Principle Of Virtual Power Research Articles

Rigorous amendment of Vlasov's theory for thin elastic plates on elastic Winkler foundations, based on the Principle of Virtual Power

Deflection modes relevant for plates with rigidly supported edges are commonly used as kind of “approximation” for the deformation behavior of plates which are freely swimming on an elastic foundation. However, this approach entails systematic errors at the boundaries. As a remedy to this problem, we here rigorously derive a theory for elastically supported thin plates with arbitrary boundary conditions, based on the Principle of Virtual Power. Somewhat surprisingly, it appears that the well-known Laplace-type differential equation for the deflections needs to be extended by additional boundary integrals entailing moments and shear forces, so as to actually “release” the boundaries from “spuriously” acting external moments and shear forces. When approximating the deflections through 2D Fourier series, the Principle of Virtual Power yields an algebraic system of equations, the solution of which provides the Fourier coefficients of the aforementioned series representation. The latter converges, with increasing number of series members, to the true solution for the plate deflections. The new method is applied to relevant problems in pavement engineering, and it is validated through comparison of the numerical results it provides, with predictions obtained from Finite Element analysis. With respect to the latter, the new series-based method reduces the required computer time by a factor ranging from one and a half to almost forty.

A thermodynamically consistent framework to derive local/nonlocal generalized nonassociative plasticity/viscoplasticity theories

We present a rigorous and consistent thermodynamic-based approach to derive generalized nonassociative rate-independent and rate-dependent plasticity theories. It is demonstrated that the nonassociative plasticity theories are a straightforward consequence of the principle of virtual power. Besides, the relationship between the plastic potential function and yield function is established on the basis of the principle of virtual power. All thermodynamic conjugate forces are decomposed into two parts referred to as the energetic and dissipative components. The energetic part is derived from the Helmholtz free energy while the dissipative component is derived from the rate of the dissipated energy by postulating the principle of maximum dissipation. It is also shown that the nonlocality energy residual, arising from plastic strain gradients, has two components: namely, the energetic nonlocality energy residual, which is stored at the interfaces; and the dissipative nonlocality energy residual, which is dissipated through the interfaces. Finally, several examples are provided focusing on the derivation of associative/nonassociative local/nonlocal plasticity and viscoplasticity models based on the presented thermodynamic framework.

A beam formulation based on RKPM for the dynamic analysis of stiffened shell structures

A beam formulation based on reproducing kernel particle method (RKPM) for the dynamic analysis of stiffened shell structures is presented in this paper. The kinematic description of a beam is obtained based on the Timoshenko beam theory. By using the principle of virtual power, the governing equations of a three-dimensional beam are derived. To obtain the numerical model of stiffened shell structures, two schemes are adopted: one is model stiffeners by the RKPM beam formulation, the other one is model the entire stiffened shell by the RKPM shell formulation. In the first scheme, the coupling model of RKPM shell and beam formulation is obtained by adding the corresponding quantities in their governing equations. In the second scheme, by determining the support domain of a stress point according to which component the stress point is located, the full shell simulation is achieved. The reliability and accuracy of those two schemes are verified by several numerical examples.

Using a penalty term to deal with spurious oscillations in second gradient finite elements

It is well known that it is necessary to introduce a length scale parameter in a continuum damage mechanics model to correctly simulate strain localization. The second gradient model, a special case of kinematically enriched continua, considers an internal length parameter by taking into account the second-order derivatives of the displacements in the virtual power principle. In this paper, we show that the original second gradient finite element of Chambon and co-workers can present spurious oscillations, especially for mode I crack propagation problems. After providing the plane stress second gradient constitutive law, we propose to add a penalty term in the original formulation in order to improve numerical convergence and to avoid spurious oscillations in the local variables distributions. Two numerical examples using classical damage mechanics laws, a three-point reinforced concrete beam and a trapezoidal notched specimen are used to test the performance of the formulation. Parametrical studies are also shown on the influence of the penalty parameter. The problem of unrealistic damage spreading for damage values close to 1, occurring often in mode I crack propagation problems, is finally discussed.

Singularity-free path optimization of the parallel test mechanism for artificial hip joints

This paper focuses on the singularity-free optimization for the working path of a 3SPS+1PS type parallel mechanism which is used for the simulation of hip joint motion. Kinematic and dynamic models of the test mechanism were set up with the vector method and the principle of virtual power, respectively. Based on the Jacobian matrix, the kinematic and dynamic performance evaluation indices were defined. The working path of the test mechanism traverses the singular surface twice. Optimization was conducted with the attitude of the virtual reference coordinate system as the design variable, the mean value of the motion transmission performance index and the mean value and the standard deviation of the driving force in the working path as the optimization objectives, and the reachable workspace as the constraint condition. The equivalent working path of the test mechanism after optimization had no singularity. Moreover, the motion transmission performance improved evidently and the driving force changed smoothly.

