An advantage of the proposed model is that it presents a unified approach to study several thermo-electromechanical couplings, including electrostriction, piezoelectricity, dependence of entropy and spontaneous polarization on flexoelectricity, among others. The model may be of interest in revealing the effects of such couplings on the properties of polarizable media, in particular ferroelectrics, when simple boundary conditions prevail. For a fully dynamic description of flexoelectricity, Hamilton's principle and the variational principle for external forces should be used instead. [ABSTRACT FROM AUTHOR]

This document is a foreword for a volume of the Journal of Elasticity dedicated to the memory of Jerald LaVerne Ericksen. Ericksen made significant contributions to the field of mechanics of materials, particularly in the study of nonlinear material behavior. His research influenced various areas of continuum mechanics, including rheology, elasticity, viscoelasticity, and thermomechanics. The foreword also highlights Ericksen's personal qualities, such as his kindness, simplicity, and dedication to education. The volume includes portraits, essays, and research papers that showcase Ericksen's impact and legacy. The document concludes with acknowledgments and a note from the publishers. [Extracted from the article]

A new principle for constructing the theory of coupled dynamic electroelasticity of dielectrics that have piezoelectric and electrostrictive effects is expounded. This theory is based on the purely mechanical two-continuum description of the deformation of dielectrics as a mixture of positive and negative charges coupled into pairs neutral molecules or elementary cells provided that an elastic potential exists and the partial stresses are in linear quadratic relationship with the difference of displacements of charges. Based on the definition of the vector of polarization of an elementary dielectric macrovolume and the electric field generated by it, the equations of two-continuum mechanics are transformed into coupled dynamic equations for the macrodisplacements of neutral molecules and electric-field strength that describe the piezoelectric and electrostrictive effects. Maxwell’s equations follow from these equations as a particular case. [ABSTRACT FROM AUTHOR]

A hybrid local/non-local thermomechanical model is developed by the Morphing method to simulate cracking under different temperature increments. In the proposed hybrid model, peridynamics as a non-local model is applied to domains where fractures may occur and classical continuum mechanics as a local model is applied to the rest domains of the structure to reduce computational cost and impose boundary conditions. Two models are coupled by the Morphing method in coupling domains. The numerical method has been introduced to implement this model that enables crack initiation and propagation. The error of this hybrid model caused by temperature increment in the coupling domains is discussed by analyzing the ghost force in detail. Accuracy and validity of the proposed model have been proved by illustrating two-dimensional numerical examples with or without cracking. [ABSTRACT FROM AUTHOR]

This document is a translation of Jerald L. Ericksen's Introduction to the collection "Studies on Mechanics of Continua." Ericksen discusses his interest in the idea of continua with internal structure and the development of a theory for anisotropic fluids. He also explores the application of his theories to liquid crystals and the study of shells. Ericksen emphasizes the importance of considering different perspectives in the field of mechanics and expresses gratitude for the opportunity to publish this collection. [Extracted from the article]

Motivated by a design of a vertical axis wind turbine, we present a theory of dynamical similarity for mechanical systems consisting of interacting elastic solids, rigid bodies and incompressible fluids. Throughout, we focus on the geometrically nonlinear case. We approach the analysis by analyzing the equations of motion: we ask that a change of variables take these equations and mutual boundary conditions to themselves, while allowing a rescaling of space and time. While the disparity between the Eulerian and Lagrangian descriptions might seem to limit the possibilities, we find numerous cases that apparently have not been identified, especially for stiff nonlinear elastic materials (defined below). The results appear to be particularly adapted to structures made with origami design methods, where the tiles are allowed to deform isometrically. We collect the results in tables and discuss some particular numerical examples. [ABSTRACT FROM AUTHOR]

