The phenomenon of stress intensification

The article deals with and aims to provide a solution for the discrepancy which has been existing between the stresses calculated according to the in-plane and out-of-plane bending formulas (by von Kármán and Vigness respectively) on the one hand, and the results of stress measurements with strain gauges carried out in laboratory tests on the other hand.     

BEM formulation for von Kármán plates

This work deals with nonlinear geometric plates in the context of von Kármán's theory. The formulation is written such that only the boundary in-plane displacement and deflection integral equations for boundary collocations are required. At internal points, only out-of-plane rotation, curvature and in-plane internal force representations are used. Thus, only integral representations of these values are derived. The nonlinear system of equations is derived by approximating all densities in the domain integrals as single values, which therefore reduces the computational effort needed to evaluate the domain value influences. Hyper-singular equations are avoided by approximating the domain values using only internal nodes. The solution is obtained using a Newton scheme for which a consistent tangent operator was derived.     

Linear buckling analysis of orthotropic inhomogeneous rectangular plates under uniform in-plane compression

Summary A linear buckling analysis is carried out for orthotropic inhomogeneous rectangular plates under uniform in-plane compression. It is assumed that material inhomogeneities of Young's modulus and shear modulus of elasticity are continuously changed in the thickness direction with the power law of the coordinate variable, while Poisson's ratio is assumed to be constant. The buckling equation can be successfully constructed as the linearized von Kármán plate model by introducing the newly defined position of the reference plane which enables us to easily deal with the problem. The critical buckling loads of the simply supported rectangular plate are presented using the derived fundamental relations. Effects of material inhomogeneity, material orthotropy, aspect ratio, width-to-thickness ratio and load ratio are discussed.     

Post-buckling analysis of skew plates subjected to combined in-plane loadings

Post-buckling behavior of laminated composite, sandwich and functionally graded skew plates is analyzed in the present work. The problem formulation is based on higher-order shear deformation theory and von Kármán’s nonlinear kinematics. Linear mapping is used to transform the physical domain into the computational domain. Chebyshev polynomials are used for spatial discretization of governing differential equations and boundary conditions. The nonlinear terms are linearized using quadratic extrapolation technique. The effect of the skew angle on the buckling and post-buckling response of the composite, sandwich and FGM-clamped skew plates is investigated for different combinations of in-plane compressive loadings.     

Nonlinear free vibration analysis of generic coupled induced strain actuated piezo-laminated beams

Piezo-laminated thin beams have been analyzed with induced strain actuation using Kirchhoff’s hypothesis and von Kármán strain displacement relations. Extremizing the Lagrangian of the system derives the governing nonlinear partial differential equations for the beam. Eliminating the in-plane displacement, an integro-partial differential equation of motion is obtained in terms of the transverse displacement. A deflection function that satisfies the simply supported boundary conditions is assumed to get the system equation as a nonlinear second order ordinary differential equation in time, which is of Duffing’s type. The solution of the problem is obtained through exact integration. Results are presented for frequency and amplitude for surface bonded PZT-5A layer in composite beams with various stacking sequences.     

Surface and nonlocal effects on the nonlinear free vibration of non-uniform nanobeams

The surface and nonlocal effects on the nonlinear flexural free vibrations of elastically supported non-uniform cross section nanobeams are studied simultaneously. The formulations are derived based on both Euler–Bernoulli beam theory (EBT) and Timoshenko beam theory (TBT) independently using Hamilton’s principle in conjunction with Eringen’s nonlocal elasticity theory. Green’s strain tensor together with von Kármán assumptions are employed to model the geometrical nonlinearity. The differential quadrature method (DQM) as an efficient and accurate numerical tool in conjunction with a direct iterative method is adopted to obtain the nonlinear vibration frequencies of nanobeams subjected to different boundary conditions. After demonstrating the fast rate of convergence of the method, it is shown that the results are in excellent agreement with the previous studies in the limit cases. The influences of surface free energy, nonlocal parameter, length of non-uniform nanobeams, variation of nanobeam width and elastic medium parameters on the nonlinear free vibrations are investigated.     

Large-amplitude vibrations of nanowires with dissipative surface stress effects

In this study, nonlinear free vibrations of doubly clamped Euler–Bernoulli nanowires (NWs) have been considered. The von Kármán strain–displacement relationships along with the classic Zener model are implemented to derive the nonlinear differential equation of the flexural motion of NW. Nonlinear natural frequencies are calculated using the computer package Mathematica. The effects of size-dependent surface dissipation, mode numbers, and amplitude of vibrations on the nonlinear natural frequencies are investigated. It is shown that the surface dissipation effect on the normalized nonlinear natural frequencies depends on the amplitudes of vibrations. Also, comparisons are made with the results published in previous studies.     

Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading

The nonlinear dynamic behavior of simply supported tapered sandwich plates subjected to air blast loading is investigated theoretically and numerically. The plate is supposed to have both tapered core and tapered laminated face sheets and be subjected to uniform air blast load. The theory is based on a sandwich plate theory, which includes von Kármán large deformation effects, in-plane stiffnesses, inertias and shear deformations. The sandwich plate theory for plates with constant thickness which have one-layered face sheets found in the literature is developed to analyze the tapered sandwich plates with multi-layered face sheets. The equations of motion are derived by the use of the virtual work principle. Approximate solution functions are assumed for the space domain and substituted into the equations. The Galerkin method is used to obtain the nonlinear differential equations in the time domain. The finite difference method is applied to solve the system of coupled nonlinear equations. The tapered sandwich plate subjected to air blast load is also modelled by using the finite element method. The displacement–time and strain–time histories are obtained. The theoretical results are compared with finite element results and are found to be in an agreement.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Nonlinear Free Vibration of In-Plane Functionally Graded Rectangular Plates

Nonlinear free vibration of functionally graded (FG) plates with in-plane material inhomogeneity subjected to different boundary conditions is presented. The nonlinear equations of motion and the related boundary conditions are extracted based on the classical plate theory. Green's strain tensor together with von Kármán assumptions is employed to model the geometrical nonlinearity. The differential quadrature method as an efficient and accurate numerical tool is employed to discretize the governing equations in spatial domain. After validating the presented approach, parametric studies are performed to clarify the effects of different parameters on the nonlinear frequency parameters of the in-plane FG plates.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Postbuckling of functionally graded cylindrical shells with tangential edge restraints and temperature-dependent properties

This paper presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded cylindrical shells subjected to thermal and axial compressive loads. Material properties are assumed to be temperature dependent and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The governing equations are established within the framework of classical thin shallow shell theory taking both geometrical nonlinearity in von Kármán–Donnell sense and initial imperfection into consideration. Thermal stability analysis also incorporates the effects of tangential edge constraints. A Galerkin procedure is applied to derive expressions of load-deflection relations from which the thermal buckling loads and postbuckling curves of the shells are obtained by an iteration. Effects played by material and geometrical properties, tangential stiffness, imperfection and buckling modes are discussed.     

Postbuckling of nanotube-reinforced composite cylindrical shells under combined axial and radial mechanical loads in thermal environment

A postbuckling analysis is presented for nanocomposite cylindrical shells reinforced by single-walled carbon nanotubes (SWCNTs) subjected to combined axial and radial mechanical loads in thermal environment. Two types of carbon nanotube-reinforced composite (CNTRC) shells, namely, uniformly distributed (UD) and functionally graded (FG) reinforcements, are considered. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction, and are estimated through a micromechanical model. The governing equations are based on a higher order shear deformation shell theory with a von Kármán-type of kinematic nonlinearity. The thermal effects are also included and the material properties of CNTRCs are assumed to be temperature-dependent. A boundary layer theory and associated singular perturbation technique are employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, FG-CNTRC cylindrical shells under combined action of external pressure and axial compression for different values of load-proportional parameters. The results for UD-CNTRC shell, which is a special case in the present study, are compared with those of the FG-CNTRC shell.     

Nonlinear analysis of forced vibration of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic nanobeams

This paper deals with the forced vibration behavior of nonlocal third-order shear deformable beam model of magneto–electro–thermo elastic (METE) nanobeams based on the nonlocal elasticity theory in conjunction with the von Kármán geometric nonlinearity. The METE nanobeam is assumed to be subjected to the external electric potential, magnetic potential and constant temperature rise. Based on the Hamilton principle, the nonlinear governing equations and corresponding boundary conditions are established and discretized using the generalized differential quadrature (GDQ) method. Thereafter, using a Galerkin-based numerical technique, the set of nonlinear governing equations is reduced into a time-varying set of ordinary differential equations of Duffing type. The pseudo-arc length continuum scheme is then adopted to solve the vectorized form of nonlinear parameterized equations. Finally, a comprehensive study is conducted to get an insight into the effects of different parameters such as nonlocal parameter, slenderness ratio, initial electric potential, initial external magnetic potential, temperature rise and type of boundary conditions on the natural frequency and forced vibration characteristics of METE nanobeams.     

