Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Continuum Mechanics of Space Seen from the Aspect of General Relativity An Interpretation of the Gravity Mechanism

A mechanical structure of space is suggested. On the supposition that a space as vacuum has a physical fine structure like continuum, it enables us to apply a continuum mechanics to the so-called ＂vacuum＂ of space. A space is an infinite continuum and its structure is determined by Riemannian geometry. Assuming that space is an infmite continuum, the pressure field derived from the geometrical structure of space is newly obtained by applying both continuum mechanics and General Relativity to space. A fundamental concept of space-time is described that focuses on theoretically innate properties of space including strain and curvature. As a trial consideration, gravity can be explained as a pressure field induced by the curvature of space.

A Hybrid Local/Nonlocal Continuum Mechanics Modeling of Damage and Fracture in Concrete Structure at High Temperatures

This paper proposes a hybrid peridynamic and classical continuum mechanical model for the high-temperature damage and fracture analysis of concrete structures.In this model,we introduce the thermal expansion into peridynamics and then couple it with the thermoelasticity based on the Morphing method.In addition,a thermomechanical constitutive model of peridynamic bond is presented inspired by the classic Mazars model for the quasi-brittle damage evolution of concrete structures under high-temperature conditions.The validity and effectiveness of the proposed model are verified through two-dimensional numerical examples,in which the influence of temperature on the damage behavior of concrete structures is investigated.Furthermore,the thermal effects on the fracture path of concrete structures are analyzed by numerical results.

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.

The renaissance of continuum mechanics

Continuum mechanics, just as the name implies, deals with the mechanics problems of all continua, whose physical （or mechanical） properties are assumed to vary continuously in the spaces they occupy. Continuum mechanics may be seen as the symbol of modem mechanics, which differs greatly from current physics, the two often being mixed up by people and even sci- entists. In this short paper, I will first try to give an illustration on the differences between （modem） mechanics and physics, in my personal view, and then focus on some important current research activities in continuum mechanics, attempting to identify its path to the near future. We can see that continuum mechanics, while having a dominating impact on engineering design in the 20th century, also plays a pivotal role in modem science, and is much closer to physics, chemistry, biology, etc. than ever before.

Analysis of a Prototypical Multiscale Method Coupling Atomistic and Continuum Mechanics:the Convex Case

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have been recently proposed that aim at coupling an atomistic model （discrete mechanics） with a macroscopic model （continuum mechanics）. We provide here a theoretical basis for such a coupling in a one-dimensional setting, in the case of convex energy.

New approach based on continuum damage mechanics with simple parameter identification to fretting fatigue life prediction

A new continuum damage mechanics model for fretting fatigue life prediction is established. In this model, the damage evolution rate is described by two kinds of quantities. One is associated with the cyclic stress characteristics obtained by the finite element （FE） analysis, and the other is associated with the material fatigue property identified from the fatigue test data of standard specimens. The wear is modeled by the energy wear law to simulate the contact geometry evolution. A two-dimensional （2D） plane strain FE implementation of the damage mechanics model and the energy wear model is presented in the platform of ABAQUS to simulate the evolutions of the fatigue damage and the wear scar. The effect of the specimen thickness is also investigated. The predicted results of the crack initiation site and the fretting fatigue life agree well with available experimental data. Comparisons are made with the critical plane Smith- Watson-Topper （SWT） method.

Ductile Fracture Characterization for Medium Carbon Steel Using Continuum Damage Mechanics

This paper presents the ductility characterization for a medium carbon steel, for two microstructural conditions, that has been evaluated using the continuum damage mechanics theory, as proposed by Kachanov and developed by Lemaitre. Tensile tests were carried out using loading-unloading cycles in order to capture the gradual deterioration of the elastic modulus, which may be linked to the ductile damage increase with increasing plastic strain. The mechanical parameters for the isotropic damage evolution equation were obtained and then used as inputs for a plasticity-damage coupled nu- merical algorithm, validated through numerical simulations of the experimental tensile tests. A comparison between the SAE 1050 steels studied and two carbon steel alloys (obtained from the literature), provided some basic understanding of the influence of the carbon level on the evolution of the damage parameters. An empiric relationship for this set of parameters, which can provide useful data for preliminary studies envisaging prediction of ductile failure in carbon steels, is also presented.

