Generalized limit analysis in poroplasticity by mathematical programming

Classical limit analysis of structures by the statical approach computationally means maximization of a load multiplier under equilibrium and yield condition constraints, namely convex mathematical programming. In elastoplasticity, generalizations of limit analysis had been proposed in order to achieve, still by load factor constrained optimization, the safety factor with respect to plastic collapse. This paper presents similar generalization in two-phase poroelastoplasticity. A method is here developed (and validated by numerical application to a masonry dam) apt to assess the safety factor of a structure interpretable as a poroplastic system, with respect to both plastic collapse and critical thresholds on deformations, by solving a nonconvex nonsmooth constrained optimization problem usually referred to in the literature as “mathematical program under equilibrium constraints”. Piece-wise linearization of yield surfaces and reduction of yield planes by a “sifting” procedure are adopted to reduce computing efforts.     

A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.     

Asymptotic analysis of an end-loaded, transversely isotropic, elastic, semi-infinite strip weak in shear

We use linear elasticity to study a transversely isotropic (or specially orthotropic), semiinfinite slab in plane strain, free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction. The governing equations (which are identical to those for a strip in plane stress, free of traction on its long sides and at infinity, and subject to tractions or displacements on its short side) are reduced, in the standard way. to a fourth-order partial differential equation with boundary conditions for a dimensionless Airy stress function ƒ. We study the asymptotic solutions to this equation for four sets of end conditions—traction, mixed (two), displacement—as g3, the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli, approaches zero. In all cases, the solutions for ƒ consist of a “wide” boundary layer that decays slowly in the axial direction (over a distance that is long compared to the width of the strip) plus a “narrow” boundary layer that decays rapidly in the axial direction (over a distance that is short compared to the width of the strip). Moreover, we find that the narrow boundary layer has a “sinuous” part that varies rapidly in the transverse direction, but which, to lowest order, does not enter the boundary conditions nor affect the transverse normal stress or the displacements. Because the exact biorthogonality condition for the cigenfunctions associated with ƒ can be replaced by simpler orthogonality conditions in the limit as →b 0, we are able to obtain, to lowest order, explicit formulae for the coeflicients in the eigenfunction expansions of ƒ for the four different end conditions.     

Energy approach to crack growth in a fiber-reinforced brittle material modeled by an inclusion

Brittle materials randomly reinforced with a low volume fraction of strong, stiff and ductile fibers are considered, with specific reference to fiber-reinforced cements and concrete. Visible cracks in such materials are accompanied by a surrounding damage zone – together these constitute a very complex “crack system”. Enormous effort has been put into trying to understand the micromechanics of such systems. Almost all of these efforts do not deal with the “crack system” propagation behavior as a whole. The propagation process of such a “crack system” includes propagation of the visible crack and the growth of the damage zone. Propagation may take place by lengthening of the visible crack together with the concomitant lengthening of the surrounding damage zone, or simply by broadening of the damage zone while the visible crack length remains unchanged – or simultaneously by growth of both types. A phenomenological completely theoretical model (for an ideal material) is here proposed which can serve to examine the propagation process by means of energy principles, without recourse to the microscopic details of the process. An application of this theoretical approach is presented for the case of a damage zone evolving with a rectangular shape. This shape is chosen because it is expected that it will illustrate the nature of damage evolution and because the computational procedure necessary to follow the growth is the most straightforward.     

A new Finite Element Method for magneto-dynamical problems: two-dimensional results

A mixed Lagrange finite element technique is used to solve the Maxwell equations in the magneto-hydrodynamic (MHD) limit in an hybrid domain composed of vacuum and conducting regions. The originality of the approach is that no artificial boundary condition is enforced at the interface between the conducting and the insulating regions and the non-conducting medium is not approximated by a weakly conducting medium as is frequently done in the literature. As a first evaluation of the performance of the method, we study two-dimensional (2D) configurations, where the flow streamlines of the conducting fluid are planar, i.e., invariant in one direction, and either the magnetic field (“magnetic scalar” case) or the electric field (“electric scalar” case) is parallel to the invariant direction. Induction heating, eddy current generation, and magnetic field stretching are investigated showing the usefulness of finite element methods to solve magneto-dynamical problems with complex insulating boundaries.     

