Stress, strain and energy splittings for anisotropic elastic solids under volumetric constraints

We define stress and strain splittings appropriate to linearly elastic anisotropic materials with volumetric constraints. The treatment includes rigidtropic materials, which develop no strains under a stress pattern that is a null eigenvector of the compliance matrix. This model includes as special case incompressible materials, for which the eigenvector is hydrostatic stress. The main finding is that pressure and volumetric strain must be redefined as effective quantities. Using this idea, an energy decomposition that exactly separates deviatoric and volumetric energy follows.     

Optimization of a hyper-elastic structure with multibody contact using continuum-based shape design sensitivity analysis

In this paper, a continuum-based shape design sensitivity formulation is presented for a hyper-elastic structure with multibody frictional contact. A nearly incompressible constraint is treated using the pressure projection method that projects a hydrostatic pressure into a lower order space to avoid a volumetric locking. The variational formulation for multibody frictional contact is developed using a penalty method that regularizes the solution of the variational inequality. The material derivative of continuum mechanics is utilized to develop the continuum-based shape design sensitivity analysis for the hyper-elastic constitutive relation and penalized contact formulation. The sensitivity equation is solved at each converged load step using the same tangent stiffness of response analysis due to the path dependency of the sensitivity of the frictional contact problem. A very accurate and efficient sensitivity results are shown through shape optimization of a windshield wiper. Received August 8, 1999     

Multiscale computational analysis in mechanics using finite calculus: an introduction

The paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus (FIC) formulation. The FIC approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size. This introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters. The discrete nodal values obtained with the FIC formulation and the finite element method (FEM) can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched FEM. Typical multiscale problems in mechanics which can be solved with the FIC method include convection–diffusion-reaction problems with high localized gradients, incompressible problems in solid and fluid mechanics, localization problems such as prediction of shear bands in solids and shock waves in compressible fluids, turbulence, etc. The paper presents an introduction of the treatment of multiscale problems using the FIC approach in conjunction with the FEM. Examples of application of the FIC/FEM formulation to the solution of simple multiscale convection–diffusion problems are given.     

Optimization of parameters of nonbonded interactions in a spectroscopically determined force field

A procedure is given by which parameters of nonbonded interactions in a molecular mechanics energy function can be optimized for maximum compatibility with ab initio force fields and structures. The method is based on a previously derived transformation of ab initio valence parameters to the molecular mechanics formalism. Explicit analytical expressions for the derivatives of the molecular mechanics force constants and reference geometry parameters with respect to the parameters of the nonbonded interactions are derived. The form of the goodness-of-fit function is discussed. A first application to a set of alanine dipeptides is described.     

Incremental formulation in nonlinear mechanics and large strain elasto-plasticity — Natural approach. Part 1

The paper discusses numerical solution techniques of problems in continuum mechanics in the presence of finite elastic as well as plastic strain components. The paper is based on the natural finite element formulation of large deformations. The proposed model requires the determination of an intermediate (stress-free) configuration at each step of the loading process. Certain approximations are adopted and comparisons are made to assess the difference between the natural approach with an intermediate reference configuration and conventional models for large displacements and small strains. Some numerical illustrations are given.     

Panel flutter analysis of plate element based on the absolute nodal coordinate formulation

In this study, aeroelastic analysis of a plate subjected to the external supersonic airflow is carried out. A 3-D rectangular plate element of variable thickness based on absolute nodal coordinate formulation (ANCF) has been developed for the structural model. In the approach to the problem, a continuum mechanics approach for the definition of the elastic forces within the finite element is considered. Both shear strain and transverse normal strain are taken into account. Linearized first-order potential (piston) theory is coupled with the structural model to account for pressure loading. Aeroelastic equations using ANCF are derived and solved numerically. Values of critical dynamic pressure are obtained by a modal approach, in which the mode shapes are obtained by ANCF. All the formulations and the computations are built up in a FORTRAN 90 computer program after it was confirmed by Mathematica^?, ver. 5. The results of free vibration analysis and flutter are compared with the available references and reasonable good agreement has been found. However, some results indicate that the known problem of locking (ANCF with uniform thickness) still persist in the current developed formulation.     

Physical modeling with orthotropic material based on harmonic fields

Although it is well known that human bone tissues have obvious orthotropic material properties, most works in the physical modeling field adopted oversimplified isotropic or approximated transversely isotropic elasticity due to the simplicity. This paper presents a convenient methodology based on harmonic fields, to construct volumetric finite element mesh integrated with complete orthotropic material. The basic idea is taking advantage of the fact that the longitudinal axis direction indicated by the shape configuration of most bone tissues is compatible with the trajectory of the maximum material stiffness. First, surface harmonic fields of the longitudinal axis direction for individual bone models were generated, whose scalar distribution pattern tends to conform very well to the object shape. The scalar iso-contours were extracted and sampled adaptively to construct volumetric meshes of high quality. Following, the surface harmonic fields were expanded over the whole volumetric domain to create longitudinal and radial volumetric harmonic fields, from which the gradient vector fields were calculated and employed as the orthotropic principal axes vector fields. Contrastive finite element analyses demonstrated that elastic orthotropy has significant effect on simulating stresses and strains, including the value as well as distribution pattern, which underlines the relevance of our orthotropic modeling scheme.     

