Perturbation growth and localization in fluid-saturated inelastic porous media under quasi-static loadings

Instabilities in inelastic saturated porous media are investigated here for general three-dimensional states under quasi-static loadings using a perturbation approach and focussing in particular on the two limiting cases of the onset of growth and of blowing up of perturbations.For associative flow rules for the skeleton, both onset of growth and blowing up of perturbations depend only on the underlying drained properties. Unbounded growth is obtained when the condition of localization for the underlying drained deformation (singularity of the drained acoustic tensor) is approached or just passed. Onset of growth has always a divergence growth character and critical conditions are always associated to the shortwavelength regime leading to the fact that the failure mode is expected to be a localized one.For non-associative behaviour of the skeleton we show in contrast that the onset of growth and unbounded growth may be defined either by the drained or undrained properties. One or the other depends on the details of the constitutive behaviour but also on the type of loadings. In particular, unbounded growth occurs when either the condition of localization under drained or undrained conditions is first passed. Transition from decaying to growing behaviour may have a divergence character or flutter-type character. Here the critical conditions are associated either to the shortwavelength or to the longwavelength regimes and therefore the failure mode may be localized or diffuse.The hierarchy between criticality of drained and undrained properties is analysed for a general class of constitutive equations and the results are fully and explicitly illustrated for saturated porous media with skeleton obeying Drucker-Prager like constitutive model.     

Perturbation methods for the analysis of the dynamic behavior of damaged plates

This paper presents an analytical method for the analysis of the dynamic behavior of damaged plates. The proposed approach allows the derivation of mode shapes and corresponding curvature modes for plates with various kinds of defects. Damage is modeled as a localized reduction in the plate thickness. Both point and line defects are considered to model notches or line cracks and delaminations in the plate. Small thickness reductions are considered so that the dynamic behavior of the damage plate can be analyzed through perturbations with respect to the undamaged modes. Results are presented to demonstrate the sensitivity of the curvature modes with respect to the considered low damage levels. Also, the curvature modes are used for the estimation of the strain energy of the plate and for the formulation of a damage index which can be used to provide damage location and extent information.     

Seismic behavior of insulated hollow-brick cavity walls using quasi-static experimentation

This paper focuses on an experimental investigation of different types of insulated hollow-brick cavity walls, i.e., walls with a door opening, with a window opening and without any opening, characterized with different tie bar arrangements, subjected to slowly applied cyclic loads. The cracking and damage patterns, strength and stiffness degradation and deterioration, energy dissipation capacity and hysteretic feature are analyzed. Based on the experimental results, the formulae of calculation for the cracking load and the ultimate load of cavity wall are mathematically established. The evaluation equation of strength and stiffness degradation of walls is presented and its parameters are numerically given from regression results. The original characteristic curves of recovery force-displacement of actual specimens under cyclic loads are discussed, and then a standard recovery force-displacement model is suggested with convenient forms for implementation.     

建筑结构显式拟静力推覆分析方法研究

基于结构的精细化非线性有限元模型和显式拟静力求解方法,提出了结构拟静力推覆分析方法(EQPA);为控制结构惯性效应,给出了加载模式和基于能量的评价方法。在自主研发的CPU+GPU并行有限元软件中开发实现EQPA,可有效克服大规模复杂结构隐式Push-Over分析中计算量大和收敛性差的问题。使用EQPA对某超高层剪力墙结构的抗震性能进行分析,并与三组人工地震动的简化动力增量分析(sIDA)对比。结果表明,EQPA的结构惯性效应得到有效抑制;EQPA耗时仅为sIDA耗时的4.97%;两种分析方法的结果有一定差异,但两者的结构能力谱曲线、层间变形规律、薄弱楼层和构件损伤规律较为接近。     

Near tip quasi-static crack growth behavior in strain hardening and softening material

Analyzed in this work is a semi-infinite crack that grows slowly in a steady-state. The assumed constitutive relation for the material permits strain hardening and softening as it is damaged in time. Four distinct regions divided angularly are identified for the asymptotic expressions of the quasi-static crack-tip stress field. They refer to material degraded in front of the crack; undergone elastic unloading; reloading of degraded material; and material completely by exhausted in its load carrying capacity.     

