Saint-Venant End Effects Research Articles

Experimental realization of tunable Poisson’s ratio in deployable origami metamaterials

Origami metamaterials are known to display highly tunable Poisson’s ratio depending on their folded state. Most studies on the Poisson effects in deployable origami tessellations are restricted to theory and simulation. Experimental realization of the desired Poisson effects in origami metamaterials requires special attention to the boundary conditions to enable nonlinear deformations that give rise to tunability. In this work, we present a novel experimental setup suitable to study the Poisson effects in 2D origami tessellations that undergo simultaneous deformations in both the applied and transverse directions. The setup comprises a gripping mechanism, which we call a Saint-Venant fixture, to eliminate Saint-Venant end effects during uniaxial testing. Using this setup, we conduct Poisson’s ratio measurements of the Morph origami pattern whose configuration space combines features of the Miura-ori and Eggbox parent patterns. We experimentally observe the Poisson’s ratio sign switching capability of the Morph pattern, along with its ability to display either completely positive or negative values of Poisson’s ratio by virtue of topological transformations. To demonstrate the versatility of the novel setup we also perform experiments on the standard Miura-ori and the standard Eggbox patterns. Our results demonstrate the agreement between the theory, the simulations, and the experiments on the Poisson’s ratio measurement and its tunability in origami metamaterials. The proposed experimental technique can be adopted for investigating other tunable properties of origami metamaterials in static and in dynamic regimes, such as elastic thermal expansion, and wave propagation control.

AXISYMMETRIC PROBLEM OF THE ELASTICITY THEORY FOR THE RADIALLY INHOMOGENEOUS CYLINDER WITH A FIXED LATERAL SURFACE

By the method of the asymptotic integration of the equations of elasticity theory, the axisymmetric problem of elasticity theory is studied for a radially inhomogeneous cylinder of small thickness. It is considered that the elasticity moduli are arbitrary positive continuous functions of the radius of the cylinder. It is also assumed that the lateral surface of the cylinder is fixed, and stresses are imposed at the end faces of the cylinder, which leave the cylinder in equilibrium. The analysis is carried out when the cylinder thickness tends to zero. It is shown that solutions corresponding to the first and second iterative processes that determine the internal stress-strain state of the radially inhomogeneous cylinder with a fixed surface do not exist. The third iterative process defines solutions that have the boundary layer character equivalent to the Saint-Venant end effect on the theory of inhomogeneous plates. The stresses determined by the third iterative process are localized at the ends of the cylinder and decrease exponentially with distance from the ends. The asymptotic integration method is used to study the problem of torsion of the radially inhomogeneous cylinder of small thickness. The nature of the stress-strain state is analyzed.

Saint-Venant End Effects in the Plane Problem for Linearly Elastic Functionally Graded Materials

An isotropic elastic strip, with a continuous inhomogeneity profile for Young’s modulus is considered, subject to a self-equilibrated load on one of its axial ends and free of traction on the remainder of its surface boundaries. By taking advantage of the analytical flexibility of an exponential inhomogeneity profile, the full equations of linear theory of elasticity are employed to find the two-dimensional Saint-Venant decay rate in terms of an inhomogeneity parameter that measures gradation steepness in a Functionally Graded Material (FGM). The results show that softening axial inhomogeneity may be introduced to engineered FGM strips and beams to accelerate the decay of stresses. For a hardening axial inhomogeneity on the other hand, the end effects extend beyond a one-width-size distance from the loaded end. The plane problem end effects are shown to decay faster compared to the anti-plane shear counterparts, consistent with what is observed in homogeneous media. For inhomogeneity in the lateral direction from the core toward the outer edges, the qualitative behaviour changes with the degree of inhomogeneity. Whether the core is softer or harder relative to the outer edges, a steep lateral gradation of elastic modulus can significantly increase the decay length of end effects.

