On the Contact Problem of an Inflated Spherical Nonlinear Membrane

1973 ◽  
Vol 40 (1) ◽  
pp. 209-214 ◽  
Author(s):  
W. W. Feng ◽  
W.-H. Yang
Keyword(s):  
Contact Problem ◽  
Contact Region ◽  
Material Behavior ◽  
Elastic Membrane ◽  
Rigid Plate ◽  
Constraint Condition ◽  
Governing Equations ◽  
First Order ◽  
Stress Strain Relation ◽  
Nonlinear Elastic

The contact problem of an inflated spherical nonlinear elastic membrane between two large rigid plates is formulated in terms of three first-order ordinary differential equations for the region where the spherical membrane is not in contact with the rigid plates. The constraint condition introduced by the rigid plate on part of the spherical membrane reduces the number of governing equations to two for the contact region. A general stress-strain relation for the spherical membrane is used in the formulation. The results presented in this paper assume that the material behavior of the spherical membrane is that described by the Mooney model. Nonlinear spring characteristics and the instability phenomena of the inflated membrane are discussed.

On Axisymmetrical Deformations of Nonlinear Membranes

1970 ◽  
Vol 37 (4) ◽  
pp. 1002-1011 ◽  
Author(s):  
W. H. Yang ◽  
W. W. Feng
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Elastic Stress ◽  
Material Behavior ◽  
System Of Equations ◽  
Initial Value ◽  
First Order ◽  
Stress Strain Relation ◽  
Flat Membrane

The mechanics problem concerning large axisymmetric deformations of nonlinear membranes is reformulated in terms of a system of three first-order ordinary differential equations with explicit derivatives. With a set of proper boundary conditions, arrangements are made to change the boundary-value problem into the form of an initial value problem such that the solution can be obtained by standard numerical methods for integrating ordinary differential equations. The system of equations derived applies to the class of all axisymmetric deformations of membranes with a general elastic stress-strain relation. Three examples are given on inflating of a flat membrane, longitudinal stretching of a tube, and flattening of a semispherical cap. In the examples, the Mooney model are assumed to describe the material behavior of the membranes. The solution on the flat membrane serves to compare with an existing one in literature. The solutions on the tube and the cap are new.

scholarly journals Analysis of a Unilateral Contact Problem with Normal Compliance

2014 ◽  
Vol 52 (1) ◽  
Author(s):  
Arezki Touzaline ◽  
Rachid Guettaf
Keyword(s):  
Contact Problem ◽  
Order Differential Equation ◽  
Uniqueness Result ◽  
Unilateral Contact ◽  
Normal Compliance ◽  
Mechanical Problem ◽  
First Order ◽  
Coulomb’S Friction ◽  
The Coefficient Of Friction ◽  
Nonlinear Elastic

AbstractThe paper deals with the study of a quasistatic unilateral contact problem between a nonlinear elastic body and a foundation. The contact is modelled with a normal compliance condition associated to unilateral constraint and the Coulomb's friction law. The adhesion between contact surfaces is taken into account and is modelled with a surface variable, the bonding field, whose evolution is described by a first-order differential equation. We establish a variational formulation of the mechanical problem and prove an existence and uniqueness result in the case where the coefficient of friction is bounded by a certain constant. The technique of the proof is based on arguments of time-dependent variational inequalities, differential equations and fixed-point theorem.

Vibration characteristics of rotating truncated conical shells reinforced with agglomerated carbon nanotubes

2021 ◽  
pp. 107754632110004
Author(s):  
Hassan Afshari ◽  
Hossein Amirabadi
Keyword(s):  
Carbon Nanotubes ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Conical Shells ◽  
First Order ◽  
Effective Mechanical Properties ◽  
Comprehensive Study

In this article, a comprehensive study is conducted on the free vibration analysis of rotating truncated conical shells reinforced with functionally graded agglomerated carbon nanotubes The shell is modeled based on the first-order shear deformation theory, and effective mechanical properties are calculated based on the Eshelby–Mori–Tanaka scheme along with the rule of mixture. By considering centrifugal and Coriolis accelerations and initial hoop tension, the set of governing equations is derived using Hamilton’s principle and is solved numerically using the differential quadrature method Convergence and accuracy of the presented model are confirmed and the effects of different parameters on the forward and backward frequencies of the rotating carbon nanotube-reinforced truncated conical shells are investigated.

