Saint-Venant Problem Research Articles

A 3D distributed plasticity beam–column element for metal structures considering tangential stresses and warping with minimal DOFs

The fiber model evaluates the normal stress at a number of points over the section for a given strain increment following the plane section assumption and, by integration, axial force and bending moments. The interaction with tangential stresses is usually neglected at the point level due to the inaccurate tangential strains of the kinematics. This work proposes a generalization of the fiber model able to capture automatically the interaction among all stress components. A preliminary cross-section analysis based on the Saint Venant problem provides an accurate 3D strain as a function of the section generalized strains. This field, accurate also in the inelastic case, is exploited to impose at each section point a 3D von Mises elasto-plastic law, obtaining by integration all the resultants and moments with a full interaction. Non-uniform warping is also easily included. The section model is implemented in a mixed 3D beam–column finite element with equilibrated stress field, accurate with a minimal mesh. Numerical tests show the excellent prediction of the proposal compared to analytical and solid FEM solutions also for structures not flexure-dominated. Its efficiency, on the same order as a standard fiber model, makes the approach suitable also for large buildings.

Identification and validation of constitutive parameters of a Hencky-type discrete model via experiments on millimetric pantographic unit cells

Pantographic metamaterial design benefits from model identification procedures starting from what can be considered as the elementary unit cell of larger pantographic structures. Results from a tensile experiment and digital image correlation are utilized to identify the constitutive parameters of a discrete Hencky-type model for a millimetric pantographic cell. In the performed calibration, two different cost functions are formulated. First, the cost function is based upon measured resultant forces on the specimen boundaries. Then, the second cost function is based upon the measured pivot displacements in addition to reaction forces. The second cost functions thus exploits the pivot kinematics, which is a key feature of the deformation of pantographic structures. The identified model is further validated by predicting the reaction forces and pivot displacements of the same specimen subjected to compression. It is shown that the identification with the cost function incorporating pivot displacements is superior. It is also noted that the calibrated parameters deviate considerably from their initial guess derived from the linear Saint Venant problem, thereby indicating microscale nonlinear affects in otherwise linear reaction force-prescribed displacement responses at the macroscale.

The Saint-Venant problem for general anisotropic piezoelectric cylinders with applications to smart metamaterials design

The Saint-Venant problem for general anisotropic linear piezoelectric cylinders is addressed. Under the assumption of material homogeneity along the cylinder axis, the Voigt hypothesis is shown to be a necessary condition for the solution, thus allowing for the reduction of the three-dimensional piezoelectric problem to a pair of boundary-value problems on the cylinder cross section. A non-standard choice of the unknown independent fields is adopted for simplifying such a task, and the mixed-type variational structure underlying the proposed formulation is investigated. A first application deals with the analytical derivation of the piezoelectric sensor/actuator vector of a general anisotropic, homogeneous, piezoelectric cylinder, subject to mechanical/electrical loadings on its bases. Then, the solution of the relaxed Saint-Venant problem is derived in closed form in the case of a homogenous cylinder with circular cross section. On the basis of such reference solution, the accuracy and efficiency of the proposed variational formulation is proven, also in comparison to three-dimensional finite element solutions. Finally, the potentialities of the obtained results for the rational design of new smart lattice metamaterials with optimized piezoelectric properties are proven through a numerical example.

