Crack separation energy-rates for the DBCS model under biaxial modes of loading

A Modified version of the Dugdale-Bilby-Cottrell-Swinden (DBCS) model simulating the effect of plasticity at the tip of a crack in an infinite region was used by kfouri and rice (1978) to calculate the crack separation energy-rate G^Δ corresponding to a finite crack growth step Δa during plane strain mode I crack extension. The loading consisted of a remote uniaxial tension σp applied normally to the plane of the crack. Using Rice's path-independent integral J to characterize the applied load in the crack tip region, and assuming the length R of the crack tip plastic zone to be small compared with the length of the crack, an analytical expression was derived relating the ratios (J/G^Δ) and (2a/R) for small values of (2a/R). The analytical solution was incomplete in itself in that the value assumed in the plastic region of the DBCS model for the normal stress Y acting on the extending crack surfaces was the product of the yield stress in uniaxial tension σ_Y and an unknown parameter C, the value of which depends on the effect of the local hydrostatic stresses in the case of plane strain conditions. The analytical solution was compared with a numerical solution obtained from a plane strain elastic-plastic finite element analysis on a centre-cracked plate (CCP) of material obeying the von Mises yield criterion. The value used for the yield stress was 310 MN/m² and moderate isotropic linear hardening was applied with a tangent modulus of 4830 MN/m². A uniaxial tension σp was applied on the two appropriate sides of the plate. The comparisons showed that the analytical and finite element solutions were mutually consistent and they enabled the value of C to be established at 1.91. In the present paper similar comparisons are made between the analytical solution and the finite element solutions for the CCP of the same material under different biaxial modes of loading. By assuming further that the form of the analytical solution does not depend on the details of the geometry and of the loading at remote boundaries, a comparison has also been made with the results of a finite element analysis on a compact tension specimen (CTS) made of the same material as the CCP. The different values of C obtained in each case are consistent with investigations by other authors on the effect of load biaxiality on crack tip plasticity.     

Heat transfer characteristics of a continuous stretching surface

The similarity solutions for the governing ordinary differential equations of the boundary layer corresponding to a stretching surface have been reported. Power law velocity and temperature distribution were assumed for velocity exponent 3≥m≥?0.41176, ?1.1≥m≥?3, and for temperature exponent 3≥n≥?3. Solutions have been found forn=0 and allm where heat transferred from the stretching surface to the ambient. The direction and amount of heat flow were found to be dependent on the magnitude ofn andm for the same Prandtl number. Nusselt number increases with increasingm andPr for uniform and variable surface temperature however, for uniform surface heat flux it decreases with increasingm for constantPr.     

Annular extrudate swell of pseudoplastic and viscoplastic fluids

Numerical simulations have been undertaken for the benchmark problem of annular extrudate swell present in pipe extrusion and parison formation in blow molding. The finite element method (FEM) is used to provide numerical results for different inner/outer diameter ratios κ under steady-state conditions. The Herschel-Bulkley model of viscoplasticity is used with the Papanastasiou regularization, which reduces with appropriate parameter choices to the Bingham–Papanastasiou, power-law and Newtonian models. The present results provide the shape of the extrudate, and in particular the thickness and diameter swells, as a function of the dimensionless power-law index (in the case of pseudoplasticity) and the dimensionless yield stress (in the case of viscoplasticity). The pressures from the simulations have been used to compute the excess pressure losses in the system (exit correction). While shear-thinning leads to reduced swelling relative to the Newtonian values for all κ-values, the opposite is true for shear-thickening fluids, which exhibit considerable swelling. Viscoplasticity leads to decreased extrudate swell as the dimensionless yield stress goes from zero (Newtonian behaviour) to an asymptotic value of fully plastic behaviour. The exit correction decreases to zero with a decrease in the power-law index to zero and an increase in the dimensionless yield stress to its asymptotic limit. However, the decrease is not monotonic: for power-law fluids it has maxima in the range of power-law indices between 0.8 and 0.6, while for viscoplastic fluids it has maxima around Bingham number values of 5.     

Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet

An analysis is made for the steady two-dimensional magneto-hydrodynamic flow of an incompressible viscous and electrically conducting fluid over a stretching vertical sheet in its own plane. The stretching velocity, the surface temperature and the transverse magnetic field are assumed to vary in a power-law with the distance from the origin. The transformed boundary layer equations are solved numerically for some values of the involved parameters, namely the magnetic parameter M, the velocity exponent parameter m, the temperature exponent parameter n and the buoyancy parameter λ, while the Prandtl number Pr is fixed, namely Pr = 1, using a finite difference scheme known as the Keller-box method. Similarity solutions are obtained in the presence of the buoyancy force if n = 2m−1. The features of the flow and heat transfer characteristics for different values of the governing parameters are analyzed and discussed. It is found that both the skin friction coefficient and the local Nusselt number decrease as the magnetic parameter M increases for fixed λ and m. For m = 0.2 (i.e. n = −0.6), although the sheet and the fluid are at different temperatures, there is no local heat transfer at the surface of the sheet except at the singular point of the origin (fixed point).     