Variational-based higher-order accurate energy–momentum schemes for thermo-viscoelastic fiber-reinforced continua

In this paper, new higher-order accurate energy–momentum schemes are presented, which rely on a discrete mixed principle of virtual power. The schemes are designed for simulating an uni-directional fiber-reinforced continuum, considered as transversally isotropic nonlinear continuum with independent material behavior in and normal to fiber direction. The matrix material is considered as an isotropic thermo-viscoelastic material and the fibers behave thermo-elastic. Hence, the model takes into account an independent conduction of heat according to Duhamel’s law with a transversally isotropic conductivity tensor as well as an independent heat expansion and heat capacity of the matrix and the fibers. The energy–momentum schemes preserve each balance law of the continuous problem also in the discrete setting, independent of the chosen time step size and the prescribed Neumann and Dirichlet boundary conditions. Therefore, the implemented time step size control with the iteration number as target function does not influence the structure-preservation of the schemes. The balance laws are also preserved together with different time scales in the mechanical, thermal and viscous time evolution. By calculating the generalized reactions on the boundary, numerical examples show energy–momentum consistent dynamic simulations of different transient Dirichlet and Neumann boundary conditions.

Computationally Efficient Inverse Dynamics of a Class of Six-DOF Parallel Robots: Dual Quaternion Approach

Computationally efficient inverse dynamics is crucial to the real-time application of parallel robots. This paper provides a computationally more efficient solution to the inverse dynamics of a class of six-DOF parallel robots based on the dual quaternion approach under the principle of virtual power. A unit dual quaternion is selected as the generalized coordinates of the system. The equations of motion are then constructed by the principle of virtual power. The dual quaternion constraints are eliminated by the null space formulation to obtain the inverse dynamic solution. It is revealed that in the new solution, the Jacobian matrices and the orthogonal complement matrix are all linear with respect to the generalized coordinates. Additionally, the positions, velocities and accelerations of all bodies are quadratic with respect to the generalized coordinates, velocities and accelerations. Such succinct expressions render the new solution computationally more efficient. The execution time of the dual quaternion approach and the traditional one are compared under the same condition by two different six-DOF parallel robots: 6-UPS and 6-PUS. The results show that the new solution can save the computational cost by 43.45% and 38.45% respectively for the two robots, illustrating the effectiveness of the proposed approach.

Dynamic analysis of the rope-sheave system of polar crane for nuclear power plant

A systematic method for analyzing the dynamic of the rope-sheave system of polar crane for nuclear power plant is proposed taking physical structure of the system into account based on the virtual power principle in this paper. The dynamical equations of the spatial ropes are established by spline interpolation based on the spatial description parameters, which could be applied to the rope with variable length. For sheaves of different types, the corresponding essential parameters and the virtual power equations are given. Properties and formulations of the variables involved in these equations are studied in detail. In order to improve the solving efficiency, the high-frequency items are filtered out by replacing the instant strain by time- average strain in integral and the degree of freedoms (DOFs) of the spatial ropes are condensed to the boundary nodes of the ropes. Hence, the total virtual power equation in terms of the sheaves’ parameters is obtained. Two numerical examples, including lowering the load and hoisting the load for different types of rope-sheave systems, are given to verify the rationality and feasibility of the method.

Modeling of surface effects in crystalline materials within the framework of gradient crystal plasticity

A finite-deformation gradient crystal plasticity theory is developed, which takes into account the interaction between dislocations and surfaces. The model captures both energetic and dissipative effects for surfaces penetrable by dislocations. By taking advantage of the principle of virtual power, the surface microscopic boundary equations are obtained naturally. Surface equations govern surface yielding and hardening. A thin film under shear deformation serves as a benchmark problem for validation of the proposed model. It is found that both energetic and dissipative surface effects significantly affect the plastic behavior.

Symbolic multibody methods for real-time simulation of railway vehicles

In this work, recently developed state-of-the-art symbolic multibody methods are tested to accurately model a complex railway vehicle. The model is generated using a symbolic implementation of the principle of virtual power. Creep forces are modeled using a direct symbolic implementation of the standard linear Kalker model. No simplifications, such as base parameter reduction, partial-linearization or lookup tables for contact kinematics, are used. An Implicit–Explicit integration scheme is proposed to efficiently deal with the stiff creep dynamics. Real-time performance is achieved: the CPU time required for a very robust $1~\text{ms}$ integration time step is 203 μs.

