Principle Of Virtual Power Research Articles

Energy analysis of face stability of deep rock tunnels using nonlinear Hoek-Brown failure criterion

The nonlinear Hoek-Brown failure criterion was introduced to limit analysis by applying the tangent method. Based on the failure mechanism of double-logarithmic spiral curves on the face of deep rock tunnels, the analytical solutions of collapse pressure were derived through utilizing the virtual power principle in the case of pore water, and the optimal solutions of collapse pressure were obtained by using the optimization programs of mathematical model with regard of a maximum problem. In comparison with existing research with the same parameters, the consistency of change rule shows the validity of the proposed method. Moreover, parametric study indicates that nonlinear Hoek-Brown failure criterion and pore water pressure have great influence on collapse pressure and failure shape of tunnel faces in deep rock masses, particularly when the surrounding rock is too weak or under the condition of great disturbance and abundant ground water, and in this case, supporting measures should be intensified so as to prevent the occurrence of collapse.

Upper bound analytical solution for surrounding rock pressure of shallow unsymmetrical loading tunnels

By combining the results of laboratory model tests with relevant flow rules, the failure mode of shallow unsymmetrical loading tunnels and the corresponding velocity field were established. According to the principle of virtual power, the upper bound solution for surrounding rock pressure of shallow unsymmetrical loading tunnel was derived and verified by an example. The results indicate that the calculated results of the derived upper bound method for surrounding rock pressure of shallow unsymmetrical loading tunnels are relatively close to those of the existing “code method” and test results, which means that the proposed method is feasible. The current code method underestimates the unsymmetrical loading feature of surrounding rock pressure of shallow unsymmetrical loading tunnels, so it is unsafe; when the burial depth is less or greater than two times of the tunnel span and the unsymmetrical loading angle is less than 45°, the upper bound method or the average value of the results calculated by the upper bound method and code method respectively, is comparatively reasonable. When the burial depth is greater than two times of the tunnel span and the unsymmetrical loading angle is greater than 45°, the code method is more suitable.

A generalized continuum theory with internal corner and surface contact interactions

We consider a classical derivation of a continuum theory, based on the fundamental balance laws of mass and momenta, for a body with internal corner and surface contact interactions. The balances of mass and linear and angular momentum are applied to the arbitrary parts of a continuum which supports non-classical internal corner and surface contact interactions. The form of the specific corner contact interaction force measured per unit length of the corner is derived. A generalized form of Cauchy’s stress theorem is obtained, which shows that the surface traction on an oriented surface depends in a specific way on both the oriented unit normal as well as the curvature of the surface. An explicit form of the surface-couple traction which acts on every oriented surface is obtained. Two fields in the continuum, which are denoted as stress and hyperstress fields, are shown to exist, and their role in representing the surface traction and the surface-couple traction is identified. Finally, the field equations for this theory are determined, and a fundamental power theorem is derived. In the absence of internal corner and surface-couple traction interactions, the equations of classical continuum mechanics are recovered. There is no appeal to any ‘principle of virtual power’ in this work.

From the Euler–Bernoulli beam to the Timoshenko one through a sequence of Reddy-type shear deformable beam models of increasing order

Abstract A sequence of elastic Reddy-type shear deformable beams of increasing (odd) order is envisioned, which starts with the Euler–Bernoulli beam (first order) and terminates with the Timoshenko beam (infinite order). The kinematics of the generic beam, including the warping mode of the cross sections, is specified in terms of three deformation variables (two curvatures, one shear angle), work-conjugate of as many stress resultants (two bending moments, one shear force). The principle of virtual power is used to determine the (static) equilibrium equations and the boundary conditions. The equations relating the bending moments and shear force to the curvatures and shear angle are also reported. Two governing differential equations (equivalent to a deflection differential equation of the sixth order) are provided, along with their general solution to within six unknown constants; the solutions for the Euler–Bernoulli and Timoshenko beams are recovered for first and infinite orders, respectively. A principle of minimum total potential energy is shown to hold for the generic beam. Extended forms of the classical Navier formula for the computation of the (axial) normal stress, and of the Jourawski formula for the computation of the “equilibrium” shear stress, are provided. A simple cantilever beam is used for an illustrative application.