A quasi-static stability analysis for Biot's equation and standard dissipative systems

In this paper, an extended version of Biot's differential equation is considered in order to discuss the quasi-static stability of a response for a solid in the framework of generalized standard materials. The same equation also holds for gradient theories since the gradients of arbitrary order of the state variables and of their rates can be introduced in the expression of the energy and of the dissipation potentials. The stability of a quasi-static response of a system governed by Biot's equations is discussed. Two approaches are considered, by direct estimates and by linearizations. The approach by direct estimates can be applied in visco-plasticity as well as in plasticity. A sufficient condition of stability is proposed and based upon the positivity of the second variation of energy along the considered response. This is an extension of the criterion of second variation, well known in elastic buckling, into the study of the stability of a response. The linearization approach is available only for smooth dissipation potentials, i.e. for the study of visco-elastic solids and leads to a result on asymptotic stability. The paper is illustrated by a simple example.     

结构多点随机地震响应分析及拟静位移计算

给出了大跨度结构多点随机地震响应虚拟激励算法的若干实施细节,并讨论了将拟静位移项用自振振型和广义坐标来表达的做法。表明将拟静位移用自振振型和广义坐标来表达时,可能导致结构位移和内力响应非常大的误差,在实际应用时应予重视。     

Perturbation analysis for the postbuckling of rectangular orthotropic plates

In this paper, applying perturbation method to von Kármán-type nonlinear large deflection equations of orthotropic plates by taking deflection as perturbation parameter, thé postbuckling behavior of simply supported rectangular orthotropic plates under inplane compression is investigated. Two types of in-plane boundary conditions are now considered and the effects of initial imperfections are also studied. Numerical results are presented for various cases of orthotropic composite plates having different elastic properties. It is found that the results obtained are in good agreement with those of experiments.     

PERTURBATION ANALYSIS FOR WAVE EQUATION OF NONLINEAR ELASTIC ROD

The longitudinal oscillation of a nonlinear elastic rod with lateral inertia was studied. Based on the far field and simple wave theory, a nonlinear Schrodinger (NLS) equation was established under the assumption of small amplitude and long wavelength. It is found that there are NLS envelop solitons in this system. Finally the soliton solution of the NLS equation was presented.     

Formal solution of quasi-static problems

When the quasi-static problem is defined by a set of differential equations complemented by initial and boundary conditions, the resulting quasi-static solutions may exhibit a limited reach over the time domain. On the other hand, the infinity of equilibrium paths that can be obtained in a general non-linear problem also indicates that a proper definition of the quasi-static solution must be provided. In inelasticity problems, this infinite number of equilibrium paths occur even when no dissipative bifurcations are present. In the present paper, a general solution for quasi-static problems in Solid Mechanics is defined and explored. Special attention is addressed to material non-linearities though geometric non-linearities are also covered by the definition. Earlier concepts of path and state stability are recovered in order to reduce the number of solutions to those that are physically acceptable. The important link with the original dynamic problem is accounted for by enforcing a preferential load direction. The resulting definition relies on a time-objective criterion with straightforward applicability to the most common numerical models. In the final part of the paper, simple 1D problems are used to illustrate some of the concepts introduced in the present developments.     

Experimental analysis of the quasi-static and dynamic torsional behaviour of shape memory alloys

This paper investigates experimentally the quasi-static and dynamic torsional behaviour of shape memory alloys wires under cyclic loading. A specifically designed torsional pendulum made of a Ni–Ti wire is described. Results on the quasi-static behaviour of the wire obtained using this setup are presented, giving an overall view of the damping capacity of the material as function of the amplitude of the loading (imposed torsional angle), the frequency and the temperature. The dynamical behaviour is then presented through measured frequency response function between forcing angle at the top of the pendulum and the difference between top and bottom rotation angles in the vicinity of the first eigenfrequency of the wire, i.e. in the range [0.3 Hz, 1 Hz]. The softening-type non-linearity and its subsequent jump phenomenon, predicted theorically by the decrease of the effective stiffness when martensite transformation starts is clearly evidenced and analysed.     