Multiaxial deformations of ionic polymer metal composites

Ionic polymer metal composites (IPMCs) are a promising class of soft active materials. Their high compliance, low actuation voltage, and ability to operate in wet environments have motivated two decades of intensive research on IPMC actuators. While we have witnessed several breakthroughs in the technology of IPMCs, from additive manufacturing to IPMC-based robots, our understanding of the physical underpinnings of their actuation remains elusive. There is a paucity of continuum physically-based models to investigate IPMC actuation, where the literature relies on structural models that postulate admissible mechanical deformations from classical beam or plate theories. In this sense, we know little about multiaxial deformations elicited by counterions’ diffusion and electromigration through the ionomer. Here, we demonstrate that macroscopic actuation of IPMCs is accompanied by localized mechanical deformations in the vicinity of the electrodes, caused by osmotic pressure and Maxwell stress. Toward this aim, we put forward a comprehensive nonlinear finite element analysis, conducted in Abaqus™ through a newly developed user element that allows for testing hypotheses on the inner workings of IPMC actuation and exploring complex configurations. Alongside with computational advances, we establish an exact solution for the two-dimensional Saint-Venant problem of plane-strain actuation of an IPMC, based on linear elasticity and nonlinear electrochemistry. Verified against finite element results, the exact solution offers a mathematically-tractable treatment of localized phenomena in the vicinity of the electrodes. Our results unveil a rich dependence of through-the-thickness deformation on the electric double layers that are formed in the vicinity of the electrodes. Due to the asymmetry of the boundary layers in the vicinity of the cathode and the anode, the ionomer deforms asymmetrically with respect to its mid-axis that also experiences an axial stretch. The Poisson ratio of the ionomer is found to have a critical role in shaping the response of the IPMC, from the onset of actuation to its back-relaxation. This study constitutes a first modeling step toward illuminating complex, multiaxial deformations of IPMCs, whose understanding is critical toward the design and manufacturing of high performance IPMC actuators.

Closed-Form Saint-Venant Solutions in the Koiter Theory of Shells

In this paper we investigate the deformation of cylindrical linearly elastic shells using the Koiter model. We formulate and solve the relaxed Saint-Venant's problem for thin cylindrical tubes made of isotropic and homogeneous elastic materials. To this aim, we adapt a method established previously in the three-dimensional theory of elasticity. We present a general solution procedure to determine closed-form solutions for the extension, bending, torsion and flexure problems. We remark the analogy and formal resemblance of these solutions to the classical Saint-Venant's solutions for solid cylinders. The special case of circular cylindrical shells is also discussed.

Saint-Venant torsion of non-homogeneous orthotropic circular cylinder

The object of this paper is the Saint-Venant torsion of a radially non-homogeneous, hollow and solid circular cylinder made of orthotropic piezoelectric material. The elastic stiffness coefficients, piezoelectric constants and dielectric constants have only radial dependence. This paper gives the solution of the Saint-Venant torsion problem for torsion function, electric potential function, Prandtl’s stress function and electric displacement potential function.

On the solutions to the Saint–Venant problem of heterogeneous beam-like structures with periodic microstructures

This paper presents displacement solutions of the Saint-Venant problem for linear elastic heterogeneous beam structures with arbitrarily shaped periodic microstructures. The solutions are generated by introducing unknown axially periodic displacement functions in place of the axially invariant warping functions in Ieşan's rational scheme, due to the periodic characteristics of the beam structure, while retaining the displacement fields integrated from rigid-body displacements, which represent basic kinematics in beam theories. The governing equations of the unknown functions, defined on a base cell, are then developed from equilibrium equations and boundary conditions of the periodic beam structure. Moreover, an improved FE formulation of the governing equations is presented and its efficient numerical implementation approach, which can be readily realized using FE software as a black box, is proposed with the aid of NIAH (Novel Numerical Implementation of Asymptotic homogenization) approach. To illustrate the validity of the Saint-Venant solution and the effectiveness of the numerical implementation approach, several numerical examples are presented and compared with detailed three-dimensional FE analysis for the heterogeneous beam structures in terms of stress components.

Validation of Bredt\u2019s formulas for beams with hollow cross sections by the method of asymptotic splitting for pure torsion and their extension to shear force bending

The equations governing pure torsion of prismatic beams with thin-walled closed cross sections, known as Bredt’s formulas, are verified using the method of asymptotic splitting. In particular, the strong formulation of the Saint-Venant problem of a straight beam is expanded asymptotically. We begin by validating well-known technical assumptions for the shear stress distribution. Furthermore, the influence of a transverse force acting on the beam is considered. This shear force causes a deformation of the cross section, and therefore an adaption of Bredt’s formulas is needed. Two distinct formulations of the shear center, called the kinematic and the energetic shear center, are obtained. The latter is verified in numerical experiments.

