Relation between homogeneous and inhomogeneous stretching of an elastic fluid

Homogeneous and inhomogeneous (steady) noninertial stretching of an elastic fluid is experimentally investigated, and a method is given for the calculation of one problem from experimental data on the other.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 4, pp. 629–635, April, 1978.     

Radial propagation of axial shear waves in an incompressible elastic material under finite deformation

Equations derived via Biot's mechanics of incremental deformations are employed as a basis for investigation of radial propagation of axial shear waves through an initially radially deformed elastic solid. In the first instance, reduction of the propagation equation is sought to one of the two canonical forms. One is soluble by finite Hankel transform techniques while the other is associated with the conventional one-dimensional wave equation. Two specific initial value problems involving a neo-Hookean solid are solved explicitly by means of these techniques. Finally, an alternative approach is presented which utilises formal asymptotic wavefront expansions.     

Approximation of incompressible large deformation elastic problems: some unresolved issues

Several finite element methods for large deformation elastic problems in the nearly incompressible and purely incompressible regimes are considered. In particular, the method ability to accurately capture critical loads for the possible occurrence of bifurcation and limit points, is investigated. By means of a couple of 2D model problems involving a very simple neo-Hookean constitutive law, it is shown that within the framework of displacement/pressure mixed elements, even schemes that are inf-sup stable for linear elasticity may exhibit problems when used in the finite deformation regime. The roots of such troubles are identified, but a general strategy to cure them is still missing. Furthermore, a comparison with displacement-based elements, especially of high order, is presented.     

Some dynamic properties of incompressible,transversely isotropic elastic media

Summary This paper investigates various dynamic properties of incompressible, transversely isotropic elastic media. The propagation condition for such materials allows the wave speeds to be obtained in explicit form. An examination of the slowness surface and direction of energy flux as the extensional modulus along the fibre tends to infinity is then easily carried out. The paper also includes an investigation of the dynamic response of such materials to a particular line impulsive force. This is done using integral transforms. These transforms are invertible in closed form.     

A monolithic approach for interaction of incompressible viscous fluid and an elastic body based on fluid pressure Poisson equation

This paper describes a new monolithic approach based on the fluid pressure Poisson equation (PPE) to solve an interaction problem of incompressible viscous fluid and an elastic body. The PPE is derived so as to be consistent with the coupled equation system for the fluid‐structure interaction (FSI). Based on this approach, we develop two kinds of efficient monolithic methods. In both methods, the fluid pressure is derived implicitly so as to satisfy the incompressibility constraint, and all other unknown variables are derived fully explicitly or partially explicitly. The coefficient matrix of the PPE for the FSI becomes symmetric and positive definite and its condition is insensitive to inhomogeneity of material properties. The arbitrary Lagrangian–Eulerian (ALE) method is employed for the fluid part in order to take into account the deformable fluid‐structure interface. To demonstrate fundamental performances of the proposed approach, the developed two monolithic methods are applied to evaluate the added mass and the added damping of a circular cylinder as well as to simulate the vibration of a rectangular cylinder induced by vortex shedding. Copyright © 2005 John Wiley & Sons, Ltd.     

A projection semi‐implicit scheme for the coupling of an elastic structure with an incompressible fluid

We address the numerical simulation of fluid–structure systems involving an incompressible viscous fluid. This issue is particularly difficult to face when the fluid added‐mass acting on the structure is strong, as it happens in hemodynamics for example. Indeed, several works have shown that, in such situations, implicit coupling seems to be necessary in order to avoid numerical instabilities. Although significant improvements have been achieved during the last years, solving implicit coupling often exhibits a prohibitive computational cost. In this work, we introduce a semi‐implicit coupling scheme which remains stable for a reasonable range of the discretization parameters. The first idea consists in treating implicitly the added‐mass effect, whereas the other contributions (geometrical non‐linearities, viscous and convective effects) are treated explicitly. The second idea, relies on the fact that this kind of explicit–implicit splitting can be naturally performed using a Chorin–Temam projection scheme in the fluid. We prove (conditional) stability of the scheme for a fully discrete formulation. Several numerical experiments point out the efficiency of the present scheme compared to several implicit approaches. Copyright © 2006 John Wiley & Sons, Ltd.     

On the linear problem of swelling porous elastic soils with incompressible fluid

In this paper we consider the linear theory of swelling porous elastic soils in the case that the fluid is incompressible. The formulation belongs to the theory of mixtures for porous elastic solids filled with fluid and gas. It proposes some new mathematical difficulties. Continuous dependence results on initial conditions and supply terms are obtained in the general case. Logarithmic convexity method is used in case of fluid saturation. Structural stability with respect two constitutive coefficients is also obtained in the case of fluid saturation.     