Energy analysis of stability on shallow tunnels based on non-associated flow rule and non-linear failure criterion

On the basis of upper bound theorem, non-associated flow rule and non-linear failure criterion were considered together. The modified shear strength parameters of materials were obtained with the help of the tangent method. Employing the virtual power principle and strength reduction technique, the effects of dilatancy of materials, non-linear failure criterion, pore water pressure, surface loads and buried depth, on the stability of shallow tunnel were studied. In order to validate the effectiveness of the proposed approach, the solutions in the present work agree well with the existing results when the non-associated flow rule is reduced to the associated flow rule and the non-linear failure criterion is degenerated to the linear failure criterion. Compared with dilatancy of materials, the non-linear failure criterion exerts greater impact on the stability of shallow tunnels. The safety factor of shallow tunnels decreases and the failure surface expands outward when the dilatancy coefficient decreases. While the increase of nonlinear coefficient, the pore water pressure coefficient, the surface load and the buried depth results in the small safety factor. Therefore, the dilatancy as well as non-linear failure criterion should be taken into account in the design of shallow tunnel supporting structure. The supporting structure must be reinforced promptly to prevent potential mud from gushing or collapse accident in the areas with abundant pore water, large surface load or buried depth.

FEM Techniques Comparison for SIF Computing of Cracked Plate

The main purpose of this paper is to perform a computation comparison of stress intensity factor ‘SIF’ evaluation in case of cracked thin plate with aluminum alloy 7075-T6 and 2024-T3 used in aeronautics structure under uniaxial loading. This evaluation is based on finite element method with a virtual power principle through two techniques: the extrapolation and \({G - \theta}\) . The first one consists to extrapolate the nodal displacements near the cracked tip using a refined triangular mesh with T3 and T6 special elements, while the second, consists to determine the energy release rate G through \({G - \theta}\) method by potential energy derivation which corresponds numricaly to elastic solution post-processing of a cracked solid by a contour integration computation via Gauss points. The SIF obtained results from extrapolation and \({G - \theta}\) methods will be compared to analytical solution in particular case. To illustrate the influence of the meshing kind and the size of integration contour position, simulations are presented and analyzed.

A multi-mechanism model for cutting simulations based on the concept of generalized stresses

In this work we extend an existing model for strain rate- and temperature-dependent asymmetric plastic material behavior accompanied by phase transformation with a gradient term of phase fraction based on the concept of generalized stresses as proposed by Gurtin and Forest. To this end a chemical variable, representing the austenite volume fraction, is treated as an extra degree of freedom, and the influence of its first gradient will be studied. A generalized principle of virtual power is postulated involving generalized stresses and used to derive the constitutive equations. The bulk model is formulated within a thermodynamic framework. For the scenario of a cutting process we have a martensite-austenite–martensite transformation. For the asymmetric visco-plastic part we use a modified strain rate form (Huh–Kang form) with a quadratic dependence of strength on the logarithm of the strain rate as a replacement for the linear dependence of the Johnson–Cook model. Furthermore, two numerical examples are given. Firstly, we identify the material parameters for the material DIN 100Cr6 in a hardened state by using experimental data under tension, compression and torsion taking the SD-effect into account. Secondly, a cutting simulation is investigated to test our model and the different mechanisms are illustrated.

Determination of Flow Curve by Cylindrical Upsetting Based on the Principle of Virtual Power

In the design of metalworking technologies, one of the most important amount is the flow stress of the material. A few experimental method can be used to measure it. The most commonly used tests to determine the flow curve of materials include upsetting of cylindrical or prismatic specimens, tension or torsion tests. In our work, the determination method of flow curve were investigated based on the principle of virtual power with cold working conditions. The experimental method, cylindrical upsetting, was executed on the Gleeble 3800 thermo-mechanical simulator. The goal was to determine reliable flow curve for large plastic strain. The results showed that reliable flow stress values were obtainable in equivalent plastic strain range of 0 to 0.7 by continuous cylindrical upsetting.