The computational efficiency has been largely hindering the application of the absolute nodal coordinate formulation (ANCF). To improve the computational efficiency of the absolute nodal coordinate formulation, the collocation strategy is employed to establish the ANCF beam element. A novel ANCF beam element based on multi-node collocation (ANCF_C) is proposed. Zero points of the second-order derivative of pth-order Legendre polynomial and the boundary points in the element domain are used as nodes to discretize the beam node-wisely. Accordingly, the (p - 1) th-order Lagrange interpolation is employed for the longitudinal displacement interpolation. The elastic force and stiffness matrix are deduced based on the enhanced continuum mechanics formulation (ECMF) to avoid the locking problem. By using p-point Lobatto quadrature for the numerical integration of the elastic force and the mass matrix, the quadrature points coincide with the discretized nodes in the element. The performance of the ANCF_C is verified by both static and dynamic examples. [ABSTRACT FROM AUTHOR]

Mechanical-diffusion coupling analysis at micro/nanotemporal and spatial scale has aroused great research interests with flourishing development of nanobattery system and fast rising of rapid charging technology, where the spatial nonlocal effects of mass transfer and elastic deformation as well as the influences of temporal nonlocal effects of mass transport (i.e., the phase laggings of diffusion flux vector and molar concentration gradient) will remarkably increase. In such cases, however, the accurate prediction of mechanical-diffusion responses is challenged: Firstly, the existing non-Fick diffusion–elasticity models are established by merely introducing mass diffusion model associated with the time rate of diffusion flux (i.e., phase laggings of diffusion flux); secondly, the spatial nonlocal effect of mass transfer is still not considered in the current on dual-phase-lag diffusion model. This work aims to develop a non-Fick diffusion–elasticity based on a new nonlocal dual-phase-lag diffusion model, which fully incorporates spatial and temporal nonlocal effects of mass transport. New constitutive and field equations are strictly derived via nonlocal continuum mechanics. To illustrate its application values, a one-dimensional isotropic homogeneous thin layer of finite thickness subjected to transient shock loadings of molar concentration is investigated. Dimensionless results are graphically presented to illustrate the effects of both nonlocal mass transfer and nonlocal elasticity on diffusive wave propagation and mechanical-diffusion responses. [ABSTRACT FROM AUTHOR]

Soft materials that can demonstrate on demand reconfigurability and changing compliance are highly sought after as actuator materials in many fields such as soft robotics and biotechnology. Whilst there are numerous proof of concept materials and devices, rigorous predictive models of deformation have not been well-established or widely adopted. In this paper, we discuss programming complex three-dimensional deformations of a soft intrinsically anisotropic material by controlling the orientation of the contractile units and/or direction of the applied electric field. Programming is achieved by patterning contractile units and/or selectively activating spatial regions. A new constitutive model is derived to describe the soft intrinsic anisotropy of soft materials. The model is developed within a continuum mechanics framework using an invariant-based formulation. Computational implementation allows us to simulate the complex three-dimensional shape response when activated by electric field. Several examples of the achievable Gauss-curved surfaces are demonstrated. Our computational analysis introduces a mechanics-based framework for design when considering soft morphing materials with intrinsic anisotropy, and is meant to inspire the development of new soft active materials. [ABSTRACT FROM AUTHOR]

Martensitic steels are widely used as a structural material in critical components of fossil fuel and nuclear power plants, such as boilers, pipes, and fittings. Martensitic steels are known to have a hierarchical microstructure that follows the Kurdjumov–Sachs (K–S) orientation relationship, where a prior austenite grain is composed of packets separated by high angle grain boundaries or packet boundaries, which are, in turn, divided into blocks or variants segregated by high angle grain boundaries called block boundaries. Blocks themselves are an agglomeration of laths divided by low angle grain boundaries named lath boundaries which have precipitates scattered on them. This work seeks to examine, using a couple dislocation dynamics—continuum mechanics approach called multiscale dislocation dynamics plasticity (MDDP), the interactions between dislocations and packet, block, lath boundaries, and precipitates under uniaxial tension loading and their effect on the mechanical response of the material. The simulations are conducted at a strain rate of 105 s−1 at room temperature. The main crystallographic features that arise during the deformation process were extracted and analyzed in terms of their contribution to the mechanical response of the material. The orientation relationship governing the microstructure of martensitic steels, namely, the K–S orientation relationship, was incorporated in MDDP in an effort to accurately capture the deformation behavior of the material in question. The strength of lath martensitic steel was analyzed as a function of the lath width, block size, and packet size to determine the appropriate effective grain size. [ABSTRACT FROM AUTHOR]