Fractional vapour content of a liquid pool through which vapour is bubbled

Data from a large number of Russian, American and German sources are examined and found to be correlated in general by

$\text{α}\text{1?α)}\text{1}\text{2}\text{= K[F}{}_{\mathrm{D}}\text{P}{}^{\mathrm{m}}\text{]}{}^{\mathrm{n}}$

where α is voidage or fractional vapour content, K is a constant, F_D is a Froude number and P is a physical properties group. However, the exponent m is found to vary from 0 to 0.3 and the exponent n from $\text{2}\text{3}$ to 0.79, depending upon the sources of the data. The most probable value for n is $\text{2}\text{3}$ but a firm choice cannot be made for m, which is either 0.16 or 0.3. The different values of m depend chiefly upon the method of measurement of the voidage.     

Limit analysis and conic programming: ‘porous Drucker–Prager’ material and Gurson's modelAnalyse limite et optimisation conique : étude d'un matériau de Drucker–Prager poreux

Extending a previous work on the Gurson model for a ‘porous von Mises’ material, the present study first focuses on the yield criterion of a ‘porous Drucker–Prager’ material with spherical cavities. On the basis of the Gurson micro-macro model and a second order conic programming (socp) formulation, calculated inner and outer approaches to the criterion are very close, providing a reliable estimate of the yield criterion. Comparison with an analytical criterion recently proposed by Barthélémy and Dormieux—from a nonlinear homogenization method—shows both excellent agreement when considering tensile average boundary conditions and substantial improvement under compressive conditions. Then the results of an analogous study in the case of cylindrical cavities in plane strain are presented. It is worth noting that obtaining these results was made possible by using mosek, a recent commercial socp code, whose impressive efficiency was already seen in our previous works. To cite this article: M. Trillat et al., C. R. Mecanique 334 (2006).     

Numerical study for influences of material parameters on hydrostatic bulging of metal sheet

In this paper, the influences of various material parameters, the hardening exponent (n), the rate sensitivity (m). the thickness anisotropy parameter (R) and the index M in the Hosford and Hill yield function, on the hydrostatic bulging of a circular clamped sheet of ductile metal materials are analysed by introducting a rigid-viscoplastic finite element method. By numerical studies, an empirical relationship within the average limit thickness strain – ₃ ^* and the material parameters (n andm) is obtained. Besides, it has been found that the influences of surface shapes of the yield function on the average limit thickness strain can be reflected by the Barlat'sP value which represents the effects ofR andM values.This work is supported by the National Natural Science Foundation and Natural Science Foundation of the Youth of China.     

Orientation effects in extensional and squeezing flows of certain anisotropic fluids

The Ericksen-Leslie continuum theory of anisotropic fluids is here used to examine the behaviour of the orientation pattern within the bulk of a fluid that is undergoing an extensional-type of flow. The flow (which is irrotational and generally unsteady) is considered to be generated by application of prescribed normal stresses, and the orientation pattern (represented by a director field n) is taken to be spatially homogeneous but time-dependent. By means of a phase-plane analysis it is shown that, in stretching flows, the director eventually aligns parallel to the direction of imposed stretch, whereas in squeezing-type flows it eventually lies in the plane normal to the direction of squeezing. In both cases the lateral components of n may vary non-monotonically in time, before approaching their asymptotic values; also the lateral components of velocity may change sign during flow.A two-parameter classification is given of all possible modes of behaviour of these model fluids in these flows. Also analytical solutions are obtained for certain special cases, such as axisymmetric flow.     

Global and local approaches to fracture normal to interfaces

The problem of a crack perpendicularly approaching a bimaterial interface is examined using both global and localapproaches to fracture. The global approach is based on the J-integral with a second parameter, Q, which scales the stress triaxiality ahead of the crack. The local approach is based on either brittle fracture(Beremin model ) or ductile fracture (Rice and Tracey model ). In the first case, the Weibull stress over the plasticzone is calculated. In the second case, the void growth rate is calculated at the tip of the crack over a representativevolume (generally associated with a characteristic length of the material ). After a brief summary of each approach,the results for a crack near an elastically homogeneous, plastically mismatched interface are presented. Thebehaviour of the bimaterial is expressed in relation to the behavior of the homogeneous material. It is shown thatthere is an effect on the crack behavior which depends on the direction of crack propagation, i.e. from the hardermaterial to the softer material or vice versa. This effect is examined as a function of change in yield strength ratioand hardening exponent, n. For the case of brittle fracture, the effect of changing the Weibull modulus, m, is also examined. The models based on the local approach show that both stress- and strain-controlledfracture mechanisms must be accounted for. This implies the necessity of using the two parameters J and Q in the global approach. This is due to the fact that the stress–strain fields ahead of the crack tip areaffected by the nature of the second material.     