Macroscopic and mesoscopic modeling based on the concept of generalized stresses for cutting simulation

AbstractBased on the concept of generalized stresses proposed by GURTIN [2] and FOREST et al. [1] macro‐ and meso‐scopic modelling are presented. For the macroscopic modelling we develop a multi‐mechanism model for strain rate and temperature dependent asymmetric plastic material behavior accompanied by phase transformation with consideration of the trip‐strain. Furthermore, we extend the multi‐mechanism model with the gradient of phase fraction, which is considered as an extra degree of freedom. For mesoscopic modelling a phase field model is implemented for describing phase transformations. For the scenario of a cutting process we have a martensite‐austenite‐martensite transformation. A generalized principle of virtual power is postulated involving generalized stresses and used to derive the constitutive equations for both approaches. Furthermore, parameters of the multi‐mechanism model related to visco‐plasticity with SD‐effect and the trip‐strain are identified for the material DIN 100Cr6. In the examples a cutting simulation for testing the multi‐mechanism model and a phase‐transformation simulation are shown. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Vehicle and powertrain dynamics analysis with an automatic gearbox

This paper presents an effective formulation to model an automatic transmission for evaluating the vehicle performance in the early development stage of a powertrain design. The corresponding equations of motion are obtained with the help of the virtual power principle, involving all rotating parts of the gearbox. Del Castillo's method is used to efficiently establish the analytical expression of partial gear ratios. A set of penalty functions are associated with the clutches and brakes allowing a continuous simulation of the gear shift. The acceleration of the vehicle is calculated by including the gearbox model in the equations governing the longitudinal behaviour of a passenger car, including the motor, a realistic model of the tire and classical road loads. This formulation offers a powertrain/vehicle dynamic model developed for the sake of simplicity (rigid interconnected bodies with kinematic constraints), and sufficiently efficient to simulate continuously the gear shifts. Two realistic cases are presented, namely Chrysler 45RFE and Aisin Warner 55-50 SN powertrains. For the latter, a validation is made through a comparison with measured data. The developed model reveals itself as a valuable tool for simulating the implementation of different control laws governing the gear shifts.

The stored energy of cold work, thermal annealing, and other thermodynamic issues in single crystal plasticity at small length scales

This paper develops a thermodynamically consistent gradient theory of single-crystal plasticity using the principle of virtual power as a paradigm to develop appropriate balance laws for forces and energy. The resulting theory leads to a system of microscopic force balances, one balance for each slip system, and to an energy balance that accounts for power expended during plastic flow via microscopic forces acting in concert with slip-rates and slip-rate gradients. Central to the theory are an internal energy and entropy, plastic in nature, dependent on densities that account for the accumulation of glide dislocations as well as geometrically necessary dislocations – and that, consequently, represent quantities associated with cold work. Our theory allows us to discuss – within the framework of a gradient theory – the fraction of plastic stress-power that goes into heating, as well as the reduction of the dislocation density in a cold-worked material upon subsequent (or concurrent) thermal annealing.

Upper bound limit analysis of support pressures of shallow tunnels in layered jointed rock strata

