Elastic Plate Theory Research Articles

Theoretical study on the influence of residual stress on adhesion-induced instability in MEMS

Adhesion and residual stress play a critical role in the performance and reliability of MEMS. The influence of residual stress on the adhesion-induced instability in MEMS is examined within the framework of thin elastic plate theory. The results show that the adhesion-induced instability will be mitigated if the residual stress exists in certain component of MEMS. Moreover, we find that the influence is significant only when the residual stress is under a proper magnitude (β⩽20).

Edge-Stress-Induced Warping of Graphene Sheets and Nanoribbons

We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We also demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Our results underscore the importance of accounting for edge stresses in thermal theories and electronic structure calculations for freestanding graphene sheets.

Book Review: J. N. Reddy, Theory and Analysis of Elastic Plates and Shells

ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und MechanikVolume 88, Issue 10 p. 827-827 Book Review Book Review: J. N. Reddy, Theory and Analysis of Elastic Plates and Shells Holm Altenbach, Holm Altenbach Halle (Saale)Search for more papers by this author Holm Altenbach, Holm Altenbach Halle (Saale)Search for more papers by this author First published: 25 September 2008 https://doi.org/10.1002/zamm.200890020Citations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat No abstract is available for this article.Citing Literature Volume88, Issue10October 2008Pages 827-827 RelatedInformation

Implications of simple flexure theory for the formation of saucer-shaped sills

Saucer-shaped dolerite sills have been identified in several sedimentary basins. They are characteristically sub-circular in plan, with an inner sill connected to a steeply inclined arcuate sheet extending upwards from the edge of the inner sill, and in some cases have raggedly developed flat outer rims. By applying elastic plate theory to the overburden, we obtain an expression for the maximum radius of a circular sill in terms of emplacement depth, excess magma pressure, elastic constants of the overburden and the tensile strain at failure. This expression predicts that the radius of the flat base of a saucer-shaped sill increases with emplacement depth, as observed in the western Karoo Basin, South Africa. Some saucer-shaped sills display an elliptical morphology. From consideration of the flexural strains around the periphery of a horizontal elliptical sill, we infer that saucer-shaped elliptical sills have been fed from below by feeder dykes oriented along the major axis of the ellipse. Flat outer rims would have developed at the final stage of each intrusive episode, when the excess magma pressure was decreasing towards equilibrium with the overburden load. Consequently, flexural strains would also have decreased, so continued intrusion might preferentially occur along a bedding plane.

A refined theory of elastic thick plates for extensional deformation

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich–Neuber solution and the Lur’e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.

Molecular mechanics modeling for properties of carbon nanotubes

Molecular mechanics calculations for in-plane stiffness, shear modulus, and the bending rigidity of single-walled carbon nanotubes are reported in this work through the calculations of the strain energy for carbon nanotubes and graphite sheets subjected to various types of loading. Elastic rod and plate theories are employed to link the material properties of carbon nanotubes directly to the molecular mechanics calculations. The in-plane stiffness of carbon nanotubes is about 372–376J∕m2. The bending rigidity is found to be around 1.78eV for relatively large tubes and graphite sheets.

A semi-analytical approach for the non-linear large deflection analysis of laminated rectangular plates under general out-of-plane loading

A semi-analytical approach for the geometrically non-linear analysis of rectangular laminated plates with general inplane and out-of-plane boundary conditions under a general distribution of out-of-plane loads is developed. The analysis is based on the elastic thin plate theory with geometrically non-linear von Kármán strains. The solution of the non-linear partial differential equations is reduced to an iterative sequential solution of non-linear ordinary differential equations using the multi-term extended Kantorovich method. The efficiency, accuracy, and convergence of the proposed method are examined through a comparison with other semi-analytical methods and with finite element analyses. The capabilities of the approach and its applicability to the non-linear large deflection analysis of plate structures are demonstrated through various numerical examples. Emphasis is placed on combinations of lamination, boundary, and loading conditions that cannot be analyzed using alternative semi-analytical methods.

Non-local elastic plate theories

A non-local plate model is proposed based on Eringen's theory of non-local continuum mechanics. The basic equations for the non-local Kirchhoff and the Mindlin plate theories are derived. These non-local plate theories allow for the small-scale effect which becomes significant when dealing with micro-/nanoscale plate-like structures. As illustrative examples, the bending and free vibration problems of a rectangular plate with simply supported edges are solved and the exact non-local solutions are discussed in relation to their corresponding local solutions.

