Principle Of Virtual Power Research Articles

Quelques considérations sur le flambement des anneaux et des arcs

ABSTRACT This article deals with elastic instabilities of arches and rings. The proposed approach is based on the Principle of Virtual Powers, which leads to express the equilibrium equations on the deformed geometry, these equations being projected on the local axis of the undeformed geometry. It is important to well precise the hypothesis concerning the orders of magnitude in the frame of the small perturbations analysis, which leads to appropriate approximations. The general case is analysed, as well as the case of the funicular arches. A resolution method is proposed and used for the resolution of some examples. To highlight certain difficulties, the classical case of the ring submitted to pressure is detailed in particular, which leads to confirm the value of the critical pressure given by Timoshenko. This focuses on the precautions which should be taken in the analysis of the instabilities of a structure submitted to pressure, when using a finite element analysis.

Nonlocal Gradient-Dependent Thermodynamics for Modeling Scale-Dependent Plasticity

This article is concerned with formulating the thermodynamics of nonlocal gradient-dependent plasticity based on the nonlocality energy residual introduced by Eringen and Edelen (1972). A thermodynamic based theory for small strain gradient plasticity is developed by introducing gradients for variables associated with kinematic and isotropic hardening. This theory is a three-nonlocal-parameter theory that takes into consideration large variations in the plastic strain, large variations in the accumulated plastic strain, and accumulation of plastic strain gradients. It is shown that the presence of higher-order gradients in the plastic strain enforces the presence of a corresponding history variable brought by the accumulation of the plastic strain gradients. Gradients in the plastic strain introduce anisotropy in the form of kinematic hardening and are attributed to the net Burgers vector, whereas gradients in the accumulation of the plastic strain introduce isotropic hardening attributed to the additional storage of geometrically necessary dislocations. The equilibrium, or so-called microforce balance, between the internal Cauchy stress and the microstresses that are conjugates to the higher-order gradients turns out to be the yield criterion, which can be simply retrieved from the principle of virtual power. The classical macroscopic boundary conditions are supplemented by nonclassical microscopic boundary conditions associated with plastic flow. The developed nonlocal theory preserves the classical assumption of the local plasticity theory such that the plastic flow direction is governed by the deviatoric Cauchy stress. However, it is also argued here that plastic flow direction is the same as if it is governed by the nonlocal microstress. This is not in line with Gurtin (2003), who argued that the plastic flow direction is governed by a microstress and not the deviatoric Cauchy stress. Some generalities and the utility of this theory are discussed, and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.

A gradient theory for single-crystal plasticity

This study develops a continuum theory of small-deformation single crystal plasticity that includes ‘strain gradients’ and ‘strain rate gradients’ in the spirit of our recent papers (Gurtin and Anand 2005a, 2005b) for isotropic plasticity. Specifically, we develop a constitutive theory that allows forenergetic microscopic stresses that arise as a consequence of an assumed dependence of the free energy on edge and screw dislocations densities induced by slip gradients; anddissipative microscopic stresses that depend on slip-rate gradients.Our theory is based on a system of microscopic force balances—one balance for each slip system—with derivation based on the principle of virtual power. When augmented by constitutive relations for the microscopic stresses these force balances result in non-local flow rules in the form of second-order partial differential equations for the slips.

A new model to predict the behavior at the interfaces of multilayer structures

One of the current problems connected with multilayer composite structures concerns the analysis of the distribution of the stresses around peculiarities (free edge and loaded edge) and at the interfaces of each layer. This work presents a new shear stress function in the form of the exponential function to predict the mechanical behavior of multilayered laminated composite structures. As a case study, the mechanical behavior of a laminated composite beam (90 /0 /0 /90 ) is examined. The results are compared with the Touratier model sine and the two-dimensional finite element method studied. Results show that this new model is more precise than older ones when compared with results obtained by finite element analysis. To introduce continuity on the interfaces of each layer, the new exponential model is used with Ossadzow kinematics. The equilibrium equations and natural boundary conditions are derived from the principle of virtual power.