A Phase-Field Damage Model for Orthotropic Materials and Delamination in Composites

A phase-field damage model for orthotropic materials is proposed and used to simulate delamination of orthotropic laminated composites. Using the deviatoric and hydrostatic tensile components of the stress tensor for elastic orthotropic materials, a degraded elastic free energy that can accommodate damage is derived. The governing equations follow from the principle of virtual power and the resulting damage model, by its construction, conforms with the physical relevant condition of no matter interpenetration along the crack faces. The model also dispenses with the traction separation law, an extraneous hypothesis conventionally brought in to model the interlaminar zones. The model is assessed through numerical simulations on delaminations in mode I, mode II, and another such problem with multiple initial notches. The present method is able to reproduce nearly all the features of the experimental load displacement curves, allowing only for small deviations in the softening regime. Numerical results also show forth a superior performance of the proposed method over existing approaches based on a cohesive law.

A thick shell model based on reproducing kernel particle method and its application in geometrically nonlinear analysis

A meshfree approach to the simulation of the large deformation of a curved shell by the reproducing kernel particle method (RKPM) is presented. Since the kinematic description is based on the Mindlin–Reissner shell theory, only one layer of particles is needed to model the shell and the time increment is not limited by the shell thickness. The reproducing interpolation function is adopted to discretize the kinematic quantities of the shell; thus, the spatial discretization is independent of the finite element mesh, so it can address large deformations without mesh distortion. The governing equation of an arbitrary curved shell is derived in detail based on the principle of virtual power, for which reasonable simplifications have been taken. The Lagrangian kernel and stress points are adopted in the calculation, which are sufficient to eliminate instability. Several numerical examples are performed, verifying the reliability and numerical accuracy of the RKPM shell model. No locking is observed in the numerical solutions.

Type synthesis and analysis of parallel mechanisms with sub-closed-loops

For the special needs of attitude adjustment and vibration isolation of precise instruments, parallel mechanisms with sub-closed-loops (PMWS) are proposed to eliminate the interferences that vary in frequency. First, the multi-drive units composed by sub-closed-loops are synthesized by two derivative methods, and the PMWS with diffident DOF are synthesized based on screw theory. Furthermore, proposing the concept of the sub generalized coordinate, and the first and the second order influence coefficients of each link of PMWS are established under the generalized coordinates and the sub generalized coordinates. Besides, based on the principle of virtual power and the Newton-Euler formula, meanwhile combining the kinematics analysis of PMWS, the dynamics formulas of PMWS are established separately in joint space and operational space. Finally, taking 6-(PRRR)US PMWS as numerical example, by comparing the results of theoretical calculation and simulation, the kinematics and dynamics formulas of PMWS are verified.

Deployment and control of cable-driven flexible solar arrays

PurposeThe purpose of this study is to investigate the deployment and control of cable-driven flexible solar arrays.Design/methodology/approachFirst, dynamic model of the system is established by using the Jourdain’s velocity variation principle and the single direction recursive construction method, including the dynamic equation of a single flexible body, the kinematical recursive relation of two adjacent flexible bodies and the dynamic equation of the solar array system. Then, the contribution of joint friction to the dynamic equation of the system is derived based on the virtual power principle. A three-dimensional revolute joint model is introduced and discussed in detail. Finally, a proportion-differentiation (PD) controller is designed to control the drift of the system caused by the deployment.FindingsSimulation results show that the proposed model is effective to describe the deployment of flexible solar arrays, joint friction may affect the dynamic behavior of the system and the PD controller can effectively eliminate the spacecraft drift.Practical implicationsThis model is useful to indicate the dynamics behavior of the solar array system with friction.Originality/valueThe relationship between ideal constraint force and Lagrange multipliers is derived. The contribution of joint friction to the dynamic equation of the system is derived based on the virtual power principle. A PD controller is designed to control the drift of the system caused by the deployment of solar arrays.

A new strain-gradient theory for an isotropic plastically deformed polycrystalline solid body

This study considers strain-gradient plasticity theory in the context of small deformations for an isotropic solid body with a view to investigating the distortion effects associated with the divergence of plastic strain through the Burgers tensor. The principle of virtual power is employed and the constraint of irrotationality is imposed on the plastic component of the gradient of the displacement vector. It is obtained that the gradient, curl, and divergence of the plastic strain in the body are mutually related. This relation establishes the existence of work done through the divergence of plastic strain as distinct from the work done through the gradient of the plastic strain. Consequently, a polycrystalline solid body undergoing distortion associated with the divergence of plastic strain exhibits new internal microstresses; and the obtained model, consisting of the microforce balance, constitutive relations, and plastic flow rule, extends the known Gurtin–Anand model in a natural fashion. Furthermore, in the governing flow rule, it is revealed that the internal microstresses associated with the divergence of plastic strain act as opposing agents to the internal microstresses associated with the gradient of the plastic strain via the length scales Q, L, and the gradient of the divergence of the plastic strain. This work shows the distortion effects associated with the divergence of plastic strain which the Gurtin–Anand strain-gradient plasticity theory in literature does not apprehend.

