On the general theory of thermo-elastic friction

Summary A theory of the thermo-elastic dissipation in vibrating bodies is developed, starting from the three-dimensional thermo-elastic equations. After a discussion of the basic thermodynamical foundations, some general considerations on the problem of the conversion of mechanical energy into heat are given. The solution of the coupled thermo-elastic equations is found in the form of series expansions in terms of normalised orthogonal eigenfunctions. For the coefficients an infinite system of algebraic equations with constants, which are complicated field integrals, is derived. An approximate solution of the infinite system is given. In some cases the coupling-constants can be calculated exactly, in other cases they have to be determined on the base of approximate theories.     

The Theory of Thin-Walled Rods of Open Profile as a Result of Asymptotic Splitting in the Problem of Deformation of a Noncircular Cylindrical Shell

A new asymptotic approach to the theory of thin-walled rods of open profile is suggested. For the problem of linear static deformation of a noncircular cylindrical shell we consider solutions, which are slowly varying along the axial coordinate. A small parameter is introduced in the equations of the modern theory of shells. Conditions of compatibility for the shell strain measures are employed. The principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. We conclude the procedure with the subsequent solution for the field of displacements. The analysis shows that the known equations of thin-walled rods, which were previously obtained with some approximate methods using hypotheses and approximations of displacements, are asymptotically exact. The presented semi-numerical analysis of the shell equations allows us to estimate the accuracy of the obtained solution. The results of the paper constitute a sound basis to the equations of the theory of thin-walled rods and provide trustworthy information concerning the distribution of stresses in the cross-section.     

Finite deformation of elastic membranes with application to the stability of an inflated and extended tube

A theory is formulated for the finite deformation of a thin membrane composed of homogeneous elastic material which is isotropic in its undeformed state. The theory is then extended to the case of a small deformation superposed on a known finite deformation of the membrane. As an example, small deformations of a circular cylindrical tube which has been subjected to a finite homogeneous extension and inflation are considered and the equations governing these small deformations are obtained for an incompressible material. By means of a static analysis the stability of cylindrically symmetric modes for the inflated and extended cylinder with fixed ends is determined and the results are verified by a dynamic analysis. The stability is considered in detail for a Mooney material. Methods are developed to obtain the natural frequencies for axially symmetric free vibrations of the extended and inflated cylindrical membrane. Some of the lower natural frequencies are calculated for a Mooney material and the methods are compared.     

Spectral/hp penalty least‐squares finite element formulation for unsteady incompressible flows

In this paper, we present spectral/hp penalty least‐squares finite element formulation for the numerical solution of unsteady incompressible Navier–Stokes equations. Pressure is eliminated from Navier–Stokes equations using penalty method, and finite element model is developed in terms of velocity, vorticity and dilatation. High‐order element expansions are used to construct discrete form. Unlike other penalty finite element formulations, equal‐order Gauss integration is used for both viscous and penalty terms of the coefficient matrix. For time integration, space–time decoupled schemes are implemented. Second‐order accuracy of the time integration scheme is established using the method of manufactured solution. Numerical results are presented for impulsively started lid‐driven cavity flow at Reynolds number of 5000 and transient flow over a backward‐facing step. The effect of penalty parameter on the accuracy is investigated thoroughly in this paper and results are presented for a range of penalty parameter. Present formulation produces very accurate results for even very low penalty parameters (10–50). Copyright © 2008 John Wiley & Sons, Ltd.     

Stability of Axially Compressed Noncircular Cylindrical Shells Consisting of Panels of Constant Curvature

The stability problem is solved for an axially compressed cylindrical shell. Its cross section is formed by circular arcs of radius r with ends supported on a closed circle of radius R. The solution is based on the Flügge equations of the classic theory of deep cylindrical shells. It is shown that the critical axial load for shells of medium length and appropriately chosen cross-sectional profile can be increased by a factor of R/r approximately, compared with the circular shell. The shells length affects considerably the efficiency of noncircular shells of this type. This design model allows us to find out how the local properties of the shell and its stiffness are related     

