The analysis of cyclically loaded creeping structures for short cycle times

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

A comparison between analytical calculations of the shakedown load by the bipotential approach and step-by-step computations for elastoplastic materials with nonlinear kinematic hardening

The class of generalized standard materials is not relevant to model the nonassociative constitutive equations. The bipotential approach, based on a possible generalization of Fenchel’s inequality, allows the recovery of the flow rule normality in a weak form of an implicit relation. This defines the class of implicit standard materials. For such behaviours, this leads to a weak extension of the classical bound theorems of the shakedown analysis. In the present paper, we recall the relevant features of this theory. Considering an elastoplastic material with nonlinear kinematic hardening rule, we apply it to the problem of a sample in plane strain conditions under constant traction and alternating torsion in order to determine analytically the interaction curve bounding the shakedown domain. The aim of the paper is to prove the exactness of the solution for this example by comparing it to step-by-step computations of the elastoplastic response of the body under repeated cyclic loads of increasing level. A reliable criterion to stop the computations is proposed. The analytical and numerical solutions are compared and found to be closed one of each other. Moreover, the method allows uncovering an additional ‘2 cycle shakedown curve’ that could be useful for the shakedown design of structure.     

The creep deformation of vibrating structures

In a recent paper [1] it was shown that the evaluation of certain bounding solutions for a structure subjected to cyclic loading was equivalent to assuming that the cycle time Δt was short compared with a stress redistribution time. Comparisons between values which are likely to occur in creep design situations indicated that Δt may often be assumed to be small and the bounding solution may be expected to closely approximate the actual stress history. In this paper the solution for the limiting case when Δt → 0 is evaluated for a class of constitutive relationships which may be expressed in terms of a finite number of state variables. Strain-hardening viscous, visco-elastic and Bailey-Orowan equations are discussed and particular solutions for which the residual stresses remain constant in time are derived. The solution for a non-linear visco-elastic model indicates that, for the stationary cyclic state, the constitutive equation need only predict the creep strain over a discrete number of cycles and need not predict the strains during a cycle. This observation should considerably simplify creep analysis.The solution of a simple example demonstrates the similarity between the predicting of the various constitutive relationships for isothermal problems. In fact they provide virtually identical solutions when expressed in terms of reference stress histories. The finite element solution of a plate containing a hole and subjected to variable edge loading is also presented for a viscous material. The solutions show behaviour which is similar to that of the two bar structure.     

Effects of thermo-mechanical coupling in a plastic cylinder

The paper describes results from numerical experiments on the effects of thermomechanical coupling in a polypropylene homopolymer plastic cylinder of infinite length. The specimen is subjected to cyclic straining of fixed amplitude and cycle time. Thermal boundary conditions, in which convective heat loss only is considered, cover the full range of boundaries, from the ideally thermally insulated to the ideally convective case. The temperature change distributions obtained from the model are in accordance with expected behaviour and are quanlitatively confirmed by preliminary industrial tests.     

Consistency between independence theorems and generalized self-consistent method

Recently Zheng & Hwang established a series of independence theorems concerning with planar effective elastic properties. It is manifested that the estimation of the effective elastic properties of microcracked solids through the generalized self-consistent method (GSCM) contradicts with these independence theorems. In this paper it is shown that such contradiction is actually caused by the approximate algorithm adopted, while the exact solution of GSCM is consistent with these rigorously established independence theorems. Since only an approximate algorithm in GCSM is available in dealing with problems involving non-circular inclusions or holes, an intrinsic GSCM is proposed, which can be performed based on an approximate algorithm and the corresponding estimations are consistent with the independence theorems.     

Non-linear boundary value problems for the circular membrane

Existence and uniqueness theorems are proved for two boundary value problems for the axisymmetric deformation of a circular membrane subjected to normal pressure. The nonlinear Föppl membrane theory is employed. The shooting method is used to establish these results. It is also shown that if the edge is free to move in the plane of the membrane then a necessary and sufficient condition for the existence of a unique solution is that the pressure is self-equilibrating.     

Dynamic response of an elastic medium containing crack

A fundamental solution for an infinite elastic medium containing a penny-shaped crack subjected to dynamic torsional surface tractions is attempted. A double Laplace–Hankel integral transform with respect to time and space is applied both to motion equation and boundary conditions yielding dual integral equations. The solution of the derived dual integral equations is based on an analytic procedure using theorems of Bessel functions and ordinary differential equations. The dynamic displacements’ field is obtained by inversion of the corresponding Laplace–Hankel transformed variable. Results of a representative example for a crack subjected to pulse surface tractions are obtained and discussed.     

