A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

A Bending-Gradient model for thick plates,Part II: Closed-form solutions for cylindrical bending of laminates

In the first part (Lebée and Sab, 2010a) of this two-part paper we have presented a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called Bending-Gradient plate theory is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case when the plate is homogeneous. Moreover, we demonstrated that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In this paper, the Bending-Gradient theory is applied to laminated plates and its predictions are compared to those of Reissner–Mindlin theory and to full 3D (Pagano, 1969) exact solutions. The main conclusion is that the Bending-Gradient gives good predictions of deflection, shear stress distributions and in-plane displacement distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

Homogenization of thick periodic plates: Application of the Bending-Gradient plate theory to a folded core sandwich panel

In a previous paper from the authors, the bounds from Kelsey et al. (1958) were applied to a sandwich panel including a folded core in order to estimate its shear forces stiffness (Lebée and Sab, 2010b). The main outcome was the large discrepancy of the bounds. Recently, Lebée and Sab (2011a) suggested a new plate theory for thick plates – the Bending-Gradient plate theory – which is the extension to heterogeneous plates of the well-known Reissner–Mindlin theory. In the present work, we provide the Bending-Gradient homogenization scheme and apply it to a sandwich panel including the chevron pattern. It turns out that the shear forces stiffness of the sandwich panel is strongly influenced by a skin distortion phenomenon which cannot be neglected in conventional design. Detailed analysis of this effect is provided.     

On the Generalization of Reissner Plate Theory to Laminated Plates,Part I: Theory

This is the first part of a two-part paper presenting the generalization of Reissner thick plate theory (Reissner in J. Math. Phys. 23:184–191, 1944) to laminated plates and its relation with the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and in Int. J. Solids Struct. 48(20):2889–2901, 2011). The original thick and homogeneous plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) is based on the derivation of a statically compatible stress field and the application of the principle of minimum of complementary energy. The static variables of this model are the bending moment and the shear force. In the present paper, the rigorous extension of this theory to laminated plates is presented and leads to a new plate theory called Generalized-Reissner theory which involves the bending moment, its first and second gradients as static variables. When the plate is homogeneous or functionally graded, the original theory from Reissner is retrieved. In the second paper (Lebée and Sab, 2015), the Bending-Gradient theory is obtained from the Generalized-Reissner theory and comparison with an exact solution for the cylindrical bending of laminated plates is presented.     

On the Generalization of Reissner Plate Theory to Laminated Plates,Part II: Comparison with the Bending-Gradient Theory

In the first part of this two-part paper (Lebée and Sab in On the generalization of Reissner plate theory to laminated plates, Part I: theory, doi: 10.1007/s10659-016-9581-6, 2015), the original thick plate theory derived by Reissner (J. Math. Phys. 23:184–191, 1944) was rigorously extended to the case of laminated plates. This led to a new plate theory called Generalized-Reissner theory which involves the bending moment, its first and second gradients as static variables. In this second paper, the Bending-Gradient theory (Lebée and Sab in Int. J. Solids Struct. 48(20):2878–2888, 2011 and 2889–2901, 2011) is obtained from the Generalized-Reissner theory and several projections as a Reissner–Mindlin theory are introduced. A comparison with an exact solution for the cylindrical bending of laminated plates is presented. It is observed that the Generalized-Reissner theory converges faster than the Kirchhoff theory for thin plates in terms of deflection. The Bending-Gradient theory does not converge faster but improves considerably the error estimate.     

Fracture mechanics of plates and shells applied to fail-safe analysis of fuselage Part I: Theory

In this paper, a general theory on the asymptotic field near the crack tip for plates and shells with and without shear deformation effect is established. It is found that four stress intensity factors, two for symmetrical and antisymmetrical stretching and two for symmetrical and antisymmetrical bending, are required to describe arbitrary asymptotic fields near the crack tip for plates without shear deformation. An additional stress intensity factor is required for the transverse shearing force induced by antisymmetrical bending when the shear deformation is included in the analysis. It is also proven by means of the complex variable technique that for problems of plates with shear deformation, there exist similarities in the asymptotic expressions of moments and membrane forces and also in the asymptotic expressions of in-plane displacements and rotations of the mid-surface. The energy release rate associated with crack growth in the direction of the crack line can be expressed in terms of stress intensity factors by means of Irwin's method of work and energy associated with a virtual crack extension. A combined stress intensity factor can be defined through the total energy release rate. The theory of the fracture of plates is generalized and applied to the study of problems in the fracture of shells. An example of an infinitely long cylindrical shell with a circumferential crack subjected to remote axial tension is given to demonstrate the application of the theory and to test the accuracy of the numerical analysis used for the problem.     