Decay rates of Saint-Venant type for a functionally graded heat-conducting hollowed cylinder

In this paper we consider the case of a functionally graded heat-conducting hollowed cylinder. Our purpose is to investigate the consequences of the material inhomogeneity on the decay of Saint-Venant end effects in the case of linear isotropic rigid solids. The mathematical issues involve the implications of spatial inhomogeneity on the decay rates of solutions to Dirichlet boundary-value problems. The rate of decay is characterized in terms of the smallest eigenvalue of a Sturm–Liouville problem. We first consider the case where the inhomogeneity depends on the radius of the cross-section, but later we also consider the case where the inhomogeneity also depends on the axial variable. The last section considers the case where the cross-section is increasing. Some tables and figures illustrate our estimates.

Decay rates in nano tubes with consideration of surface elasticity

In the present paper we examine the Saint-Venant end effect in the nano tubes via a continuum mechanics with consideration of surface elasticity. The Saint-Venant end effect is quantitatively described by the decay rate. By analytically solving an axial-symmetric torsion in a circular cross-section tube configuration, we demonstrate that the decay rate decreases as the radius of the nano wire/tube decreases with consideration of the surface effect.

The eigenvalue problem and Saint-Venant decay rate for a nonhomogeneous semi-infinite strip

The eigenvalue problem about a nonhomogeneous semi-infinite strip is investigated using the methodology proposed by Papkovich and Fadle for homogeneous plane problems. Two types of nonhomogeneity are considered: (i) the elastic modulus varying with the thickness coordinate x exponentially, (ii) it varying with the length coordinate y exponentially. The eigenvalues for the two cases are obtained numerically in plane strain and plane stress states, respectively. By considering the smallest positive eigenvalue, the Saint-Venant Decay rates are estimated, which indicates material nonhomogeneity has a significant influence on the Saint-Venant end effect.

Decay rate of saint-venant end effects for plane deformations of piezoelectric-piezomagnetic sandwich structures

This paper is concerned with the decay of Saint-Venant end effects for plane deformations of piezoelectric (PE)-piezomagnetic (PM) sandwich structures, where a PM layer is located between two PE layers with the same material properties or reversely. The end of the sandwich structure is subjected to a set of self-equilibrated magneto-electro-elastic loads. The upper and lower surfaces of the sandwich structure are mechanically free, electrically open or shorted as well as magnetically open or shorted. Firstly the constitutive equations of PE materials and PM materials for plane strain are given and normalized. Secondly, the simplified state space approach is employed to arrange the constitutive equations into differential equations in a matrix form. Finally, by using the transfer matrix method, the characteristic equations for eigenvalues or decay rates are derived. Based on the obtained characteristic equations, the decay rates for the PE-PM-PE and PM-PE-PM sandwich structures are calculated. The influences of the electromagnetic boundary conditions, material properties of PE layers and volume fraction on the decay rates are discussed in detail.

Saint-Venant End Effects of Transversely Isotropic Piezoelectric Materials

In this paper, Saint-Venant end effects for plane deformations of transversely isotropic piezoelectric materials are investigated. The stress decay rates in linear piezoelectric strips that are traction free with two kinds of electrical boundary conditions are considered. The characteristic equations for decay rate are obtained for symmetric and antisymmetric deformations. Numerical values are given for the roots with the smallest positive real part, which are associated with the slowest decay. Saint-Venant end effects of piezoelectric materials of crystal class 6mm penetrate much further into strip than those of elastic isotropic materials.

Saint-Venant decay rates for an anisotropic and non-homogeneous mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects in anti-plane shear deformations of linear mixtures of elastic solids. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear anisotropic mixtures of elastic solids. As our first results have a very general point of view, we study some examples in detail.

Saint-Venant decay analysis of FGPM laminates and dissimilar piezoelectric laminates

