Two-dimensional Theory Of Elasticity Research Articles

Determination of Viral Capsid Elastic Properties from Equilibrium Thermal Fluctuations

We apply two-dimensional elasticity theory to viral capsids to develop a framework for calculating elastic properties of viruses from equilibrium thermal fluctuations of the capsid surface in molecular dynamics and elastic network model trajectories. We show that the magnitudes of the long wavelength modes of motion available in a simulation with all atomic degrees of freedom are recapitulated by an elastic network model. For the mode spectra to match, the elastic network model must be scaled appropriately by a factor which can be determined from an icosahedrally constrained all-atom simulation. With this method we calculate the two-dimensional Young's modulus Y, bending modulus κ, and Föppl-von Kármán number γ, for the T=1 mutant of the Sesbania mosaic virus. The values determined are in the range of previous theoretical estimates.

A Macro-scale Approximation for the Running-in Period

The article presents asymptotic modeling of the running-in wear process with fixed contact zone under a prescribed constant normal load or an imposed contact displacement. The wear contact problem is formulated within the framework of the two-dimensional theory of elasticity in conjunction with Archard’s law of wear. The running-in process is considered at the macro-scale level, while the micro-processes associated with roughness changes, tribomaterial evolution, and microstructural alteration in the subsurface layers as a first approximation are neglected. The setting of the steady-state regime for the macro-contact pressure evolution is chosen as the criterion to characterize the completion of running-in. Simple closed-form approximations are derived for the running-in period and running-in sliding distance. The obtained results can be used for estimating the running-in period in wear processes where the evolution of the macro-shape deviations at the contact interface plays a dominant role.

Closed-form solutions for elastic stress–strain analysis in unbalanced adhesive single-lap joints considering adherend deformations and bond thickness

A theoretical model is developed for the stress analysis in adhesive-bonded single-lap joints under tension, for which the two adherends could have different thicknesses and consist of different materials. A two-dimensional (2D) elasticity theory is adopted in the analysis, which simultaneously incorporates the complete strain–displacement and the complete stress–strain relationships for the adherends and adhesive. The approach provides a unified treatment for any possible adhesive layer flexibility and capable of satisfying the stress-free condition at the ends of the bondline. An explicit closed-form analytical solution is formulated for upper and lower adherends/adhesive stresses (strains) and tensile, shear and bending loads acting on the adherends along the overlap and then simplified for practical applications, and simple design formulae for adhesive stresses are produced. The results predicted by the present full and simplified solutions were compared with the previously theoretical solution by Bigwood and Crocombe (1989) [35], and the 2D geometrically nonlinear finite element model using MSC/NASTRAN. The agreement validates the present formulation and solutions for unbalanced bonded joints. The effects of the stiffness unbalanced parameters on the adhesive stress distributions were also discussed.

On the Principal Boundary Value Problem in the Two-Dimensional Anisotropic Theory of Elasticity

By definition, the principal problem of the two-dimensional theory of elasticity consists in solving the equation for the Airy’s stress function in a region with its first order derivatives assigned at a boundary. In this paper, an indirect formulation of this problem based on integral equations with weakly singular kernels is proposed. In a bounded region with a Lyapunov boundary it is reduced to the solution of weakly singular integral equations. Differential properties of its solution are investigated.

Identification of Adhesive Bond in A Multi-Layered Structure Via Sound Insulation Characterestics

AbstractIn this paper, adhesive bonds in multi-layered plates are identified based on experimental values of their sound insulation characteristics. An exact model based on two-dimensional elasticity theory is formulated. The problem is a time harmonic plane acoustic progressive wave interaction with an isotropic multi-layered infinite elastic plate with interlaminar bonding imperfections. The T-matrix solution technique, which involves a system global transfer matrix, is formed as the product of individual transfer matrices. This is accomplished by applying continuity of the displacement and stress components at the interfaces of neighboring layers along with the relevant boundary conditions at the left and right interfaces of the plate with the surrounding acoustic fluid (air). The resulting equations are then solved for the unknown plane wave reflection and transmission coefficients. The experimental values of sound transmission loss (TL) are measured by a modified B&K impedance tube. Results are presented for a double-layered (lead-steel) plate while the layers are bonded together with metal glue. The normal and transverse adhesive spring constants of the metal glue are then identified in an inverse manner. The agreement of experiments with the analytical TL values predicted for a new triple-layered plate (based on the identified bond properties) confirms the validity of the method.