A new simple third-order shear deformation theory of plates

An improved simple third-order shear deformation theory for the analysis of shear flexible plates is presented in this paper. This new plate theory is composed of three parts: the simple third-order kinematics of displacements reduced from the higher-order displacement field derived previously by the author; a system of 10th-order differential equilibrium equations in terms of the three generalized displacements of bending plates; five boundary conditions at each edge of plate boundaries. Although the resulting displacement field is the same as that proposed by Murthy, the variational consistent governing equations and the associated proper boundary conditions are derived and identified in this work for the first time in the literature. The applications and accuracy of the present shear deformation theory of plates are demonstrated by analytically solving the differential governing equations of a twisting plate, a bending beam and two bending plates to which the 3-D elasticity solutions are available, and excellent agreements are achieved even for the torsion of a plate with square cross-section as well the local effects of stresses at plate boundaries can be characterized accurately. These analytical solutions clearly show that the simple third-order shear deformation theory developed in this work indeed gives better results than the first-order shear deformation theories and other simple higher-order shear deformation theories, since the present third-order shear flexible theory is based on a more rigorous kinematics of displacements and consists of not only a system of variational consistent differential equations, but also a group of consistent boundary conditions associated with the differential equations. The present simple third-order shear deformation theory can easily be applied to the static and dynamic finite element analysis of laminated plates just like the applications of other popular shear flexible plate theories, and improved results could be obtained from the present simple third-order shear deformable theories of plates.     

Steady-state creep of bent reinforced metal-composite plates with consideration of their reduced resistance to transverse shear 2. Analysis of calculated results

Deformation of annular plates with different structures of helical reinforcement is studied. It is demonstrated that the use of the classical theory for calculating steady-state creep for thick reinforced plates subjected to bending leads to underprediction of the compliance of thin-walled metal-composite structures. It is also shown that there are significant shear strain rates in the binder of such plates, which has to be taken into account and which is mainly responsible for creep strain accumulation. Results calculated by two different models, which take into account the composite structure, are compared.     

非饱和颗粒材料的多孔连续体有效压力与有效广义Biot应力

多孔连续体理论框架下的非饱和多孔介质广义有效压力定义和Bishop参数的定量表达式长期以来存在争议,这也影响了对与其直接相关联的非饱和多孔介质广义Biot有效应力的正确预测.基于随时间演变的离散固体颗粒-双联液桥-液膜体系描述的Voronoi胞元模型,利用由模型获得的非饱和颗粒材料表征元中水力-力学介观结构和响应信息,文章定义了低饱和度多孔介质局部材料点的有效内状态变量:非饱和多孔连续体的广义Biot有效应力和有效压力,导出了其表达式.所导出的有效压力公式表明,非饱和多孔连续体的有效压力张量为各向异性,它不仅对非饱和多孔连续体广义Biot有效应力张量的静水应力分量的影响呈各向异性,同时也对其剪切应力分量有影响.文章表明,非饱和多孔连续体中提出的广义Biot理论和双变量理论的基本缺陷在于它们均假定反映非混和两相孔隙流体对固相骨架水力-力学效应的有效压力张量为各向同性.此外,为定义各向同性有效压力张量和作为加权系数而引入的Bishop参数并不包含对非饱和多孔连续体中局部材料点水力-力学响应具有十分重要效应的基质吸力.所导出的非饱和多孔介质广义Biot有效应力和有效压力公式(包括反映有效压力...     

Accurate modelling of piezoelectric plates: single-layered plate

Summary  The paper presents an efficient two-dimensional approach to piezoelectric plates in the framework of linear theory of piezoelectricity. The approximation of the through-the-thickness variations accounts for the shear effects and a refinement of the electric potential. Using a variational formalism, electromechanically coupled plate equations are obtained for the generalized stress resultants as well as for the generalized electric inductions. The latter are deduced from the conservative electric charge equation, which plays a crucial role in the present model. Emphasis is placed on the boundary conditions at the plate faces. The model is used to examine some problems for cylindrical bending of a single simply supported plate. Number of situations are examined for a piezoelectric plate subject to (i) an applied electric potential, (ii) a surface density of force, and (iii) a surface density of electric charge. The through-thickness distributions of electromechanical quantities (displacements, stresses, electric potential and displacement) are obtained, and compared with results provided by finite element simulations and by a simplified plate model without shear effects. A good agreement is observed between the results coming from the present plate model and finite element computations, which ascertains the effectiveness of the proposed approach to piezoelectric plates. Received 17 July 2000; accepted for publication 26 September 2000     