The effect of suction or injection on the boundary layer flows induced by continuous surfaces stretched with prescribed skin friction

The boundary layer flow and heat transfer on a stretched surface moving with prescribed skin friction is studied for permeable surface. Three major cases are studied for isothermal surface (n=0) stretched corresponding to different dimensional skin friction boundary conditions namely; skin friction at the surface scales as (x ^?1/2) at m=0, constant skin friction at m=1/3 and skin friction scales as (x) at m=1. The constants m and n are the indices of the power law velocity and temperature exponent respectively. Similarity solutions are obtained for the boundary layer equations subject to power law temperature and velocity variation. The effect of various governing parameters, such as Prandtl number Pr, suction/injection parameter f _w , m and n are studied. The results show that for isothermal surface increasing m enhances the dimensionless heat transfer coefficient for fixed f _w at the suction case and the reverse is true at the injection case. Furthermore, for fixed m, as f _w increases the dimensionless heat transfer coefficient increases. Large enhancements are observed in the heat transfer coefficient as the temperature boundary condition along the surface changes from uniform to linear where the dimensional skin friction is of order (x) at m=1. This enhancement decreases as the suction increases.     

A computational study on the instrumented sharp indentations with dual indenters

Finite element analysis was performed to investigate the indentation response of elasto-plastic solids for conical indenters of half included angles of 60° and 70.3°. The interdependence indentation parameters resulting from a single indentation load–depth curve is considered. Regarding dimensional analysis, several dimensionless relationships are constructed as functions of the reduced elastic modulus-loading curvature ratio E¹/C and the strain hardening exponent n. Further, the duality between corresponding parameters with dual indenters is explored. Finally, a new method based on dual indenters is proposed to extract the strain hardening exponent and the reduced elastic modulus of an indented material. The accuracy of this method is verified and discussed with experimental data from the literature and representative materials.     

Mixed convection cooling of a heated, continuously stretching surface

 An numerical study of the flow and heat transfer characteristics associated with a heated, continuously stretching surface being cooled by a mixed convection flow has been carried out. The relevant heat transfer mechanisms are of interest in a wide variety of practical applications, such as hot rolling, continuous casting, extrusion, and drawing. The surface velocity of the continuously stretching sheet was assumed to vary according to a power-law form, that is, u _w (x)=Cx ^p . Two conditions of surface heating were considered, a variable wall temperature (VWT) in the form T _w (x)−T _∞=Ax ⁿ and a variable surface heat flux (VHF) in the form q _w (x)=Bx ^m . The governing differential equations are transformed by introducing proper nonsimilarity variables and solved numerically using a procedure based on finite difference approximations. Results for the local Nusselt number and the local friction coefficient are obtained for a wide range of governing parameters, such as the surface velocity parameter p, the wall temperature exponent n, the surface heat flux exponent m, the buoyancy force parameters (ξ for the VWT case and χ for the VHF case), and Prandtl number of the fluid. It is found that the local Nusselt number is increased with increasing the velocity exponent parameter p for the VWT case, while the opposite trend is observed for the VHF case. The local friction coefficient is increased for a decelerated stretching surface, while it is decreased for an accelerated stretching surface. Also, appreciable effects of the buoyancy force on the local Nusselt number and the local friction coefficient are observed for both VWT and VHF cases, as expected. Received on 11 January 1999     

Fission of solitons in a symmetric triangular channel with variable cross section

In this paper, we study the disintegration of a soliton in a symmetric triangular channel when it propagates from one uniform cross section of the channel into another through a transition region. A criterion under which a soliton is split into n solitons is given. Numerical results for n = 3 are presented to confirm the analytical predictions.     

Mixed convection boundary layer flow past a wedge with permeable walls

The effects of suction/injection on steady laminar mixed convection boundary layer flow over a permeable horizontal surface of a wedge in a viscous and incompressible fluid is considered in this paper. The similarity solutions of the governing boundary layer equations are obtained for some values of the suction/injection parameter f ₀, the constant exponent m of the wall temperature as well as the mixed convection parameter λ. The resulting system of nonlinear ordinary differential equations is solved numerically for both assisting and opposing flow regimes using an implicit finite-difference scheme known as the Keller-box method. Numerical results for the reduced skin friction coefficient, the local Nusselt number, and the velocity and temperature profiles are obtained for various values of parameters considered. Dual solutions are found to exist for the case of opposing flow.     