The decrease in apparent relative viscosity that occurs when blood is made to flow through a tube whose diameter is less than about 0.3 mm is a well-known and documented phenomenon in physiology, known as the Fåhræus-Lindqvist effect. However, since the historical work of Fåhræus and Lindqvist (Amer. J. Physiol. 96(3): pp. 562–568, 1931), the underlying physical mechanism has remained enigmatic. A widely accepted qualitative explanation was provided by Haynes (Amer. J. Physiol. 198, pp. 1193–1200, 1960) according to which blood flows in microvessels with a core-annulus structure, where the erythrocytes concentrate within a central core surrounded by a plasma layer. Although sustained by observations, this conjecture lacks a rigorous deduction from the basic principles of continuum dynamics. Moreover, relations aimed to reproduce the blood apparent relative viscosity, extensively used in micro-circulation, are all empirical and not derived from the analysis of the fluid mechanical phenomena involved. In this paper, we apply the recent results illustrated in Guadagni and Farina (Int. J. Nonlinear Mech. 126, p. 103587, 2020), with the purpose of showing that Haynes' conjecture, slightly corrected to make it more realistic, can be proved and can be used to reach a sound explanation of the Fåhræus-Lindqvist effect based on continuum mechanics. We propose a theoretical model for the blood apparent relative viscosity which is validated by matching not only the original experimental data reported by Fåhræus and Lindqvist (Amer. J. Physiol. 96(3), pp. 562–568, 1931), but also those provided by several subsequent authors. [ABSTRACT FROM AUTHOR]

In this work, the mass-sensing ability and the corresponding detection efficiency of a fully clamped single-walled carbon nanotube (SWCNT) are investigated through a free vibration analysis, by utilizing mainly a molecular mechanics (MM) formulation and secondarily a continuum mechanics (CM) analytical approximation. The MM method is based on representing the SWCNT as a three-dimensional (3D) finite element frame of point masses and linear springs, while the CM one is grounded on the Euler-Bernoulli beam theory. The overall effort is focused on obtaining detection maps, based on natural frequency data which are capable of leading to the straightforward identification of an unknown mass attached to the external surface of the candidate SWCNT sensor. For this reason, the SWCNT relative natural frequency shifts due to the mass addition, regarding specific modes of vibration and for a variety of mass magnitude and position combinations, are calculated beforehand. Then, the magnitude as well as the position of the added mass may be graphically found by superposing the arisen relative natural frequency shift variations in common contour diagrams. Some results from the open literature are utilized to confirm the predictive performance of the proposed MM method concerning the free vibration of pure SWCNTs, while the MM and CM established detection maps are set in contrast to examine whether a mass nanosensor may be satisfactorily treated in a continuum manner. [ABSTRACT FROM AUTHOR]

Hill's first-order lemma in continuum mechanics has been derived in the literature for quasi-static cases and for dynamic cases in the absence of body forces. In this manuscript, a generalized first-order Hill identity is first derived in an Eulerian dynamic description including body forces. Then, a second-order Hill identity using a Lagrangian description is obtained, involving the variation in time of the associated kinetic energy. Simple examples, that allow for analytical analysis, of one-dimensional elastic deformation of bars are considered that involve dynamics and body forces to illustrate the generalization of Hill's identities proposed here. [ABSTRACT FROM AUTHOR]

