Saint-Venant Principle Research Articles

Numerical study of the applicability of the Saint-Venant principle

The limits of the applicability of the Saint-Venant principle are investigated while using the LIRA-CAD app. A large number of bars loaded at the end with different loads are considered. A graph of the distribution of the zones of influence of the Saint-Venant principle is plotted, depending on the size of the bars. It can be concluded that the stress-strain state near the perturbation zones can be represented by three components, two of which are linear, and the third is self-balanced in force and moment.

Nonlinear Dynamic Response of Carbon Nanotube Nanocomposite Microbeams

The nonlinear dynamic response of nanocomposite microcantilevers is investigated. The microbeams are made of a polymeric hosting matrix (e.g., epoxy, polyether ether ketone (PEEK), and polycarbonate) reinforced by longitudinally aligned carbon nanotubes (CNTs). The 3D transversely isotropic elastic constitutive equations for the nanocomposite material are based on the equivalent inclusion theory of Eshelby and the Mori–Tanaka homogenization approach. The beam-generalized stress resultants, obtained in accordance with the Saint-Venant principle, are expressed in terms of the generalized strains making use of the equivalent constitutive laws. These equations depend on both the hosting matrix and CNTs elastic properties as well as on the CNTs volume fraction, geometry, and orientation. The description of the geometry of deformation and the balance equations for the microbeams are based on the geometrically exact Euler–Bernoulli beam theory specialized to incorporate the additional inextensibility constraint due to the relevant boundary conditions of microcantilevers. The obtained equations of motion are discretized via the Galerkin method retaining an arbitrary number of eigenfunctions. A path following algorithm is then employed to obtain the nonlinear frequency response for different excitation levels and for increasing volume fractions of carbon nanotubes. The fold lines delimiting the multistability regions of the frequency responses are also discussed. The volume fraction is shown to play a key role in shifting the linear frequencies of the beam flexural modes to higher values. The CNT volume fraction further shifts the fold lines to higher excitation amplitudes, while it does not affect the backbones of the modes (i.e., oscillation frequency–amplitude curves).

Multiscale GFEM with superposition of crack enrichment functions driven by finite fracture mechanics: Theory, first computation and open problems

This work presents a direct extension of the multiscale GFEM (MS-GFEM) where damage and fracture mechanisms are now taken into account. Using a pattern-based description of the microscale, the construction of the model falls into two main parts: the modeling of a multiphase composite yarn, and the modeling of crack events inside the microstructure. The classical difficulties associated with the implementation and computation cost of the GFEM are dealt through a multiscale approach based on the Saint–Venant principle. This multiscale vision enables the microstructure and the microkinematics to be handled on the scale of the pattern independently of the macroscale discretization. Additionally, numerical enrichment functions are superimposed between the existing enrichment functions in a local computational domain. These enrichment functions deliver accurate solutions of crack interactions among local features, like inclusions, voids, etc, and are specifically appealing to evolution type problems like the crack events defined under the finite fracture mechanics framework. Different crack patterns can be introduced at different positions in the global domain when certain fracture criteria are met. The multiscale approach is implemented inside the finite element software Cast3M in order to use the existing infrastructure of the object oriented code for the development of MS-GFEM.

Dynamic Saint-Venant principle for rotor system with undetectable initial crack

This paper dedicates on the analysis of the initially cracked rotor system and the ratiocination on the reason why the initial crack is undetectable via signal processing method using dynamic Saint-Venant principle (DSVP). A numerical simulation is intensively conducted to justify the theoretical analysis. The conclusion has been obtained that the dynamic resultant of the initial crack force equals to zero in one period and the stress wave excited by the initial crack force will not interfere the lubrication of the supporting bearings when the frequency of the initial crack force preserves to be less than the limited frequency. This conclusion indicates that the undetectable initially cracked rotor system satisfies the criterions of DSVP, and the health monitoring method using ultrasonic wave should be adopted for initial crack diagnosis.