Thin elastic shells with variable thickness for lithospheric flexure of one-plate planets

Planetary topography can either be modeled as a load supported by the lithosphere, or as a dynamical effect due to lithospheric flexure caused by mantle convection. In both cases the response of the lithosphere to external forces can be calculated with the theory of thin elastic plates or shells. On one-plate planets the spherical geometry of the lithospheric shell plays an important role in the flexure mechanism. So far the equations governing the deformations and stresses of a spherical shell have only been derived under the assumption of a shell of constant thickness. However local studies of gravity and topography data suggest large variations in the thickness of the lithosphere. In this article we obtain the scalar flexure equations governing the deformations of a thin spherical shell with variable thickness or variable Young's modulus. The resulting equations can be solved in succession, except for a system of two simultaneous equations, the solutions of which are the transverse deflection and an associated stress function. In order to include bottom loading generated by mantle convection, we extend the method of stress functions to include loads with a toroidal tangential component. We further show that toroidal tangential displacement always occurs if the shell thickness varies, even in the absence of toroidal loads. We finally prove that the degree-one harmonic components of the transverse deflection and of the toroidal tangential displacement are independent of the elastic properties of the shell and are associated with translational and rotational freedom. The flexure equations for a shell of variable thickness are useful not only for the prediction of the gravity signal in local admittance studies, but also for the construction of stress maps in tectonic analysis.

The Contact Problems of the Mathematical Theory of Elasticity for Plates with an Elastic Inclusion

The contact problems of the theory of elasticity and bending theory of plates for finite or infinite plates with an elastic inclusion of variable rigidity are considered. The problems are reduced to integro-differential equations or to systems of integro-differential equations with variable coefficient and singular operator. If such coefficient varies according to power law, we investigate the obtained equations and get exact or approximate solutions and study behavior of unknown contact stresses at the ends of the elastic inclusion.

Residual Stresses in Laminar Functionally Graded Ceramic Materials

A simple analytical model of residual stresses far from edges in symmetrical planar functionally graded material (FGM) is presented. The model is based on elastic plate theory and neglects the influence of edges and free surfaces. The results are compared to analytical model of laminar ceramics and to finite element model of FGM. The influence of various geometrical and material parameters on the internal stress state is discussed.

Deformation of bowed silicon chips due to adhesion and applied pressure

One potential method to accomplish high-rate nanomanufacturing is to develop processes which allow for rapid transfer of nano-scaled devices from a template to a device wafer. In order to accomplish this transfer, the device wafer must make intimate contact with the template. A similar situation exists in wafer bonding, except that in that case the two wafers remain in bonded contact due to the work of adhesion even after the applied pressure is removed. In high-rate nanomanufacturing intimate contact must be maintained during transfer, while allowing for easy separation afterwards. Wafers typically have waviness and bow which cause a deviation of many micrometers from flatness over the 15-mm length scale of a typical chip. This non-flatness can be a serious problem in the transfer of nanometer-scale elements. In this investigation, a model is developed to examine the effects of applied pressure, bow radius and the work of adhesion on the flattening of a spherically/cylindrically bowed chip. This model uses elastic plate theory and the work of adhesion. An operating window is found which provides intimate contact while allowing for separation once the pressure is removed. It is also shown that the effect of adhesion is to produce a discontinuity in the internal bending moment, at the separation boundary, which is proportional to the square-root of the product of the work of adhesion and the flexural rigidity. This "moment-discontinuity" method can be applied to other problems involving adhesion of elastic plates.

General Solutions for the Statics of Anisotropic, Transversely Inhomogeneous Elastic Plates in Terms of Complex Functions

This paper develops the general solution of high-order partial differential equations (PDEs) that govern the static behavior of transversely inhomogeneous, anisotropic, elastic plates, in terms of complex functions. The basic development deals with the derivation of such a form of general solution for the PDEs associated with the most general, two-dimensional (“equivalent single-layered”), elastic plate theory available in the literature. The theory takes into consideration the effects of bending–stretching coupling due to possible un-symmetric forms of through-thickness material inhomogeneity. Most importantly, it also takes into consideration the effects of both transverse shear and transverse normal deformation in a manner that allows for a posteriori, multiple choices of transverse strain distributions. As a result of this basic and most general development, some interesting specializations yield, as particular cases, relevant general solutions of high-order PDEs associated with all of the conventional, elastic plate theories available in the literature.

Modeling of Back Pressure Distribution on the Wafer Loaded in a Multi-Zone Carrier in Chemical Mechanical Polishing

The within-wafer non-uniformity (WIWNU) of material removal rate in chemical mechanical polishing (CMP) is important for IC manufacture. The non-uniform distributions of polishing pressure and the relative speed between the wafer and the polishing pad are main factors affecting the WIWNU. In this paper，based on the contact mechanics and the elastic plate theory, a compensate pressure computing model is presented, in which the effects of kinematic parameters are taken into acount. By modelling and calculating, the desired compensate back pressure distribution is obtained. In the last section the design of a schematic carrier with multi-zone, in which the compensate back pressure can be applied, is presented. The model and the design can be used for providing theoretical guide to the development of CMP equipments and selection of the kinematic variables in CMP process.

On the bending of micropolar viscoelastic plates

The theory of micropolar elastic plates has been established by [A.C. Eringen, Linear theory of micropolar viscoelasticity, Int. J. Engng. Sci. 5 (1967) 191–204]. The present paper is concerned with the bending of micropolar viscoelastic plates within the dynamic theory. First, the basic equations of the bending theory of viscoelastic thin plates are presented. Then, a variational characterization of solutions and a minimum principle of Reiss type are established.