In Memoriam: James K. Knowles 1931–2009

In Memoriam: James K. Knowles 1931–2009

A new analytic symplectic elasticityapproach for beams resting on Pasternak elastic foundations

Analytic solutions describing the stresses and displacements of beams on a Pasternak elastic foundation are presented using a symplectic method based on classical two-dimensional elasticity theory. Hamilton’s principle with a Legendre transformation is employed to derive the Hamiltonian dual equation, and separation of variables reduces the dual equation to an eigenequation that differs from the conventional eigenvalue problems involved in vibration and buckling analysis. Using adjoint symplectic orthonormality, a group of eigensolutions of zero eigenvalue, corresponding to the Saint-Venant problem, are derived. This approach differs from the traditional semi-inverse analysis, which requires stress or deformation trial functions in the Lagrangian system. The final solutions, which account for the effects of an elastic foundation and applied lateral loads, are approximated by an eigenfunction expansion. Comparisons with existing numerical solutions are conducted to validate the efficiency of this new approach.

Crack nucleation sensitivity analysis

A simple analytical expression for crack nucleation sensitivity analysis is proposed relying on the concept of topological derivative and applied within a two-dimensional linear elastic fracture mechanics theory (LEFM). In particular, the topological asymptotic expansion of the total potential energy together with a Griffith-type energy of an elastic cracked body is calculated. As a main result, we derive a crack nucleation criterion based on the topological derivative and a criterion for determining the direction of crack growth based on the topological gradient. The proposed methodology leads to an axiomatic approach of crack nucleation sensitivity analysis. Copyright © 2010 John Wiley & Sons, Ltd.

A variational method for the solution of biharmonic problems for a rectangular domain

We develop a variational method for the solution of biharmonic problems for a rectangular domain where, at one pair of its opposite sides, the unknown function and its normal derivative take zero values, and, at the other pair, certain inhomogeneous conditions are valid. The cases of semiinfinite and finite domain are considered. The method is based on the minimization of a quadratic functional determining the deviation of the solution from the given inhomogeneous conditions in the norm of L 2. To solve this variational problem, we apply the expansion of the solution in the systems of complex biharmonic functions (the so-called Papkovich homogeneous solutions), which satisfy identically the given homogeneous conditions at the pair of opposite sides of the rectangle. This representation of the solution is somewhat different from that proposed earlier [V. F. Chekurin, “A variational method for the solution of direct and inverse problems of the theory of elasticity for a semiinfinite strip,” Izv. Ross. Akad. Nauk, Mekh. Tverdogo Tela, No. 2, 58–70 (1999)]. We consider several variants of inhomogeneous boundary conditions arising in the problems of the two-dimensional theory of elasticity. Finally, we give an example of applying the proposed method for the determination of stress distributions in a rectangular area one of whose sides is rigidly fastened and the opposite one is subjected to the action of normal forces.

Indentation in lightweight composite sandwich beams

Indentation law is the relationship between force and deflection under the load on a sandwich beam in a fully backed state (on rigid ground), which plays an important role in low-velocity impact models. In the classical method, the Winkler model, which is based on analysing a beam on elastic foundation, is used for measuring the linear part of indentation law and core compression yielding load. This load is assumed to be the same as three-point bending (3PB) state core compression yielding load. In the present article, the sandwich beam is divided into three regions: two faces and a core. Faces are modelled with classical beam theory separately and the core is modelled with two-dimensional elasticity theory, which is known as sandwich panel higher-order theory (SPHOT). A simply supported and fully backed sandwich beam with concentrated loading is solved with Fourier series, and closed-form solutions for core compression, shear, and normal stresses of the core are obtained. The obtained core yielding load by SPHOT is compared with the classical method using the commercial finite element code ANSYS and also with recently reported researches in the literature. The results show that SPHOT predicts more accurate core yielding load and better indentation law compared with other theories and are in good agreement with the results of finite element simulation. In addition, differences between SPHOT theory and the classical method for indentation are investigated in terms of geometry and material properties of sandwich elements. The bottom face effect is determined in core compression in the 3PB state. It is shown that core compression in the 3PB state is lower than fully backed indentation and fully backed indentation is lower than classical prediction.

Two-dimensional in-plane free vibrations of functionally graded circular arches with temperature-dependent properties

Abstract In this paper, the in-plane free vibration analysis of functionally graded (FG) thick circular arches subjected to initial stresses due to thermal environment is studied. The formulations are based on the two-dimensional elasticity theory. The material properties are assumed to be temperature-dependent and graded in the thickness direction. Considering the thermal environment effects, the equations of motion are derived using the Hamilton’s principle. The initial thermal stresses are obtained by solving the two-dimensional thermoelastic equilibrium equations. Two types of temperature rise, uniform and variable through the thickness, is considered. Differential quadrature method (DQM) is adopted to solve the equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with those available in the literature for isotropic arches. The effects of temperature rise, material and geometrical parameters on the natural frequencies are investigated. The new results can be used as benchmark solutions for future researches.