A hybrid analysis method for displacement-monitored segmented circular tunnel rings

Segmented tunnel rings exhibit load-induced interfacial dislocations. In order to facilitate structural analysis, a hybrid method is developed and applied to a real-scale test of a segmented tunnel ring. Point loads, imposed on the tested ring, and measured interfacial discontinuities serve as input for the analysis. Moreover, the method accounts for the structural behavior of the individual segments by means of newly derived transfer relations. They represent analytical solutions of the first-order theory of slender circular arches, exhibiting constant cross-sectional properties. The tool for the development of this basically well-known theory is the principle of virtual power. Its involvement is motivated by the possibility of a mechanically consistent derivation of relations, some of which have been used for a long time without analyzing their scientific background. The validity and the usefulness of the transfer relations follow from a comparison of newly derived solutions with (i) alternative analytical solutions, (ii) Finite Element solutions, and (iii) experimental data. The computational efficiency and the usefulness of the developed hybrid method are demonstrated by structural analysis of a segmented tunnel ring. It provides valuable insight into the load-carrying behavior of the tested structure without the need to describe the nontrivial behavior of segment-to-segment interfaces.

Hybrid modeling for dynamic analysis of cable-pulley systems with time-varying length cable and its application

The dynamic analysis of cable-pulley systems is investigated in this paper, where the time-varying length characteristic of the cable as well as the coupling motion between the cable and the pulleys are considered. The dynamic model for cable-pulley systems are presented based on the principle of virtual power. Firstly, the cubic spline interpolation is adopted for modeling the flexible cable elements and the virtual 1powers of tensile strain, inertia and gravity forces on the cable are formulated. Then, the coupled motions between the cable and the movable or fixed pulley are described by the input and output contact points, based on the no-slip assumption and the spatial description. The virtual powers of inertia, gravity and applied forces on the contact segment of the cable, the movable and fixed pulleys are formulated. In particular, the internal node degrees of freedom of spline cable elements are reduced, which results in that only the independent description parameters of the nodes connected to the pulleys are included in the final governing dynamic equations. At last, two cable-pulley lifting mechanisms are considered as demonstrative application examples where the vibration of the lifting process is investigated. The comparison with ADAMS models is given to prove the validity of the proposed method.

Mathematical model for isometric and isotonic muscle contractions

A new mathematical model is presented to describe both the active and passive mechanics of muscles. In order to account for the active response, a two–layer kinematics that introduces both the visible and rest lengths of the muscle is presented within a rational mechanics framework. The formulation is based on an extended version of the principle of virtual power and the dissipation principle. By using an accurate constitutive description of muscle mobility under activation, details of microscopic processes that lead to muscle contraction are glossed over while macroscopic effects of chemical/electrical stimuli on muscle mechanics are retained. The model predictions are tested with isometric and isotonic experimental data collected from murine extensor digitorum muscle. It is shown that the proposed model captures experimental observations with only three scalar parameters.

Finite element procedure for one-dimensional viscoplastic problems of an isotropic slab

This study, based on a modified form of the Gurtin–Anand model (J Mech Phys Solids 53:1624–1649, 2005), investigates the effect of an internal microforce on the flow rule of viscoplastic bodies. A generalized microforce balance is obtained via the virtual power principle, and using the free-energy imbalance to suitably restrict the constitutive relations, a flow rule which accommodates two energetic length scales is obtained. This leads to a nonlinear second-order partial differential equation in the plastic strain. Finite element models for one-dimensional problems arising from a pure shear state of a viscoplastic slab are derived, while solutions for linear rate-dependent and nonlinear small strain-gradient rate-independent problems are presented.

An RVE-based multiscale modeling method for constitutive relations

This article reports an efficient method to characterize constitutive responses based on multiscale modeling for fluid flow in heterogeneous media based on the concept of representative volume element (RVE). Between different scales, it is considered as the basic principles for down-scaling information the conservation of velocity and of the strain rate tensor. Within this context, we formulate (i) the problem to be solved at the micro-scale, (ii) the up-scaling procedure which involves homogenization rules, and (iii) the generalized principle of multiscale virtual power. The complete theory for constitutive modeling is revisited and shown that when employing multiscale analysis among the suitable variational arguments we are able to obtain, in a straightforward manner, new constitutive behavior between kinematic motions and actions. Some examples of application of fluid flow in heterogeneous media with obstacles are presented to show the consequences of the proposed approach.