Random region function and its applications

This paper establishes some basic concepts of the random region, and random region function. From these concepts, and the existence of a random stable point of a random region function of the random region the necessary and suficient conditions for the existence of a random stable centre of any random region are defined.     

Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions

This article concerns an extension of the topological derivative concept for 2-D potential problems involving penetrable inclusions, whereby a cost function J is expanded in powers of the characteristic size ε of a small inclusion. The O(ε⁴) approximation of J is established for a small inclusion of given location, shape and conductivity embedded in a 2-D region of arbitrary shape and conductivity, and then generalized to several such inclusions. Simpler and more explicit versions of this result are obtained for a centrally-symmetric inclusion and a circular inclusion. Numerical tests are performed on a sample configuration, for (i) the O(ε⁴) expansion of potential energy, and (ii) the identification of a hidden inclusion. For the latter problem, a simple approximate global search procedure based on minimizing the O(ε⁴) approximation of J over a dense search grid is proposed and demonstrated.     

Two arc cracks around a circular rigid inclusion

Summary The problem of two symmetric arc cracks lying between a rigid circular inclusion embedded in an infinite matrix under biaxial loading at infinity is considered. By using the complex variable tecnique, the boundary value problem is solved and stress and displacement components are calculated along the inclusion boundary. Moreover, investigating the local stress field, a stress criterion, already proposed by authors, allowing either the crack extension at the interface or its deviation into the matrix to be taken into account, is applied to study the fracture response of the elastic system. The critical applied loads as well as the angle of the incipient crack extension are expressed in terms of the central angle subtended by the crack arcs.Finally the biaxial load effects are graphically shown and discussed.

---

Sommario Si considera il problema piano di due «craks» simmetrici all'interfaccia tra una inclusione circolare rigida e la circostante matrice, in regime di carico biassiale. Facendo uso della tecnica delle variabili complesse, viene risolto il problema dei valori al contorno, ricavando altresi le espressioni delle tensioni e degli spostamenti lungo il contorno dell'inclusione. Inoltre, applicando un criterio tensionale, già proposto dagli autori, che consente di studiare sia la estensione del crack all'interfaccia che la sua deviazione nella matrice, si analizza la risposta del sistema considerato. I valori critici dei carichi applicati, nonchè l'angolo di frattura, risultano espressi in funzione dell'angolo al centro sotteso dai cracks.Infine, vengono illustrati graficamente e discussi gli effetti del carico laterale.

---

Financial support of the National Research Council (C.N.R.) (research contribution N. 79.01625.07) is gratefully acknowledged.     

Fatigue growth modeling of cracks emanating from a circular hole in infinite plate

This paper presents a numerical approach of fatigue growth analysis of cracks emanating from a hole in infinite elastic plate subjected to remote loads. It involves a generation of Bueckner’s principle and a hybrid displacement discontinuity method (a boundary element method) proposed recently by the senior author of the paper. Because of an intrinsic feature of the boundary element method, a general crack growth problem can be solved in a single region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is modeled conveniently by adding new boundary elements on the incremental crack extension to the previous crack boundaries. As an example, fatigue growth process of an inclined crack in an infinite plate under uniaxial cycle load is modeled to illustrate the effectiveness of the numerical approach. In addition, fatigue growth of cracks emanating from a circular hole in infinite elastic plate subjected to remote loads is investigated by using the numerical approach. Many numerical results are given     

Some New Implications of Certain Constitutive Inequalities in Plane Isotropic Elasticity

In plane isotropic elasticity a strengthened form of the Ordered–Forces inequality is shown to imply that the restriction of the strain-energy function to the class of deformation gradients which share the same average of the principal stretches is bounded from below by the strain energy corresponding to the conformal deformations in this class. For boundary conditions of place, this property (together with a certain version of the Pressure–Compression inequality) is then used (i) to show that the plane radial conformal deformations are stable with respect to all radial variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with arbitrary plane radial deformations. For the same type of boundary conditions and together with a different version of the Pressure–Compression inequality, an analogous property in plane isotropic elasticity (established in [3] under the assumption that the material satisfies a strengthened form of the Baker–Ericksen inequality and according to which the restriction of the strain-energy function to the class of deformation gradients which share the same determinant is bounded from below by the strain energy corresponding to the conformal deformations in that class) is used (i) to show that the plane radial conformal deformations are stable with respect to all variations of class C ¹ and (ii) to obtain explicit lower bounds for the total energy associated with any plane deformation.     