Sur le remodelage des tissus osseux anisotropes

Growth (change of relaxed lengths) and remodelling (change of mechanical properties) are both involved in the morphogenesis of biological tissues. To model them is of paramount import for progressing both in scientific understanding and health technologies. We model bone tissue as a microstructured continuum, whose mechanical properties at the macroscopic scale are described by a linear, anisotropic elastic response that evolves in time. Our kinematics is rich enough to allow for the microstructural evolution, as well as for the interplay between stress, growth and remodelling. This is a unified approach to the mechanics of growth and remodelling, in which all balance laws derive from one virtual-power principle. As a first application, we study the problem of stiffness remodelling due to planar rotation of the microstructure, excluding bulk growth and all physiological response to mechanical stimuli (passive remodelling). To cite this article: A. DiCarlo et al., C. R. Mecanique 334 (2006).

A thermodynamic based higher-order gradient theory for size dependent plasticity

A physically motivated and thermodynamically consistent formulation of small strain higher-order gradient plasticity theory is presented. Based on dislocation mechanics interpretations, gradients of variables associated with kinematic and isotropic hardenings are introduced. This framework is a two non-local parameter framework that takes into consideration large variations in the plastic strain tensor and large variations in the plasticity history variable; the equivalent (effective) plastic strain. The presence of plastic strain gradients is motivated by the evolution of dislocation density tensor that results from non-vanishing net Burgers vector and, hence, incorporating additional kinematic hardening (anisotropy) effects through lattice incompatibility. The presence of gradients in the effective (scalar) plastic strain is motivated by the accumulation of geometrically necessary dislocations and, hence, incorporating additional isotropic hardening effects (i.e. strengthening). It is demonstrated that the non-local yield condition, flow rule, and non-zero microscopic boundary conditions can be derived directly from the principle of virtual power. It is also shown that the local Clausius–Duhem inequality does not hold for gradient-dependent material and, therefore, a non-local form should be adopted. The non-local Clausius–Duhem inequality has an additional term that results from microstructural long-range energy interchanges between the material points within the body. A detailed discussion on the physics and the application of proper microscopic boundary conditions, either on free surfaces, clamped surfaces, or intermediate constrained surfaces, is presented. It is shown that there is a close connection between interface/surface energy of an interface or free surface and the microscopic boundary conditions in terms of microtraction stresses. Some generalities and utility of this theory are discussed and comparisons with other gradient theories are given. Applications of the proposed theory for size effects in thin films are presented.

Hydrogen Storage: Modeling and Analytical Results

We present a thermomechanical model describing hydrogen storage by use of metal hydrides. The problem is considered as a phase transition phenomenon. The model is recovered by continuum mechanics laws, using a generalization of the principle of virtual power accounting for microscopic movements related to the phase transition. The resulting nonlinear PDE system is investigated from the point of view of existence, uniqueness, and regularity of solutions.

A finite element formulation for dynamic parameter identification of robot manipulators

In this paper a finite element based approach is described for the automatic generation of models suitable for dynamic parameter identification. The method involves a nonlinear finite element formulation in which both links and joints are considered as specific finite elements [6, 7]. Since the identification procedure considers rigid-link robot models, the inertial properties of the link elements are described using a lumped mass formulation. The parameters to be identified are masses, first-order moments and inertial tensor components of the links. The equations of motion are written in a form which is linear in the dynamic parameters. This formulation is obtained by employing Jourdain’s principle of virtual power. The parameters are estimated using a linear least squares technique. Singular value decomposition of the regression matrix is used to find the minimum parameter set. Simulation results obtained from the 6 DOF PUMA 560 robot based on the estimated parameters show that the method yields accurate responses.

Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales

Using a nonstandard version of the principle of virtual power, we develop general balance equations and boundary conditions for second-grade materials. Our results apply to both solids and fluids as they are independent of constitutive equations. As an application of our results, we discuss flows of incompressible fluids at small-length scales. In addition to giving a generalization of the Navier–Stokes equations involving higher-order spatial derivatives, our theory provides conditions on free and fixed boundaries. The free boundary conditions involve the curvature of the free surface; among the conditions for a fixed boundary are generalized adherence and slip conditions, each of which involves a material length scale. We reconsider the classical problem of plane Poiseuille flow for generalized adherence and slip conditions.

WELL-POSEDNESS OF A PHASE TRANSITION MODEL WITH THE POSSIBILITY OF VOIDS

The paper deals with a phase transition model applied to a two-phase system. There is a wide literature on the study of phase transition processes in case that no voids nor overlapping can occur between the two phases. The main novelty of our approach is the possibility of having voids during the phase change. This aspect is described in the model by the mass balance equation whose effects are included by means of the pressure of the system in the dynamical relations. The state variables are the absolute temperature (whose evolution is ruled by the entropy balance equation), the strain tensor (satisfying a quasi-static macroscopic equation of motion), and the volume fractions of the two phases (whose evolutions are described by a vectorial equation coming from the principle of virtual power and related to the microscopic motions). Well-posedness of the initial-boundary value problem associated to the PDEs system resulting from this model is proved.

The Concept of a Minimal State in Viscoelasticity: New Free Energies and Applications to PDEs

We show here the impact on the initial-boundary value problem, and on the evolution of viscoelastic systems of the use of a new definition of state based on the stress-response (see, e.g., [48, 16, 41]). Comparisons are made between this new approach and the traditional one, which is based on the identification of histories and states. We shall refer to a stress-response definition of state as the minimal state [29]. Materials with memory and with relaxation are discussed. The energetics of linear viscoelastic materials is revisited and new free energies, expressed in terms of the minimal state descriptor, are derived together with the related dissipations. Furthermore, both the minimum and the maximum free energy are recast in terms of the minimal state variable and the current strain. The initial-boundary value problem governing the motion of a linear viscoelastic body is re-stated in terms of the minimal state and the velocity field through the principle of virtual power. The advantages are (i) the elimination of the need to know the past-strain history at each point of the body, and (ii) the fact that initial and boundary data can now be prescribed on a broader space than resulting from the classical approach based on histories. These advantages are shown to lead to natural results about well-posedness and stability of the motion. Finally, we show how the evolution of a linear viscoelastic system can be described through a strongly continuous semigroup of (linear) contraction operators on an appropriate Hilbert space. The family of all solutions of the evolutionary system, obtained by varying the initial data in such a space, is shown to have exponentially decaying energy.

WHOLE WORKSPACE OPTIMIZATION BASED PARAMETER TUNING OF HIGH-SPEED PARALLEL MANIPULATOR CONTROLLER

A new tuning method of constant PID parameters for parallel manipulator is investigated. Considering the mechatronics coupling, characteristics of the whole workspace and variable inertia load along with the pose, three trajectories with different weights are selected. Then inverse kinematics and inverse dynamic model are set up by using vector method and virtual power principle. Making the minimized following trajectory errors of end effector as optimized goal,the controller parameters could be tuned. An example is adopted with one parallel manipulator-Diamond which is developed by Tianjin University, and the values of controller parameters can be given according to the above theory. The problem that the multiobjective functions contradict each other is also solved. At last, a series of experiments are done for validating the correctness and validity of the theory.

