An energy formulation of continuum magneto-electro-elasticity with applications

We present an energy formulation of continuum electro-elasticity and magneto-electro-elasticity. Based on the principle of minimum free energy, we propose a form of total free energy of the system in three dimensions, and then systematically derive the theories for a hierarchy of materials including dielectric elastomers, piezoelectric ceramics, ferroelectrics, flexoelectric materials, magnetic elastomers, magnetoelectric materials, piezo-electric–magnetic materials among others. The effects of mechanical, electrical and magnetic boundary devices, external charges, polarizations and magnetization are taken into account in formulating the free energy. The linear and nonlinear boundary value problems governing these materials are explicitly derived as the Euler–Lagrange equations of the principle of minimum free energy. Finally, we illustrate the applications of the formulations by presenting solutions to a few simple problems and give an outlook of potential applications.     

Trans-scale mechanics： looking for the missing links between continuum and micro/nanoscopic reality

Problems involving coupled multiple space and time scales offer a real challenge for conventional frame-works of either particle or continuum mechanics. In this paper, four cases studies （shear band formation in bulk metallic glasses, spallation resulting from stress wave, interaction between a probe tip and sample, the simulation of nanoindentation with molecular statistical thermodynamics） are provided to illustrate the three levels of trans-scale problems （problems due to various physical mechanisms at macro-level, problems due to micro-structural evolution at macro/micro-level, problems due to the coupling of atoms/ molecules and a finite size body at micro/nano-level） and their formulations. Accordingly, non-equilibrium statistical mechanics, coupled trans-scale equations and simultaneous solutions, and trans-scale algorithms based on atomic/molecular interaction are suggested as the three possible modes of trans-scale mechanics.     

A continuum damage model for delaminations in laminated composites

Delamination, a typical mode of interfacial damage in laminated composites, has been considered in the context of continuum damage mechanics in this paper. Interfaces where delaminations could occur are introduced between the constituent layers. A simple but appropriate continuum damage representation is proposed. A single scalar damage parameter is employed and the degradation of the interface stiffness is established. Use has been made of the concept of a damage surface to derive the damage evolution law. The damage surface is constructed so that it combines the conventional stress-based and fracture-mechanics-based failure criteria which take account of mode interaction in mixed-mode delamination problems. The damage surface shrinks as damage develops and leads to a softening interfacial constitutive law. By adjusting the shrinkage rate of the damage surface, various interfacial constitutive laws found in the literature can be reproduced. An incremental interfacial constitutive law is also derived for use in damage analysis of laminated composites, which is a non-linear problem in nature. Numerical predictions for problems involving a DCB specimen under pure mode I delamination and mixed-mode delamination in a split beam are in good agreement with available experimental data or analytical solutions. The model has also been applied to the prediction of the failure strength of overlap ply-blocking specimens. The results have been compared with available experimental and alternative theoretical ones and discussed fully.     

Kinematic variables bridging discrete and continuum granular mechanics

It is known that there is wide, and at present, unbridgeable, gap between discrete and continuum granular mechanics. In this contribution, first, microscopic kinematic variables neglected in classical continuum granular mechanics are investigated based on the kinematics of discs in contact. Then, a kinematic variable called the averaged pure rotation rate (APR) is proposed for an assembly of circular discs of different sizes, which is then used to produce another two kinematic tensors with one equal to the deformation rate tensor and the other unifying the spin tensor and the APR. As an example, the kinematic variables are incorporated into the unified double-slip plasticity model. Finally, these theoretical analyses are verified using a two-dimensional discrete element method. The study shows that these kinematic variables can be used to bridge discrete and continuum granular mechanics.     

A theoretical and computational framework for anisotropic continuum damage mechanics at large strains

The main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains. Based on the concept of a fictitious, isotropic, undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient. Then, the corresponding Finger tensor – denoted as damage metric – constructs a second order, internal variable. Due to the principle of strain energy equivalence with respect to the fictitious, effective space and the standard reference configuration, the free energy function can be computed via push-forward operations within the nominal setting. Referring to the framework of standard dissipative materials, associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation. The numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account. Finally, some numerical examples demonstrate the applicability of the proposed framework.     