On thermal boundary layer of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection

 Heat transfer characteristics of a non-Newtonian fluid on a power-law stretched surface of variable temperature with suction or injection were investigated. Similarity solutions of the laminar boundary layer equations describing heat transfer and fluid flow in a quiescent fluid were obtained and solved numerically. Velocity and temperature profiles as well as the Nusselt number, Nu, were studied for two thermal boundary conditions; uniform surface temperature and variable surface temperature, for different parameters; Prandtl number Pr, temperature exponent b, velocity exponent m, injection parameter d and power-law index n. It was found that decreasing injection parameter d, and power-law index n and increasing Prandtl number Pr and surface temperature exponent b enhance the heat transfer coefficient. Received on 27 April 2000     

Elastic properties of model random three-dimensional open-cell solids

Most cellular solids are random materials, while practically all theoretical structure-property relations are for periodic models. To generate theoretical results for random models the finite element method (FEM) was used to study the elastic properties of open-cell solids. We have computed the density (ρ) and microstructure dependence of the Young's modulus (E) and Poisson's ratio (ν) for four different isotropic random models. The models were based on Voronoi tessellations, level-cut Gaussian random fields, and nearest neighbour node-bond rules. These models were chosen to broadly represent the structure of foamed solids and other (non-foamed) cellular materials. At low densities, the Young's modulus can be described by the relation E∝ρⁿ. The exponent n and constant of proportionality depend on microstructure. We find 1.3<n<3, indicating a more complex dependence than indicated by periodic cell theories, which predict n=1 or 2. The observed variance in the exponent was found to be consistent with experimental data. At low densities we found that ν≈0.25 for three of the four models studied. In contrast, the Voronoi tessellation, which is a common model of foams, became approximately incompressible (ν≈0.5). This behaviour is not commonly observed experimentally. Our studies showed the result was robust to polydispersity and that a relatively large number (15%) of the bonds must be broken to significantly reduce the low-density Poission's ratio to ν≈0.33.     

基于双剪应力屈服准则的受内压管道爆破压力分析

采用双剪应力屈服准则,对在内压作用下的无缺陷管道进行了塑性极限分析,得到了管道爆破压力的计算公式;并且将结果与基于Tresca、Mises、平均剪应力屈服准则得到的爆破压力进行了比较.研究结果表明:爆破压力随着管道材料的应变硬化指数的增大而减小,随着管道厚径比的增大而增大;此外,基于双剪应力屈服准则得到的管道爆破压力为爆破压力的上限,而基于Tresca屈服准则得到的爆破压力为管道爆破压力的下限.     

On a vibrating plate with a concentrated mass

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^−m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. Depending on the value of m (m<2, m=2 and m>2) we describe the asymptotic behaviour, as ϵ→0, of the eigenvalues and eigenfunctions of the corresponding spectral problem. For m>2 the vibrations associated with the low frequencies affect asymptotically only a neighbourhood of the concentrated mass; we also consider the asymptotic behaviour of the eigenfunctions associated with the high frequencies.     

Instability of channel flow of a shear-thinning White–Metzner fluid

We consider the inertialess planar channel flow of a White–Metzner (WM) fluid having a power-law viscosity with exponent n. The case n = 1 corresponds to an upper-convected Maxwell (UCM) fluid. We explore the linear stability of such a flow to perturbations of wavelength k⁻¹. We find numerically that if n < n_c ≈ 0.3 there is an instability to disturbances having wavelength comparable with the channel width. For n close to n_c, this is the only unstable disturbance. For even smaller n, several unstable modes appear, and very short waves become unstable and have the largest growth rate. If n exceeds n_c, all disturbances are linearly stable. We consider asymptotically both the long-wave limit which is stable for all n, and the short-wave limit for which waves grow or decay at a finite rate independent of k for each n.The mechanism of this elastic shear-thinning instability is discussed.     

双均值逼近屈服准则解析含腐蚀缺陷管道爆破压力

通过对Tresca和TSS屈服边长和边心距的均值同时进行逼近,建立了一个线性屈服准则,称为双均值逼近屈服准则.该准则在π平面上是一个等边非等角的十二边形,位于Mises圆内部.利用该准则对受内压作用的管道进行塑性极限分析,导出了含腐蚀缺陷管道爆破压力的解析解.该解析解是管材屈强比(σY/σT)、原始管道厚径比(t0/D0)、抗拉强度σT以及缺陷深度比(d0/t0)的函数.对比表明,该解析解所预测爆破压力与已有模拟和实验数据吻合较好.影响参数的定量分析表明,爆破压力随着屈强比或原始管道厚径比的增大而增大,随着缺陷深度比的增加而减小.所建立的爆破压力解析解对于管道的选材、设计以及安全评估具有重要意义.