Development of a surface-cracked specimen

A surface-cracked specimen has been developed that features a web which facilitates fatigue precracking. In addition, when the web is removed, the remaining semi-elliptical surface crack forms a “natural” intersection with the free surface. The specimen, which resembles an elongated compact tension specimen, is quite efficient in its load requirements; relatively low loads are necessary to produce a given level of stress intensity factor.     

Fatigue crack growth in high and low strength steel under torsional loading

During loading of a crack in mode III the crack surfaces in contact slide against each other giving rise to friction, abrasion and mutual support, thereby reducing the effective stress at the crack tip (“sliding mode crack closure”). This phenomenon was investigated in a high strength steel (AISI 4340) and in a low strength steel (AISI C1018) in circumferentially notched specimens under pure cyclic torsion and combined loading (cyclic torsion plus static axial load). The influence of sliding mode crack closure on fatigue crack propagation is shown and “true” crack growth values (without the sliding mode crack closure influence) are determined on the basis of an extrapolation procedure. Explanations are given for causes of the various fracture modes observed, such as “factory roof” fracture, macroscopically flat mode III fracture and “lamella” fracture. Finally the scientific and technical importance of sliding mode crack closure is demonstrated.     

Analysis of prestressed mechanisms

A new theory is presented for the matrix analysis of prestressed structural mechanisms made from pin-jointed bars. The response of a prestressed mechanism to any external action is decomposed into two almost separate parts, which correspond to extensional and inextensional modes. A matrix algorithm which treats these two modes separately is developed and tested. It is shown that the equilibrium requirements for the assembly, in its initial configuration as well as in deformed configurations which are obtained through infinitesimal inextensional displacements, can be fully described by a square equilibrium matrix. It is also shown that any set of extensional nodal displacements has to satisfy some equilibrium conditions as well as standard compatibility equations, and that the resulting system of linear equations defines a square kinematic matrix. Theoretical as well as experimental evidence supporting this approach is given in the paper ; two simple experiments which were of crucial importance in arriving at the equilibrium conditions on the extensional displacements are described.The interaction between the two modes of action of a prestressed mechanism is discussed, together with a rapidly converging iterative procedure to handle it. A study of the non-linear effect by which the self-stress level in a statically indeterminate assembly rapidly increases if an inextensional mode is excited, supported by further experimental results, concludes the paper. This work is relevant to the analysis of most cable systems, pneumatic domes, fabric roofs, and “Tensegrity” frameworks.     

Fatigue strength of welded joints based on local, semi-local and nominal approaches

Analysis based on the so-called “local approach” is made to estimate the fatigue strength of welded joints. Numerical analyses or strain gauges are employed for finding the stress and/or strain state in the vicinity of the weld toe. The notch stress intensity factor (NSIF) approach applied to fillet welded joints, as far as the opening angle between the weld and the main plate surface is constant (e.g. 135°, typical for many fillet welds), is able to rationalise the fatigue strength data both for different joint geometries and absolute dimensions. The NSIF approach has been previously developed as an extension of the Linear Elastic Fracture Mechanics (LEFM) to open V-notches and is based on the exponential local stress field around the V-notch tip. Several different “local approaches”, although simpler and more practical than the NSIF, are based on the stress (or strain) values determined beyond the exponential local one. To distinguish such approaches from the NSIF based one, we define the former as semi-local or nominal approaches while the latter is a local approach. The paper underlines that the local approaches, differently from the other ones, are able to unify in a single scatter band the fatigue strength data obtained from welded joints having different geometry and absolute dimensions.     

Kinematics and dynamics of small-scale vorticity and strain-rate structures in the transition from isotropic to shear turbulence