Saint-Venant Problems for Two-dimensional Elastic Solids

Using the symplectic system method, the classical Saint-venant problems in elasticity are studied. Based on the Laplace transformation and integral transformation and the symplectic character, all the Saint-Venant solutions, including the tension, bending and local solutions, are obtained. The final solution of the problem can be directly analysed in the time domain solution space. As applications of the proposed method, some stress concentration boundary condition problems are discussed.

Symplectic Solutions in Two-dimensional Viscoelastic Polymers

As applications of viscoealstic polymers increase, a comprehensive understanding of physical properties of these kind of materials has been proved to be critical in the modern science and technology. In this paper, a new symplectic system method is proposed to solve the classical Saint-venant problem for viscoelastic polymers. Based on the Laplace integral transformation and the symplctic character, the dual governing equations are established, and all the general eigensolutions are obtained in analytical form. In numerical examples, some boundary condition problems are studied, which well exhibit the viscoelastic behavior of polymers.

Validation of Bredt's formulas by the method of asymptotic splitting

AbstractThe equations describing torsion of prismatic bars with thin‐walled closed cross sections, known as Bredt's formulas, are verified using the method of asymptotic splitting. In particular, the strong formulation of the Saint Venant problem of a straight beam is expanded asymptotically. At first, well‐known technical assumptions of the shear stress distribution are validated. Further, the influence of a transverse force acting on the beam is considered. This shear force causes a deformation of the cross section and therefore an adaption of Bredt's formulas. Two distinct formulations of the shear center, called the kinematic and the energetic shear center, are obtained. The latter are verified in numerical experiments.

Displacement and Stress Function-based Linear and Quadratic Triangular Elements for Saint-Venant Torsional Problems

Torsional problems commonly arise in frame structural members subjected to unsym­metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy.

Saint-Venant torsion of a circular bar with radial cracks incorporating arbitrarily varied surface elasticity

ABSTRACTWe analytically investigate the contribution of arbitrarily varied surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The varied surface elasticity is incorporated by using a modified version of the continuum-based surface/interface model of Gurtin and Murdoch. In our discussion, the surface shear modulus is assumed to be arbitrarily varied along the crack surfaces. Both internal and edge cracks are studied. By using Green's function method, the boundary value problem is reduced to the Cauchy singular integro-differential equation of first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials, and the collocation method. The torsion problem of a cylinder containing two symmetric collinear radial cracks of equal length with symmetrically varied surface elasticity is also solved by using a similar method. Our numerical results indicate that the variation of the surface elasticity exerts a significant influence on the strengths of the logarithmic stress singularity at the crack tips, the torsional rigidity, and the jump in warping function.

On the deformation of almost cylindrical elastic beams

BackgroundThe Saint-Venant problem for porous elastic cylinders is of interest both from the technical and mathematical point of view. The intended applications of solution are to mechanics of bone and to some engineering structures.MethodThis work investigates the Saint-Venant problem for almost prismatic bars made of an isotropic porous material. We express the solution in terms of the solutions of two problems concerning the deformation of a right cylinder.Results and ConclusionWe use the method to study the extension of an almost prismatic conical frustum. It is found that the displacement vector is a polynomial of two degree in the cartesian coordinates. The volume fraction field depends linearly on the axial coordinate. The solution contains terms characterizing the influence of the material porosity and the dependence on the lateral surface.

Saint-Venant torsion of a circular bar with a bridged radial crack incorporating surface elasticity

We examine the contribution of crack bridging and surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The surface effect on the crack faces is incorporated by using the continuum-based surface/interface model of Gurtin and Murdoch. The bridging force is assumed proportional to the crack opening displacement, while the bridging stiffness is allowed to vary arbitrarily along the crack. Both an internal crack and an edge crack are studied. By using the Green’s function method, the boundary value problem is reduced to a first-order Cauchy singular integro-differential equation. After the utilization of the Gauss–Chebyshev integration formula and the expansion of both the unknown dislocation density and the known variable bridging stiffness into Chebyshev polynomials, the singular integro-differential equation is solved numerically by using the collocation method. The stresses exhibit a weak logarithmic singularity rather than a strong square root singularity at the crack tips due to the surface effect. The strengths of the logarithmic singularity at the crack tips, the reduction in torsional rigidity and the jump in warping function across the crack faces are calculated. Our results also show that both surface elasticity and crack bridging will stiffen the cracked cylinder and that the stiffening effect of crack bridging is remarkable in the case when the crack is very long or is extremely close to the lateral surface of the cylinder.