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.     

Investigation of the damage variable basic issues in continuum damage and healing mechanics

Consistent mathematics and mechanics are used here to properly interpret the damage variable within the confines of the concept of reduced area due to damage. In this work basic issues are investigated for the damage variable in conjunction with continuum damage and healing mechanics. First, the issue of the additive decomposition of the damage variable into damage due to voids and damage due to cracks in continuum damage mechanics is discussed. The accurate decomposition is shown to be non-additive and involves a term due to the interaction of cracks and voids. It is shown also that the additive decomposition can only be used for the special case of small damage. Furthermore, a new decomposition is derived for the evolution of the damage variable. The second issue to be discussed is the new concept of independent and dependent damage processes. For this purpose, exact expressions for the two types of damage processes are presented. The third issue addressed is the concept of healing processes occurring in series and in parallel. In this regard, systematically and consistently, the equations of healing processes occurring either consecutively or simultaneously are discussed. This is followed by introducing the new concept of small healing in damaged materials. Simplified equations that apply when healing effects are small are shown. Finally, some interesting and special damage processes using a systematic and original formulation are presented.     

Intermediate asymptotics,scaling laws and renormalization group in continuum mechanics

Scaling laws and self-similar solutions are very popular concepts in modern continuum mechanics. In the present paper these concepts are analyzed both from the viewpoint of intermediate asymptotics, known in classical mathematical physics and fluid mechanics, and from the viewpoint of the renormalization group technique, known in modern theoretical physics. The definition of the renormalization group is proposed, related to the intermediate asymptotics with incomplete similarity. The general presentation is illustrated by examples of essentially non-linear problems where all analytical properties of the solutions and their asymptotics are rigorously proved, as well by an example from turbulence, where the rigorous problem statement is missing. General lecture delivered at the 11th Italian National Congress of Theoretical and Applied Mechanics (AIMETA), Trento, Sept./Oct. 1992.     

Closed-form expressions for the macroscopic flexural rigidity coefficients of periodic brickwork

Approximate expressions for the macroscopic out-of-plane elastic coefficients of brick masonry with a regular pattern are derived in closed form using a homogenization approach for periodic media. Following an approach similar to the Method of Cells for fiber reinforced composites, a (piecewise-)differentiable expression depending on very a limited number of degrees of freedom and fulfilling suitable periodicity conditions is proposed for the microscopic transverse displacement field over any Representative Volume Element (RVE). Some of the equilibrium conditions at the interfaces between bricks and mortar joints are also fulfilled. By averaging the moment and curvature fields over the RVE, the macroscopic bending stiffness coefficients can be explicitly obtained. Using the FE solution of a masonry panel subjected to elementary load conditions as a benchmark, the proposed approach is found to accurately match the numerically obtained stiffness coefficients, for masonry elements of different geometry and different mechanical properties. In several instances, the proposed expressions agree with the numerical predictions better than other analytical expressions available in the literature.     

Functional Concepts in Continuum Mechanics

This paper is an attempt to give a concise presentation of the main concepts of continuum mechanics and to show their articulation. Functional definitions have been favoured.The first section is devoted to a review of continuum mechanics.The second section deals with the mechanics of materials.Constitutive equations of the material are given first by equations of state and then by complementary equations written in order to fulfil the fundamental inequality concerning the production of entropy and the physical properties of the material (viscosity, plasticity, damage etc.).Section 3 gives the Lagrangian and the Hamiltonian formulations for a moving body. Section 4 is devoted to the motion of surfaces through which discontinuities appear, to show briefly two examples of application of the previous concepts. One can easily define the source of intrinsic inhomogeneity, of heat, of irreversible entropy on a surface of phase transition and also for a shockwave.SommarioIn questo lavoro si cerca di dare una presentazione concisa dei concetti fondamentali della meccanica dei continui e di mostrare le loro articolazioni,con particolare rilievo alle definizioni funzionali. La prima sezione è dedicata ad un esame generale introduttivo. La seconda alla meccanica dei materiali, le cui equazioni costitutive sono date come equazioni di stato e come equazioni complementari scritte al fine di soddisfare le diseguaglianze concernenti la produzionedi entropia e le proprietà fisiche dei materiali (qualiviscosità, plasticità, guasti...).La terza sezione dà le formulazioni Lagrangiana ed Hamiltoniana per il moto di un corpo.La quarta sezione è dedicata al moto di superfici di discontinuità per mostrare brevemente due esempi di applicazione dei concetti precedenti.Viene quindi definita la sorgente di disomogeneità intrinseca, di calore, di entropia irreversibile su di una superficie, di transizione di fase ed anche di un' onda d'urto.     

