Eulerian formulation of transport equations for three-dimensional shock waves in simple elastic solids

The propagation of a three-dimensional shock wave in an elastic solid is studied. The material is assumed to be a simple elastic solid in which the Cauchy stress depends on the deformation gradient only. It is shown that the growth or decay of a discontinuity ψ depends on (i) an unknown quantity φ^? behind the shock wave, (ii) the two principal curvatures of the shock surface, (iii) the gradient on the shock surface of the shock wave speeds and (iv) the inhomogeneous term which depends on the motion ahead of the shock surface and vanishes when the motion ahead of the shock surface is uniform. If a proper choice is made of the propagation vectorb along which the growth or decay of the discontinuity is measured, the dependence on item (iii) can be avoided. However,b assumes different directions depending on the choice of discontinuity ψ with which one is concerned and the unknown quantity φ^? behind the shock wave on which one chooses to depend. As in the case of one-dimensional shock waves, the growth (or decay) of one discontinuity may not be accompanied by the growth (or decay) of other discontinuities. A universal equation relating the growth or decay of discontinuities in the normal stress, normal velocity and specific volume is also presented.     

Universal relations for instantaneous deformations of viscoelastic fluids

The note considers viscoelastic fluids which undergo an instantaneous homogeneous deformation consisting of shear superposed on triaxial extension. Two relations involving the stress and deformation components are presented, which are valid for all such fluids, and hence are termed “universal relations”. The first contains the Lodge-Meissner relation as a special case; the second arises when a block is deformed by shear traction only. It relates dimensional changes to the amount of shear.     

Consistency tests for start-up and cessation deformations of viscoelastic fluids

The time-dependent shear stress and first normal stress difference were measured for a polystyrene solution for start-up and cessation-flow experiments over a relatively wide range of shear rate. Consistency tests for the K-BKZ model were applied to the data, and it was concluded that the K-BKZ equation generally does not satisfactorily describe the start-up and cessation data. Modified consistency tests were developed using a strain-coupling constitutive equation, and the evidence suggests that most of the differences between the predictions of the K-BKZ theory and experiment can be explained by including a strain-coupling effect in the rheological constitutive equation.     

Inhomogeneous rectilinear shear deformations for electro-active elastic solids

Using the inverse method we consider the admissibility of a family of inhomogeneous rectilinear shear deformations in isotropic electroactive materials. Moreover, several boundary value problems related to these deformations are investigated numerically.     

Thin-plate theory for large elastic deformations

Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff-Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate.     

Four-field Galerkin/least-squares formulation for viscoelastic fluids

A new Galerkin/Least-Squares (GLS) stabilized finite element method is presented for computing viscoelastic flows of complex fluids described by the conformation tensor; it extends the well-established GLS method for computing flows of incompressible Newtonian fluids. GLS methods are attractive for large-scale computations because they yield linear systems that can be solved easily with iterative solvers (e.g., the Generalized Minimum Residual method) and because they allow simple combinations of interpolation functions that can be conveniently and efficiently implemented on modern distributed-memory cache-based clusters.Like other state-of-the-art methods for computing viscoelastic flows (e.g., DEVSS-TG/SUPG), the new GLS method introduces a separate variable to represent the velocity gradient; with the aid of this variable, the conservation equations of mass, momentum, conformation, and the definition of velocity gradient are converted into a set of first-order partial differential equations in four unknown fields—pressure, velocity, conformation, and velocity gradient. The unknown fields are represented by low-order (continuous piecewise linear or bilinear) finite element basis functions.The method is applied to the Oldroyd-B constitutive equation and is tested in two benchmark problems—flow in a planar channel and flow past a cylinder in a channel. Results show that (1) the mesh-convergence rate of GLS is comparable to the DEVSS-TG/SUPG method; (2) the LS stabilization permits using equal-order basis functions for all fields; (3) GLS handles effectively the advective terms in the evolution equation of the conformation tensor; and (4) GLS yields accurate results at lower computational costs than DEVSS-type methods.     

Finite deformations of an elastic anisotropic body

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 166–174, May–June, 1989.     

Coupling of elastic and plastic deformations of bulk solids

Summary In such solids like rocks, soils, ceramics, grain en masse the plastic deformation strongly aects the current unloading modulus. The consequences of this effect referred to as the elastoplastic coupling both to the elastic and the plastic part of the constitutive law are examined. Particularly, it appears that such phenomenon induces a specific kind of the non-normality in the plastic flow law. The departure from the normality is studied in connection with the form of the elastic modulus variation basing on the notion of a coupling potential.