With more and more large-span shallow underground projects being built in layered jointed rock strata, there is an increasing interest in the study of stability problems of such projects. The upper bound theorem of limit analysis has been found to be a valid method for analyzing such problems and it is crucial to ensure the accuracy of the failure mode when applying this method. In the present paper, for tunnels in layered jointed rock strata, 8 factors including the thickness of the weathered overburden, the states of the rock joints, and the depth of the tunnel were considered by analyzing the failure mode. By using the orthogonal array testing strategy and the distinct element method, 64 different numerical simulation models based on an actual tunnel project were simulated. The test results showed that the range of the loose zone was determined by the stretch, slippage, and fracture area of the joints. Two kinds of loose zone were obtained, i.e. the arch collapse loose zone and the caving collapse loose zone. The loose zone boundary was analyzed and fitted with linear and non-linear models; such as parabola and power function curves, respectively. The rational failure mode was studied by analyzing the recognition method, the classification, and the failure mechanism of the caving collapse loose zone. By using the liner least squares method, an empirical formula of the boundary of the caving collapse loose zone was obtained, and its boundary were divided equally with identical angles, then a kinematically admissible velocity field was established. Based on the virtual power principle and virtual work principle, the upper bound theorem of limit analysis was adopted to study the support pressures and a limit analysis model was established. A real tunnel project case was studied as an example, and the results were compared with the Terzaghi theory and the Fraldi theory, so as to demonstrate the applicability of the proposed method.

Mechanics Modeling of Multisegment Rod-Driven Continuum Robots

This paper presents a novel modeling approach for the mechanics of multisegment, rod-driven continuum robots. This modeling approach utilizes a high-fidelity lumped parameter model that captures the variation in curvature along the robot while simultaneously defined by a discrete set of variables and utilizes the principle of virtual power to formulate the statics and dynamics of the continuum robot as a set of algebraic equations for the static model and as a set of coupled ordinary differential equations (ODEs) in time for the dynamic model. The actuation loading on the robot by the actuation rods is formulated based on the calculation of contact forces that result in rod equilibrium. Numerical optimization calculates the magnitudes of these forces, and an iterative solver simultaneously estimates the robot's friction and contact forces. In addition, modeling considerations including variable elastic loading among segments and mutual segment loading due to rods terminating at different disks are presented. The resulting static and dynamic models have been compared to dynamic finite element analyses and experimental results to validate their accuracy.

Continuum thermomechanics of the nonlocal granular rheology

We formulate a nonlocal, or scale-dependent, elasto-viscoplastic continuum model for granular materials, consistent with the principles of modern continuum thermomechanics. Importantly, the theory contains a scalar, energetic order parameter, referred to as the granular fluidity. We assume power to be expended over the rate of change of the fluidity and its gradient and undertake a derivation based upon the principle of virtual power in the style of Gurtin (1996). This approach results in a non-standard microforce balance, which when combined with our choice of specific constitutive equations, takes the form of a partial differential relation that the fluidity must obey. Finally, we simplify the equations into a form appropriate for steady granular flows. The resulting boundary-value problem was previously shown to be capable of describing a wide array of experimental granular flow data.

A non-linear one-dimensional model of cross-deformable tubular beam

A direct non-linear one-dimensional model of an elastic, thin-walled, planar beam is formulated. The model accounts for changes in shape of the cross-section, in particular the ovalization (or flattening) occurring in tubular beams. The deformation of the cross-section is described in the spirit of the Generalized Beam Theory, as a linear combination of known deformation modes and unknown amplitude functions, said to be distortions. Kinematics calls for introducing distortional and bi-distortional strains, in addition to the usual strain measures of rigid cross-section beams. The balance equations are derived through the Virtual Power Principle, in which distortional and bi-distortional stresses, as well as distortional forces, are defined as conjugate quantities of distortional strain-rates and velocities, respectively. A non-linear, fully coupled, hyperelastic law is assumed. All the distortional quantities and the constitutive law are identified, via energy equalities, from a three-dimensional fiber-model of thin-walled beam where, for simplicity, just a distortion mode is considered. The model is specialized to a Euler–Bernoulli tubular beam, in which only constitutive non-linearities are retained, while kinematics is linearized. The relevant non-linear equations are solved, via a perturbation method, for several static loadings and for large-amplitude free vibrations. The interaction occurring between global bending and cross-section distortion is analyzed.