Time-dependent crack behavior in an integrated structure

Devices in modern technologies often have complex architectures, dissimilar materials, and small features. Their long-term reliability relates to inelastic, time-dependent mechanical behavior of such structures. This paper analyzes a three-layer structure consisting of, from top to bottom, an elastic film, a power-law creep underlayer, and a rigid substrate. The layers are bonded. Initially, the film is subject to a uniform biaxial tensile stress. A channel crack is introduced in the elastic film. As the underlayer creeps, the stress field in the film relaxes in the crack wake, but intensifies around the crack tip. We formulate nonlinear diffusion-like equations that evolve the displacement field. When the crack is stationary, the region in which the stress field relaxes increases with time. We identify the length scale of the region as a function of time. The stress intensity factor is proportional to the square-root of the length scale. For the power-law creep underlayer, this newly identified length depends on the film stress, and corrects an error in a previous paper by Huang, Prévost and Suo (Acta Materialia 50, 4137, 2002). When the crack advances, its velocity can reach a steady state. We identify the scaling law for the steady velocity. An extended finite element method (X-FEM) is used to simultaneously evolve the creep strain and crack length. Numerical results are presented for the stress intensity factors of stationary cracks, and the steady velocities of advancing cracks.     

Second-order effects on the wave propagation in elastic, isotropic, incompressible, and homogeneous media

In this paper we propose a perturbation method to investigate the propagation of the ordinary waves in second-order elastic, isotropic, incompressible, and homogeneous materials. This method allows us to determine the first-order terms of the speeds and the amplitudes both of the principal waves and the waves in any propagation direction, when the undisturbed region is subjected to an arbitrary isochoric deformation. Another application of this method is presented when the undisturbed region is subjected to a simple shear. Finally, some numerical results are presented in Mooney-Rivlin materials.     

Axially symmetrical jet mixing of an incompressible dusty fluid

Summary Incompressible laminar jet mixing of a dustry fluid issuing from a circular jet has been considered by using Saffman's model for dusty fluid. Assuming the velocity and temperature in the jet to differ only slightly from that of the surrounding stream, a perturbation method has been employed to linearize the basic equations. The linearized equations have been solved by using Hankel transform, technique. Numerical computations have been made to discuss the veoocity and temperature profiles. It is observed that the particles move faster than the fluid in the down stream direction and the particles temperature is lower than the fluid temperature.With 6 Figures     

The second order deformation of a finite incompressible isotropic elastic annulus subjected to circular shearing

Summary This work investigates the second order deformation of a uniformly thick incompressible isotropic elastic annulus with an axial cylindrical hole. The annulus is clamped at its outer edge and is subjected to a constant angular deformation on the interior boundary of the hole. The implicit mathematical solution is formulated in terms of finite Hankel transforms with Weber-Orr kernel functions which are then numerically inverted.     

Statistical theory of the nonuniform turbulence of an incompressible fluid

The nonuniform turbulence problem is treated by a statistical approach based on the use of a finite number of equations for the higher-order single-point correlations. Additional differential equations are derived for the unknown moments in the single-point correlation equations. The equations are closed by means of approximate expressions for the anisotroplc two-point correlation tensors for near points. A closed system of seven tensorial differential equations is given, describing the variation of the fundamental characteristics of nonuniform turbulence.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 19, No. 5, pp. 789–802, November, 1970.     

Rotary oscillations of a spheroid in an incompressible micropolar fluid

The paper discusses the flow generated by rotary oscillations of a spheroid (prolate and oblate) in incompressible micropolar fluid. The velocity and microrotation components are determined explicitly in terms of spheroidal wave functions and are expressed in infinite series form. The couple on the oscillating spheroid is evaluated and numerical studies are undertaken to examine the effects of the geometric parameter and material constant parameters of the fluid.     

Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

Summary The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.     

Residual stresses due to diffusion in an elastic homogeneous plate

The stress state due to nonuniform distribution of a diffusing element was analyzed, and approximate formulas were derived for determining the residual stresses under various diffusion saturation conditions.     

Thermoconvective waves in a horizontal bounded layer of an incompressible fluid

The propagation of finite-amplitude thermoconvective waves in a horizontal fluid layer with rigid boundaries is investigated.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.31, No.1, pp.79–85, July, 1976.     

Isothermal axisymmetric flow of an incompressible fluid in radial contact apparatus

A method is proposed for the calculation of the axisymmetric flow of an incompressible fluid in a radial apparatus with a nonmoving layer of granular material. An engineering method is developed to evaluate the degree of flow nonuniformity in equipment of this type, thus making it possible operationally to choose among structural solutions in the design of this equipment.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 56, No. 4, pp. 555–562, April, 1989.