Geometrically nonlinear bending of thin-walled shells and plates under creep-damage conditions

Summary A phenomenological constitutive model for characterization of creep and damage processes in metals is applied to the simulation of mechanical behaviour of thin-walled shells and plates. Basic equations of the shell theory are formulated with geometrical nonlinearities at finite time-dependent deflections of shells and plates in moderate bending. Numerical solutions of initial/boundary-value problems have been obtained for rectangular thin plates (two-dimensional case) and axisymmetrically loaded shells of revolution (one-dimensional case). Based on the numerical examples for the two problems, the influence of geometrical nonlinearities on the creep deformation and damage evolution in shells and plates is discussed. Accepted for publication 30 October 1996     

Geometrically nonlinear isogeometric analysis of laminated composite plates based on higher-order shear deformation theory

In this paper, we present an effectively numerical approach based on isogeometric analysis (IGA) and higher-order shear deformation theory (HSDT) for geometrically nonlinear analysis of laminated composite plates. The HSDT allows us to approximate displacement field that ensures by itself the realistic shear strain energy part without shear correction factors (SCFs). IGA utilizing basis functions namely B-splines or non-uniform rational B-splines (NURBS) enables to satisfy easily the stringent continuity requirement of the HSDT model without any additional variables. The nonlinearity of the plates is formed in the total Lagrange approach based on the small strain assumptions. Numerous numerical validations for the isotropic, orthotropic, cross-ply and angle-ply laminated plates are provided to demonstrate the effectiveness of the proposed method.     

Geometrically nonlinear,steady state vibration of viscoelastic beams

The problem of geometrically non-linear steady state vibrations of beams excited by harmonic forces is considered in this paper. The beams are made of a viscoelastic material defined by the classic Zener rheological model - the simplest model that takes into account all the basic properties of real viscoelastic materials. The constitutive stress-strain relationship for this type of material is given as a differential equation containing derivatives of both stress and strain. This significantly complicates the solution to the problem. The von Karman theory is applied to describe the effects of geometric nonlinearities of beam deformations. The equations of motions are derived using the finite element methodology. A polynomial approximation of bending moments is used. The order of basis functions is set so as to obtain a coherent approximation of moments and displacements. In the steady-state solution of equations of motion, only one harmonic is taken into account. The matrix equations of amplitudes are derived using the harmonic balance method and the continuation method is applied for solving them. The tangent matrix of equations of amplitudes is determined in an explicit form. The stability of steady-state solution is also examined. The resonance curves for beams supported in a different way are shown and the results of calculation are briefly discussed.     

Geometrically nonlinear finite-element models for thin shells with geometric imperfections

A finite-element method to analyze the stress–strain state and stability of thin shells with geometric imperfections is proposed. An arbitrary curvilinear finite element with vector approximation of the displacement function is used. To solve the systems of nonlinear algebraic equations by iteration methods, linearized stiffness matrices of finite elements and residual and load vectors are formed. The stress–strain state of a thin-walled shell with real geometric imperfections under surface pressure and axial compression is analyzed. The effect of geometric imperfections on the critical combination of loads is evaluated     

Geometrically nonlinear large deformation analysis of triangular CNT-reinforced composite plates

A first known investigation on the geometrically nonlinear large deformation behavior of triangular carbon nanotube (CNT) reinforced functionally graded composite plates under transversely distributed loads is investigated. The analysis is carried out using the element-free IMLS-Ritz method. In this study, the first-order shear deformation theory (FSDT) and von Kármán assumption are employed to account for transverse shear strains, rotary inertia and moderate rotations. A convergence study is conducted by varying the supporting size and number of nodes. The effects of transverse shear deformation, CNT distribution and CNT volume fraction on the nonlinear bending characteristics under different boundary conditions are examined.     

Nonclassical models of the theory of plates and shells

Publications dealing with the study of methods of reducing a three-dimensional problem of the elasticity theory to a two-dimensional problem of the theory of plates and shells are reviewed. Two approaches are considered: the use of kinematic and force hypotheses and expansion of solutions of the three-dimensional elasticity theory in terms of the complete system of functions. Papers where a three-dimensional problem is reduced to a two-dimensional problem with the use of several approximations of each sought function (stresses and displacements) by segments of Legendre polynomials are also reviewed.     

Forced vibrations and vibroheating of shallow viscoelastic laminated shells with the piezoelectric effect

The vibrations and dissipative heating of a hinged shallow shell made of viscoelastic piezoelectric material and subject to harmonic electric loading are considered. The basic relations are obtained by using the Kirchhoff-Love mechanical hypotheses supplemented with the respective hypotheses for electric quantities. Analytical solutions of both electromechanical and thermal problems are derived for the case where the temperature is constant along the shell thickness. Translated from Prikladnaya Mekhanika, Vol. 36, No. 6, pp. 78–87, June, 2000.     

Geometrically nonlinear analysis of the stressstrain state of toroidal shells under pure bending

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 1, pp. 139–145, January–February, 1995.     

Dynamical behaviors of nonlinear viscoelastic thick plates with damage

Based on the deformation hypothesis of Timoshenko's plates and the Boltzmann's superposition principles for linear viscoelastic materials, the nonlinear equations governing the dynamical behavior of Timoshenko's viscoelastic thick plates with damage are presented. The Galerkin method is applied to simplify the set of equations. The numerical methods in nonlinear dynamics are used to solve the simplified systems. It could be seen that there are plenty of dynamical properties for dynamical systems formed by this kind of viscoelastic thick plate with damage under a transverse harmonic load. The influences of load, geometry and material parameters on the dynamical behavior of the nonlinear system are investigated in detail. At the same time, the effect of damage on the dynamical behavior of plate is also discussed.