The dynamical interaction between solids and fluids is a subject of paramount importance in Mechanics with a wide range of applications to engineering problems. It is, however, still a challenging topic of theoretical investigation. With a view to case studies of dynamical behaviour of rockets, turbines, jets and sprinklers, we develop here a treatment that, in the full respect of the principle of conservation of mass and under suitable simplifying assumptions, leads to evaluate the thrusting force exerted by the fluid on the solid. The goal is reached by applying the Euler-d'Alembert law of continuum dynamics to the trajectory of a skeleton whose motion is an extension of the one of the solid. It is shown that the formulation in the context of continuum mechanics is essential to get a full understanding of the dynamical problem and for grasping the meaning and range of validity of the results. This is a distinctive feature from treatments in literature where particles or control windows with variable mass are considered. The statement of the von Buquoy-Meshchersky law as a governing principle in the dynamics of particles with variable mass, in substitution of Newton's second law, is critically addressed. Under the assumption of low mass and high momentum time rate, the formula for the thrusting force is validated as a simplified expression fulfilling Galilei's principle of relativity. [ABSTRACT FROM AUTHOR]

This article is intended to present an overview of dynamic analyses of single-/multi-layered graphene sheets (SLGSs/MLGSs) and nanoplates with combinations of simply supported, free, and clamped edge conditions embedded in an elastic medium using various two-dimensional (2D) nonlocal continuum mechanics-based plate theories. Based on Hamilton's principle incorporating Eringen's nonlocal constitutive relations, the authors derive strong formulations of assorted 2D nonlocal continuum mechanics-based plate theories for the free vibration analysis of embedded SLGSs/MLGSs, including the nonlocal classical plate theory, the nonlocal first-order shear deformation plate theory (SDPT), the nonlocal refined higher-order SDPT, the nonlocal sinusoidal SDPT, the nonlocal exponential SDPT, and the nonlocal hyperbolic SDPT. Navier-type solutions based on these 2D nonlocal plate theories for free vibration problems of simply supported, SLGSs/MLGSs and nanoplates embedded in an elastic medium are obtained. Articles examining various mechanical behaviors of SLGSs/MLGSs and nanoplates using different nonlocal plate theories available in the literature are tabulated into categories, including bending, free vibration, buckling, dynamic instability, wave propagation, geometrically nonlinear bending, geometrically nonlinear vibration, and forced vibration analyses. [ABSTRACT FROM AUTHOR]

Inspired by biology and engineered soft active material systems, we propose a new constitutive formulation for a soft material consisting of soft contractile fibers embedded in a soft matrix. The mathematical implementation of the model is based on a multi-field invariant formulation within a nonlinear continuum mechanics framework. The coupled constitutive formulation highlights a new electromechanical coupling term that describes the intrinsic (or active) anisotropy due to the contractile units. The model demonstrates the relative role that intrinsic anisotropy plays in the overall stress response. The resulting formulation could be used to design and inspire the development of new soft material systems that seek to replicate three dimensional biological motion. [ABSTRACT FROM AUTHOR]