On the Constituent Structure of a Classical Mixture

Thiw work is concerned with the formulation of constituent interactions and corresponding balance relations in classical mixture theory as based on a model for the (classical) constituent structure of such a mixture.     

A continuum model of the blood flow in the lung microcirculation

An one-dimensional continuum model and the corresponding governing equations are proposed for the blood flow in lung microcirculation, the analytical solutions in closed form are presented. It is shown that the obtained results coincide with Fung's in sheet flow. Project is supported by the National Natural, Science Fundation of China.     

An atomistic-based continuum theory for carbon nanotubes: analysis of fracture nucleation

Carbon nanotubes (CNTs) display unique properties and have many potential applications. Prior theoretical studies on CNTs are based on atomistic models such as empirical potential molecular dynamics (MD), tight-binding methods, or first-principles calculations. Here we develop an atomistic-based continuum theory for CNTs. The interatomic potential is directly incorporated into the continuum analysis through constitutive models. Such an approach involves no additional parameter fitting beyond those introduced in the interatomic potential. The atomistic-based continuum theory is then applied to study fracture nucleation in CNTs by modelling it as a bifurcation problem. The results agree well with the MD simulations.     

Size-dependent elastic properties of a single-walled carbon nanotube via a molecular mechanics model

An analytical model based on a molecular mechanics approach is presented to relate the elastic properties of a single-walled carbon nanotube to its atomic structure. We derive closed-form expressions for elastic modulus and Poisson's ratio as a function of the nanotube diameter. Properties at different length scales are directly connected via these expressions. The analytically calculated elastic properties for achiral nanotubes using force constants obtained from experimental data of graphite are compared to those based on tight binding numerical calculations. This study represents a preliminary effort to develop analytical methods of molecular mechanics for applications in nanostructure modeling.     

Smectic liquid crystals as continua with latent microstructure

Smectic liquid crystals are quasi solids, which possess microstructure both of the material and local type (the nematic mictrostructure and the lamellae, respectively). They also exhibit phase transitions: from smectic B to smectic A, to nematic, to isotrtopic liquid (see [1], Ch. 7). Their study, though special, might show up general advantages and drawbacks of different types of mathematical models. Here, while intent on proving the property announced in the title, we set forth the groundwork for the continuum model (see, in a more specific and concrete mode [2]).

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Sommario I cristalli liquidi smettici sono quasi-solidi che possiedono una microstruttura sia di tipo materiale che locale (la microstruttura nematica e le lamelle, rispettivamente). Essi esibiscono anche transizioni di fase: smettici B a smettici A, a nematici, a liquidi isotropi. Il loro studio, sebbene speciale, potrebbe evidenziare vantaggi generali e difetti di vari tipi di modelli matematici. Qui, nel provare la proprietà annunciata nel titolo, si prepara il lavoro di base per il modello continue.

     

On the Collapse of Masonry Arches

In this paper a constitutive equation for masonry arches is defined and its main properties are proven; in this equation to each pair of generalized strains (, ), with the extensional strain and the curvature change of the centre line, is assigned the pair of generalized internal forces (N,M), where N is the normal force and M the bending moment. Subsequently, the collapse of masonry arches is characterized and the static and kinematic theorems proven. Finally, a method for determining the collapse load in the case of circular arches subjected to their own weight and a vertical point load applied at a point of the extrados is presented. The results obtained, of interest in some applications, are summarized in a series of graphs.