A Thermal Stress Analysis of Beams with Out-of-Plane Warping

In this paper, a methodology, which is able to predict the thermal stresses accurately yet efficiently, is presented for beam structures via Saint-Venant's principle. In general, higher-order beam theories have been known to be effective for the prediction of thermal stresses. In contrast to this, we propose the method to predict the thermal stresses of beam structures by post-processing the classical beam theory via Saint-Venant's principle. The approach includes an out-of-plane warping displacement to account for the through-the-thickness thermal deformation. With this, one can accurately recover the thermal stresses as compared to the elasticity solutions. In fact, they are identical for the beams made of isotropic materials. The effect of out-of-plane warping is also investigated, it turns out that the effect is negligible in mechanical stress analysis but not in thermal stress analysis.

Three-dimensional analytical solutions for the axisymmetric bending of functionally graded annular plates

Based on the three-dimensional theory, the axisymmetric bending behavior of functionally graded annular plates made of magneto-electro-elastic, piezoelectric and elastic materials respectively, was investigated. The transverse loads were expanded in terms of a Fourier–Bessel series and the solutions corresponding to each item of the series were found by the direct displacement method. Then the superposition principle was utilized to determine the mechanical, electric and magnetic quantities of the annular plates subjected to the arbitrary transverse pressures. It was found that four unusual boundary conditions can be rigorously satisfied at the inner and outer circumferential edges by this method. Meanwhile, the common boundary conditions such as simply supported, clamped and free boundary conditions were approximately satisfied using the Saint-Venant's principle. An innovative isoparametric element was also developed to calculate the axisymmetric behavior of the functionally graded magneto-electro-elasticity for comparison. Numerical examples were finally given to validate the present method.

On the determination of local residual stress gradients by the slit milling method

This paper concentrates on an extension of a rather new methodology to determine local residual gradients at an enhanced lateral resolution using the so-called slit milling technique. The method is based on stress relaxation by making a slit using a focused ion beam and the displacements are measured through digital image correlation so as to calculate the residual stress. Our novel approach consists of a multiple fitting procedure along the length of the slit instead of the commonly applied averaging method. The applicability of our approach is demonstrated when stress gradients exist. In accordance to the Saint-Venant principle in linear elasticity, longer slits are better than shorter slits because of end effects. The proof-of-principle is supported by measurements on steel under controlled bending and by finite element modeling.

Spatial behavior in phase-lag heat conduction

In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and three-phase-lag theories, reflecting Saint-Venant's principle. Depending on the relative order of derivation, with respect to the time, we propose different arguments. One is inspired by the arguments for parabolic problems and the other is inspired by the arguments for hyperbolic problems. In the first case, we obtain a Phragmén-Lindelöf alternative for the solutions. In the second case, we obtain an estimate for the decay as well as a domain of influence result. The main tool to manage these problems is the use of an exponentially weighted Poincaré inequality.

Asymptotic analysis on flexural dynamic characteristics for a sandwich plate with periodically perforated viscoelastic damping material core

The flexural dynamic characteristics of the cored sandwich plate with periodically perforated viscoelastic damping material (VDM) are analyzed by considering the frequency and temperature relationship in this paper. First, the mechanism of the sandwich plates is developed by using the Saint–Venant principle and the flexural and shearing deformation compatible equations. Next, the deflection equation of the sandwich plate is established based on the principle of flexural deflection of isotropy plate, and then, it is expanded for easy solution by the second order “rapid” asymptotic method. The opening ratio and the thickness of the core layer which affects the loss factor of sandwich plate are thoroughly researched. Finally, the validity is performed and the advantage of the perforated structure in industrial application is presented.