The virtual power principle in fluid mechanics

AbstractA conceptual framework on analytical mechanics for continuous fluid medium, which connects the fluid motion and all of the (internal and external) forces with mechanical power, is proposed by using the virtual power and the virtual velocity. Based on this framework, it is found that the internal virtual power is equal to the external virtual power in fluid dynamics, which is called the virtual power principle. This framework is also proved to be equivalent to the vector dynamics (Cauchy’s equation or Navier–Stokes equation). Furthermore, based on the virtual power principle, a theorem is introduced for continuous fluid medium, which indicates the relationship between the force (or torque) acting on a body immersed in a fluid and the specified virtual power. Subsequently, according to Galilean invariance, the detailed relationship for Newtonian fluids in incompressible flows is derived and used to illustrate the mechanisms on instantaneous forces: the added inertial effects, the boundary energy flux and dissipation effects, the vortex contribution, and the explicit body force contribution. As an application of the principle, the advantage of the V formation flight of geese is preliminarily discussed in the view of aerodynamics. Specifically, the total drag of the flock is reduced by contrast with the simple sum of the drag in solo fight and the optimal angle of V ranges from$60^{\circ }$to$120^{\circ }$. The principle could be a useful approach to reveal the contributions of the flow structures and the moving or deforming boundaries to the force and torque acting on a body, especially in a multibody system.

Continuum Robot Dynamics Utilizing the Principle of Virtual Power

Efficient formulations for the dynamics of continuum robots are necessary to enable accurate modeling of the robot's shape during operation. Previous work in continuum robotics has focused on low-fidelity lumped parameter models, in which actuated segments are modeled as circular arcs, or computationally intensive high-fidelity distributed parameter models, in which continuum robots are modeled as a parameterized spatial curve. In this paper, a novel dynamic modeling methodology is studied that captures curvature variations along a segment using a finite set of kinematic variables. This dynamic model is implemented using the principle of virtual power (also called Kane's method) for a continuum robot. The model is derived to account for inertial, actuation, friction, elastic, and gravitational effects. The model is inherently adaptable for including any type of external force or moment, including dissipative effects and external loading. Three case studies are simulated on a cable-driven continuum robot structure to study the dynamic properties of the numerical model. Cross validation is performed in comparison to both experimental results and finite-element analysis.

On Forces in Magnetized Matter

In magnetoelastic interactions, the virtual power principle is powerful enough to solve all force problems, but only if the coupled constitutive laws are completely known. We use it to derive force densities in magnetized matter for a few such laws, insisting on the fact that a full knowledge of the magnetic field may not be enough to determine the force density when the local B-H law depends on the local strain of the material.

Remodelling in statistically oriented fibre-reinforced materials and biological tissues

We present a mathematical model of structural reorganisation in a fibre-reinforced composite material in which the fibres are oriented statistically, i.e. obey a probability distribution of orientation. Such a composite material exemplifies a biological tissue (e.g. articular cartilage or a blood vessel) whose soft matrix is reinforced by collagen fibres. The structural reorganisation of the composite takes place as fibres reorient, in response to mechanical stimuli, in order to optimise the stress distribution in the tissue. Our mathematical model is based on the Principle of Virtual Powers and the study of dissipation. Besides incompressibility, our main hypothesis is that the composite is characterised by a probability density distribution that measures the probability of finding a family of fibres aligned along a given direction at a fixed material point. Under this assumption, we describe the reorientation of fibres as the evolution of the most probable direction along which the fibres are aligned. To test our theory, we compare our simulations of a benchmark problem with selected results taken from the literature.

The configuration space and principle of virtual power for rough bodies

In the setting of an n-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat (n − 1)-chains. Initially, bodies are modeled as normal n -currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of (n − 1)-currents and locally Lipschitz mappings. A version of Cauchy’s postulates implies that a Cauchy flux may be uniquely extended to an n-tuple of flat (n − 1)-cochains. Thus, the class of admissible bodies is extended to include flat n-chains and a generalized form of the principle of virtual power is presented. Wolfe’s representation theorem for flat cochains enables the identification of stress as an n-tuple of flat (n − 1)-forms representing the flat (n − 1)-cochains associated with the Cauchy flux.