Nonlinear dynamic stability of laminated composite shells integrated with piezoelectric layers in thermal environment

This paper reports the nonlinear dynamic stability characteristics of laminated composite cylindrical (CYL) and spherical (SPH) shells integrated with piezoelectric layers using the finite element method. The shells are subjected to a thermal environment in addition to the in-plane periodic load and the electric load. The theoretical formulation considers Sanders?? approximation for doubly curved shells, and von Kármán type nonlinear strains are incorporated into the first-order shear deformation theory (FSDT). The formulation includes the effects of transverse shear, in-plane and rotatory inertia. The in-plane periodic load is taken as the parametric excitation in the governing equation. The nonlinear matrix amplitude equation is obtained by employing Galerkin??s method. The correctness of the formulation is established by comparing the authors?? results with those available in the published literature. Detailed parametric studies are carried out to investigate the effects of different parameters on the dynamic stability characteristics of laminated composite shells.     

Error bounds in nonlinear elastostatics

The boundary value problem of place and traction in nonlinear hyper-elastostatics is considered. As a consequence of convexity of the strain energy function in some neighborhood of a nondegenerate critical point in a quotient space the constitutive equations are invertible. Complementary functionals and a generalized interaction energy lead to estimates for the error energy and error norm without use of the orthogonality conditions. Introduction of extended singular Green states leads formally to pointwise estimates for some field quantities. Numerical results for the von Kármán plate are presented.     

An n-sided polygonal finite element for nonlocal nonlinear analysis of plates and laminates

In this study, a locking-free n-sided C¹ polygonal finite element is presented for nonlinear analysis of laminated plates. The plate kinematics is based on Reddy's third-order shear deformation theory (TSDT). The in-plane displacements are approximated using barycentric form of Lagrange shape functions. The weak-form Galerkin formulation based on the kinematics of TSDT requires the C¹ approximation of the transverse displacement over the polygonal element. This is achieved by embedding the C⁰ Lagrange interpolants over a cubic Bernstein-Bezier patch defined over the n-sided polygonal element. Such an approach ensures the continuity of the derivative field at the inter-element edges. In addition, Eringen's stress-gradient nonlocal constitutive equations are used in the present formulation to account for nonlocality. The effect of geometric nonlinearity is taken by considering the von Kármán geometric nonlinearity. Examples are presented to show the effect of nonlocality, geometric nonlinearity, and the lamination scheme on the bending behavior of laminated composite plates. The results are compared with analytical solutions, conventional FEM results, and with those available in the literature. Shear locking is addressed considering reduced integration and consistent interpolation techniques. The patch test is used to check the convergence of the element developed.     

A simple finite element model for the geometrically nonlinear analysis of thin shells

A triangular flat finite element for the analysis of thin shells which undergo large displacements is proposed. It is based upon the geometrically nonlinear theory of von Kármán for thin plates and the total Lagrangian approach. It has a total of only twelve degrees of freedom, namely, three translations at each vertex and one rotation at each mid-side. The stiffness matrix and the tangent stiffness matrix are derived explicitly. The element is tested against nonlinear patch test solutions and its performance is evaluated by solving several standard problems. The directional derivatives of the potential energy function required for the stability analysis are also provided. Received 10 September 1997     

Linearly elastic annular and circular membranes under radial, transverse, and torsional loading. Part II: Vibrations about deformed equilibria

Equilibrium configurations of annular and circular membranes involving large displacements were determined in Part I of this study. Now small vibrations about such equilibrium states are analyzed for annular membranes. The membrane is linearly elastic, initially flat, and taut. Föppl-von Kármán theory is used to obtain the linearized vibration equations. The problems considered include inward and outward radial stretching, transverse displacement at the inner edge, transverse pressure, a vertical distributed load with a vertically sliding outer membrane edge, torsion at the inner edge along with outward stretching, and in-plane vibrations for cases having a flat equilibrium shape. In many of the cases the transverse motion is coupled with radial and circumferential motions. A shooting method is used to obtain vibration frequencies and corresponding vibration modes with different numbers of nodal diameters and nodal circles. The effects of Poisson’s ratio, in-plane radial and circumferential inertias, the ratio of the radii of the inner and outer edges, and the loading magnitude on the vibration frequencies are investigated.