Using a similar procedure to the Hamiltonian system based model [Wang, J.S., Qin, Q.H., 2007. A symplectic model for piezoelectric wedges and its application to analysis of electroelastic singularities, Philosophical Magazine 87 (2), 225–251], the mixed-variable state space formulation is developed for functionally graded piezoelectric material (FGPM) strips and laminates, in which the material inhomogeneity is considered. For dissimilar homogeneous piezoelectric laminates, the state space formulation degenerated to a Hamiltonian system. Applying the developed model, we analyzed the decay of Saint-Venant end effects in a single FGPM strip and a FGPM laminate. The numerical results show that the decay rate depended strongly on the eigenvalue of the proposed operator matrix for the single FGPM strip. By using the coordinate transformation technique and the continuity conditions on the interface between two dissimilar materials (different piezoelectric properties or different material inhomogeneous parameters), the decay rates are also determined for multi-layered FGPM laminates, including dissimilar piezoelectric laminates (without material inhomogeneity) as a special case. Numerical results are presented to show the applicability of the proposed state space model to piezoelectric and FGPM laminates. In addition, the variation of the decay rate with the thickness has also been investigated for the dissimilar homogeneous piezoelectric laminates. This study indicates that material inhomogeneity plays an important role in Saint-Venant end effects for FGPM laminates, as in the case of a single FGPM strip [Borrelli, A., Horgan, C.O., Patria, M.C., 2004. Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials. Proceedings of the Royal Society of London Series A 460, 1193–1212].

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6 mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.

Saint-Venant decay rates for a non-homogeneous isotropic mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity on the decay of Saint-Venant end effects in anti-plane shear deformations of linear isotropic mixtures of elastic solids. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e. materials with spatially varying properties tailored to satisfy particular engineering applications. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear isotropic mixtures of elastic solids.

Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials

This paper is concerned with the investigation of the effect of material inhomogeneity on the decay of Saint–Venant end effects in functionally graded linear piezoelectric materials. Saint–Venant's principle and related results for elasticity theory have received considerable attention in the literature but it is only recently that analogous issues in piezoelectricity have been investigated. The current rapidly developing smart–structures technology provides motivation for the investigation of such problems. We examine the decay of Saint–Venant end effects in the context of anti–plane shear deformations for linear inhomogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations (PDEs) of equilibrium are shown to be a coupled system of second–order PDEs with variable coefficients for the mechanical displacement u and electric potential φ . The traction boundary–value problem with prescribed surface charge is formulated as an oblique derivative boundary–value problem for this elliptic system. The axial decay of solutions on a semi–infinite strip subjected to non–zero boundary conditions only at the near end is investigated. This analysis is carried out for a subclass of special graded materials, namely, laterally inhomogeneous solids. Two specific piezoceramics that have received widespread application are considered. For the first (corresponding to the hexagonal 6 mm crystal–symmetry class), it is shown that the mechanical and electrical problems essentially decouple, and that the decay rates for mechanical and electrical Saint–Venant end effects coincide and are equal to that of linear inhomogeneous isotropic anti–plane elasticity, for which results on the decay rate have been obtained previously. For the second class of materials (corresponding to the cubic 4∼3m symmetry) the situation is quite different. The boundary–value problem involves a full coupling of mechanical and electrical effects. Energy decay estimates using differential–inequality methods are used to obtain an explicit estimated decay rate (a lower bound for the actual decay rate) in terms of a single dimensionless piezoelectric coupling constant d and the decay rate for an associated linear inhomogeneous isotropic anti–plane shear elasticity problem. For fixed material inhomogeneity, the decay rate is shown to be monotone decreasing with increasing values of d In the limit as d→0 we recover the purely mechanical case. Thus, for this class of functionally graded materials, piezoelectric end effects are predicted to penetrate further into the strip than their functionally graded isotropic elastic counterparts , confirming recent results obtained in other contexts in linear piezoelectricity. For both classes of materials, the influence of material inhomogeneity on the decay rate is reflected solely by the shear modulus of an associated inhomogeneous isotropic anti–plane shear elasticity problem. It is shown that, for both material classes, material inhomogeneity has a significant influence on the decay of end effects.

End effects for pre-stressed and pre-polarized piezoelectric solids in anti-plane shear

This paper is concerned with investigation of the influence of pre-stress and pre-polarization on the decay of Saint-Venant end effects in piezoelectric solids. This issue is examined within the framework of incremental anti-plane shear states for prestressed and prepolarized solids. Attention is confined to a specific class of materials, namely those of hexagonal 6mm or 4mm symmetry. It is shown that the mechanical and electrical fields uncouple in this case, and the governing system of coupled second-order linear partial differential equations uncouple to a single homogeneous second-order equation with constant coefficients for the displacement and a Poisson equation for the electric potential. The exact decay rate for mechanical and electrical end effects is determined and the influence of pre-stress and pre-polarization is assessed. Motivation for this work arises from the current rapid developments in smart materials technology.