Computational model of three-dimensional cardiac electromechanics

We coupled a continuum model of impulse propagation with a three-dimensional model of regional ventricular mechanics. Equations for action potential propagation, myofilament activation and active contraction were solved in an anatomically detailed finite element model of canine left and right ventricular geometry, muscle fiber architecture and Purkinje fiber network anatomy. Finite element equations for time-dependent excitation and recovery variables were assembled using a collocation method and solved using adaptive Runge-Kutta integration. Resting tissue mechanics were modeled as nonlinear, orthotropic and hyperelastic. A Windkessel model for arterial impedance was coupled to ventricular pressure and volume to compute hemodynamic boundary conditions during ejection. Ventricular volume constraints were imposed during the isovolumic phases. This model showed good agreement in the time course of regional systolic strains with experimental measurements during normal sinus rhythm and demonstrates the importance of the Purkinje fiber system in determining the mechanical activation sequence.     

On the Theories and Numerics of Continuum Models for Adaptation Processes in Biological Tissues

Computational continuum mechanics have been used for a long time to deal with the mechanics of materials. During the last decades researches have been using many of the theoretical models and numerical approaches of classical materials to deal with biological tissue which, in many senses, are a much more sophisticated material. We aim to review the last achievements of continuum models and numerical approaches on adaptation processes in biological tissues. In this review, we are looking, in particular, at growth in terms of changes of density and/or volume as, e.g., in collagen remodeling, wound healing, arterial thickening, etc. Furthermore, we point out some of the most relevant limitations of the current state-of-the-art in terms of these well established computational continuum models. In connection with these limitations, we will finish by discussing the trend lines of future work in the field of modeling biological adaptation, focusing on the computational approaches and mechanics that could overcome the current drawbacks. We would also like to attract the attention of all those researchers in classical materials (metal, alloys, composites, etc), to point out how similar the continuum and computational models between our fields are. We hope we can motivate them for getting their expertize in this challenging field of research.     

Exploring the applicability of gradient elasticity to certain micro/nano reliability problems

It has long been assessed that continuum mechanics can be used successfully to address a variety of mechanical problems at both macroscopic and microscopic scales. The term “micromechanics”, in particular, has been used in considering elasticity, plasticity, damage, and fracture mechanics problems at the micron scale involving metallic, ceramic and polymeric materials, as well as their composites. Applications to automobile, aerospace, and concrete industries, as well as to chemical and microelectronic technologies have already been documented. The recent developments in the field of nanotechnology have prompted a substantial literature in nanomechanics. While this term was first introduced by the author in the early 90’s to advance a generalized continuum mechanics framework for applications at the nanoscale, it is mainly used today in considering “hybrid” ab-initio/molecular dynamics/finite element simulations, usually based on elasticity theory, to interpret the mechanical response of nano-objects (nanotubes, nanowires, nanoaggregates) and extract information on nano-configurations (dislocation cores, crack tips, interfaces). The modest goal of this article is to show that continuum elasticity can indeed be extended to describe a variety of problems at the micro/nano regime. The resultant micro/nanoelasticity theory includes long-range or nonlocal material point interactions and surface effects in the form of (phenomenological) higher-order stress/strain gradients. Coupled thermo-diffuso-chemo-mechanical processes can also be considered within such a higher-order theory. Size effects on micro/nano holes and micro/nano cracks can conveniently be modeled, and some standard strength of materials formulas routinely used for micro/nano beams can be improved, with potential applications to MEMS/NEMS devices and micro/nano reliability components.     

Thermodynamic-based interface model for cohesive brittle materials: Application to bond slip in RC structures

The main objectives of this work concern a new formulation for an inelastic constitutive model for bond-slip phenomena between brittle cohesive materials and reinforcement through a nondimensional zero-thickness joint finite element. The bond-slip behavior assumes the small deformation and plane strains case. It is firmly placed within the framework of thermodynamics and continuum damage mechanics accounting for frictional sliding. The finite element interface model is based on a 2D-degenerated four-node quadrilateral element which is able to account for interface normal stress as well as for the nonlinear hysteretic tangential-normal coupling of bonding. One of the most important advantages of the proposed model consists in a sufficiently rich kinematics structure for contact allowing for the implementation of refine stress-strain constitutive relations.     