Some issues related to growth and goal functions for continuum biological systems

The growth of biological tissues is described as the volumic production of mass of so-called germs, considered as potential sites for tissue renewal/resorption. The growth is characterized by a growth transformation gradient. The continuous change of the domain occupied by each germ during growth is analysed, hence, the mechanical balance laws are written considering the additional terms due to the domain variation. The principle of virtual power is then expressed, leading to the balance of momentum, assuming that the power of internal forces originates from both volumic and surfacic potentials. The equilibrium equations of the growing germ split into volumic, surfacic and lineic contributions. Viewing growth as an irreversible phenomenon, the writing of the second principle of thermodynamics for a germ continuously receiving matter due to transport phenomena allows to identify the driving forces linked to growth. In the second part, the growth and evolution are viewed as finding the best configuration of the growing solid in terms of its metric. Thereby, the four-dimensional space–time is structured in the form of isogrowth surfaces, the evolution of growth being described as a motion from one slice to the next. A Lagrangian formalism is established in the space of metric, whereby both the geometrical and physical contents of the growing solid are considered. Adopting this higher level of description, the stationnarity of the total action integral renders the dynamics of the metric field, while the kinematic compatibility condition prescribes its spatial form.

A Unified Approach to Analytical and Numerical Modelling of Sintering

This paper presents a unified framework to model the sintering process of fine powders. The framework is based on classical virtual power principle and its corresponding variational principle. Firstly, the classical models of solid state, viscous and liquid phase sintering are reproduced assuming single matter re‐distribution mechanism and using the virtual power principle as the starting point. Then we demonstrate how to obtain the governing equations for microstructural evolution using the variational principle. These provide a common thread through the existing sintering models. Finally a numerical solution scheme is briefly outlined for computer simulation of microstructural evolution using the variational principle as the starting point. The computer simulation can follow the entire sintering process from powder compact to fully dense solid and deal with fully couple multi‐physics processes involving all the possible underlying matter re‐distribution mechanisms. Several examples are provided to demonstrate the deep insights that can be gained into the sintering process by using the numerical tool.

On the physical meaning of the dynamical equations governing geometrically-exact multilayer shells

In this paper, the physical meaning of the equations governing geometrically-exact multilayer shells is provided through a totally new derivation by means of dynamic equilibrium consideration, starting from a 3-D continuum setting in curvilinear coordinates. The theory is valid for large deformation and large overall motion, as characteristic of geometrically exact formulations. The dynamic equilibrium derivation offers a clear physical insight into the meaning of the terms in the equations of motion; such insight was not afforded by previous derivations based on the principle of virtual power. In particular, we obtain directly the balance of angular momentum in true resultant couples, without the need to use the constitutive restriction as in our previous papers. Moreover, we also obtain here new expressions for the inertia operators that will greatly simplify the computational formulation, when compared to our previous work. In addition, an analytical justification of the inertial operators is also provided. The present formulation is independent of any kinematic assumption along the shell director, and is in particular valid for the case of shells with variable-length director (e.g., for modeling through-the-thickness deformation) and higher-order kinematic assumptions.

Mechanical modeling of growth considering domain variation. Part I: constitutive framework

The growth of biological tissues is described as the volumetric production of mass within tissue elements, considered as one component potential sites for tissue renewal/resorption at a mesoscopic scale of description. The growth is characterized by a growth transformation gradient, which is accompanied by an additional accommodation transformation, so that the total transformation gradient defines a compatible global displacement field. The continuous change of the domain occupied by each tissue element during growth is considered, hence the mechanical balance laws are written, accounting for the additional terms due to the domain variation. The principle of virtual power is then expressed, considering that the power of internal forces originates from both volumic and surface potentials. The surface potential is thought to express configurational forces tied to the interface motion, extent and orientation. The equilibrium equations of the growing tissue element then follow, associated to surface and line boundary conditions. The writing of the second principle of thermodynamics for a tissue element continuously receiving matter due to transport phenomena allows to identify the driving forces linked to growth, and provides the evolution laws for the growth velocity. The large strains compatibility conditions that characterize the accommodation tensor are further considered as an important aspect of growth. A first insight into the numerical solutions of the compatibility conditions is given, envisaging the situation of radial growth.