The study of energy transport in granular systems can involve a number of different angles to view the problem; for example, one can propagate sinusoidal waves within the granular assembly, which makes the particles vibrate; besides the large wavelength low-amplitude elastic limit, this can be at very large frequencies and medium-large amplitudes, thereby posing the particles in perturbations of different modes, like resulting in cyclic shear, which can be translational and/or rotational as well as oblique collisions between the particles to occur. If these particles are naturally occurring grains, they will have a far from classical "elastic" response and their morphologies will be evolving during these perturbations. If a viscous fluid is added, then the dynamics of these perturbations and the way the energy is transferred among the particles may be substantially different. One may wish to see this problem even at a smaller scale, examining only two perturbating particles in contact, or allowing them to impact each other in the presence of a fluid. If you load the granular system in a cyclic mode, but this time at a very low frequency, some mechanisms will be altered, and the way the energy will be dissipated may also be expected to be altered, thereby the interpretations made from such analysis. Of course, a granular assembly is often part of a larger system that we are interested in to study by stability analysis, as e.g. internal erosion, or the dynamics of a submarine landslide involving an extraordinarily large span of particle sizes and morphologies. Taking as example research works in soil dynamics, the rate of stiffness decrease in a granular system, caused by the nonlinearity of that system, is proportional to the rate of energy dissipation increase as macroscopically measured in medium-frequency torsional shear dynamic excitation. However, if the excitation amplitude is reduced enough to lead to measurements of elastic stiffness, some small dissipation of energy might still be observed, which generally contradicts principles of classic continuum mechanics. These, and many others are interesting and exciting, though challenging areas of research in granular matter, in which scientists from a wide span of expertise are working to provide answers, and perhaps raise more questions about what is happening in a granular system. The topical collection "Energy transport and dissipation in granular systems" aimed to provide a forum bringing together scientists and engineers from different disciplines to answer some simple, though challenging questions about what the involved mechanisms of energy transport and dissipation in granular systems are, and extending these towards understanding, how micromechanical-based features influence the macroscopic behavior of larger-scale systems involving particles or powders. Finally, we could see very proudly that a total of 18 high-quality articles were contributed and published under this topical collection in Granular Matter, which attempted to examine systems at various scales, from the scale of two interacting particles in contact, to element-size numerical and laboratory samples, to the behavior of a submarine landslide, or a system which reaches a fluidization state. The outcome is very rich and diverse, and we can be very confident that these works will comprise a state-of-the-art for future research in granular matter. We had a diversity of contributions in terms of demographics, from institutes in the U.S.A, Canada, and Europe, to Middle-East and East-Asia all the way to Australia-based researchers. The contributions have different perspectives, from physics, mechanical-aerospace engineering, civil and environmental engineering. The diversity was also present in the scientific methodology, from purely analytical works to experimental-based studies at various scales, with novel laboratory setups developed and presented which allow the examination of energy transport and dissipation analysis in granular systems. The majority of these works considered a granular system in which gravitational forces are dominant, i.e., frictional materials, but we also had contributions on adhesive-type granular micro-systems. Below is a summary of major outcomes from these contributions which are divided in different sub-sections based on some common characteristics of the different contributing works. [ABSTRACT FROM AUTHOR]

Dilational materials, for which the angles between pairs of material fibers are preserved under deformations, are an important class of metamaterials. Although these materials are typically made by assembling discrete elemental building blocks in repeating patterns, continuum mechanics provides a powerful tool for exploring their macroscopic properties and response. We present an analysis of the constraint, the constitutive relation, and the equilibrium equations for homogeneous and isotropic dilational elastic material surfaces. We also describe the possibility of penalizing deviations from local area preservation to yield a framework for approximating isometric deformations of unstretchable elastic material surfaces. [ABSTRACT FROM AUTHOR]

After deriving a novel Principle of Stationary Correlated Action for the general bi-anisotropic case of size-dependent materials governed by consistent couple stress theory, a Ritz Spline Method is created for free vibration analysis. The underlying formulation is then specialized for isotropic materials requiring the specification of only a single couple stress material parameter that can be defined in terms of an effective intrinsic material length scale. The framework permits both two- and three-dimensional analysis using a multivariate B-spline basis that enables the flexibility to select a range of piecewise polynomial orders and higher levels of continuity, as required in couple stress theory. To validate the formulation and numerical implementation, two example problems involving a cantilever with rectangular cross section and a free-free cylindrical tube are studied in detail. Interestingly, one finds that while couple stress effects tend to stiffen the structures, the effect is not uniform. The natural frequencies of some modes are significant enhanced as the couple stress effective length scale increases, while other modes are nearly or completely unaffected. In addition, the important three-dimensional nature of several aspects of consistent couple stress continuum mechanics is clearly observed from these two basic problems of free vibration. [ABSTRACT FROM AUTHOR]