A New Class of Simple,General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper,a new efficient,and at the same time,very simple and general class of thermodynamically compatiblefinite volume schemes is introduced for the discretization of nonlinear,overdetermined,and thermodynamically compatiblefirst-order hyperbolic systems.By construction,the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm.A very peculiar feature of our approach is that entropy is discretized directly,while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization.The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs,including both,conservative and non-conservative products,as well as potentially stiff algebraic relaxation source terms,provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law,such as the conservation of total energy density.The proposed family offinite volume schemes is based on the seminal work of Abgrall[1],where for thefirst time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented.We apply our new approach to three particular thermodynamically compatible systems:the equations of ideal magnetohydrodynamics(MHD)with thermodynamically compatible generalized Lagrangian multiplier(GLM)divergence cleaning,the unifiedfirst-order hyperbolic model of continuum mechanics proposed by Godunov,Peshkov,and Romenski(GPR model)and thefirst-order hyperbolic model for turbulent shallow waterflows of Gavrilyuk et al.In addition to formal mathematical proofs of the properties of our newfinite volume schemes,we also present a large set of numerical results in order to show their potential,efficiency,and practical applicability.

Curved Space-Time at the Planck Scale

This paper presents a physically plausible and somewhat illuminating first step in extending the fundamental principles of mechanical stress and strain to space-time. Here the geometry of space-time, encoded in the metric tensor, is considered to be made up of a dynamic lattice of extremely small, localized fields that form a perfectly elastic Lorentz symmetric space-time at the global (macroscopic) scale. This theoretical model of space-time at the Planck scale leads to a somewhat surprising result in which matter waves in curved space-time radiate thermal gravitational energy, as well as an equally intriguing relationship for the anomalous dispersion of light in a gravitational field.

Lie symmetry and Mei continuum conservation law of system＊

Lie symmetry and Mei conservation law of continuum Lagrange system are studied in this paper. The equation of motion of continuum system is established by using variational principle of continuous coordinates. The invariance of the equation of motion under an infinitesimal transformation group is determined to be Lie-symmetric. The condition of obtaining Mei conservation theorem from Lie symmetry is also presented. An example is discussed for applications of the results.

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.

A CONTINUUM DAMAGE MODEL OF AGING CONCRETE

There is up to now no constitutive model in the current theoriesof CDM that could give a description for the degradation of agingconcrete. The two internal state variable β and ω are intro- ducedin this paper β is called cohesion variable as an additionalkinematic parameter, reflecting the cohesion state among materialparticles. ω is called damage factor for micro-defects such s voids.Then a damage model and a series of constitutive equations aredeveloped on Continuum Mechanics. The model proposed could give avalid description for the whole-course-degradation of aging concretedue to chemical and mechanical actions. Finally, the validity of themodel is evaluated by an example and experimental results.

Finite Deformation, Finite Strain Nonlinear Dynamics and Dynamic Bifurcation in TVE Solids

This paper presents the mathematical model consisting of conservation and balance laws (CBL) of classical continuum mechanics (CCM) and the constitutive theories derived using entropy inequality and representation theorem for thermoviscoelastic solids (TVES) matter without memory. The CBL and the constitutive theories take into account finite deformation and finite strain deformation physics. This mathematical model is thermodynamically and mathematically consistent and is ideally suited to study nonlinear dynamics of TVES and dynamic bifurcation and is used in the work presented in this paper. The finite element formulations are constructed for obtaining the solution of the initial value problems (IVPs) described by the mathematical models. Both space-time coupled as well as space-time decoupled finite element methods are considered for obtaining solutions of the IVPs. Space-time coupled finite element formulations based on space-time residual functional (STRF) that yield space-time variationally consistent space-time integral forms are considered. This approach ensures unconditional stability of the computations during the entire evolution. In the space-time decoupled finite element method based on Galerkin method with weak form for spatial discretization, the solutions of nonlinear ODEs in time resulting from the decoupling of space and time are obtained using Newmark linear acceleration method. Newton’s linear method is used to obtain converged solution for the nonlinear system of algebraic equations at each time step in the Newmark method. The different aspects of the deformation physics leading to the factors that influence nonlinear dynamic response and dynamic bifurcation are established using the proposed mathematical model, the solution method and their validity is demonstrated through model problem studies presented in this paper. Energy methods and superposition techniques in any form including those used in obtaining solutions are neither advocated nor used in the present work as these are not supported by calculus of variations and mathematical classification of differential operators appearing in nonlinear dynamics. The primary focus of the paper is to address various aspects of the deformation physics in nonlinear dynamics and their influence on dynamic bifurcation phenomenon using mathematical models strictly based on CBL of CCM using reliable unconditionally stable space-time coupled solution methods, which ensure solution accuracy or errors in the calculated solution are always identified. Many model problem studies are presented to further substantiate the concepts presented and discussed in the paper. Investigations presented in this paper are also compared with published works when appropriate.