基于J-C模型的镁合金MB2动静态拉伸破坏行为

为了研究不同应力状态和应变率条件下镁合金MB2的拉伸破坏行为,利用材料试验机和分离式Hopkinson拉杆(SHTB),对镁合金MB2的光滑及缺口圆柱试件进行了动静态拉伸加载;拟合得到了镁合金MB2的动静态拉伸本构关系,建立了其修正的Johnson-Cook失效破坏准则,并对不同试件的拉伸破坏行为进行了数值模拟;利用SEM对宏观破坏模式对应的微观损伤机理进行了分析。结果表明,随着应力三轴度的增加,镁合金MB2的等效破坏应变先增大后减小,宏观破坏模式由剪切转为正拉断,微观损伤机制由混合断裂转变为韧窝断裂;而随着应变率的增加,等效破坏应变不断减小,破坏模式不发生改变。Johnson-Cook本构关系和修正后的Johnson-Cook失效破坏准则能较好地拟合动态静态拉伸实验结果并预测不同试件的杯锥形破坏特征。

     

Theoretical analysis on quasi-static lateral compression of elliptical tube between two rigid plates

Based on the rigid plastic theory, the load-deflection functions with and without considering the effect of strain hardening are respectively derived for an elliptical tube under quasi-static compression by two parallel rigid plates. The non-dimensional load-deflection responses predicted by the present theory and the finite element simulations are compared, and the favorable agreement is found. The results show that strain hardening may have a noticeable influence on the load-deflection curves of an elliptical tube under quasi-static compression. Compared with the circular counterpart, the elliptical tube exhibits different energy absorption behavior due to the difference between the major axis and the minor axis. When loaded along the major axis of a slightly oval tube, a relative even and long plateau region of the load-deflection curve is achieved, which is especially desirable for the design of energy absorbers.     

Perturbation analysis for periodic heat transfer in radiating fins

A perturbation analysis is presented for periodic heat transfer in radiating fins of uniform thickness. The base temperature is assumed to oscillate around a mean value. The perturbation expansion is carried out in terms of dimensionless amplitude of the base temperature oscillation. The zero-order problem which is nonlinear, and corresponds to the steady state fin behaviour, is solved by quasilinearization. A method of complex combination is used to reduce both the first and the second order problems to two, coupled linear boundary value problems which are subsequently solved by a noniterative numerical scheme. The second-order term is composed of an oscillatory component with twice the frequency of base temperature oscillation and a time-independent term which causes a net change in the steady state values of temperature and heat transfer rate. Within the range of parameters used, the net effect is to decrease the mean temperature and increase the mean heat transfer rate. This is in constrast to the linear case of convecting fins where the mean values are unaffected by base temperature oscillations. Detailed numerical results are presented illustrating the effects of fin parameter N and dimensionless frequency B on temperature distribution, heat transfer rate, and time-average fin efficiency. The time-average fin efficiency is found to reduce significantly at low N and high B.

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Störungsanalyse für periodische Wärmeübertragung an Strahlungsrippen

Zusammenfassung Eine Störungsanalyse wird für periodische Wärmeübertragung in Strahlungsrippen gleicher Dicke vorgelegt. Die Fußtemperatur wird als um einen Mittelwert schwingend angenommen. Die Störungsentwicklung wird in Termen einer dimensionslosen Amplitude e dieser Schwingung angesetzt. Das Problem nullter Ordnung, das nichtlinear ist und dem stationären Verhalten der Rippe entspricht, wird durch Quasilinearisierung gelöst. Eine Methode der komplexen Kombination wird angewandt, um die Probleme erster und zweiter Ordnung auf zwei gekoppelte Grenzwertprobleme zu reduzieren, die nacheinander nach einem nichtiterativen Schema gelöst werden. Der Term zweiter Ordnung besteht aus einer Schwingungskomponente mit der doppelten Frequenz der Schwingung der Fußtemperatur und einem zeitunabhängigen Term, der eine Nettoänderung der stationären Werte der Temperatur und der Wärmeübertragung verursacht. Im verwendeten Bereich der Parameter tritt eine Abnahme der mittleren Temperatur und eine Zunahme der mittleren Wärmeübertragung auf. Das steht im Gegensatz zum linearen Fall der Konvektionsrippe, bei dem die Mittelwerte durch Schwingungen der Fußtemperatur nicht beeinflußt werden. Detaillierte numerische Ergebnisse zeigen die Einflüsse des Rippenparameters N und der dimensionslosen Frequenz B auf Temperatur Verteilung, Wärmeübertragung und zeitliches Mittel des Rippengütegrades. Dieses zeitliche Mittel nimmt merklich ab bei kleinem N und hohem B.