Among different hallmarks of cancer, understanding biomechanics of tumor growth and remodeling benefits the most from the theoretical framework of continuum mechanics. Tumor remodeling initiates when cancer cells seek new homeostasis in response to the microenvironmental stimuli. Cells within a growing tumor are capable to remodel their inter- and intra-connections and become more mobile to achieve a new homeostasis. This mobility enables the tumor to undergo large deformation. In this work, we studied the remodeling of homogeneous tumors, at their early stage of growth, in the context of continuum mechanics. We developed an evolution law for the remodeling-associated deformation which correlates the remodeling to a characteristic tensor of external stimuli. The asymmetric remodeling and the induced mechanical stresses were analyzed for different types of biochemical distributions. To experimentally investigate the model, we studied the remodeling of human glioblastoma (hGB) tumoroids in response to the gradient of nutrients. Using a tumoroid-on-a-chip platform, the degree of remodeling was estimated for the ellipsoidal tumoroids over time. It was observed that higher gradient of nutrients induces higher degree of ellipticity suggesting that the gradient of nutrient is a characteristic property of nutrient distribution that derives the remodeling. We also showed that remodeling gives rise to heterogeneity in cell distribution forming circumferentially aligned cells within the tumors. Compared to the existing studies on tumor growth, our work provides a biomechanical module that relates the remodeling to biochemical stimuli, and allows for large deformation. It also includes experimental component, a necessary but challenging step, that connects the theory and reality to evaluate the practicability of the model. [ABSTRACT FROM AUTHOR]

It is reported that there exist deformable media which display quantum effects just as quantum entities do. As such, each quantum entity usually treated as a point particle may be represented by a deformable medium, the dynamic behavior of which is prescribed by four dynamic state variables, including mass density, velocity, internal pressure, and intrinsic angular momentum. In conjunction with the finding of the characteristic equation characterizing the physical nature of such media, it is found that a complex field quantity may be introduced to uncover a perhaps unexpected correlation, i.e., the governing dynamic equations for such media may be exactly reduced to the Schrödinger equation, from which the closed-form solutions for all the four dynamic state variables can be obtained. It turns out that this complex field quantity is just the wavefunction in the Schrödinger equation. Moreover, the dynamic effects peculiar to spin are derivable as direct consequences. It appears that these results provide a missing link in quantum theory, in the sense of disclosing the physical origin and nature of both the wavefunction and the wave equation. Now, the inherent indeterminacy in quantum theory may be rendered irrelevant. The consequences are explained for certain long-standing fundamental issues. [ABSTRACT FROM AUTHOR]

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials. [ABSTRACT FROM AUTHOR]

The simulation of fracture in large-scale structures made of porous media remains a challenging task. Current techniques either assume a homogeneous model, disregarding the microstructure characteristics, or adopt a micro-mechanical model, which incurs an intractable computational cost due to its complex stochastic geometry and physical properties, as well as its nonlinear and multiscale features. In this study, we propose a multiscale analysis-based dual-variable-horizon peridynamics (PD) model to efficiently simulate macroscopic structural fracture. The influence of microstructures in porous media on macroscopic structural failure is represented by two PD parameters: the equivalent critical stretch and micro-modulus. The equivalent critical stretch is calculated using the microscale PD model, while the equivalent micro-modulus is obtained through the homogenization method and energy density equivalence between classical continuum mechanics and PD models. Numerical examples of porous media with various microstructures demonstrate the validity, accuracy, and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