Hamiltonian systems of propagation of elastic waves and localized vibrations in the strip plate

In this paper, based on Lagrange–Germanian theory of elastic thin plates, applying the method in Hamiltonian state space, the elastic waves and vibrations when the boundary of the two lateral sides of the strip plate are free of traction are investigated, and the process of analysis and solution are proposed. The existence of all kinds of vibration modes and wave propagation modes is also analyzed. By using eigenfunction expansion method, the dispersion relations of waveguide modes in the strip plate are derived, and the comparisons with the dispersion relations directly obtained by the classical theory of thin plates are also presented. At last, the results are analyzed and discussed.

An asymptotic analysis of initial-value problems for thin elastic plates

An asymptotic technique is developed to analyse initial-value problems in application to the three-dimensional theory of thin elastic plates. Various sets of long-wave initial data are considered, with arbitrary distribution along the plate thickness. To account for the initial data, composite asymptotic expansions are employed, utilizing both two-dimensional low-frequency and high-frequency models. The former correspond to the classical theories of plate bending and extension, and their refinements, whereas the latter are associated with the long-wave motions occurring in the vicinities of thickness stretch and shear resonance frequencies. Six cases of iteration process are revealed depending on the symmetry of the initial data and their thickness variation. For each case, approximate two-dimensional initial conditions are derived, including higher-order corrections for the two-dimensional low-frequency refined plate theories. The validity of the proposed approach is justified by comparison with the exact solution of the model plane problem for initial data with uniform distribution along the thickness and sinusoidal distribution along the mid-plane. The methodology of the paper has potential for more general initial-value problems specified over a narrow domain, including many of those commonly met in physical applied mathematics.

Dynamic Stress Concentration of a Circular Cutout Buried in Semi-Infinite Plates Subjected to Flexural Waves

In this paper, based on the theory of elastic thin plates, applying the image method and the wave function expansion method, multiple scattering of elastic waves and dynamic stress concentration in semi-infinite plates with a circular cutout are investigated, and the general solutions of this problem are obtained. As an example, the numerical results of dynamic stress concentration factors are graphically presented and discussed. Numerical results show that the analytical results of scattered waves and dynamic stress in semi-infinite plates are different from those in infinite plates when the distance ratio b∕a is comparatively small. In the region of low frequency and long wavelength, the maximum dynamic stress concentration factors occur on the illuminated side of scattered body with θ=π, but not on the side of cutout with θ=π∕2. As the incidence frequency increases (the wavelength becomes short), the dynamic stress on the illuminated side of cutout becomes little, and the dynamic stress on the shadow side becomes great.

Limiting Slurry Pressures to Control Hydraulic Fracturing in Directional Drilling Operations in Purely Cohesive Soil

Hydraulic fracturing during horizontal directional drilling can permit drilling mud to travel to the ground surface, to collect under overlying pavements, or to be released into the riverbed during river crossings. These events can decrease drilling efficiency, damage adjacent infrastructure, and cause environmental damage. Current design equations for maximum drilling pressure focus on shear failure around the borehole drilled to house the new pipe. The present research examines, instead, the potential for tensile fracture of cohesive soils surrounding the borehole. Two-dimensional nonlinear finite element modeling is used to study the effectiveness of closed-form solutions for calculation of the drilling slurry pressures that fracture the surrounding soil. Tangential stresses are calculated for a borehole with a 0.2-m diameter in an undrained clay soil with various K0 values, construction depths, and drilling slurry pressures. Elastic plate theory is found to provide effective values of ( a) tangential crown and spring line stresses when the soil responds elastically and ( b) the decreases in tangential crown stress that occur as drilling slurry pressures increase. Closed-form plasticity solutions provide good values of tangential stresses once the soil yields. Following yield, increases in mud pressure result in increases in tangential stress, so hydraulic fracture from tension is no longer an issue. The study indicates that mud loss in low-strength clays or those with K0 close to unity is most likely the result of unconfined plastic flow (blowout). Mud loss for stiff clays and those with other K0 values is more likely the result of tensile fracture.

Limiting Slurry Pressures to Control Hydraulic Fracturing in Directional Drilling Operations in Purely Cohesive Soil

Hydraulic fracturing during horizontal directional drilling can permit drilling mud to travel to the ground surface, to collect under overlying pavements, or to be released into the riverbed during river crossings. These events can decrease drilling efficiency, damage adjacent infrastructure, and cause environmental damage. Current design equations for maximum drilling pressure focus on shear failure around the borehole drilled to house the new pipe. The present research examines, instead, the potential for tensile fracture of cohesive soils surrounding the borehole. Two-dimensional nonlinear finite element modeling is used to study the effectiveness of closed-form solutions for calculation of the drilling slurry pressures that fracture the surrounding soil. Tangential stresses are calculated for a borehole with a 0.2-m diameter in an undrained clay soil with various K0 values, construction depths, and drilling slurry pressures. Elastic plate theory is found to provide effective values of (a) tangential c...