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Nomenclature b fin thickness - B dimensionless frequency, L²/ - E emissivity - f₀, f₁ functions of X - g₀, g₁, g₂ functions of X - h₀, h₁, h₂ functions of X - k thermal conductivity - L fin Length - N fin parameter, 2EL²T_bm/bk - q heat transfer rate - Q dimensionless heat transfer rate, qL/kbT_bm - t time - T temperature - T_b fin base temperature - T_S effective sink temperature - T_bm mean fin base temperature - x axial distance - X dimensionless axial distance, x/L - dimensionless amplitude of base temperature (s. Eq.2) - thermal diffusivity - instantaneous fin efficiency - time-average fin efficiency - _ss steady state fin efficiency - dimensionless temperature, T/T_bm - ₀ zero-order approximation - ₁ first-order approximation - ₂ second-order approximation - _2s steady component of ₂ - , ₁, ₂ constants - complex function of X - ₁ real part of - ₂ imaginary part of - complex function of X - ₁ real part of Y - ₂ imaginary part of - dimensionless time, t/L² - frequency of base temperature oscillation     

Nanocrystalline aluminum and iron: Mechanical behavior at quasi-static and high strain rates,and constitutive modeling

The responses of nanocrystalline aluminum powder of different grain sizes, prepared by ball milling and consolidated into bulk specimens by hot pressing, were determined under quasi-static and dynamic compression. The experiments demonstrated that the reduction in grain size resulted in several-fold increase in hardness and strength; the responses of nanocrystalline aluminum was found to be strain-rate-dependent. Using these measurements, Khan, Huang and Liang (KHL) viscoplastic model was modified by including a bi-linear Hall–Petch type relation to correlate with the response of nanocrystalline aluminum, including the variation of work hardening with grain size. The modified constitutive equation gives results very close to the experimental observations on nanocrystalline aluminum. In addition, the response of nanocrystalline iron, previously published, was also correlated with proposed model. Both correlations of strain-rate-dependent responses for different grain sizes were in good agreement with the experimental results over a wide range of grain sizes (micrometer to nanometer) and strain rates.     

Perturbation analysis of a continuous system for desalination by reverse osmosis

Blasius perturbation solutions are developed for the incompressible laminar boundary layer at high Schmidt numbers. The eigenvalues have been found to be multiples of 3/2 and the eigenfunctions can be expressed as confluent hypergeometric functions. The eigenfunctions are employed in analyzing a continuous desalination system. The reverse-osmosis boundary condition leads to an integral equation which is, in turn, solved by iterative techniques. The first-order calculations show that the results are most accurate for large ratios of osmotic to applied pressures. For medium ratios, the results are reasonably good.The present analysis should be useful, in general, for high Schmidt number flows with linear or nonlinear surface conditions provided that the transverse wall velocity is vanishingly small.     

Perturbation analysis of an undulating free surface in a multi-layered structure

This paper presents an analysis of the effect of an undulating free surface on the stress state in a multi-layered structure with an arbitrary number of layers. We present a simple procedure to derive in terms of stress functions the first-order perturbation solution for a multi-layered structure subjected to residual stresses and to external edge loads. The solution is valid for undulation amplitudes that are small compared to undulation wavelength, and it reproduces results reported earlier for homogeneous materials and bi-layers. Using this perturbation solution, the stability of a stressed undulating surface with a given surface energy is evaluated. This analysis leads naturally to the definition of a critical undulation wavelength above which undulations are preferred energetically over a flat surface. The results obtained in the perturbation analysis can be applied to surface diffusion/etching problems in multi-layered structures and to a range of other engineering problems related to undulating surfaces and surface stability.