Two-dimensional elasticity solutions for temperature-dependent in-plane vibration of FGM circular arches

Abstract Temperature-dependent in-plane vibration of functionally graded (FGM) circular arches based on the two-dimensional theory of elasticity is investigated. An analytical solution using the state space formulation and Fourier series expansion is obtained for a simply supported circular arch. For such functionally graded arches, the state equation has variable coefficients. Because a definite, continuously varying FG model through the thickness is impractical if not impossible, an approximate laminate model is constructed to derive an asymptotic solution through the thickness direction. The transfer relationship between the state vectors at the inner and outer surfaces is ultimately obtained by considering the continuity conditions at the artificial interfaces. The new formulation is validated by comparing some numerical solutions with established results in open literature, such as functionally graded straight beams, curved sandwich beams and laminated thick circular arches. Effective material properties are predicted using the Mori–Tanaka model and taken as temperature-dependent. Effects of the gradient index, temperature and geometric parameters, i.e. the curvature, length-to-thickness ratio, subtended angle, on the vibration frequency are analyzed and discussed.

Topological aspect of disclinations in two-dimensional crystals

By using topological current theory, this paper studies the inner topological structure of disclinations during the melting of two-dimensional systems. From two-dimensional elasticity theory, it finds that there are topological currents for topological defects in homogeneous equation. The evolution of disclinations is studied, and the branch conditions for generating, annihilating, crossing, splitting and merging of disclinations are given.

A Two-Dimensional Approach of Single-Lap Adhesive Bonded Joints

A general two-dimensional (2D) theoretical approach capable of providing an explicit closed-form solution is developed for the calculation of elastic stresses in single-lap adhesive bonded joints, assuming linear distribution of a longitudinal normal stress in the joint thickness direction. By treating the adhesive layer in the same way as the adherends, the two-dimensional stress and strain distributions at any point, and the tensile force, shearing and bending moment at any cross section can be predicted accurately, in both the adhesive and adherends. This makes it possible for the subsequent failure analysis of the bonded joints to predict all the main modes such as adherend failure, cohesive failure and adhesive failure. The analysis is based on a two-dimensional elasticity theory that both includes the complete stress-strain and the complete strain-displacement relationships for the adhesive and adherends. Then, through the use of the displacement continuity conditions at the interfaces between the adhesive and adherends, four coupled, fifth-order ordinary differential equations with constant coefficients for the determination of the stresses are developed. The proposed method is capable of satisfying all the boundary stress conditions of the joint, including the stress-free surface condition at the ends of the bondline. The calibration of the new theoretical approach was verified by comparing it with the previously theoretical solutions, and the 2D geometrically nonlinear finite element models with the rotation and non-rotation boundary conditions. It is indicated that the present solution can provide a good prediction for the stress and strain distributions in the adhesive and adherends. A different approach is also established for determining bending moment factor of the short joints wherein the consideration of overall joint moment equilibrium is emphasized. It was shown that when the proposed bending moment factor is adopted instead of the original one, the classical solution by Goland and Reissner (1944) [4] might still remain applicable to the adherends and adhesive even with the parameter E3t1 /( E 1t3 ) > 0.1.

In-plane free vibration of circular annular disks

This paper presents a generalized formulation for the in-plane modal characteristics of circular annular disks under combinations of all possible classical boundary conditions. The in-plane free vibration of an elastic and isotropic disk is studied on the basis of the two-dimensional linear plane stress theory of elasticity. The boundary characteristic orthogonal polynomials are employed in the Rayleigh–Ritz method to obtain the natural frequencies and associated mode shapes. Two approaches have been used to represent a clamped boundary condition. The first approach assumes a polynomial expression that satisfies the clamped conditions, while the second approach uses a disk with free boundary supported on artificial springs with stiffness tending to infinity. The natural frequencies are tabulated and compared with data available in the literature. Mode shapes are presented to illustrate the free vibration behavior of the disk.

Two-Dimensional Stress Analysis of Adhesive Butt Joints Filled With Elastic Circular Fillers in the Adhesive Subjected to Static Tensile Loadings

This paper deals with a two-dimensional stress analysis of adhesive butt joints with elastic circular fillers in the adhesive subjected to external tensile loadings. Similar adherends and an adhesive including elastic circular fillers were replaced with finite strips in the analyses. Stress distributions in the adhesive butt joints were analyzed exactly using a two-dimensional theory of elasticity. The effects of stiffness, location of filler particles and number of filler particles on the interface stress distributions and at the periphery of the filler particles were examined in the numerical calculations. It was seen that as the amount of filler particles increased, the stiffness of filler increased and the location of filler approached the edges of interfaces, and the joint strengths were assumed to increase. For verification of the analyses, experiments were carried out to measure the strains. The analytical results were in fairly good agreement with the experimental results. In addition, the analytical results were also compared with those obtained from 2-D and 3-D FEM calculations. Fairly good agreements were seen among the analytical, the 2-D and 3-D FEM results. However, it was observed that the difference in the interface stress distributions between the present analysis and 3-D FEM calculation was substantial due to the stress singularity at the interfaces in the z-direction obtained from 3-D FEM. Rupture tests were conducted, and joint strengths were shown to be improved by adding the elastic fillers.