常用假设下黏弹组合模型两类表述参数间转换及适用性

黏弹性组合模型通常有两类表述，一类为基于拉压模量和拉压黏性系数的表述，另一类为基于剪切模量和剪切黏性系数的表述。对于广义Kelvin模型，这两类表述参数间的转换已经建立。但存在理论基础较薄弱、转换的适用范围和适用条件不够明确的问题。从线黏弹性理论出发，考虑岩土工程两种常用的三维假设（常泊松比假设和常体积模量假设），给出了这些假设下黏弹性组合模型蠕变柔量及其复柔量在两类表述之间的转换关系，然后将其应用于广义Kelvin 模型和Poynting-Thomson模型，分别推导出了两个模型在两类表述参数间的转换公式，明确了参数转换的适用范围和适用条件，以及应用于实际工程时须注意的问题。     

Elastic Wave Scattering and Dynamic Stress Concentrations in Stretching Thick Plates with Two Cutouts by Using the Refined Dynamic Theory

Based on the refined dynamic equation of stretching plates, the elastic tensioncompression wave scattering and dynamic stress concentrations in the thick plate with two cutouts are studied. In view of the problem that the shear stress is automatically satisfied under the free boundary condition, the generalized stress of the first-order vanishing moment of shear stress is considered. The numerical results indicate that, as the cutout is thick, the maximum value of the dynamic stress factor obtained using the refined dynamic theory is 19% higher than that from the solution of plane stress problems of elastic dynamics.     

Multimodulus elasticity theory

A variant of the multimodulus elasticity theory for isotropic materials is proposed under the assumption that the shear modulus in Hooke’s law is a constant and the volume modulus depends on the sign of the first invariant of the stress tensor. Plane problems (plane strain and generalized plane stressed state) and problems of plate bending are considered. Some examples are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 1, pp. 157–164, January–February, 2008.     

On the Equivalence of Energetic and Geometric Shear Factors Based on Saint Venant Flexure

The equivalence of the shear compliance tensor obtained via an energy approach and the shear compliance tensor obtained via a geometric/kinematic approach is proven in the context of Saint Venant flexure. Specifically, it is shown that such an equivalence holds, as a general result, for sections of arbitrary geometry when both these tensors are computed on the basis of the long wavelength shear warpage, i.e., of the shear warpage independent from the longitudinal abscissa, provided all long-wavelength terms are included. The equivalence of energetic and kinematic shear factors stems as an immediate consequence of the equivalence of shear compliance tensors. A general analytical proof of this result is provided by analyzing in full detail the 3-dimensional elastostatic solution, inclusive of short-wavelength terminal fields, for a tip loaded cantilever. In particular, the developments reported exploit the objective tensor representations of stress and displacement fields solution to Saint Venant problem recently presented in a companion paper (Serpieri and Rosati, J. Elast., 2013). The well posedness of the concepts of energetic principal shear axes and geometric principal shear axes is shown as a further direct consequence. In particular, principal shear axes are shown to deserve the same legitimacy of principal bending axes both under an energetic and a geometric/kinematic viewpoint.     

An engineering theory of soil behaviour in unloading and reloading

Summary A constitutive law is proposed for describing the stress-strain characteristic of soils in unloading-reloading. The constitutive equations are valid piecewisely between subsequent, appropriately formulated, stress reversal loci. The stress-strain relationships are formulated in terms of generalized stress and strain differences referred to the last stress reversal point and connected through a variable compliance tensor. The shear compaction effect is modelled by a suitable formulation of the compliance tensor.Specialization to conventional triaxial condition is given. As well as fitting available experimental data in unloading-reloading of normally consolidated and overconsolidated clays, the proposed constitutive relation can model the dependence on OCR of the shape of the undrained effective stress paths, the phenomenon of cyclic mobility of clay in undrained compression and the unloading-reloading stress paths in the oedometer. The theory requires the identification of only three material constants in addition to those pertinent to the usual elastoplastic theory of soil with which it may be easily combined.

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Sommario Si propone una legge costitutiva per descrivere il legame sforzi-deformazioni dei terreni sottoposti a processi di scarico e ricarico. Le equazioni costitutive sono formulate a tratti e definite su un dominio limitato dai luoghi di inversione di carico. Nella formulazione della legge costituitiva verranno introdotte delle variabili generalizzate di sforzo riferite all'ultimo punto di inversione di carico. Queste variabili sono legate alle deformazioni, riferite anche esse allo stato relativo all'ultimo punto di inversione di carico, da un tensore di cedevolezza variabile. Un'adeguata formulazione di questo tensore permette di modellare l'efferio di densificazione sotto carico deviatorico ciclico.Questa legge costitutiva interpreta bene i risultati sperimentali su argille normalmente consolidate e sovraconsolidate. La teoria permette anche di descrivere la dipendenza del percorso degli sforzi efficaci in condizioni non drenate dal grado di sovraconsolidazione, la mobilità ciclica dell'argilla in condizioni non drenate e il percorso degli sforzi efficaci in un processo di scarico e ricarico in un edometro.Per identificare il modello sono necessari solo tre parametri oltre a quelli necessari per identificare il comportamento del terreno vergine.