Stability of Periodic Solutions¶of Conservation Laws with Viscosity:¶Pointwise Bounds on the Green Function

We investigate stability of periodic traveling-wave solutions of systems of conservation laws with viscosity within the abstract Evans function framework established by R. A. Gardner. Our main result, generalizing the work of Zumbrun and Howard in the traveling-front or -pulse setting, is to establish sharp pointwise bounds on the Green function for the linearized evolution equations, provided that an appropriate Evans function condition applies to the linearized operator about the wave. This condition is equivalent to a spectral stability criterion introduced by Schneider in the context of periodic reaction-diffusion waves. An immediate consequence is that strong spectral stability (in the sense of Schneider) implies linearized L¹ → L ^p asymptotic stability for all p > 1. On the other hand, we show that the strict version of Schneider's condition genericallyfails in the conservation law setting, leading to complicated new “metastable” behavior reminiscent of that seen for degenerate, neutrally stable families in the traveling-front or -pulse case. Our results apply also to the reaction-diffusion setting, sharpening (at the linearized level) results obtained by Schneider using weighted-norm and Bloch-decomposition methods.As in the traveling-front or -pulse case, the basic approach is to mimic in the Laplace-transform setting the elementary Fourier-transform analysis of the constant-coefficient case. However, the technical issues involved are rather different in the two cases. Somewhat surprisingly, we find the analogy to the constant-coefficient case to be rather stronger in the periodic-coefficient case, permitting a more standard approach involving the explicit construction of “continuous” spectral measure as in the self-adjoint case. This is equivalent to the method of Zumbrun and Howard in this special case.     

Numerical Modelling of Swelling Pressure in Unsaturated Expansive Elasto-Plastic Porous Media

The focus of this work is to provide a new concept for accessing the swelling stress in expansive porous media, especially in highly compacted bentonite. The key to the new approach is the simulation with a chemical swelling model of an infinitesimal volume change followed by a back compaction Process. Free extension is allowed in the first step, to calculate the interlayer porosity change (micro) and the induced volume change potential (macro). The object-oriented FEM simulator GeoSys/RockFlow allows the combination of different processes. The hydro-mechanic/chemical (H²M/C) model takes into consideration two phase flow and deformation, as well as chemical swelling effects. The negative displacements on each boundary after the free extension simulation are taken as Dirichlet boundary conditions of the back compaction problem. The deformation step is simulated in the context of elasto-plasticity using the modified Cam-Clay model. The stresses obtained by back compaction represent the swelling pressure. A 2D example of compacted bentonite is analyzed with the new H²M/C model.     

Fatigue and fracture of bovine dentin

In this paper, the fatigue and fracture properties of bovine dentin are evaluated usingin vitro experimental analyses. Double cantilever beam (DCB) specimens were prepared from bovine maxillary molars and subjected to zeroto-tension cyclic loads. The fatigue crack growth rate was evaluated as a function of the dentin tubule orientation using the Paris law. Wedge-loaded DCB specimens were also prepared and subjected to monotonic opening loads. Moiré interferometry was used to acquire the in-plane displacement field during stable crack growth, and the instantaneous wedge load and crack length were acquired to evaluate the crack growth resistance and crack tip opening displacement (CTOD) with crack extension. The rate of fatigue crack growth was generally larger for crack propagation occurring perpendicular to the dentin tubules. The Moiré fringe fields documented during monotonic crack growth exhibited non-linear deformation occurring within a confined region adjacent to the crack tip. Both the wedge load and CTOD response provided evidence that a fracture process zone contributes to energy dissipation during crack extension and that dentin exhibits a risingR-curve behavior. Results from this preliminary investigation are being used as a guide for an evaluation of the fatigue and fracture properties of human dentin.     