A semi-analytical elastic solution for stress field of lined non-circular tunnels at great depth using complex variable method

In this paper, a semi-analytical elastic plane strain solution was provided for stress field around a lined non-circular tunnel subjected to uniform ground load. Concrete lining and the surrounding rock mass were assumed as linearly elastic materials. Due to complexity of the problem for non-circular geometric configurations, complex variable method introduced by Muskhelishvili and conformal mapping functions were used to determine stress components within concrete lining and the surrounding rock mass. Finally, the solution was validated by ABAQUS finite element software through an example. Very good agreement was demonstrated between semi-analytical and numerical solution although some discrepancies were found at tunnel corners where large curvature existed. It was demonstrated that the solution predicted stress components more accurately around the tunnels, especially the corners with large stress concentration. Practical significance of the solution was placed in the fact that it could be used as a quick-solver with high accuracy.     

The stationary solution of a random dynamical model

This paper studies the stationary probability density function (PDF) solution of a nonlinear business cycle model subjected to random shocks of Gaussian white-noise type. The PDF solution is controlled by a Fokker–Planck–Kolmogorov (FPK) equation, and we use exponential polynomial closure (EPC) method to derive an approximate solution for the FPK equation. Numerical results obtained from EPC method, better than those from Gaussian closure method, show good agreement with the probability distribution obtained with Monte Carlo simulation including the tail regions.     

Heat transfer in vertical annular laminar flow of two immiscible liquids

The paper investigates heat transfer in annular laminar undisturbed flow of two immiscible liquids, with constant heat-flux generated at the wall of the tube. It presents an analytical solution for the fully developed temperature field. This is used to obtain a more general solution from a model, describing the temperature field as a superposition of the fully developed and the developing fields. This superposition model is solved by an orthogonal collocation method. An asymptotic model for short entry lengths is also described. Calculations for a kerosene-water system, show that the superposition solution converges to the entrance solution below 100 diameters and converges asymptotically to the solution of the fully developed temperature field beyond 5000 diameters. The effect of the wavy interface is assessed experimentally for annular kerosene-water flow, by comparing predicted and measured temperature profiles. It is found that experimental profiles are considerably flatter and measured Nusselt numbers for the kerosene phase are accordingly higher by 40–320% as compared to the undisturbed flow analyses.     

Transient thermoelastoplastic stress analysis for a hollow sphere using the incremental theory of plasticity

An analysis based on the incremental strain theory is formulated for solving the problem of an elastoplastic hollow sphere subjected to a transient temperature distribution. Thermal and material properties are assumed to be temperature dependent and the behaviour of the medium to be characterized by the Ramberg-Osgood stress-strain relation. A method of successive elastic solutions is used to obtain a numerical solution. An illustrative example shows that the effective stress is not a monotonie function of the radius, but is much dependent on the history, gradient, and distribution of the temperature in the hollow sphere. In addition, unloading in the plastically deformed region is confirmed from the detailed discussion on the distribution of strains. As a result, the analysis based on the total strain theory is not permissible for solving this kind of elastoplastic problems subjected to transient thermal loading. In the following analysis the problem is treated in a quasi-static sense and the inertia terms in the thermoelastoplastic equations are neglected.     

Probabilistic character of the microcrack growth in brittle layered materials

Characteristics of microcrack initiation, multiplication and saturation in layered materials are discussed. A probabilistic-analytical method, the ‘characteristic curve method (CCM)’ is developed to correlate the initial defects and the microcrack evolution under static and cyclic loadings. The ‘equivalent applied loading’ and the ‘equivalent crack density’ concepts are introduced to describe different microcrack multiplication features in different layered materials. Microcrack multiplication processes in many layered materials with brittle matrices subjected to static and cyclic loadings can be easily predicted.     

Solution for the thermoelastic problem in vertically inhomogeneous media

The fundamental transient-thermoelastic problem with body forces and a heat source in vertically inhomogeneous media is investigated by a method presented in this paper. The basic equations in Fourier transforms and Laplace transform are obtained in the form of two sets of first order linear ordinary differential equations inz, Eq. (7). Furthermore, forN-layered media, the general solution in the transformed spaces of thej-th layer is given for fully connected interface between layers, Eq. (11). Finally, under general condition, a closed-form solution for the quasi-static transient displacements, stresses, temperature in the body can be obtained by the convolution theorems for the two integral transforms. In the final solution, the Green's functions can be expressed in terms of Hankel transforms of order zero and unity as well as inverse Laplace transform, and come out rather neatly. Comprehensive Institute of Geotechnical Investigation and Surveying, Ministry of Urban and Rural Construction and Environmental Protection     

Thermo-mechanical measurement of elasto-plastic transitions during cyclic loading

The surface temperature of stainless steel SS304 low cycle fatigue specimens subjected to cyclic loading was studied using infrared thermography technique. The thermal data mapped onto the various stages of cyclic stress-strain curve shows the ability of these measurements to identify the yield points in both the compression and tension loading. Based on the results of this study, it is possible to identify the state of stress for materials such as elastic tension, plastic tension, elastic compression, plastic compression during cyclic loading using infrared thermographic data. The thermo-elastic slope and thermo-plastic slope was observed to be dependent on the prior loading cycles.     