Existence of a Solution for a Nonlinearly Elastic Plane Membrane Subject to Plane Forces

The two-dimensional nonlinear ‘membrane’ equations for a plate made of a Saint Venant–Kirchhoff material have been justified by D. Fox, A. Raoult and J.C. Simo (1993) by means of the method of formal asymptotic expansions applied to the three-dimensional equations of nonlinear elasticity. This model, which retains the material-frame indifference of the original three dimensional problem in the sense that its energy density is invariant under the rotations of R³, is equivalent to finding the critical points of a functional whose nonlinear part depends on the first fundamental form of the unknown deformed surface. We establish here a local existence result for these equations in the case of the membrane subject to forces parallel to its plane and we give qualitative properties of the solutions found in this fashion in terms of injectivity and of minimization. This revised version was published online in August 2006 with corrections to the Cover Date.     

Constitutive Models for Compressible Nonlinearly Elastic Materials with Limiting Chain Extensibility

Constitutive models are proposed for compressible isotropic hyperelastic materials that reflect limiting chain extensibility. These are generalizations of the model proposed by Gent for incompressible materials. The goal is to understand the effects of limiting chain extensibility when the compressibility of polymeric materials is taken into account. The basic homogeneous deformation of simple tension is considered and simple closed-form relations for the deformation characteristics are obtained for slightly compressible materials. An explicit first-order approximation is obtained for the lateral contraction and for the Poisson function in terms of the axial extension which is shown to be valid for each of two specific compressible versions of the Gent model. One of the main results obtained is that the effect of limiting chain extensibility is to stiffen the material relative to the neo-Hookean compressible case. Mathematics Subject Classifications (2000) 74B20, 74G55.     

Stress state of two collinear stamps over the surface of orthotropic materials

An exact contact analysis of two collinear stamps over the surface of orthotropic materials is performed. Eigenvalue analyses related to the governing equations are conducted, and real fundamental solutions are obtained for each eigenvalue distribution. Using Fourier sine or cosine transforms, a singular integral equation of Cauchy type is obtained. In particular, the exact solution of the reduced singular integral equation is obtained in the case of two collinear flat or cylindrical stamps. Explicit expressions for various stresses are obtained in terms of elementary functions. Numerical results are presented, and interesting observations are found. For two collinear flat stamps, detailed calculations are provided to show influences of distance between two flat stamps on the contact stress and in-plane stress. For a cylindrical stamp, influences of distance between two cylindrical stamps, mechanical loading P, and the cylindrical stamp radius R on the surface normal stress and surface in-plane stress are detailed.     

Approximate theories for wave propagation and vibrations in elastic rings and helical coils of small pitch

From the equations of linear elasticity, three levels of approximate theories are derived for in-plane deformation and motion of thin, circular rings. The accuracy of each theory is determined by comparison with harmonic wave solutions of the elasticity solution. Boundary conditions for uniqueness are established. The results may also be applied to helical coils of small pitch and to cylindrical shells when the equations are converted to plane strain.     

Coefficient of discharge and spray cone angle of a pressure nozzle with combined axial and tangential entry of power-law fluids

Flows of incompressible, time-independent purely viscous power-law fluids through pressure nozzle with combined axial and tangential entry are analysed. Theoretical predictions of coefficient of discharge and spray cone angle are made through an approximate analytical solution of hydrodynamics of flow inside the nozzle. In the converging section of the nozzle, the boundary layer equations have been derived with modified order approximation [O(δ/R)≈1, O(δ ²/R ²)≪1] of Navier-Stokes equations for a better accuracy. Smoother attainment of the free-stream condition at the edge of the boundary layer is ensured by requiring the appropriate shear rate terms, compatible with the above order analysis, to be zero. The pertinent independent input parameters which govern the flow field are the generalized Reynolds number at inlet to the nozzle based on the tangential velocity of injection , the ratio of the axial-to-tangential velocity at the inlet to the nozzle V _R , the flow behaviour index of the fluid n, the length-to-diameter ratio of the swirl chamber L ₁/D ₁, the spin chamber angle 2α and the orifice-to-swirl-chamber-diameter ratio D ₂/D ₁. Experiments reported in the paper corroborate the qualitative trends of analytical results.     