Low Reynolds number flow inside straight micro channels with irregular cross sections

This paper presents an analysis of the compound effect of finite temperature differences and fluid friction on the existence of an optimum laminar flow regime in singly connected micro channels with complex free flow area cross sections. A widespread conviction has been established that the two competing irreversibility sources in a channel flow with heat transfer lead to the existence of an optimum flow regime. The results presented in this paper clearly shows the opposite. When an objective function is represented by the entropy generation rate per unit heat capacity rate of the fluid stream, the thermodynamic optimum flow regime represents a rather rare occurrence in the laminar region of irregularly shaped ducts. The presence of an extremum is more probable for very small diameters, the ones of an order of magnitude of O(≤10⁻³ m). The analysis is performed for selected ranges of relevant geometric, flow, and thermal parameters of a set of straight micro channels with irregular free flow area cross-sections. The following geometries of the free flow area cross section were investigated: (i) sine duct, (ii) circular duct, (iii) elliptical duct, (iv) moon-shaped ducts, and (v) four-cuspped duct. The range of Reynolds numbers has been established between O(10²) and O(10⁴). The existence of the objective function minimum is confirmed for ducts with an irregular cross section only for very small hydraulic diameters. These minima are relatively weak, and as a general rule, the sets of optimum parameters are close to the onset of turbulence or possibly even in the transitional or turbulent regions. Received on 10 November 1998     

High-rate start-up behaviour of unvulcanised rubber compounds

The recently developed machines, which are shown to be comparable with “established” devices, are used to perform the start-up of uniaxial extension and simple shear at high constant deformation rates. Henky strains in excess of 8 at constant strain rates of over 100 s^?1 are achieved for two highly filled unvulcanised rubber compounds of Butyl and Chloroprene rubbers. In general no steady state is attained in uniaxial extension; the stress reaches a maximum and then declines. It is shown that the time—temperature superposition principle holds for start-up flows resulting in a family of master curves. It is established that the extensional stresses are one or two orders of magnitude greater than the shear stresses at corresponding deformation rates over a wide rate range.     

Crack paths in three-dimensional elastic solids. i: two-term expansion of the stress intensity factors—application to crack path stability in hydraulic fracturing

The aim of this work is to lay theoretical foundations for the prediction of crack paths within the theory of quasistatic LEFM under the most general hypotheses: arbitrary three-dimensional geometry, arbitrary loading. This objective requires to derive the expression of the stress intensity factors along the crack front after an arbitrary infinitesimal propagation. Only the first two terms of their expansion in powers of the crack extension length δ, proportional to δ⁰ = 1 and δfn1fn2, are considered in this paper. Fully general formulae for these terms are obtained by combining arguments of dimensional analysis (scale changes) and regularity properties (continuity, differentiability) of the stresses at a fixed, given point with respect to δ for δ = 0 derived from the Bueckner–Rice weight function theory. This notably allows to extend the Cotterell–Rice criterion for stable rectilinear propagation of a mode I crack under plane strain conditions to the three-dimensional case. As an application, a penny-shaped crack induced by hydraulic fracturing is considered. Conclusions concerning the influence of the orientation and depth of such a crack upon the stability of its coplanar propagation seem to be compatible with experimental evidence.     

Experimental study of turbulence near a duct wall

A specially developed stroboscopic instrument has been used to measure the instantaneous velocities in the wall region (including the viscous sublayer) in a turbulent liquid (water) duct flow. Values of the average velocity and the turbulent fluctuations are found as a function of the distance from the wall. The method employed is much simpler than hot-wire anemometry, although somewhat less accurate.The author thanks V. M. Karsten and E. S. Mikhailov for their assistance with the experiments and E. M. Khabakhpashev for his useful advice.     

Collision dynamics of bubble pairs moving through a perfect liquid

Equations are derived describing the inertial motion of a bubble pair through a perfect liquid. The relative bubble motion is driven by an interactional force induced by the centre of mass motion. This force can be derived from a potential that is proportional tos ⁿ (n3) and that depends on the bubble pair orientation. The path of two bubbles passing each other is investigated. The angle of deflection of the relative velocity in a two-bubble encounter is calculated numerically as a function of the impact parameter, the relative velocityg and the ratio of the centre of mass velocity componentsc ₂/c ₁. The specific conditions necessary for two bubbles to collide are determined. Ifc ₂/c ₁>1 there is a region with irregular behaviour of the deflection angle. The collision cross-section is calculated and depends smoothly ong, approximately proportional tog ^–1, and has a weak dependence onc ₂/c ₁.     

The eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation

The eigenvalue problem and expansion theorems associated with Orr-Sommerfeld equation are fundamental for the investigation of the stability of laminar fluid motion and have been studied by many authors. Nevertheless, the results are still incomplete^[6]. In this paper, this problem is investigated again and some new results have been obtained, namely: (1) The expansion series converges uniformly and absolutely. (2) The coefficients of the expansion series satisfy an inequality of Paley-Wiener type, which is the natural extension of the well-known Bessel equality of a complete orthogonal set.     

Extension of the non-uniform warping theory to an orthotropic composite beam

This Note proposes an extension to composite section of the non-uniform (out-of-plane) warping beam theory recently established for homogeneous and isotropic beam by R. El Fatmi (C. R. Mecanique 335 (2007) 467–474). For the present work, which constitutes a first step of this extension, the cross-section is assumed to be symmetric and made by orthotropic materials; however, Poisson's effects (called here in-plane warping) are also taken into account. Closed form results are given for the structural behavior of the composite beam and for the expressions of the 3D stresses; these ones, easy to compare with 3D Saint Venant stresses, make clear the additional contribution of the new internal forces induced by the non-uniformity of the (in and out) warpings. As first numerical applications, results on torsion and shear-bending of a cantilever sandwich beam are presented.     

Polydomain model predictions of liquid crystalline polymer orientation in mixed shear/extensional channel flows

 The Larson-Doi (LD) polydomain model is used to simulate orientation development along the centerline of slit-expansion and slit-contraction flows of liquid crystalline polymers (LCPs). Orientation is computed using the LD structural evolution equations, subject to an imposed velocity field that accounts for the spatial variation of both shear and extension rates characteristic of this class of flows. Computed axial distributions of orientation averaged through the sample thickness are qualitatively similar to birefringence and X-ray scattering measurements of molecular orientation in similar flows of lyotropic and thermotropic LCPs. In slit-expansion flows, the simulations predict a 90^∘ flip in orientation direction near the midplane due to transverse stretching in the expansion region. Far away from the midplane where shear gradients dominate, orientation remains primarily along the flow direction. Within the LD model, tumbling and flow aligning materials respond in a qualitatively similar manner to mixed shear and extension, although tumbling materials are systematically more susceptible to the effects of extension. Received: 22 October 1999/Accepted: 13 January 2000     

Experimental Measurement and Numerical Validation of Bone Cement Mantle Strains of an In Vitro Hip Replacement Using Optical FBG Sensors

Experimental pre-clinical tests associated with numeric models of cemented implants are important for screening of new implants in the market. The aim of this study was to measure strain profiles and maximum temperature polymerization inside a cement mantle of an in vitro cemented hip reconstruction using optical fiber Bragg grating (FBG) sensors. For this purpose, a hip femoral prosthesis was instrumented with 12 FBG sensors, three in each aspect of the femur, anterior, posterior, medial and lateral. These were positioned at the proximal, middle and distal part of the cement mantle relatively to the stem. Another sensor was placed in the lateral-proximal region of the mantle to measure the maximum temperature of cement polymerization. The strains measured were compared with those obtained with a Finite Element model, both for quaistatic mechanical loading. The results show that the experimental technique used can measure strains inside the cement mantle with good correlation, R²?=?0.970, with the numerical model results. The results present a maximum temperature of polymerization around 110°C inside of cement at proximal region. It was also observed strain concentration in lateral aspect of the femur in polymerization process. The procedure hereby explained can be used to improve experimental pre-clinical tests to measure the strain distribution inside the cement mantle as well as residual strain and temperature variation along with time, as a result of the curing process of cement.     