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Sommario Il modulo di scarico elastico di solidi come le rocce, i suoli, i materiali ceramici e granulari dipende dalla entità delle deformazioni plastiche. Le conseguenze di questo fenomeno, chiamato nel seguito accoppiamento elastoplastico, vengono esaminate sia in relazione alla parte elastica che alla parte plastica della legge costitutiva. In particolare si dimostra che l'accoppiamento elastoplastico determina la non-normalità della legge di scorrimento plastico. La deviazione della normalità legata alla variazione del modulo elastico è studiata per mezzo di un potenziale di accoppiamento derivato da quello elastico.

     

Non-singular terms for multiple cracks in anisotropic elastic solids

The non-singular and bounded terms for the stresses near the crack tip are investigated. This paper deals with the multiple crack problem for an infinite plate. The original problem is decomposed into three elementary subproblems: (1) the problem for remote uniform stress filed without cracks; (2) the single crack problem with traction applied along the crack face and (3) the problem for the influence of the other cracks. Several examples for collinear and non-collinear cracks are discussed and the results are shown.     

Conservation properties for plane deformations of anisotropic linearly elastic curved strips

Plane deformations of a curved strip, composed of an homogeneous cylindrically anisotropic linearly elastic material, are considered. The strip is in equilibrium under the action of end loads, with the lateral sides traction-free. Two conservation properties for certain cross-sectional stress measures are established, generalizing previously known results for the case of a rectangular strip. Such conservation properties are useful in assessing the influence of material anisotropy on Saint-Venant's principle, as well as in establishing convexity properties for cross-sectional stress measures. In particular, it is anticipated that the results should be useful in determining the extent of edge effects in the testing of anisotropic and composite curved strips.     

Conservation properties for plane deformations of isotropic and anisotropic linearly elastic strips

Plane deformations of a rectangular strip, composed of an homogeneous fully anisotropic linearly elastic material, are considered. The strip is in equilibrium under the action of end loads, with the lateral sides traction-free. Two conservation properties for certain cross-sectional stress measures are established, generalizing previously known results for the isotropic case. It is noteworthy that in the first of these conservation laws only one of the off-axis elastic constants appears explicitly while in the second only the opposite off-axis constant appears explicitly. Such conservation properties are useful in assessing the influence of material anisotropy on Saint-Venant's principle, as well as in establishing convexity properties for cross-sectional stress measures. In particular, it is anticipated that the results should be useful in determining the extent of edge effects in the off-axis testing of anisotropic and composite materials.     

End effects for plane deformations of an elastic anisotropic semi-infinite strip

In the linear theory of elasticity, Saint-Venant's principle is used to justify the neglect of edge effects when determining stresses in a body. For isotropic materials, the validity of this is well established. However for anisotropic and composite materials, experimental results have shown that edge effects may persist much farther into the material than for isotropic materials and as a result cannot be neglected. This paper further examines the effects of material anisotropy on the exponential decay rate for stresses in a semi-infinite elastic strip. A linearly elastic semi-infinite strip in a state of plane stress/strain subject to a self-equilibrated end load is considered first for a specially orthotropic material and then for the general anisotropic material. The problem is governed by a fourth-order elliptic partial differential equation with constant coefficients. In the former case, just a single dimensionless material parameter appears, while in the latter, only three dimensionless parameters are required. Energy methods are used to establish lower bounds on the actual stress decay rate. Both analytic and numerical estimates are obtained in terms of the elastic constants of the material and results are shown for several contemporary engineering materials. When compared with the exact stress decay rate computed numerically from the eigenvalues of a fourth-order ordinary differential equation, the results in some cases show a high degree of accuracy. In particular, for strongly orthotropic materials, an asymptotic estimate provides extremely accurate estimates for the decay rate. Results of the type obtained here have several important practical applications. For example, they provide physical insight into the mechanical testing of anisotropic and laminated composite structures (including the off-axis tension test), are useful in assessing the influence of fasteners, joints, etc. on the behavior of composite structures and allow for tailoring a material with specific properties to ensure that local stresses attenuate at a desired rate.     

