Axisymmetric Equations Of Motion Research Articles

Nonlinear Vibration Behavior of Rapidly Heated Temperature-Dependent FGM Shallow Spherical Shells

In the present research, thermally induced vibration of shallow spherical shells made of functionally graded materials is investigated. The thermomechanical properties of the FGM media are assumed to be temperature dependent. Based on the uncoupled thermoelasticity theory, the one-dimensional heat conduction equation is established. The top and bottom surfaces of the shell are subjected to several types of rapid heating boundary condition. Because of the temperature dependency of the material properties, solution of nonlinear heat conduction equation needs a numerical method. In the first step, generalized differential quadrature method (GDQM) is applied to discretize the heat conduction equation across the shell thickness. Afterward, the governing system of time-dependent ordinary differential equations is solved using the successive Crank–Nicolson scheme. The obtained thermal force and thermal moment resultants at each time step are applied to the equations of motion. The axisymmetric equations of motion, using the first-order shear deformation theory (FSDT) based on the Kármán type of geometrical nonlinearity, are derived with the aid of the Hamilton principle. Using the harmonic differential quadrature method, shell domain is divided into a number of nodal points, and differential equations are turned into a system of ordinary differential equations. To obtain the displacement components at any time, time marching scheme based on the Newmark method is implemented. The obtained highly nonlinear algebraic equations are solved by means of the Newton–Raphson method. Comparison studies are performed to validate the formulation and solution method of the present research. Various examples are illustrated to discuss the influences of effective parameters such as power law index in the FGM formulation, thickness of the shell, temperature dependency, shell opening angle, in-plane boundary conditions, type of thermal boundary conditions, and geometrical nonlinearity on thermally induced response of the FGM shell under thermal shock.

Analyzing calculation results of non-stationary axisymmetric thermoelasticity task for a circular isotropic plate

The paper releases results of numerical calculation of axisymmetric dynamic thermoelasticity task for a fixed circular isotropic plate in case of temperature change on its front faces (boundary conditions of the 1st type). The calculated ratios are obtained by using the GL-theory of thermoelasticity (classical theory), which determines the dependence of the vector of heat flux on the velocity of change and temperature gradient. The mathematical model of the task in question includes differential equations of axisymmetric motion and thermal conductivity, formulated as regard to the component of the movement vector and the function of temperature change. Not self-adjoint system is investigated independently. For its solution, a mathematical apparatus technique of separation of variable in the form of finite integral transformations is used, that is transformations of Fourier, Hankel and generalized integral transformation (GIT). The constructed calculation ratios give an opportunity to define stress and strain state and character of distribution of a thermal field of rigidly fixed circular plate with arbitrary axially symmetrical temperature external influence. It is shown, that elastic inertial characteristics of a plate influence the law of change of movement over time only while investigating very thin plates at high-speed temperature impact.

On the equations of axisymmetric motion of a viscous incompressible fluid

where u, v, and w are the radial, azimuthal, and axial velocity components, respectively; p is the fluid pressure; and ν is the kinematical viscosity. Without restriction of generality, the fluid density is taken to be equal to unity, and the fluid is subjected to potential external forces. In this case, p is the modified pressure that is equal to the sum of the original pressure and externalforce potential. As is known [1], system (1) reduces to two equations relating the azimuthal velocity v component to the current function ψ defined by the expressions

Frequency-dependent open-circuit acoustic sensitivity of fluid-filled, coated, radially polarized piezoelectric ceramic cylindrical shells of arbitrary thickness and infinite length

The frequency-dependent free-field open-circuit voltage generated by radially polarized piezoelectric ceramic fluid-filled shells of infinite length coated with an elastic layer and excited by a plane acoustic wave is studied. The shell and the coating are of arbitrary thickness and are anisotropic. It is shown that the charge density in the piezoelectric shell is zero by using the electrostatic and open-circuit conditions. General expressions for the pressures and radial velocities in the interior and exterior fluids are obtained by using the finiteness and radiation conditions, respectively. General expressions for the radial stresses and velocities in the piezoelectric shell and elastic coating are also determined by using two-dimensional equations of state and axisymmetric equations of motion. The coefficients in the general expressions are then determined by using the continuity conditions at the various interfaces. Finally, an expression for the open-circuit sensitivity is obtained by using a piezoelectric equation of state. Numerical results are presented for air, water, and vacuum in the interior, two types of elastic coating, and piezoelectric shells with various thicknesses. The effect of neglecting anisotropy on the sensitivity is also illustrated.

Exact solutions of the axisymmetric equations of motion of a viscous heat-conducting perfect gas described by systems of ordinary differential equations

All invariant solutions of rank 1 of the two-dimensional equations of motion of a heat-conducting perfect gas with a polytropic equation of state are described. A sufficient condition for reducibility of regular, partially invariant solutions of rank 1 and defect 1 to invariant solutions is given.

Dynamic breakup of liquid–liquid jets

The axisymmetric, dynamic breakup of a Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from jet start-up to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using an algorithm based on the Volume of Fluid (VOF) method that was previously proven successful in simulations of steady-state liquid jets (i.e., of the jet region close to the nozzle before breakup). The algorithm has been further refined here based on its performance on transient problems such as the solution of the free liquid–liquid capillary jet breakup problem. The comparison of the simulation results with previous experimental measurements of jet length under conditions where all forces, i.e., viscous, inertial, buoyancy, and surface tension, are important, can be judged satisfactory given the sensitive dependence of the results on details of the experimental setup that are not available. The comparison involves the jet length till breakup as well as the jet and drop shapes, often far from regular. In comparison with experiment, the results of the present numerical method show a greater sensitivity of the jet length to the Reynolds number than the best predictions previously available based on the linear stability analysis of the free liquid–liquid capillary jet breakup problem.

Axisymmetric instability model for shaped charge jets

As a shaped charge jet stretches it breaks up into a series of particles along the jet axis of symmetry. These particles are frequently of approximately the same length, suggesting the existence of a critical wavelength for which the growth of an initial surface disturbance is greatest. Following the failure of earlier one-dimensional models to reproduce this critical wavelength effect, this paper addresses the problem using the axisymmetric equations of motion and boundary conditions of continuum mechanics. Certain simplifying assumptions are made, which enable the analysis of the growth of a small initial sinusoidal disturbance for a range of wavelengths including that suggested by observation of break-up by flash radiographic experiments. The disturbance growth equation derived is of a complicated and singular nature. Under certain initial conditions it can approximately predict from the experimental data the observed critical wavelength.