The development of reliable miniaturized devices capable of performing sensing, actuation, and computing functions in a robust manner strongly depends on the mechanical performance of arrays of micromachines subjected to alternating stresses during operation. Consequently, to enhance the functionality and longevity of these micromachines, it is essential to accurately determine the structural behavior under both static and dynamic loadings. However, this requires knowledge of the stress distribution and variation due to texture anisotropy effects in critical micromachine regions and insight into the material behavior at micrometer length scales. Accordingly, the objective of this study was to develop an effective computational mechanics framework for performing stress analysis of micromachines at resonance. Dynamic finite element analysis of resonating micromachines was performed to elucidate the effect of material damping on the dynamic response. A Poisson–Voronoi diagram was incorporated in the critical region of finite element models of polysilicon micromachines to mimic the local inhomogeneity and anisotropy of the polycrystalline microstructure. A computationally effective static finite element analysis of the steady-state resonant response, validated by the results of the dynamic analysis, was used to obtain the steady-state stress distribution in the critical region of polysilicon micromachines with different textures. Moreover, a probabilistic stress analysis of hundreds of simulations was carried out to examine the stochastic nature of the grain geometry, size, and orientation and the beam width on the maximum equivalent von Mises stress in the critical micromachine region. The present study provides an efficient computational methodology, which integrates local texture into continuum mechanics modeling, for performing stress analysis of dynamic microdevices exhibiting different polycrystalline microstructures. [ABSTRACT FROM AUTHOR]

This work compares two different computational approaches to geometrically exact elastoplastic rods. The first approach applies an elastoplastic constitutive model in terms of stress resultants, i.e. forces and moments. It requires knowledge of the rod's elasticity and yield-criterion in terms of stress resultants. Furthermore a resultant-type hardening expression must be formulated. These are obtained by integrating elastoplastic stress and hardening measures from three-dimensional continuum mechanics over the rod's deformed cross-section, which is performed in an offline stage. The second approach applies an F E 2 approach as established in computational homogenization. Therein, the macro-scale describing the geometrically exact rod is coupled to the micro-scale, i.e., the cross-section of the rod. A novelty of the presented work is the determination of a hardening tensor for use in the stress resultant approach. The mechanical response of both approaches is first compared on the material point level, a single cross-section of a uniformly strained rod. Later, also the mechanical response and the deformation of finitely and non-uniformly strained rods are investigated. [ABSTRACT FROM AUTHOR]

The virtual power principle (VPP) of continuum mechanics states a celebrated variational equality between external and internal virtual powers for any virtual velocity field conforming with linear kinematic constraints. The topic is here addressed to investigate how the original ideas born in the early XIX century are modelled by modern formulations based on Functional Analysis and Differential Geometry. These notions are able to provide an effective mathematical context for proving existence of Lagrange multipliers associated with the constraint of rigidity on velocity fields. The VPP stands as privileged tool for giving to stress fields a consistent definition based on duality with conforming virtual stretching fields. By complementarity, the VPP generates a variational condition for integrability of stretching fields, with self-equilibrated stresses as test fields. Progress is got by the formulation of the rate virtual power principle (RVPP) by time derivation of the VPP along the motion, with internal virtual power integrated per unit mass. The basic distinction between spatial and material fields according to the geometric paradigm is prompted to replace the one previously adopted in the literature. The need for a non-redundant implicit formulation of the rigidity constraint is emphasised to contrast degeneracy. This logical demand avoids proliferation of multipliers, in the spirit of Ockham's Razor, a celebrated philosophical motto with multiform applications. The shining mathematical theory set out by Leonhard Euler, Jean-Baptiste Le Rond d'Alembert, Joseph Louis Lagrange, and Augustin Cauchy is in this respect a point of optimality. A geometric rate theory of elasticity meets the call for no-dissipation in push-closed elastic cycles, with non need of any finite strain elastic energy functional, thus leading to a proper statement of rate equilibrium problems, basilar for computational formulations and for investigations about instability phenomena and post-critical behaviours. [ABSTRACT FROM AUTHOR]