An Asymptotic, Thermo-Diffusive Ignition Theory of Porous Solid Fuels

1976 ◽  
Vol 98 (2) ◽  
pp. 269-275 ◽  
Author(s):  
Choong Se Kim ◽  
Paul M. Chung
Keyword(s):  
Nonlinear Partial Differential Equations ◽  
Porous Solid ◽  
Solid Fuels ◽  
Thermal Ignition ◽  
Mass Balances ◽  
Governing Equations ◽  
Order Ordinary Differential Equation ◽  
First Order ◽  
Time Space ◽  
Gaseous Oxidant

The governing equations of thermal ignition are analyzed for porous solid fuel, such as coal, of various two-dimensional and axisymmetric geometries by the Laplace asymptotic method. Mass diffusion of the gaseous oxidant through the porous fuel is included. The nonlinear partial differential equations of energy and mass balances in time-space coordinates containing the Arrhenius volumic chemical reaction terms are analyzed. By employing the Laplace asymptotic technique and by invoking a certain limit theorem, the governing equations are reduced to a first order ordinary differential equation governing the fuel surface temperature, which is readily solved numerically. Detailed discussion of the effects of the various governing parameters on ignition is presented. Because of the basically closed-form nature of the solutions obtained, many general and fundamental aspects of the ignition criteria hitherto unknown are found.

A Viscoelastic Correspondence Principle for Plane Membranes Subjected to Lateral Pressure

1983 ◽  
Vol 50 (4a) ◽  
pp. 740-742 ◽  
Author(s):  
B. Stora˚kers
Keyword(s):  
Time Dependence ◽  
Viscoelastic Material ◽  
Lateral Pressure ◽  
Correspondence Principle ◽  
Material Model ◽  
Material Behavior ◽  
Linear Elastic Material ◽  
First Order ◽  
Linear Elastic ◽  
Consistent Approximation

The classical Fo¨ppl equations, governing the deflection of plane membranes, constitute the first-order consistent approximation in the case of linear elastic material behavior. It is shown that despite the static and kinematic nonlinearities present, for arbitrary load histories a correspondence principle for viscoelastic material behavior exists if all relevant relaxation moduli are of uniform time dependence. Application of the principle is illustrated by means of a popular material model.

Application of Generalized Differential Quadrature Method to the Bending of Thick Laminated Plates with Various Boundary Conditions

2006 ◽  
Vol 5-6 ◽  
pp. 407-414 ◽  
Author(s):  
Mohammad Mohammadi Aghdam ◽  
M.R.N. Farahani ◽  
M. Dashty ◽  
S.M. Rezaei Niya
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Simple Procedure ◽  
Shear Deformation Theory ◽  
Laminated Plates ◽  
Differential Quadrature ◽  
Governing Equations ◽  
First Order ◽  
Various Boundary Conditions ◽  
Generalized Differential

Bending analysis of thick laminated rectangular plates with various boundary conditions is presented using Generalized Differential Quadrature (GDQ) method. Based on the Reissner first order shear deformation theory, the governing equations include a system of eight first order partial differential equations in terms of unknown displacements, forces and moments. Presence of all plate variables in the governing equations provide a simple procedure to satisfy different boundary condition during application of GDQ method to obtain accurate results with relatively small number of grid points even for plates with free edges .Illustrative examples including various combinations of clamped, simply supported and free boundary condition are given to demonstrate the accuracy and convergence of the presented GDQ technique. Results are compared with other analytical and finite element predictions and show reasonably good agreement.