     

Complex potential formalisms for bending of inhomogeneous monoclinic plates including transverse shear deformation

This paper develops complex potential formalisms for the solution of the bending problem of inhomogenoeus anisotropic plates, on the basis of the most commonly used refined plate theories. Being an initial step in that direction, it works out such formalisms only in connection with the bending problem of shear deformable homogeneous plates as well as plates having a special type of inhomogeneity along their thickness direction. The adopted type of inhomogeneity is however still general enough to include certain classes of plates made of functionally graded material as well as the classes of cross- and angle-ply symmetric laminates as particular cases. The basic formalism, similar to that developed by Stroh in plane strain elasticity, is detailed in relation with the equilibrium equations of a generalized plate theory that accounts for the effects of transverse shear deformation and includes conventional, refined theories as particular cases. Some interesting specializations, related to the most important of those conventional plate theories, are then presented and discussed separately. Hence, the outlined formalisms provide, for the first time in analytical form, the general solution of the partial differential equations associated with the most commonly used refined, elastic plate theories.     

简化Navier—Stokes方程的IFT理论的各向异性张力有限元法

本文首先讨论简化Navier-Stokes IFT方程组的有限元离散方式,然后对其广义解进行分析,并从而利用与之相匹配的各向异性张力单元对流函数—涡量方程进行计算。通过平板层流和台阶绕流两个算例的分析,证明这种与IFT理论相匹配的有限单元算法是成功的。     

A simple mixture theory for ν Newtonian and generalized Newtonian constituents

This work presents the development of mathematical models based on conservation laws for a saturated mixture of ν homogeneous, isotropic, and incompressible constituents for isothermal flows. The constituents and the mixture are assumed to be Newtonian or generalized Newtonian fluids. Power law and Carreau–Yasuda models are considered for generalized Newtonian shear thinning fluids. The mathematical model is derived for a ν constituent mixture with volume fractions ${\phi_\alpha}$ using principles of continuum mechanics: conservation of mass, balance of momenta, first and second laws of thermodynamics, and principles of mixture theory yielding continuity equations, momentum equations, energy equation, and constitutive theories for mechanical pressures and deviatoric Cauchy stress tensors in terms of the dependent variables related to the constituents. It is shown that for Newtonian fluids with constant transport properties, the mathematical models for constituents are decoupled. In this case, one could use individual constituent models to obtain constituent deformation fields, and then use mixture theory to obtain the deformation field for the mixture. In the case of generalized Newtonian fluids, the dependence of viscosities on deformation field does not permit decoupling. Numerical studies are also presented to demonstrate this aspect. Using fully developed flow of Newtonian and generalized Newtonian fluids between parallel plates as a model problem, it is shown that partial pressures p _α of the constituents must be expressed in terms of the mixture pressure p. In this work, we propose ${p_\alpha=\phi_\alpha p}$ and ${\sum_\alpha^\nu p_\alpha = p}$ which implies ${\sum_\alpha^\nu \phi_\alpha = 1}$ which obviously holds. This rule for partial pressure is shown to be valid for a mixture of Newtonian and generalized Newtonian constituents yielding Newtonian and generalized Newtonian mixture. Modifications of the currently used constitutive theories for deviatoric Cauchy stress tensor are proposed. These modifications are demonstrated to be essential in order for the mixture theory for ν constituents to yield a valid mathematical model when the constituents are the same. Dimensionless form of the mathematical models is derived and used to present numerical studies for boundary value problems using finite element processes based on a residual functional, that is, least squares finite element processes in which local approximations are considered in ${H^{k,p}\left(\bar{\Omega}^e\right)}$ scalar product spaces. Fully developed flow between parallel plates and 1:2 asymmetric backward facing step is used as model problems for a mixture of two constituents.     

Constitutive equations for polymer fluids based on the concept of configuration-dependent molecular mobility: a generalized mean-configuration model

After a brief outline of the concept of configuration-dependent molecular mobility for the particular case of the one-mode mean-configuration theory, a generalized model is introduced in which the dependence of the mobility tensor on the configuration tensor is given by a relaxation-type functional. This model is analysed for steady and transient extensional and shear flows. In extensional flow it predicts a maximum in the steady-state uniaxial viscosity curves and stress overshoot in the stressing curves, and in shear flow it predicts even larger stress overshoot in the stressing curves. This model bridges the gap between the current molecular models and the most elaborate network models. In an appendix it is shown that for the relaxation-type dependence of the mobility it is only by using the upper Oldroyd derivative that physically acceptable results are predicted.     

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates.Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton's principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-ordex theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.