Torsional deformations in incompressible fibre-reinforced cylindrical pipes

The first part of the paper deals with an extension of the classical Rivlin's solution of the torsion problem of a neo-Hookean pipe. The second part concerns a study of the passive torsional deformation processes in a fibre-reinforced cylindrical dummy of the beating heart. Especially, the dependence of the torsional and volumetric stiffness of the cylindrical pipe on different geometric and material parameters is discussed through a set of numerical simulations.     

Plane strain problems in second-order elasticity theory

The equations of second-order elasticity are developed in polar coordinates R, θ for plane strain deformations of incompressible isotropic elastic materials. By considering a ‘displacement function’ the second-order problem is reduced to the solution of an equation of the form ⁴ψ = g(R, Θ) where ² is Laplace's differential operator and g(R, Θ) depends only on the first-order solution. The displacement function technique is then applied to obtain a second-order solution to the problem of an elastic body contained between two concentric rigid circular boundaries, when the outer boundary is held fixed and the inner is subjected to a rigid body translation.     

Circumferentially-symmetric finite eigenstrains in incompressible isotropic nonlinear elastic wedges

Eigenstrains are created as a result of anelastic effects such as defects, temperature changes, bulk growth, etc., and strongly affect the overall response of solids. In this paper, we study the residual stress and deformation fields of an incompressible, isotropic, infinite wedge due to a circumferentially symmetric distribution of finite eigenstrains. In particular, we establish explicit exact solutions for the residual stresses and deformation of a neo-Hookean wedge containing a symmetric inclusion with finite radial and circumferential eigenstrains. In addition, we numerically solve for the residual stress field of a neo-Hookean wedge induced by a symmetric Mooney–Rivlin inhomogeneity with finite eigenstrains.     

Entropy Solutions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">∞</Emphasis></Superscript> for the Euler Equations in Nonlinear Elastodynamics and Related Equations

A compactness framework is established for approximate solutions to the Euler equations in one-dimensional nonlinear elastodynamics by identifying new properties of the Lax entropies, especially the higher order terms in the Lax entropy expansions, and by developing ways to employ these new properties in the method of compensated compactness. Then this framework is applied to establish the existence, compactness, and decay of entropy solutions in L for the Euler equations in nonlinear elastodynamics with a more general stress-strain relation than those for the previous existence results. This compactness framework is further applied to solving the Euler equations of conservation laws of mass, momentum, and energy for a class of thermoelastic media, and the equations of motion of viscoelastic media with memory.     

Small deformations superposed on a finite cylindrical deformation of a cord reinforced rubber belt

An equation of admissibility is drived for small deformations superposed upon a cylindrical bending of a block, made of a Mooney-type material. Series solutions are obtained, which can be expressed in generalized hypergeometric functions and functions derivable from them. Boundary conditions which correspond to the deformation of a cord-reinforced rubber belt around a rigid cylindrical pulley are treated in the last section.Nomenclature A _nh , B _nh , C _h arbitrary constants - B reference configuration - B ^ij deformation tensor components - C ₀ constitutive constant defined at equation (18) - f _n , g _n , h _n radial dependent functions in the solution (20–23) - _p F _q generalized hypergeometric functions - g ^ij metric tensor components associated with the undeformed configuration - G ^ij do. for the deformed configuration B. - G _i base vectors - n index defined by (23) - n _i normal vector - p pressure function - Q function defined by equation (14) - R ₀, R ₁, R ₂ inner, cord, resp. outer radii in B - s _h roots of the indicial equation - t _i stress vector - T cord tension per unit width - w _i displacement vector - x ² n ² r ²/16R ₁ ⁴ - _ij ^r Christoffel symbols - axial extension ratio - coefficient of friction between belt and pulley - radius of curvature - ^ij stress tensor components - 2C ₂, a constant of the Mooney strain energy function     

On axisymmetric motion of an incompressible elastic medium under impact loading

The method of matched asymptotic expansions was employed to obtain approximate solutions to the one-dimensional boundary-value problems of nonlinear dynamic elasticity theory of impact loading on the surface of a cylindrical cavity of an incompressible medium that causes antiplane motion or torsion of the medium. The expansion of the solution in the near-front region is based on solutions of evolution equations different from the equations for quasi-simple waves. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 144–151, November–December, 2006.     

