Instabilities in elastic–plastic fluid-saturated porous media: harmonic wave versus acceleration wave analyses

A linearized analysis of the stability of the flow in elastic–plastic fluid-saturated porous media with incompressible constituents is performed. A relationship is established between the results of this analysis and the results of analyses based on acceleration waves performed by Loret and Harireche [Journal of the Mechanics and Physics of Solids 39, 569–606 (1991) ] and by Loret et al. [International Journal of Solids and Structures 34, 1583–1608 (1997) ]; a justification is found for the growth of the acceleration waves in the non-associative case when their speeds are real and a clarification is provided relative to the growth or decay of waves in the interior of flutter regions.     

动态裂纹扩展中的裂纹滞留现象

借助于Ｊ．Ｒ．Ｗｉｌｉｓ等（１９９５）的动态权函数和Ｒｉｃｅ等（１９９４）的模拟方法，本文研究了裂纹前方开始环绕和穿透介质中各种粗糙微粒而形成粗糙的裂纹表面等非稳定现象．数值结果表明：在Ⅰ型情形，振荡性裂纹前缘运动源于应力强度波的建设性—破坏性相互作用．这些波包括膨胀波、剪切波和瑞利波系，彼此相互干涉且与裂纹运动边界相互作用，导致了裂纹前方扩展路径和扩展速度的连续波动．     

Localised knife waves in a structured interface

We consider a Mode III lattice with an interface layer where the dynamic crack growth is caused by a localised sinusoidal wave. In the wave–fracture scenario, the ‘feeding wave’ (here also called the knife wave) delivers energy to the moving crack front, while the dissipative waves carry a part of this energy away from the front. The questions addressed here are:

• What are the conditions of existence of the localised knife wave?

• What is the lower bound of the amplitude of the feeding wave, which supports the crack propagation, for a given deformational fracture criterion?

• How does the crack speed depend on the amplitude of the feeding wave?

• What are the dissipative waves? How much energy is irradiated by these waves and what is the total dissipation?

• What are the conditions of existence of the steady-state regime for the propagating crack?

We consider analytically two established regimes: the steady-state regime, where the motion of neighbouring masses (along the interface) differs only by a constant shift in time, and an alternating-strain regime, where the corresponding amplitudes differ by sign. We also present the numerical simulation results for a model of a high-contrast interface structure. Along with the energy of the feeding and dissipative waves, an energy radiated to the bulk of the lattice is identified.

A centrally cracked thin circular disk, Part I: 3D Elastic–plastic finite element analysis

A full field solution, based on small deformation, three-dimensional elastic–plastic finite element analysis of the centrally cracked thin disk under mode I loading has been performed. The solution for the stresses under small-scale yielding and lo!cally fully plastic state has been compared with the HRR plane stress solution. At the outside of the 3D zone, within a distance of rσ_o/J=18, HRR dominance is maintained in the presence of a significant amount of compressive stress along the crack flanks. Ahead of this region, the HRR field overestimate the stresses. These results demonstrate a completely reversed state of stress in the near crack front compared to that in the plane strain case. The combined effect of geometry and finite thickness of the specimen on elastic–plastic crack tip stress field has been explored. To the best of our knowledge, such an attempt in the published literature has not been made yet. For the qualitative assessment of the results some of the field parameters have been compared to the available experimental results of K, gives a fair estimate of the crack opening stress near the crack front at a distance of order 10⁻² in. On the basis of this analysis, the Linear Elastic Fracture Mechanics approach has been adopted in analyzing the fatigue crack extension experiments performed in the disk (Part II).     

Numerical calculation of wall effect and ‘backflow’ on the Stokes force

We give numerical results on the modification of the drag force F_x exerted on a sphere positioned eccentrically and moving at very low Reynolds number, at constant velocity within and along a cylindrical tube. The numerical results computed by Lattice-Boltzmann method or by finite volume formulation are in good agreement with the experimental results obtained by Ambari et al. (J. Fluid Mech. 149 (1984) 235–253). In particular, they confirm the existence of a minimum of the force F_x away from the axis of the cylinder and a sharp increase when the sphere approaches the sidewall. To cite this article: T. Godin et al., C. R. Mecanique 330 (2002) 837–842.     