Fully-developed heat transfer in annuli for viscoelastic fluids with viscous dissipation

Analytical solutions are obtained for heat transfer in concentric annular flows of viscoelastic fluids modeled by the simplified Phan-Thien–Tanner constitutive equation. Solutions for thermal and dynamic fully developed flow are presented for both imposed constant wall heat fluxes and imposed constant wall temperatures, always taking into account viscous dissipation.Equations are presented for the normalized temperature profile, the bulk temperature, the inner and outer wall temperatures and, through their definitions for the inner and outer Nusselt numbers as a function of all relevant non-dimensional parameters. Some special results are discussed in detail. Given the complexity of the derived equations, for ease of use compact exact expressions are presented for the Nusselt numbers and programmes to calculate all quantities are made accessible on the internet. Generally speaking, fluid elasticity is found to increase the heat transfer for imposed heating at the wall, especially in combination with internal heat generation by viscous dissipation, whereas for imposed wall temperatures it reduces heat transfer when viscous dissipation is weak.     

A large displacement formulation for anisotropic beam analysis

Summary The displacement of a beam can be conveniently resolved into a roto-translational section displacement and a section warping. The correct second order approximation of the strain is deduced accounting for large displacements and thus for large rotations. On the basis of displacement method, both linear and nonlinear formulations are given: the first one leads to the elastic section properties and to the correct characterization of section warping; the second one leads to the so-called geometric section stiffness, accounting for prestress. Both formulations are general with respect to elastic material properties, thus allowing to deal with aniso-tropic and unhomogeneous cross-sections. Elastic and geometric section rigidities here proposed can then be easily used in second order problems on beam frames: either initial buckling eigenvalue analyses, either large displacement incremental analyses.

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Sommario E' conveniente scomporre lo spostamento di un punto di una trave in una rototraslazione della sezione cui appartiene e in uno spostamento che deforma la sezione (in-gobbamento). Si deduce la corretta approssimazione al second'ordine della deformazione per grandi spostamenti e quindi grandi rotazioni. Vengono presentate sia la formulazione lineare che quella non lineare, basate sul metodo degli spostamenti: dalla prima si ottengono le caratteristiche elastiche della sezione e la corretta caratterizzazione dell'ingob-bamento; dalla seconda la cosiddetta rigidezza geometrica della sezione che tiene conto dello stato di presforzo. Entrambe le formulazioni sono generali per quanto riguarda le proprietà del materiale elastico, potendosi così considerare anche sezioni anisotrope e non omogenee. Le caratteristiche di rigidezza elastica e geometrica della sezione possono quindi facilmente essere usate in problemi del second'ordine su strutture a travi: sia analisi agli autovalori della stabilità, sia analisi incrementali con grandi spostamenti.

     

Flow of viscous and viscoelastic fluids through and around porous bodies

The slow flow of a viscous fluid through and around porous spheres is considered. The numerical simulation uses a special mixture of computational techniques: quadratic approximation and expansion in power series. The resulting calculations predict the evolution of the main features of the flow if the boundary conditions are varying, particularly if the tangential velocity is neglected or if a viscous filtration velocity is assumed at the sphere surface. The cases of full and hollow spheres with uniform and non uniform permeabilities are considered, the external impermeable walls of the flow being concentric spheres or cylinders. Some influence of viscoelastic properties of the fluid is also given.Nomenclature AA _n , A_n, B_n, b_n, C_n, c_n, D_n constants of integration - C _n (t) Gegenbauer functions with degree n and order –1/2 - e shell thickness - K, K* permeability - P _n (t) Legendre functions - Q _v volumetric rate of flow - p, p ₀, p _e pressure, far away pressure, average pressure - R* sphere radius - r, spherical coordinates - Re Reynolds' number (see equation 37) - s, t sinus and cosinus - V ₀ ^* uniform velocity - v velocity component - We Weissenberg's number (see equation (37)) - permeability coefficient - thickness coefficient - structural coefficient - diameter ratio sphere-cylinder - * dynamic viscosity of the fluid - stream functions - normal stress ( _rr ) - tangential stress ( ) - ₀ ^* relaxation time of the fluid     

Motions of elastic solids in fluids under vibration

Motion of a rigid or deformable solid in a viscous incompressible fluid and corresponding fluid–solid interactions are considered. Different cases of applying high frequency vibrations to the solid or to the surrounding fluid are treated. Simple formulas for the mean velocity of the solid are derived, under the assumption that the regime of the fluid flow induced by its motion is turbulent and the fluid resistance force is nonlinearly dependent on its velocity. It is shown that vibrations of a fluid’s volume slow down the motion of a submerged solid. This effect is much pronounced in the case of a deformable solid (i.e., gas bubble) exposed to near-resonant excitation. The results are relevant to the theory of gravitational enrichment of raw materials, and also contribute to the theory of controlled locomotion of a body with an internal oscillator in continuous deformable (solid or fluid) media.     

Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length. To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple-stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.