Elasticity solution of multi-span beams with variable thickness under static loads

This paper studies the stress and displacement distributions of continuously varying thickness multi-span beams simply supported at two ends and under static loads. The intermediate supports of the beam may be elastic and/or rigid in one or two directions. On the basis of the two-dimensional plane elasticity theory, the general solution of stress function, which exactly satisfies the governing differential equations and the simply supported boundary conditions, is deduced. In the present analysis, the reaction forces of the intermediate supports are regarded as the unknown external forces acting on the lower surface of the beam under consideration. The unknown coefficients in the solutions are determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beam and using the linear relations between reaction forces and displacements of the beam at intermediate supports. The solution obtained is exact and excellent convergence has been confirmed. Comparing the numerical results obtained from the proposed method to those obtained from the Euler beam theory, the Timoshenko beam theory and those obtained from the commercial finite element software ANSYS, high accuracy of the present method is demonstrated.

Exact two-dimensional solutions for in-plane natural frequencies of laminated circular arches

Exact analysis on the in-plane free vibration of simply supported laminated circular arches is carried out based on the two-dimensional theory of elasticity. The method of separation of variables is employed to expand all the variables into Fourier series about the longitudinal coordinate, so that the system of partial differential equations is reduced to the ordinary one about the radial coordinate. The state space method is used to derive a series of simultaneous first-order differential equations. Due to the variable coefficients posed by the radial coordinate, analytical solutions are rather unpractical, and hence the approximate laminate model is adopted to translate the state equation into the one with constant coefficients. The relation between the state vector at the inner and outer surface of the arch is finally obtained according to the continuity conditions at interfaces and the traction free conditions at the two lateral surfaces. The formulation is validated by comparing the present results to those available in literature. Effects of geometric parameters and stacking sequences on the natural frequencies of circular arches are investigated and discussed. Numerical results presented in this paper provide benchmarks for future analysis of circular arches.

A hybrid layerwise and differential quadrature method for in-plane free vibration of laminated thick circular arches

An accurate and efficient solution procedure based on the two-dimensional elasticity theory for free vibration of arbitrary laminated thick circular deep arches with some combinations of classical boundary conditions is introduced. In order to accurately represent the variation of strain across the thickness, the layerwise theory is used to approximate the displacement components in the radial direction. Employing Hamilton's principle, the discretized form of the equations of motion and the related boundary conditions in the radial direction are obtained. The resulting governing equations are then discretized using the differential quadrature method (DQM). After performing the convergence studies, new results for laminated arches with different set of boundary conditions are developed. Additionally, different values of the arch parameters such as opening angle, thickness-to-length and orthotropy ratios are considered. In all cases, comparisons with the results obtained using the finite element software ‘ABAQUS’ and also with those of the first- and higher-order shear deformation theories available in the literature are performed. Close agreements, especially with those of ABAQUS, are achieved.

Sound Insulation Characteristics of Functionally Graded Panels

An exact treatment based on the two-dimensional elasticity theory, is employed to investigate sound transmission through an arbitrarily thick isotropic functionally graded infinite plate subjected to an oblique plane acoustic incident wave. The mechanical properties of the graded plate are assumed to vary smoothly and continuously with the change of volume concentrations of the constituting materials across the thickness of the plate according to a power law distribution. The original inhomogeneous structure is approximated by a laminate model, for which the solution is expected to gradually approach the exact one as the number of layers increases. The T-matrix solution technique, which involves a system global transfer matrix formed as the product of the individual transfer matrices by applying continuity of the displacement and stress components at the interfaces of neighboring layers along with the pertinent boundary conditions at the upper and lower interfaces of the plate with the surrounding acoustic fluid (air) are employed to solve for the unknown plane wave reflection and transmission coefficients. The analytical results are illustrated with numerical examples in which three frequently used metal-ceramic FGM panels with distinct compositional gradient profiles are subjected to plane incident waves at selected angles of incidence. Furthermore, a Gaussian weighting function for describing the directional distribution of incident energy is adopted for predicting the sound transmission loss along with the weighted sound reduction index, R w (as specified in ISO 717-1), of the FGM panels subjected to a perfectly diffuse sound field. Primary attention is focused on the effects of material compositional gradient profiles on the sound insulation characteristics of the panels. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.