Delayed fracture of a cylindrically anisotropic viscoelastic composite with a circular crack under a constant load

Conclusions The results obtained here can also be used for composites having plane circular normal-rupture cracks subjected to tension by constant forces if: I) the composite, with cylindrically anisotropic viscoelastic strain properties, can be modeled by a homogeneous, cylindrically anisotropic viscoelastic material; 2) the height of the prefracture region at the front of the moving crack is small; 3) the length of this region remains constant during propagation of the crack.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 10, pp. 53–60, October, 1991.     

Mécanique de la rupture fragile en présence de plasticité : modélisation de la fissure par une entaille

Some issues are met in elastic-plastic fracture mechanics if the crack is modelled by a sharp tipped crack. They may be overcome with a notch model. An energy release rate can be defined using the Francfort and Marigo theory and Continuum Damage Mechanics. In particular, the paradox of Rice and some spurious scale effects are removed with this approach. The results obtained in the case of the analysis of the shallow crack effect are in good agreement with the experimental results. To cite this article: Y. Wadier, E. Lorentz, C. R. Mecanique 332 (2004).     

A micromechanical analysis of damage propagation in fluid-saturated cracked mediaUne analyse micromécanique de la propagation de l'endommagement d'une fissure sous pression de fluide

We first revisit the well known framework of Linear Elastic Fracture Mechanics (LEFM) in the case of a fluid-saturated crack. We next consider a r.e.v. of cracked medium comprising a family of cracks characterized by the corresponding crack density parameter ε. Generalizing the classical energy approach of LEFM, the proposed damage criterion is written on the thermodynamic force associated with ε, which is estimated by means of standard homogenization schemes. This criterion proves to involve a macroscopic effective strain tensor, or alternatively the Terzaghi effective stress tensor. The stability of damage propagation is discussed for various homogenization schemes. A comparison with experimental results is presented in the case of a uniaxial tensile test on concrete. To cite this article: L. Dormieux et al., C. R. Mecanique 334 (2006).     

Measuring soil properties to predict tractive performance of an agricultural drive tire

The coefficient of traction for a 9.5–16 R-1 bias ply tire was measured and compared with predictions using equations developed by Janosi and Hanamoto [Proc. 1st Int. Conf. on Mechanics, p. 707–736 (1961)]; Wismer and Luth [J. Terramechanics10, 49–61 (1973)] Gee-Clough et al. [J. Terramechanics15, 81–84 (1978)] and Brixius [ASAE Paper No. 87–1622 (1987)]. For the soft soil condition, with a cone index of 120 kPa, Gee-Clough's equation predicted the coefficient of traction better, but predictions using the Brixius equation were better for soil with a cone index of 225 kPa. An experimental device was developed to simultaneously measure the horizontal and vertical stress-strain relationships of soil. The use of resultant stress from the experimental device data failed to show any improvement in coefficient of traction prediction over using the cone index. The resultant of the normal and shear stress from the experimental device data did not adequately represent the soil properties involved in terrain-vehicle mechanics.     

A numerical analysis of flat internal cracks under mixed mode loading

A numerical/analytical approach is proposed to determine the stress intensity factors K_I, K_II, and K_III of a 3D internal crack. The main point of this approach is the meshing technique that can model very sharp crack fronts. The meshing technique is based on an elliptical coordinate transformation that starts from a circular crack. It allows the obtainment of a curved crack front with elements normal to the crack front. Remarkable accuracy can be obtained for elliptical crack fronts with axes ratio smaller that 0.01. Accuracy demonstration is provided for cylindrical element with an inclined internal crack subjected to uni-axial tension. This case corresponds to crack propagation for all three modes of loading, the solution of which can checked with references’ results.     