Configuration Modeling and Experimental Verification with 3D Laser Scanning Technology for a Constrained Elastica Cable

For large complex electromechanical products, cable digital design has a problem of low accuracy because of its large deformable characteristics. To solve the problem, an efficacious method is proposed to do cable geometry physical modeling, simulation and test for two-end constrained spatial flexible cables. Euler angles are used as generalized parametric variables to describe cable configuration on the base of Kirchhoff elastic rod theory. Via establishment of a cylindrical coordinate system according to Saint-Venant principle using end constraint, Euler angles’ Analytical solutions of the Kirchhoff equations are gained. Based on an open source geometry kernel system and using numerical integration ideas, a cable configuration simulation platform is developed in C++ environments. Then, a cable geometry measurement bench is founded with the use of 3D laser scan technology. Finally, a cable configuration simulation and experimental example is given to verify the accuracy of the proposed model and relative algorithm, through 3D positional deviation analysis.

Three-Dimensional Beam Theory for Flexible Multibody Dynamics

In multibody systems, it is common practice to approximate flexible components as beams or shells. More often than not, classical beam theories, such as the Euler–Bernoulli beam theory, form the basis of the analytical development for beam dynamics. The advantage of this approach is that it leads to simple kinematic representations of the problem: the beam's section is assumed to remain plane and its displacement field is fully defined by three displacement and three rotation components. While such an approach is capable of accurately capturing the kinetic energy of the system, it cannot adequately represent the strain energy. For instance, it is well known from Saint-Venant's theory for torsion that the cross-section will warp under torque, leading to a three-dimensional deformation state that generates a complex stress state. To overcome this problem, sectional stiffnesses are computed based on sophisticated mechanics of material theories that evaluate the complete state of deformation. These sectional stiffnesses are then used within the framework of a Euler–Bernoulli beam theory based on far simpler kinematic assumptions. While this approach works well for simple cross-sections made of homogeneous material, inaccurate predictions may result for realistic configurations, such as thin-walled sections, or sections comprising anisotropic materials. This paper presents a different approach to the problem. Based on a finite element discretization of the cross-section, an exact solution of the theory of three-dimensional elasticity is developed. The only approximation is that inherent to the finite element discretization. The proposed approach is based on the Hamiltonian formalism and leads to an expansion of the solution in terms of extremity and central solutions, as expected from Saint-Venant's principle.

New insights on free energies and Saint-Venant’s principle in viscoelasticity

Explicit expressions for the minimum free energy of a linear viscoelastic material and Noll’s definition of state are used here to explore spatial energy decay estimates for viscoelastic bodies, in the full dynamical case and in the quasi-static approximation.In the inertial case, Chirita et al. obtained a certain spatial decay inequality for a space–time integral over a portion of the body and over a finite time interval of the total mechanical energy. This involves the work done on histories, which is not a function of state in general. Here it is shown that for free energies which are functions of state and obey a certain reasonable property, the spatial decay of the corresponding space–time integral is stronger than the one involving the work done on the past history. It turns out that the bound obtained is optimal for the minimal free energy.Two cases are discussed for the quasi-static approximation. The first case deals with general states, so that general histories belonging to the equivalence class of any given state can be considered. The continuity of the stress functional with respect to the norm based on the minimal free energy is proved, and the energy measure based on the minimal free energy turns out to obey the decay inequality derived Chirita et al. for the quasi-static case.The second case explores a crucial point for viscoelastic materials, namely that the response is influenced by the rate of application of loads. Quite surprisingly, the analysis of this phenomenon in the context of Saint-Venant principles has never been carried out explicitly before, even in the linear case. This effect is explored by considering states, the related histories of which are sinusoidal. The spatial decay parameter is shown to be frequency-dependent, i.e. it depends on the rate of load application, and it is proved that of those considered, the most conservative estimate of the frequency-dependent decay is associated with the minimal free energy. A comparison is made of the results for sinusoidal histories at low frequencies and general histories.