Axisymmetric Interaction Problem for a Sphere Pulsating Inside an Elastic Cylindrical Shell Filled with and Immersed Into a Liquid

An interaction problem is formulated for a spherical body oscillating in a prescribed manner inside a thin elastic cylindrical shell filled with a perfect compressible liquid and submerged in a dissimilar infinite perfect compressible liquid. The geometrical center of the sphere is on the cylinder axis. The solution is based on the possibility of representing the partial solutions of the Helmholtz equations written in cylindrical coordinates for both media in terms of the partial solutions written in spherical coordinates, and vice versa. Satisfying the boundary conditions on the sphere and shell surfaces results in an infinite system of linear algebraic equations. This system is used to determine the coefficients of the Fourier-series expansions of the velocity potentials in terms of Legendre polynomials. The hydrodynamic characteristics of both liquids and the shell deflections are determined. The results obtained are compared with those for a sphere oscillating on the axis of an elastic cylindrical shell filled with a compressible liquid (the ambient medium being neglected).     

Axisymmetric membrane in adhesive contact with rigid substrates: Analytical solutions under large deformation

The large deformation of an elastic axisymmetric membrane in adhesive contact with a rigid flat punch is studied. Detachment of membrane is analyzed using a critical energy release rate criterion. Two types of incompressible hyperelastic material models are considered: neo-Hookean and a class of materials whose elastic energy density functions are independent of the trace of the Cauchy–Green tensor (I₂-based material). We also include pre-stretch in our formulation and study the stability of detachment process. Closed form analytical solutions for the membrane stresses, deformed profiles and energy release rate are obtained in the regime of large longitudinal stretch. For the I₂-based material, we discover an interesting “pinching” instability where the contact angle suddenly increases in a displacement controlled test. The region of validity of our analytical solutions is determined by comparing them with numerical solutions of the governing equations. We found that the accuracy of our solution improves with pre-stretch; for pre-stretch ratios greater than 1.3, our analytical solution also works well in the small deformation regime.     

Non-linear vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method

Geometrically non-linear forced vibrations of a shallow circular cylindrical panel with a complex shape, clamped at the edges and subjected to a radial harmonic excitation in the spectral neighborhood of the fundamental mode, are investigated. Both Donnell and the Sanders–Koiter non-linear shell theories retaining in-plane inertia are used to calculate the elastic strain energy. The discrete model of the non-linear vibrations is build using the meshfree technique based on classic approximate functions and the R-function theory, which allows for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries; Chebyshev orthogonal polynomials are used to expand shell displacements. A two-step approach is implemented in order to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for expanding the non-linear displacements. Lagrange approach is applied to obtain a system of ordinary differential equations on both steps. Different multimodal expansions, having from 15 up to 35 generalized coordinates associated with natural modes, are used to study the convergence of the solution. The pseudo-arclength continuation method and bifurcation analysis are applied to study non-linear equations of motion. Numerical responses are obtained in the spectral neighborhood of the lowest natural frequency; results are compared to those available in the literature. Internal resonances are also detected and discussed.     

On Non-Symmetric Deformations of Neo-Hookean Solids

Deformations possible (i.e., those satisfying the governing three-dimensional equations of equilibrium and the incompressibility constraint) within a class of non-symmetric deformations for a neo-Hookean nonlinearly elastic body were determined in [1], where it was found that only three special cases of the class of deformation fields considered could be solutions. One of these is the trivial solution, one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation contained within a family of universal deformations. In this paper, the results reported in [1] are shown to hold for a substantially broadened deformation field. This revised version was published online in July 2006 with corrections to the Cover Date.