Stress intensity factors for asymmetric branched cracks in plane extension by using crack-tip displacement discontinuity elements

Stress intensity factors are important in the analysis of cracked materials. They are directly related to the fracture propagation and fatigue crack growth criteria. Based on the analytical solution (Crouch, S.L., 1976. Solution of plane elasticity problems by displacement discontinuity method, Int. J. Numer. Methods Eng. 10, pp. 301–343; Crouch, S.L., Starfield, A.M., 1983. Boundary Element Method in Solid Mechanics, with Application in Rock Mechanics and Geological Mechanics, London, Geore Allon and Unwin, Bonton, Sydney) to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid, recently, the crack-tip displacement discontinuity element which can be classified as the left and right crack-tip displacement discontinuity elements are developed by the author Yan, X., (in press. A special crack-tip displacement discontinuity element, Mechanics Research Communications) to model the crack-tip fields to more accurately compute the stress intensity factors of cracks in general plane elasticity. In the boundary element implementation the left or the right crack-tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. To prove further the efficiency of the suggested approach and provide more results of the stress intensity factors, in this study, analysis of an asymmetric branched crack bifurcated from a main crack in plane extension is carried out.     

Two instructive instabilities in non-linear elasticity: Biaxially loaded membrane,and rubber balloons

Elastic bodies buckle under compressive loads, i.e. solutions become non-unique, they bifurcate and the body becomes unstable. Similar phenomena occur in tension as is evidenced here by the symmetric biaxial loading of a square membrane. Symmetry breaking removes the non-uniqueness. Under non-symmetric loading the load-deformation curves become non-monotone, consequently a hysteresis occurs which is the reflection of a fold-type catastrophy. This instructive instability was discovered by Kearsley [1]. Here we investigate it more fully and present some additional aspects.Balloons have non-monotone pressure-radius relations which suggest non-trivial stability properties. A stability analysis is presented for two interconnected balloons. In this we follow — and expand on — the analyses presented by Dreyer et al. [2] and Kitsche et al. [3].General Invited Lecture presented at AIMETA '95 — 12th Congress of the Italian Association of Theoretical and Applied Mechanics, Napoli, 3–6 October, 1995.     

Subsonic interfacial fracture using strain gages in isotropic–orthotropic bimaterial

An experimental study has been conducted in which strain fields were used to investigate the behavior of subsonic crack propagation along the interface of an isotropic–orthotropic bimaterial system. Strain field equations were developed from available field equations and critically evaluated in a parametric study to identify optimum strain gage location and orientation. Bimaterial specimens were prepared with PSM-1 polycarbonate and Scotchply^® 1002 unidirectional, glass-fiber-reinforced, epoxy composite. Dynamic experiments were conducted using these specimens with strain gages mounted on the composite half to obtain values of the dynamic complex stress intensity factor, K=K₁+iK₂, in the region of the crack tip while photoelasticity was used on the PSM-1 half. Results show that the trend and magnitude of K obtained using strain gages compare favorably with those obtained using photoelasticity.     

Equivalence Theorems on the Propagation of Small-Amplitude Waves in Prestressed Linearly Elastic Materials with Internal Constraints

By extending the procedure of linearization for constrained elastic materials in the papers by Marlow and Chadwick et al., we set up a linearized theory of constrained materials with initial stress (not necessarily based on a nonlinear theory). The conditions of propagation are characterized for small-displacement waves that may be either of discontinuity type of any given order or, in the homogeneous case, plane progressive. We see that, just as in the unconstrained case, the laws of propagation of discontinuity waves are the same as those of progressive waves. Waves are classified as mixed, kinematic, or ghost. Then we prove that the analogues of Truesdell"s two equivalence theorems on wave propagation in finite elasticity hold for each type of wave. This revised version was published online in August 2006 with corrections to the Cover Date.     

Crack movement in layered composites under electric field affect

The variation principle is applied for defining a crack in the solid body. The methods proposed in [G. Sih, C. Chen, Non-self-similar crack growth in elastic–plastic finite thickness plate, Theoretical and Applied Fracture Mechanics 3 (1985) 125–139] extend to presence of electromagnetic fields in material. Crack propagation in non-homogeneous media has been considered. It is shown that electromagnetic fields in the material are essentially affecting the trajectory. The crack trajectory stability has been studied as function of fracture energy, phase portraits of the trajectory in different media have been built, and various attractor types have been revealed. Different crack morphologies from single straight and oscillating crack propagation to straight double crack propagation were theoretically founded. In compliance with the experimental data of [R. Niefanger, V.-B. Pham, G. Schneider, H.-A. Bahr, H. Balke, U. Bahr, Quasi-static straight and oscillatory crack propagation in ferroelectric ceramics due to moving electric field: experiments and theory, Acta Materialia 52 (1) (2004) 117–127], it has been demonstrated that periodic electromagnetic field results in trajectory stochastization. This can be used for switching the crack over from the mode of mainline propagation into the mode of development of the field of diffused microcracks.     