We applied a technique that defines and extracts “structures” from a DNS dataset of a turbulence variable in a way that allows concurrent quantitative and visual analysis. Local topological and statistical measures of enstrophy and strain-rate structures were compared with global statistics to determine the role of mean shear in the dynamical interactions between fluctuating vorticity and strain-rate during transition from isotropic to shear-dominated turbulence. We find that mean shear adjusts the alignment of fluctuating vorticity, fluctuating strain-rate in principal axes, and mean strain-rate in a way that (1) enhances both global and local alignments between vorticity and the second eigenvector of fluctuating strain-rate, (2) two-dimensionalizes fluctuating strain-rate, and (3) aligns the compressional components of fluctuating and mean strain-rate. Shear causes amalgamation of enstrophy and strain-rate structures, and suppresses the existence of strain-rate structures in low-vorticity regions between enstrophy structures. A primary effect of shear is to enhance “passive” strain-rate fluctuations, strain-rate kinematically induced by local vorticity concentrations with negligible enstrophy production, relative to “active,” or vorticity-generating strain-rate fluctuations. Enstrophy structures separate into “active” and “passive” based on the level of the second eigenvalue of fluctuating strain-rate. We embedded the structure-extraction algorithm into an interactive visualization-based analysis system from which the time evolution of a shear-induced hairpin enstrophy structure was visually and quantitatively analyzed. The structure originated in the initial isotropic state as a vortex sheet, evolved into a vortex tube during a transitional period, and developed into a well-defined horseshoe vortex in the shear-dominated asymptotic state.     

轴对称结构极限下限分析的自然单元法

作为一种基于自然邻近插值的新型无网格法,自然单元法克服了大多数无网格法难以施加本质边界条件的困难.将自然单元法与减缩基技术相结合,建立了一种轴对称结构极限下限分析的数值格式和求解算法.通过不断修正自平衡应力场,轴对称结构极限下限分析可转化为一系列的非线性数学规划子问题,并由复合形法求解.在每个非线性规划子问题中,自平衡应力场表示为一组带有待定系数的自平衡应力场基矢量的线性组合,并且这些自平衡应力场基矢量可由弹塑性增量分析的平衡迭代结果得到.算例结果表明,本文所提的轴对称结构极限下限分析方法行之有效.     

The role of heterogeneous deformation on damage nucleation at grain boundaries in single phase metals

The mechanical response of engineering materials evaluated through continuum fracture mechanics typically assumes that a crack or void initially exists, but it does not provide information about the nucleation of such flaws in an otherwise flawless microstructure. How such flaws originate, particularly at grain (or phase) boundaries is less clear. Experimentally, “good” vs. “bad” grain boundaries are often invoked as the reasons for critical damage nucleation, but without any quantification. The state of knowledge about deformation at or near grain boundaries, including slip transfer and heterogeneous deformation, is reviewed to show that little work has been done to examine how slip interactions can lead to damage nucleation. A fracture initiation parameter developed recently for a low ductility model material with limited slip systems provides a new definition of grain boundary character based upon operating slip and twin systems (rather than an interfacial energy based definition). This provides a way to predict damage nucleation density on a physical and local (rather than a statistical) basis. The parameter assesses the way that highly activated twin systems are aligned with principal stresses and slip system Burgers vectors. A crystal plasticity-finite element method (CP-FEM) based model of an extensively characterized microstructural region has been used to determine if the stress–strain history provides any additional insights about the relationship between shear and damage nucleation. This analysis shows that a combination of a CP-FEM model augmented with the fracture initiation parameter shows promise for becoming a predictive tool for identifying damage-prone boundaries.     

A methodology for studying fatigue crack propagation under spectrum loading: Application to rail steels

A testing procedure is applied to investigate the fatigue behavior of the detail fracture, in rail steel, under Mode I train stress spectrum. The results show that fatigue crack under spectrum loading depends on a “dynamic” accumulation of damage. This term is related to the load sequence in the spectrum and the gradient of the elastic stress intensity factor. It is theoretically very difficult to quantify this term. The rationale underlying the testing procedure is such that this dynamic term effect is implicitly included in the experimental results. The detail fracture was modelled by the penny-shaped crack geometry. This model has proved, in this study, to be satisfactory for simulating the fatigue growth of the detail fracture.     

L''exactitude du rendement des machines agricoles

Precision of the working height, or depth, of earth-moving equipment and other farm machinery can have a marked effect on the quality of the operation. The main factors affecting a machine's performance in maintaining the correct working height are its geometric parameters and the soil surface upon which it operates. The soil surface profile is defined by the “mean slope”, by the “root-mean-square” (RMS) of the deviations from the mean slope, and by the wavelengths calculated by Fourier analysis. The shortest wavelength with an amplitude above 1 cm was used in this study to find the effect of the soil surface waves on the accuracy of farm machinery operations. The machine's geometric parameters affecting its accuracy are its wheelbase and the lengthwise location of the working point (above or under ground) in relation to the wheels. A computer simulation analysis showed that the minimum deviations from the required height are found at the wheels of the machine. The deviations increased at the middle of the machine's span and rose steeply outside the span. The shorter the wheel-span of the machine relative to the wavelengths, the smaller the deviations will be. Smoothed surfaces have longer wavelengths and smaller deviations and as a result, higher accuracy of the working height, or depth, of the machines working on them.     