Extensional-flow-induced crystallization of isotactic polypropylene

A filament stretching extensional rheometer with a custom-built oven was used to investigate the effect of uniaxial flow on the crystallization of polypropylene. Prior to stretching, samples were heated to a temperature well above the melt temperature to erase their thermal and mechanical histories and the Janeschitz-Kriegl protocol was applied. The samples were stretched at extension rates in the range of 0.01 s^-1 ￡ [(e)\dot] ￡ 0.75 s^-10.01\,\mbox{s}^{-1}\le \dot{{\varepsilon }}\le 0.75\,{\rm s}^{-1} to a final strain of ε = 3.0. After stretching, the samples were allowed to crystallize isothermally. Differential scanning calorimetry was applied to the crystallized samples to measure the degree of crystallinity. The results showed that a minimum extension rate is required for an increase in percent crystallization to occur and that there is an extension rate for which percent crystallization is maximized. No increase in crystallization was observed for extension rates below a critical extension rate corresponding to a Weissenberg number of approximately Wi = 1. Below this Weissenberg number, the flow is not strong enough to align the contour path of the polymer chains within the melt and as a result there is no change in the final percent crystallization from the quiescent state. Beyond this critical extension rate, the percent crystallization was observed to increase to a maximum, which was 18% greater than the quiescent case, before decaying again at higher extension rates. The increase in crystallinity is likely due to flow-induced orientation and alignment of contour path of the polymer chains in the flow direction. Polarized light microscopy verified an increase in number of spherulites and a decrease in spherulite size with increasing extension rate. In addition, small angle X-ray scattering showed a 7% decrease in inter-lamellar spacing at the transition to flow-induced crystallization. Although an increase in strain resulted in a slight increase in percent crystallization, no significant trends were observed. Crystallization kinetics were examined as a function of extension rate by observing the time required for molten samples to crystallize under uniaxial flow. The crystallization time was defined as the time at which a sudden increase in the transient force measurement was observed. The crystallization time was found to decrease as one over the extension rate, even for extension rates where no increase in percent crystallization was observed. As a result, the onset of extensional-flow-induced crystallization was found to occur at a constant value of strain equal to ε _c  = 5.8.     

Correlation of the separation region length in shock wave/channel boundary layer interaction

 Results of an experimental investigation of the characteristics of a separation region induced by the interaction of an externally generated oblique shock with the turbulent boundary layer formed in a rectangular half channel are discussed. The experiments were carried out in the supersonic wind tunnel of the Institute of Theoretical and Applied Mechanics SB RAS at a free-stream Mach number M _∞=3.01 over a range of Reynolds numbers Re ₁=(9.7–47.5)×10⁶ m^-1 and at zero incidence and zero yaw of the model. Particular attention is paid to the size of the zone of the upstream propagation of disturbances (upstream influence region) under different experimental conditions: with varied values of the shock wave strength, half channel width, and Reynolds number. It is shown, in particular, that the normalized upstream influence region length as a function of inclination angle of the shock generator in a rectangular half channel is readily approximated by a simple exponential function. In support of the known reference data obtained for supersonic numbers M _∞ and moderate Re in other configurations, it is also shown that the upstream influence region length decreases with increasing Reynolds number. Generalization of experimental data on the length of the upstream influence region formed in similar geometric configurations is possible using an additional reference linear scale which is the distance from the leading edge of the shock generator to the exposed surface. A substantial dependence of the reference dimensions of separation region on the half channel width is also established. Received: 20 January 1997/Accepted: 30 June 1997     

New universal relations for nonlinear isotropic elastic materials

A nonlinear isotropic elastic block is subjected to a homogeneous deformation consisting of simple shear superposed on triaxial extension. Two new relations are established for this deformation which are valid for all nonlinear elastic isotropic materials, and hence are universal relations. The first is a relation between the stretch ratios in the plane of shear and the amount of shear when the deformation is supported only by shear tractions. The second relation is established for a thin-walled cylinder under combined extension, inflation and torsion. Each material element of the cylinder undergoes the same local homogeneous deformation of shear superposed on triaxial extension. The properties of this deformation are used to establish a relation between pressure, twisting moment, angle of twist and current dimensions when no axial force is applied to the cylinder. It is shown that these relations also apply for a mixture of a nonlinear isotropic solid and a fluid.