Non-local energetics of random heterogeneous lattices

In this paper, we study the mechanics of statistically non-uniform two-phase elastic discrete structures. In particular, following the methodology proposed in Luciano and Willis (2005) [Journal of the Mechanics and Physics of Solids 53, 1505-1522], energetic bounds and estimates of the Hashin-Shtrikman-Willis type are developed for discrete systems with a heterogeneity distribution quantified by second-order spatial statistics. As illustrated by three numerical case studies, the resulting expressions for the ensemble average of the potential energy are fully explicit, computationally feasible and free of adjustable parameters. Moreover, the comparison with reference Monte-Carlo simulations confirms a notable improvement in accuracy with respect to approaches based solely on the first-order statistics.     

On the eccentricity factor in the progressive crushing of tubes

The analysis of an axially compressed circular tube deforming in progressive axisymmetric folds carried out by Wierzbicki et al. [(1992) Int. J. Solids Structures 29, 3269–3288] assumes an eccentricity factor relating the inward and outward parts of the folds. This factor was arbitrary and was not derived from the analysis. The present work re-examines the problem and produces a value for the eccentricity factor which conforms with the experimental findings. Values of the critical angles required for the formation of the inward and outward folds obtained from the analysis were substantiated by those obtained from experiments.     

Theoretical analysis of crack front instability in mode I+III

This paper focusses on the theoretical prediction of the widely observed crack front instability in mode I+III, that causes both the crack surface and crack front to deviate from planar and straight shapes, respectively. This problem is addressed within the classical framework of fracture mechanics, where the crack front evolution is governed by conditions of constant energy-release-rate (Griffith criterion) and vanishing stress intensity factor of mode II (principle of local symmetry) along the front. The formulation of the linear stability problem for the evolution of small perturbations of the crack front exploits previous results of Movchan et al. (1998) (suitably extended) and Gao and Rice (1986), which are used to derive expressions for the variations of the stress intensity factors along the front resulting from both in-plane and out-of-plane perturbations. We find exact eigenmode solutions to this problem, which correspond to perturbations of the crack front that are shaped as elliptic helices with their axis coinciding with the unperturbed straight front and an amplitude exponentially growing or decaying along the propagation direction. Exponential growth corresponding to unstable propagation occurs when the ratio of the unperturbed mode III to mode I stress intensity factors exceeds some “threshold” depending on Poisson's ratio. Moreover, the growth rate of helical perturbations is inversely proportional to their wavelength along the front. This growth rate therefore diverges when this wavelength goes to zero, which emphasizes the need for some “regularization” of crack propagation laws at very short scales. This divergence also reveals an interesting similarity between crack front instability in mode I+III and well-known growth front instabilities of interfaces governed by a Laplacian or diffusion field.     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

Fatigue fracture model for thin isotropic plates with cracks in axial loading

Conclusion We have constructed a model of the growth of a fatigue crack in a thin, isotropic plate, taking the two-stage evolution of the fracture process into account. The model is based on concepts of the mechanics of a continuous defective state and on a schematic representation of the neighborhood of the tip of a fatigue crack as a plastic zone moving together with the crack. The model takes into account the influence of the cumulative defective state (damage level) along the crack propagation front on the speed of propagation.We have formulated solutions for the cases when the length of the plastic zone is constant and when it varies during the growth of fatigue cracks. We have established the fact that the plastic zone at the crack tip tends to disrupt the stability of the motion immediately at the time of inception or opening of the crack. The speed of crack propagation decreases as the plastic zone grows in size.We have shown that the problem of estimating the kinetics of fatigue cracks in thin plates can be reduced to calculating the growth rate as a function of the peak-to-peak amplitude of the stress intensity factor while preserving the structure of the governing equations of the model. We have also shown that the concept of a plastic zone of constant length induces a power-law dependence of the crack rate on K, the power exponent varying from 2 to 10–12. The Dugdale model gives a square-law dependence of the crack rate on K, which for the most part is applicable to plastic materials.S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 7, pp. 53–63, July, 1994.