In a previous article in this journal, it was shown that vorticity is only an approximation to the rate of rotation of an element of material, and that an exact expression for the rate of rotation consists of the vorticity plus an added term, denoted by H, but not then given a proper name. It was shown that the new formulation led rigorously to the so-called Green-Naghdi stress rate, which has been previously reported but not actually derived. In an example it was shown that Green-Naghdi stress rate leads to a stable deformation in a calculation of simple shear, while the more traditional Jaumann-Noll stress rate leads to a late-stage instability, though they are very nearly equal for small strains. Here we name the term to be added to the vorticity, the swirl, and derive an alternate, but simpler, algebraic expression. In addition, we provide examples of its use with the goal of providing added credibility. The first concerns the dynamics of a classical vortex, showing that the rate of rotation is not zero as classically claimed but is consistent with the macroscopic rate of rotation. The second discusses a formula for a test known as simple strain that is compared with an experiment by Moreira and Nunes. The analysis follows standard usage in continuum mechanics, but the special use of polar decomposition allows for a very convenient derivation of the algebraic expression for the swirl. [ABSTRACT FROM AUTHOR]

The brief history of relaxation in continuum mechanics ranges from early application of non-convex plasticity and phase transition formulations to small and large strain continuum damage mechanics. However, relaxed continuum damage mechanics formulations are still limited in the following sense that their material response lack to model strain softening and the convexification of the non-convex incremental stress potential is computationally costly. This paper presents a reduced model for relaxed continuum damage mechanics at finite strains which includes strain softening by a fiber-specific damage in the microsphere approach. Computational efficiency is achieved by novel adaptive algorithms for the fast convexification of the one-dimensional fiber material model. The algorithms are benchmarked against state-of-the-art methods, and the choice of quadrature schemes for the microsphere approach is discussed. This contribution is finalized by a mesh independence test. [ABSTRACT FROM AUTHOR]

The Temetos platform is designed to conduct computational experiments at all stages of analysis and study of continuum mechanics models. A module has been developed to study the stress–strain state of a system of bodies with allowance for inelastic strains and contact interaction. It was used to analyze a fuel element that included up to 100 fuel pellets and a shell. The platform's solvers are applied to astrophysics problems. Models of the formation of accretion disks in binary star systems that allow the interpretation of observation results are constructed. [ABSTRACT FROM AUTHOR]

Due to the complex constitutive relationship of tread rubber and the strong non-linearity of the dynamic contact relationship between tread rubber and road surface, the torsional deformation characteristics and distribution mechanism are difficult to be modeled theoretically in the tire contact patch. In order to accurately express the nonlinear flexible large deformation of tire tread block under torsional condition, the constitutive theory modeling of tread rubber based on continuum mechanics and the constitutive model verification based on improved digital image correlation are studied. A novel constitutive model considering the hyperelastic, viscoelastic and frictional property of tread rubber is established. A full-field measuring equipment for torsional deformation of tread block is established based on machine vision technology. The digital image sequence of instantaneous torsion deformation is analyzed by the improved digital image correlation to obtain the strain rate distribution characteristics and distribution law of the marker points and the whole area of the tread block. The analysis results demonstrate that the proposed constitutive model is feasible to predict the full-field distribution characteristics and distribution trend of tread rubber blocks in the contact patch. [ABSTRACT FROM AUTHOR]

The article is a personal recollection of the author's experiences with Jerry Ericksen, a renowned figure in the field of continuum mechanics. The author describes how they were advised to contact Ericksen and visit him at Johns Hopkins University, where they found him to be knowledgeable and open-minded. The author also mentions their later interactions with Ericksen and highlights the impact of his influential papers on their own research. Overall, the author expresses gratitude for Ericksen's mentorship and considers him a master of continuum mechanics. [Extracted from the article]

In this paper, the application of fractional continuum mechanics to rate independent plasticity is presented. This new formulation is non-local due to the properties of the applied fractional differential operator during the definition of kinematics. In the description, a small fractional strains assumption is used together with additive decomposition of total fractional strains into elastic and plastic parts. Classical local rate independent plasticity is recovered as a special case. [ABSTRACT FROM AUTHOR]