Legendre integrators, post-processing and quasiequilibrium

A toolbox for the development and reduction of the dynamical models of nonequilibrium systems is presented. The main components of this toolbox are: Legendre integrators, dynamical post-processing, and the thermodynamic projector. The thermodynamic projector is the tool to transform almost any anzatz to a thermodynamically consistent model. The post-processing is the cheapest way to improve the solution obtained by the Legendre integrators. Legendre integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (“maximum entropy”, MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to the FENE-P model of polymer kinetic theory. The multi-peak model of polymer dynamics is developed.     

Viscoplastic creep of rocks around a lined tunnel

A rock-support analysis is done assuming that the rock behaves elastic/viscoplastically while the support can be any kind of nonlinear support. The case of circular tunnels is considered and the formulation of the mathematical problem is discussed. Furthermore, the creep of rock around a tunnel with nonlinear supports (called “yieldable” or “self-adapting” supports) to limit the closure of the tunnel is studied. It is shown that the ultimate ground reaction is not unique since it depends on the loading history such as excavation layout and support design. For relatively small pressures exerted by the support, it is shown that the onset of failure by dilatancy can be determined. Two kinds of solutions are given: a simplified approximate one, which is easy to apply and with which the qualitative discussion of the solution is simple and revealing, and a more rigorous general elaborate numerical solution obtained by using computer programs.     

Brownian dynamics simulations of star-branched polymers in dilute solutions in extensional flow

Using Brownian dynamics (BD) simulations of FENE bead-spring models, the dynamics of star-branched polymers in dilute solutions under extensional flow have been investigated. Studies on star polymers in transient extensional flow reveal that the initial transient stress response at low strains is governed by both the number of arms and the shortest arm. On the other hand, the steady-state behavior of star polymers in elongational flow is limited by the maximum effective “contour” length of the molecules. The distribution of arm extension and birefringence of the star-branched molecule are broader and the mean is shifted to lower values, when compared to equivalent linear systems. As a result, the degree of arm extension at steady-state decreases as the number of arms in the star increases. Both an analysis of individual ensembles in Brownian dynamics simulations and a study of a simple force balance indicate that the constraint imposed on the star arms by the central branch point and contributions from “asymmetric” arm arrangements give rise to overall less extended and oriented star-branched molecules with broader arm extension and birefringence distributions. The results obtained from stress-conformation hysteresis simulation indicate that less-stretched arms exhibit more retarded relaxation, as the number of arms increases in star-branched molecules. The effect of excluded volume (EV) interactions, incorporated through the Lennard–Jones potential, on the dynamics of star polymers in extensional flow appears unimportant.     

Stretch and twist of rotating prismatic beam

A turbine blade is modelled as a uniform isotropic prismatic beam of general cross-section and “setting angle” rotating about one end, and is analysed according to the linear theory of elasticity. A semi-inverse solution is presented which reduces the three-dimensional problem to one of two dimensions, and explicit stress and strain components given for the mathematically amenable elliptic cross-section. As expected, the planar stresses σ_x,σ_y, and τ_xy arising from the two-dimensional problem are found to be small. For the general section, the theory predicts small curvature of the blade centre line, and a twisting moment which tends to reduce the “angle of set”.     

Geomechatronics – Interaction between ground and construction machinery and its application to construction robotics

“Geomechatronics” is a technical field in which “Geotechniques” is fused with “Mechatronics” that is the technical field to promote the automatic control of machines by using the electronics. In the field of “Geomechatronics”, a construction machine, which treats geotechnical materials such as soil and rock, automatically evaluates the properties and conditions of the ground and determines the optimum controlling method of itself for the ground with the base of the machine–ground interaction. Some researches for practical use in the field of geomechatronics are introduced, and then the progressing view of this research and technical filed is explained in this paper.