State Feedback Stabilization of the Linearized Bilayer Saint-Venant Model

We consider the problem of stabilizing the bilayer Saint-Venant model, which is a coupled system of two rightward and two leftward convecting transport partial differential equations (PDEs). In the stability proofs, we employ a Lyapunov function in which the parameters need to be successively determined. To the best of the authors’ knowledge, this is the first time this kind of Lyapunov function is employed, and this result is the first one on the stabilization of the linearized bilayer Saint-Venant model. Numerical simulations of the bilayer Saint-Venant problem are also provided to verify the result.

Semi-inverse method à la Saint-Venant for two-dimensional linear isotropic homogeneous second-gradient elasticity

This semi-inverse method is similar to that used in the so-called Saint-Venant problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. This semi-inverse method is similar to that used by Saint-Venant to solve the omonimus problem for cylindrical three-dimensional first-gradient linear homogeneous and isotropic materials. Two examples are also presented. It is found that wedge forces are necessary to maintain the body in equilibrium and that these are not an artefact of the double application of the divergence theorem in the second-gradient material derivations.

Torsion of an arbitrarily shaped nanosized bar

We investigate the Saint-Venant torsion problem of a nanosized bar with arbitrary smooth cross section by enhancing the classical continuum-based model with a theory of surface elasticity based on a version of the Gurtin–Murdoch surface/interface model. An elegant analytical solution is derived by means of conformal mapping, Faber series, Fourier series, and simple matrix algebra. Our solution leads to the establishment of a complex torsion function which, in turn, allows for a concise expression for the torsional rigidity. Our results clearly demonstrate that both the stress field and the normalized torsional rigidity are size dependent. An approximate formula for the size-dependent torsional rigidity is also proposed.

Lattice Boltzmann method for the Saint–Venant equations

The Saint–Venant equations represent the hydrodynamic principles of unsteady flows in open channel network through a set of non-linear partial differential equations. In this paper, a new lattice Boltzmann approach to solving the one-dimensional Saint–Venant equations (LABSVE) is developed, demonstrating the variation of discharge and sectional area with external forces, such as bed slope and bed friction. Our research recovers the Saint–Venant equations through deducing the Chapman–Enskog expansion on the lattice Boltzmann equation, which is a mesoscopic technique, bridging the molecular movement and macroscopic physical variables. It is also a fully explicit process, providing simplicity for programming. The model is verified by three benchmark tests: (i) a one-dimensional subcritical gradient flow; (ii) a dam-break wave flow; (iii) a flood event on the Yongding River. The results showed the accuracy of the proposed method and its good applicability in solving Saint–Venant problems.

Symplectic analysis of functionally graded beams subjected to arbitrary lateral loads

The rational analytical solutions are presented for functionally graded beams subjected to arbitrary tractions on the upper and lower surfaces. The Young's modulus is assumed to vary exponentially along the thickness direction while the Poisson's ratio keeps unaltered. Within the framework of symplectic elasticity, zero eigensolutions along with general eigensolutions are investigated to derive the homogeneous solutions of functionally graded beams with no body force and traction-free lateral surfaces. Zero eigensolutions are proved to compose the basic solutions of the Saint-Venant problem, while general eigensolutions which vary exponentially with the axial coordinate have a significant influence on the local behavior. The complete elasticity solutions presented here include homogeneous solutions and particular solutions which satisfy the loading conditions on the lateral surfaces. Numerical examples are considered and compared with established results, illustrating the effects of material inhomogeneity on the localized stress distributions.