Variational methods for the steady state response of elastic–plastic solids subjected to cyclic loads

Solids (or structures) of elastic–plastic internal variable material models and subjected to cyclic loads are considered. A minimum net resistant power theorem, direct consequence of the classical maximum intrinsic dissipation theorem of plasticity theory, is envisioned which describes the material behavior by determining the plastic flow mechanism (if any) corresponding to a given stress/hardening state. A maximum principle is provided which characterizes the optimal initial stress/hardening state of a cyclically loaded structure as the one such that the plastic strain and kinematic internal variable increments produced over a cycle are kinematically admissible. A steady cycle minimum principle, integrated form of the aforementioned minimum net resistant power theorem, is provided, which characterizes the structure’s steady state response (steady cycle) and proves to be an extension to the present context of known principles of perfect plasticity. The optimality equations of this minimum principle are studied and two particular cases are considered: (i) loads not exceeding the shakedown limit (so recovering known results of shakedown theory) and (ii) specimen under uniform cyclic stress (or strain). Criteria to assess the structure’s ratchet limit loads are given. These, together with some insensitivity features of the structure’s alternating plasticity state, provide the basis to the ratchet limit load analysis problem, for which solution procedures are discussed.     

Shakedown analysis for a class of strengthening materials within the framework of gradient plasticity

The classical shakedown theory is extended to a class of perfectly plastic materials with strengthening effects (Hall–Petch effects). To this aim, a strain gradient plasticity model previously advanced by Polizzotto (2010) is used, whereby a featuring strengthening law provides the strengthening stress, i.e. the increase of the yield strength produced by plastic deformation, as a degree-zero homogeneous second-order differential form in the accumulated plastic strain with associated higher order boundary conditions. The extended static (Melan) and kinematic (Koiter) shakedown theorems are proved together with the related lower bound and upper bound theorems. The shakedown limit load problem is addressed and discussed in the present context, and its solution uniqueness shown out. A simple micro-scale structural system is considered as an illustrative example. The shakedown limit load is shown to increase with decreasing the structural size, which is a manifestation of the classical Hall–Petch effects in a context of cyclic loading.     

Improved preconditioned conjugate gradient method and its application in F. E. A. for engineering

In this paper two theorems with theoretical and practical significance are given in respect to the preconditioned conjugate gradient method(PCCG).The theorems discuss respectively the qualitative property of the iterative solution and the construction principle of the iterative matrix.The authors put forward a new incompletely LU factorizing technique for non-M-matrix and the method of constructing the iterative matrix.This improved PCCG is used to calculate the ill-conditioned problems and large-scale three-dimensional finite element problems,and simultaneously contrasted with other methods.The abnormal phenomenon is analyzed when PCCG is used to solve the system of ill-conditioned equations,It is shown that the method proposed in this paper is quite effective in solving the system of large-scale finite element equations and the system of ill-conditioned equations.     

Two-dimensional thermoelastic problem of an infinite magneto-microstretch homogeneous isotropic plate

In this work, the magneto-thermoelastic problem of an infinite microstretch homogeneous isotropic plate placed in a transverse magnetic field is studied in the context of different theories of generalized thermoelasticity. The upper surface of the infinite plate is subjected to a zonal time-dependent heat shock. The problem is investigated by applying finite element method. The solution is obtained by solving finite element governing equations of the problem in time domain directly. The results, including temperature, stresses, displacements, microrotation, microstretch, induced magnetic field, and induced electric field, are presented graphically. Comparison is made in the results predicted by different theories of generalized thermoelasticity, to show that the micropolar effect has a slight influence on the results while the microstretch effect has a great influence on the results. Finally, a parameter study provides an idea about the influence of the respective terms of the theories.     

高温合金材料循环相关热机械疲劳寿命预测

在变温非线性运动强化规律所描述的高温合金材料热机械寿命应力－应变循环特性的基础上，讨论了应变控制的循环相关热机械疲劳寿命预测技术，所建模型采用了由应变以密度表示的损伤参数，并且引入了温度损伤系数，考虑了温度变化范围以及温度循环和应变循环相位关系对疲劳寿命的影响，在确定模型的一些参数，采用等温力学试验和疲劳试验的数据，为了把等温疲劳研究成果推广到变温疲劳分析领域，开辟了新的途径。