Fatigue Reliability Analysis of Fiber-Reinforced Laminated Composites by Continuum Damage Mechanics

This paper investigates the reliability of composite laminates with various lay-ups under fatigue loading.The prediction of failure probability of composite laminates subjected to different loads involves many uncertainties associated with mechanical properties,loading,and boundary conditions.Failure in the composite material is truly hard to trace because there are individual faults in each ply,and we face a stochastic process due to the scatter in the mechanical properties.The continuum damage mechanics(CDM),as a powerful approach,is applied to model the damage of fiber,matrix,and fiber/matrix debonding.This method defines criteria for damage detection and determines safe zones.The material constitutive equations are executed using a subroutine inAbaqus.The first-order reliability method and second-order reliability method have been applied to examine the reliability of laminated composites.The results are compared with those of the Monte Carlo simulation.Different composite laminates under different stress levels are considered for the failure probability investigation.The limit state functions and random variables have been determined based on the CDM model.Finally,the effects of the number of cycles,applied stress,and stacking sequence of the laminate on the reliability and fatigue life in fiber-reinforced laminated composites are assessed.

VARIATIONAL PRINCIPLFS OF NONLINEAR PIEZOTHERMOELASTIC MEDIA

Through the phenomenological approach,the nonlinear constitutive equations coupling the electro/magnetic therrnoelastic media are derived.Several nonlinear variational principles for piezothermoelastic continua are presented and employed to formulate the incremental variational princi- ples which are of important significance in practical applications such as the nonlinear finite element analysis,the buckling,postbuckling and dynamic stability analyses of piezoelectric smart structures.

Progressive Damage Analysis(PDA)of Carbon Fiber Plates with Out-of-Plane Fold under Pressure

The out-of-plane fold is a common defect of composite materials during the manufacturing process and will greatly affect the compressive strength as well as the service life.Making it of great importance to investigate the influence of out-of-plane defects to the compressive strength of laminate plates of composite materials,and to understand the patterns of defect evolution.Therefore,the strip method is applied in this article to create out-of-plane defects with different aspect ratios in laminated plates of composite materials,and a compressive performance test is conducted to quantify the influence of out-of-plane defects.The result shows that the compressive strength becomes weaker with a greater aspect ratio.When the highest aspect ratio is set to 0.12 in the experiment,the compressive strength reduces by 36.1%.Then we establish a 3-D progressive damage model based on continuum mechanics,and write it into the UMAT subroutine together with the 3-D Hashin criteria and the non-linear degradation criteria of materials.3-D solid modeling is performed for the samples with an out-of-plane fold based on ABAQUS,and progressive damage analysis is conducted to acquire the inplane evolution process of initial failure strength with different laminates.The experimental results agree well with the simulation results.

GUIDED CIRCUMFERENTIAL WAVES IN DOUBLE-WALLED CARBON NANOTUBES

A model of guided circumferential waves propagating in double-walled carbon nanotubes is built by the theory of wave propagation in continuum mechanics, while the van der Waals force between the inner and outer nanotube has been taken into account in the model. The dispersion curves of the guided circumferential wave propagation are studied, and some dispersion characteristics are illustrated by comparing with those of single-walled carbon nanotubes. It is found that in double-walled carbon nanotubes, the guided circumferential waves will propagate in more dispersive ways. More interactions between neighboring wave modes may take place. In particular, it has been found that a couple of wave modes may disappear at a certain frequency and that, while a couple of wave modes disappear, another new couple of wave modes are excited at the same wave number.

ELECTRIC FIELD EFFECTS ON YOUNG'S MOLULUS OF NANOWIRES

As an important component of nanodevices and nanomachine constructions, the mechanical performance of nanowires （NWs） has been a subject of intense research efforts due to gaining relevance in controlling functionality of nanoelectromechanical systems （NEMS）; meanwhile, one of the characteristics of the NEMS is the dependence of the functionality of the systems upon the applied electric field. The study of the electric effects on the Young＇s modulus of nanostructures is of certain usefulness in the design of NEMS and the precise measurement of mechanical properties of one-dimensional nanostructures. This paper reviews the origin of the size-dependence of the elastic property of NWs and the factors influencing the discrepancies and inconsistencies in the measured values of the Young＇s modulus for the NW, besides the surface effects, nonlinear effects, the electromechanical coupling effects as a possible effect responsible for the differences in quantitative and qualitative performance of the measured Young＇s modulus for the NWs versus the diameter are clarified.

THEORY OF NONLOCAL ASYMMETRIC ELASTIC SOLIDS

In this paper, a nonlinear theory of nonlocal asymmetric, elastic solids is developed on the basis of basic theories of nonlocal continuum field theory and nonlinear continuum mechanics. It perfects and expands the nonlocal elastic field theory developed by Eringen and others. The linear theory of nonlocal asymmetric elasticity developed in [1] expands to the finite deformation. We show that there is the nonlocal body moment in the nonlocal elastic solids. The nonlocal body moment causes the stress asymmetric and itself is caused by the covalent bond formed by the reaction between atoms. The theory developed in this paper is applied to explain reasonably that curves of dispersion relation of one-dimensional plane longitudinal waves are not similar with those of transverse waves.