Saint-Venant end effects in multilayered piezoelectric laminates

The Saint-Venant end effects and stress decay in multilayered laminates of piezoelectric materials are studied. A state space approach is developed in the context of generalized plane strain. By means of matrix algebra and transfer matrix, the characteristics of the stress decay in 2-D piezoelectric strips and laminates are examined. Their elastic counterparts are included as special cases. Evaluation of the characteristic decay length for typical materials shows that the Saint-Venant end effects are significant. It also indicates that the analysis based on the plane strain assumption is misleading.

Improved Analytical Models for Mixed-Mode Bending Tests of Adhesively Bonded Joints

In this paper, improved analytical models have been developed for more accurate determination of mode I and mode II compliances and strain energy release rates of adhesively bonded Double Cantilever Beam (DCB) type specimens. In these models, the effects of adhesive layer, elastic foundation and shearing deformation ahead of the crack tip due to the Saint-Venant end effect were considered. The improved analytical models were verified using finite element analysis (FEA), and excellent agreements were found for a variety of crack lengths. The comparisons of the measured and predicted testing machine loading point compliances with crack length at three different mixed-mode loading conditions for the adhesive layer thickness of 0.254 mm were also performed for further validation. In addition, contributions to the total mode I strain energy release rate from each effect are summarized.

Saint-Venant's Principle for Antiplane Shear Deformations of Linear Piezoelectric Materials

This paper is concerned with investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. We examine the decay of Saint-Venant end effects in the context of antiplane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are shown to be a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential $\varphi$. The traction boundary-value problem with prescribed surface charge is formulated as an oblique derivative boundary-value problem...

An Analysis of the Decay of Saint-Venant End Effects in a Transversely Isotropic Matrix with an Isotropic Covering

A calculation model is proposed for a numerical analysis of the decay of Saint-Venant end effects in a laminate material of irregular structure to which corresponds a transversely isotropic matrix with an isotropic covering. The elastic properties of the matrix correspond to those of a unidirectional glass-fiber-reinforced plastic. The problem is investigated within the framework of the concept of a representative element of the material. The decay of the end effect in the direction perpendicular to the isotropy plane of a transversely isotropic matrix for the case of symmetric deformation of the material is considered. The source of the end effect is simulated by a piecewise-constant periodic surface load. This load is local for the calculated region and changes within a part of the boundary comparable with the typical size of the structural heterogeneity of the material. The equations of the linear elasticity theory of orthotropic bodies, a model of piecewise-homogeneous media, and quantitative criteria of decay of the end effect are used. Starting from the base system of equations, a discrete problem is constructed and solved. The results of a computing experiment for the choice of a representative element and the results of determining the maximum extent of the end-effect zone for this element are presented.

Saint-Venant End Effects in Antiplane Shear for Functionally Graded Linearly Elastic Materials

The purpose of this research is to investigate the influence of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. The work is motivated by the recent research activity on functionally graded materials (FGMs) i.e., materials with continuously varying properties tailored to satisfy specific engineering applications. In the framework of antiplane shear deformations of a semi-infinite strip, the mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to a Dirichlet boundary value problem for a second-order linear elliptic partial differential equation with variable coefficients. In previous work, the variable coefficient (the shear modulus) was assumed to be strictly positive on the closed semi-infinite strip. The authors relax this assumption to allow for the shear modulus to be zero on either long side of the strip. Thus, the results are applicable to FGMs that are graded continuously from zero shear modulus at one side of the strip. Lower bounds for the rate of decay of end effects are obtained, which allow for assessing the influence of material inhomogeneity. The results are illustrated using several specific material models.

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ϕ. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal \({\bar 4}\) crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d0,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d0. In the limit as d0→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal \({\bar 4}\) class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.