On the numerical investigations of the stress-strain behavior of materials based on the isotropic elasticity model of the multimodulus media

The paper presents the multimodulus approach to the continuum mechanics problem. In the introduction, a number of multimodulus models are presented that were developed by V.P. Myasnikov for the isotropic elasticity case. The numerical finite-element approximation of the multimodulus equilibrium state equations forming the sophisticated Lame system is also considered. In the final part, the results of two-dimensional calculations of the stress-strain state for various multimodulus media such as steel, concrete, ceramics, and soil are presented. The effect is discussed of the influence of the different moduli in tension and compression on the strain intensity     

Comparison of different formulations of 2D beam elements based on Bond Graph technique

Bond Graphs are well suited for modelling multibody systems. In this paper modelling of planar flexible beams undergoing large overall motions are studied based on finite element (FE) technique. Two well-known approaches are used – the co-rotational (CR) and absolute nodal coordinate (ANC) formulation. Two ANC formulations are analyzed – one in which elastic forces is described using classical beam theory in a local coordinate frame, and another based on a global continuum mechanics approach. Starting from these classical formulations velocity formulations are developed and used to develop Bond Graph FE components. The effect of gravity has been considered as well. These components can be put in libraries and used for systematic Bond Graph flexible body model development. It is shown that Bond Graph technique is capable of dealing with different flexible body formulations and can be used as a general approach in parallel to other modelling approaches. Models are developed and simulations are performed using the object oriented environment of BondSim. Owing to the object oriented approach, transformation from one to the other model is relatively simply. The results are illustrated by suitable examples and they confirm accuracy of the developed models. It was shown that the CR approach offers much better performance than the both ANC formulations.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.     

Quasi-3D large deflection nonlinear analysis of isogeometric FGM microplates with variable thickness via nonlocal stress–strain gradient elasticity

Engineering with Computers - Via the nonlocal stress–strain gradient continuum mechanics, the microscale-dependent linear and nonlinear large deflections of transversely loaded composite...     

A generalized Hill’s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials

Based on the Hill’s lemma for classical Cauchy continuum, a generalized Hill’s lemma for micro–macro homogenization modeling of gradient-enhanced Cosserat continuum is presented in the frame of the average-field theory. In this context not only the strain and stress tensors defined in classical Cosserat continuum but also their gradients are attributed to assigned micro-structural representative volume element (RVE), that leads to a higher-order macroscopic Cosserat continuum modeling and enables to incorporate the micro-structural size effects. The enhanced Hill–Mandel condition for gradient-enhanced Cosserat continuum is extracted as a corollary of the presented generalized Hill’s lemma. The derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the RVE in order to ensure the satisfaction of the Hill–Mandel energy condition.With the link between the discrete particle assembly and its effective Cosserat continuum in an individual RVE, the boundary conditions prescribed on the RVE modeled as Cosserat continuum are transformed into those prescribed to the peripheral particles of the RVE modeled as the discrete particle assembly. The micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress–strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters.     

Conservation laws of (3+α)-dimensional time-fractional diffusion equation

The concept of Lie–Backlund symmetry plays a fundamental role in applied mathematics. It is clear that in order to find conservation laws for a given partial differential equations (PDEs) or fractional differential equations (FDEs) by using Lagrangian function, firstly, we need to obtain the symmetries of the considered equation.Fractional derivation is an efficient tool for interpretation of mathematical methods. Many applications of fractional calculus can be found in various fields of sciences as physics (classic, quantum mechanics and thermodynamics), biology, economics, engineering and etc. So in this paper, we present some effective application of fractional derivatives such as fractional symmetries and fractional conservation laws by fractional calculations. In the sequel, we obtain our results in order to find conservation laws of the time-fractional equation in some special cases.     

Numerical computations of tensor quantities on a curved surface by the curvilinear finite difference (CFD) method

The ability to compute explicitly certain geometric tensors such as the metric tensor, the curvature tensor and its covariant derivatives, at any point on a general curved surface is a prerequisite of any attempt to computer implement the general shell theory in a tensor code. As a preparatory paper leading towards this final goal, this paper, based on the Curvilinear Finite Difference (or abbreviated as CFD) method, describes and documents the computer algorithm for the numerical computation of the above mentioned tensors and some of the related quantities such as the Christoffel symbols. With the present developed programming routines, fundamental techniques that are often used in tensor calculus—like the raising or lowering indices of tensors and their transformation from one coordinate system to another—can be carried out directly in a numerical sense. The validity of these programming routines was justified by applying them to compute the above mentioned tensors for a hyperbolic paraboloid. Numerical results obtained were found to be identical with the analytical ones.

As further illustrations of how the programming routines developed in the present work can be applied to solve practical problems in continuum mechanics, the plate bending problem formulated within the framework of tensor calculus and the problem of finding the total surface area of a general curved surface were solved.