Mixed State Hamiltonian Element for Plane Transversely Isotropic Magnetoelectroelastic Solids

2013 ◽  
Vol 275-277 ◽  
pp. 1978-1983
Author(s):  
Xiao Chuan Li ◽  
Jin Shuang Zhang
Keyword(s):  
Mixed State ◽  
Hamiltonian Formulation ◽  
Transversely Isotropic ◽  
Laminated Plates ◽  
Governing Equations ◽  
Time Coordinate ◽  
First Order ◽  
Linear Elements ◽  
Propagation Matrix ◽  
Magnetoelectroelastic Solids

Hamiltonian dual equation of plane transversely isotropic magnetoelectroelastic solids is derived from variational principle and mixed state Hamiltonian elementary equations are established. Similar to the Hamiltonian formulation in classic dynamics, the z coordinate is treated analogous to the time coordinate. Then the x-direction is discreted with the linear elements to obtain the state-vector governing equations, which are a set of first order differential equations in z and are solved by the analytical approach. Because present approach is analytic in z direction, there is no restriction on the thickness of plate through the use of the present element. Using the propagation matrix method, the approach can be extended to analyze the problems of magnetoelectroelastic laminated plates. Present semi-analytical method of mixed Hamiltonian element has wide application area.

Higher-Order Multilevel Solvers for the EHL Line and Point Contact Problem

1994 ◽  
Vol 116 (4) ◽  
pp. 741-750 ◽  
Author(s):  
C. H. Venner
Keyword(s):  
Contact Problem ◽  
Computing Time ◽  
Point Contact ◽  
Higher Order ◽  
Second Order ◽  
Point Of View ◽  
First Order ◽  
Fundamental Point ◽  
Numerical Solver ◽  
Circular Contact

This paper addresses the development of efficient numerical solvers for EHL problems from a rather fundamental point of view. A work-accuracy exchange criterion is derived, that can be interpreted as setting a limit to the price paid in terms of computing time for a solution of a given accuracy. The criterion can serve as a guideline when reviewing or selecting a numerical solver and a discretization. Earlier developed multilevel solvers for the EHL line and circular contact problem are tested against this criterion. This test shows that, to satisfy the criterion a second-order accurate solver is needed for the point contact problem whereas the solver developed earlier used a first-order discretization. This situation arises more often in numerical analysis, i.e., a higher order discretization is desired when a lower order solver already exists. It is explained how in such a case the multigrid methodology provides an easy and straightforward way to obtain the desired higher order of approximation. This higher order is obtained at almost negligible extra work and without loss of stability. The approach was tested out by raising an existing first order multilevel solver for the EHL line contact problem to second order. Subsequently, it was used to obtain a second-order solver for the EHL circular contact problem. Results for both the line and circular contact problem are presented.

Experimental Evaluation of Material Behavior in a Wire Under Transverse Impact

1967 ◽  
Vol 34 (2) ◽  
pp. 392-396 ◽  
Author(s):  
A. B. Schultz ◽  
P. A. Tuschak ◽  
A. A. Vicario
Keyword(s):  
Aluminum Alloy ◽  
Pure Copper ◽  
Closed Form Solution ◽  
Strain Curve ◽  
Simple Wave ◽  
Form Solution ◽  
Material Behavior ◽  
Governing Equations ◽  
Transverse Impact ◽  
Behavior Consistency

Results are reported from an experiment involving photographic observation of constant-velocity transverse impact on long wires of annealed pure copper, two pure aluminums, and an aluminum alloy. Predictions of deformation are made assuming the quasi-static stress-strain curve governs behavior. Consistency with experimental observations is examined. Predictions are based on a closed-form solution to the problem, which is shown to be a compounding of two simple wave solutions of the governing equations. Predictions are consistent with observations for the aluminum alloy even under conditions of moderate or high static prestrain. The two pure aluminums and the copper show consistency at low but not at high strain levels. Highest strain levels reached were in the range 0.06–0.14.

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