Geometric nonlinear formulation and discretization method for a rectangular plate undergoing large overall motions

In the present paper, the geometric nonlinear formulation is developed for dynamic stiffening of a rectangular plate undergoing large overall motions. The dynamic equations, which take into account the stiffening terms, are derived based on the virtual power principle. Finite element method is employed for discretization of the plate. The simulation results of a rotating rectangular plate obtained by using such geometric nonlinear formulation are compared with those obtained by the conventional linear method without consideration of the stiffening effects. The application limit of the conventional linear method is clarified according to the frequency error. Furthermore, the accuracy of the assumed mode method is investigated by comparison of the results obtained by using the present finite element method and those obtained by using the assumed mode method.

Dynamics of three-dimensional beams undergoing large overall motion

In the previous linear formulation of flexible multibody system, the neglect of stiffening terms may cause significant error in case of high rotating speed. In this paper, a geometric nonlinear formulation of three-dimensional beams is proposed based on virtual power principle. Frequency results of a rotating spatial beam using the present nonlinear model are compared with those using the linear model without stiffening. An influence ratio, which is related to non-dimensional axial base acceleration and lateral angular velocity, is put forward to clarify the limit of the linear formulation. It is shown that the relative frequency error is closely related to the influence ratio. Finally, simulation of a flexible spatial manipulator is carried out to verify the effectiveness of the criterion.

Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector

This paper discusses boundary conditions appropriate to a theory of single-crystal plasticity (Gurtin, J. Mech. Phys. Solids 50 (2002) 5) that includes an accounting for the Burgers vector through energetic and dissipative dependences on the tensor G = curl H p , with H p the plastic part in the additive decomposition of the displacement gradient into elastic and plastic parts. This theory results in a flow rule in the form of N coupled second-order partial differential equations for the slip-rates γ ˙ α ( α = 1 , 2 … , N ) , and, consequently, requires higher-order boundary conditions. Motivated by the virtual-power principle in which the external power contains a boundary-integral linear in the slip-rates, hard-slip conditions in which (A) γ ˙ α = 0 on a subsurface S hard of the boundary for all slip systems α are proposed. In this paper we develop a theory that is consistent with that of (Gurtin, 2002), but that leads to an external power containing a boundary-integral linear in the tensor H ˙ ij p ɛ jrl n r , a result that motivates replacing (A) with the microhard condition (B) H ˙ ij p ɛ jrl n r = 0 on the subsurface S hard . We show that, interestingly, (B) may be interpreted as the requirement that there be no flow of the Burgers vector across S hard . What is most important, we establish uniqueness for the underlying initial/boundary-value problem associated with (B); since the conditions (A) are generally stronger than the conditions (B), this result indicates lack of existence for problems based on (A). For that reason, the hard-slip conditions (A) would seem inappropriate as boundary conditions. Finally, we discuss conditions at a grain boundary based on the flow of the Burgers vector at and across the boundary surface.

Identification of Friction and Rigid-Body Dynamics of Parallel Kinematic Structures for Model-Based Control

Until now, the experimental identification of the dynamics of parallelrobots is restricted to simple models in combination with adaptivecontrol algorithms. This gap is closed by a new approach presented inthis paper, which is suited for even complex parallel kinematicstructures. The approach consists of two steps and utilizes simplepoint-to-point (PTP) motions that lead to a separation of friction andrigid-body dynamics. In the first step, local models are determined fora lot of different configurations, i.e. end-effector positions. In thesecond step, the overall friction model and the overall rigid-bodymodel are calculated from the local models by linear Least-Squaresestimators. The use of linear estimators is based on a formulation ofthe dynamic equations, which is linear with respect to a dynamicparameter vector of minimal dimension. This formulation is automaticallyobtained by an algorithm, which utilizes Jourdain's principle of virtualpower. The experimental application of the identified model tomodel-based feedforward control of the innovative hexapod PaLiDA, whichhas been developed by the Institute of Production Engineering andMachine Tools of the University of Hannover, proves the capability andefficiency of the presented algorithms.