Investigation of dynamic Saint-Venant's principle in a cylindrical waveguide – Analytical results

An analytical approach has been carried out to analyze the wave propagation in the cylindrical waveguide which was hit by four impactors with different head shapes. The Pochhammer–Chree solution, including evanescent wave modes, was applied to describe the stress field in the vicinity of the proximal end of the waveguide. Two strategies were used to derive the Pochhammer–Chree equation convergent at the vicinity of the loaded end, i.e., 1) Fast Fourier Transformation of the loading boundary condition, and 2) derivation of complex wave-numbers for the evanescent modes. The length of the affected zone, which confirms a dynamic Saint-Venant's principle (DSVP), was about 2.2 times of radius based on the analytical results, which coincide with the experimental and numerical results. Parametric study has also been conducted with respect to different frequency ranges of the pulse wave.

Atomic to Continuum Passage for Nanotubes: A Discrete Saint-Venant Principle and Error Estimates

We consider general infinite nanotubes of atoms in \({\mathbb{R}^3}\) where each atom interacts with all the others through a two-body potential. At the equilibrium, the positions of the atoms satisfy a Euler–Lagrange equation. When there are no exterior forces and for a suitable geometry, a particular family of nanotubes is the set of perfect nanotubes at the equilibrium. When exterior forces are applied on the nanotube, we compare the nanotube to nanotubes of the previous family. In part I of the paper, this quantitative comparison is formulated in our first main result as a discrete Saint-Venant principle. As a corollary, we also give a Liouville classification result. Our Saint-Venant principle can be derived for a large class of potentials (including the Lennard-Jones potential), when the perfect nanotubes at the equilibrium are stable. The approach is designed to be applicable to nanotubes that can have general shapes like, for instance, carbon nanotubes or DNA, under the oversimplified assumption that all the atoms are identical. In part II of the paper, we derive from our Saint-Venant principle a macroscopic mechanical model for general nanotubes. We prove that every solution is well approximated by the solution of a continuum model involving stretching and twisting, but no bending. We establish error estimates between the discrete and the continuous solution. More precisely we give two error estimates: one at the microscopic level and one at the macroscopic level.

A higher order model for thin-walled structures with deformable cross-sections

A higher order model for the analysis of linear, prismatic thin-walled structures that considers the cross-section warping together with the cross-section in-plane flexural deformation is presented in this paper. The use of a one-dimentional model for the analysis of thin-walled structures, which have an inherent complex three-dimensional (3D) behaviour, can only be successful and competitive when compared with shell finite element models if it fulfills a twofold objective: (i) an enrichment of the model in order to as accurately as possible reproduce its 3D elasticity equations and (ii) the definition of a consistent criterion for uncoupling the beam equations, allowing to identify structural deformation modes.The displacement field is approximated through a linear combination of products between a set of linear independent functions defined over the cross-section and the associated weights only dependent on the beam axis; this approximation is not constrained by any ab initio kinematic assumptions. Towards an efficient application of the approximation procedure, the cross-section is discretized into thin-walled elements, being the displacement field approximated for each element independently of the displacement direction. The approximation is thus hp refined enhancing the “capture” of the 3D structural mechanics of thin-walled structures. The beam model governing equations are obtained through the integration over the cross-section of the corresponding elasticity equations weighted by the cross-section global approximation functions.A criterion for uncoupling the beam governing equations is established, allowing to (i) retrieve the classic equations of the thin-walled beam theory both for open and closed sections and (ii) derive a set of uncoupled deformation modes representing higher order effects. The criterion is based on the solution of the polynomial eigenvalue problem associated with the beam differential equations, allowing to quantify the Saint-Venant principle for thin-walled structures. In fact, the solution of the non linear eigenvalue problem yields a twelve fold null eigenvalue (representing polynomial solutions) that are verified to represent beam classic solutions and sets of pairs and quadruplets of non-null eigenvalues corresponding to higher order modes of deformation.

Elastic solutions of the end problem for a nonhomogeneous semi-infinite strip in anti-plane shear

The end problem referring to anti-plane shear deformation of a nonhomogeneous semi-infinite strip is investigated here, by using the analogous methodology proposed by Papkovich and Fadle in plane problem. Two types of nonhomogeneity are considered: (i) the shear modulus varies with the thickness coordinate x exponentially; (ii) it varies with the length coordinate y exponentially. The closed elastic solutions in trigonometric series form are derived by the eigenfunctions expansion, and the completeness of the solutions is also proved. Therefore, the elastic field caused by a self-equilibrating traction on the end could be solved in an arbitrary accuracy by taking a necessary number of terms in the series, approximatively, which is usually neglected by invoking Saint-Venant principle. By considering the biggest negative eigenvalue, the Saint-Venant Decay rates of the problem is also estimated in the last section.

Cable Anchorage System Modeling Methods for Self-Anchored Suspension Bridges with Steel Box Girders

The multiscale modeling method is newly introduced to the research on cable anchorage systems for self-anchored suspension bridges with steel box girders to improve the reliability of the conventional modeling methods. Two kinds of boundary conditions (BCs), the hard BCs and flexible BCs, are defined and illustrated. The strategy of the modeling method of the cable anchorage system for multiscale analysis is then introduced. Based on the proposed strategy, a multiscale model of the cable anchorage system is developed for the Taohuayu Bridge, which is the largest self-anchored suspension bridge in the world. For comparison, a scale model test is carried out, and two elaborate finite-element (FE) models, including a traditional scale model and a full-scale model, are established according to the traditional scale modeling method and full-scale modeling method. For further validation, a full-scale whole-bridge elaborate FE model is established, determined as the full-scale G model. In the test, a complex stress distribution of the anchor structure is newly observed. Intensive comparison study of various models demonstrates that the fuzzy region exists in the test model, scale model, and full-scale model, caused by the use of the assumption of hard BCs and the Saint-Venant principle. The fuzzy region cannot be quantitatively determined, leading to unreliable analysis results and a large amount of trial-and-error work. In the multiscale model, the complex BCs of the cable anchorage system are accurately simulated by the flexible BCs, and the influence of the fuzzy region is eliminated. The results of the multiscale model matches well with the full-scale G model. It is concluded that the multiscale modeling method is the most suitable modeling method to simulate the mechanical behaviors of the cable anchorage system of self-anchored bridges with steel box girders, reliably and efficiently.

For a wave equation involving p-Laplacian term Phragmen-Lindelöf type estimates

Behaviours of the solutions to initial-boundary value problems in terms of spatial or time variables are investigated by the theories such as Phragmen-Lindelof Theorem and Saint-Venant Principle. In this work, behaviours of solutions to a hyperbolic equation involving p-Laplacian term in terms of spatial variable is studied. It is showed that how the solutions decay to zero depending on the constant p.

Stress Calculation of Gravity Dam Based on Elastic Mechanics

Based on the elastic mechanics theory, combined with Saint-Venant principle and the superposition principle, the calculating method for the stress of gravity dam in arbitrary storage water lever were put forward by means of the equivalent stress method and the dimensional analysis method through the conversion between a Cartesian coordinate system and Polar coordinate system. The method not only offers a new idea for calculating the stress of gravity dam, but also presents research bases for evaluating synthetically the strength of gravity dam.

Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

Exact solutions for generally supported functionally graded plane beams are given within the framework of symplectic elasticity. The Young’s modulus is assumed to exponentially vary along the longitudinal direction while the Poisson’s ratio remains constant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which is limited to the Saint-Venant solution. Here, a complete rational analysis of the displacement and stress distributions in the beam is presented by exploring the eigensolutions that are usually covered up by the Saint-Venant principle. These solutions play a significant role in the local behavior of materials that is usually ignored in the conventional elasticity methods but possibly crucial to the material/structure failures. The analysis makes full use of the symplectic orthogonality of the eigensolutions. Two illustrative examples are presented to compare the displacement and stress results with those for homogenous materials, demonstrating the effects of material inhomogeneity.