Delayed Fracture of an Aging Viscoelastic Composite under Plane Strain

The theory of delayed fracture of cracked viscoelastic bodies and the method of continued fractions are used to study the problem on the delayed fracture of an unidirectional fibrous composite subjected to uniaxial tension under plane-strain conditions. The aging viscoelastic binder contains a mode I macrocrack, which is parallel to the fibers. The results of a numerical crack-resistance analysis of a composite whose aging properties are described by the Volterra operator with the Maslov–Arutyunyan kernel are presented. A convergence analysis is made of the expansion of an irrational function of the resolvent Volterra operator into a continued fraction.     

Viscoelastic Deformation of a Reinforced Plate with a Crack

The deformation of a viscoelastic reinforced composite is studied. The composite has an axis of elastic symmetry and consists of transversally isotropic fibers and a viscoelastic matrix, which differ by the volume concentration and mechanical characteristics. The material is modeled by a transversely isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. A plate fabricated from the composite in question is weakened by a through mode I crack and is subjected to constant tensile forces. The viscoelastic properties of the matrix material are described by a convolution operator. The Volterra principle is used to derive expressions for the viscoelastic characteristics and crack opening. The irrational function of the integral operator that describes the crack opening is expanded into an operator continued fraction and is represented as the sum of base operators     

The theory of static decay in computational mechanics

In this paper,a new mathematical form,matrix,continued fraction(MCF)isintroduced to describe the decay of effects of an equilibrant system of forces acting on asphere of an elastic body.By this way,the famous Saint-Venant’s principle is proved oftenbut not always valid in computational mechanics.     

A Method for Determining the Viscoelastic Characteristics of Composites

An efficient method for determining the deformation function of a composite is discussed. The method is based on a fractional exponential representation of the deformation functions of the composite components. The viscoelastic solution is obtained using the Volterra principle. The deformation function is represented as a function of a base operator. Thus, the problem is solved by approximating the deformation function by a continued fraction and applying the method of operator continued fractions. A computational procedure is detailed and illustrated using data on longitudinal relaxation of polymethylmethacrylate. As an example, the deformation of a polymethylmethacrylate-based fibrous composite with viscoelastic properties is analyzed__________Translated from Prikladnaya Mekhanika,Vol. 41, No. 5, pp. 9–21, May 2005.     

基于改进遗传算法的桥梁结构损伤识别应用研究

提出了一种基于残余力向量和改进遗传算法的桥梁结构损伤识别方法。在无噪声的情况下,使用任意一阶模态数据,残余力向量法都能够对损伤进行准确定位。但是,振动测试数据中往往包含噪声,导致运用残余力向量法进行损伤识别完全不可行。考虑到这个问题,在常规模态分析的基础上,以节点的残余力向量构造目标函数,提出了一种用于遗传搜索优化的目标函数形式。利用改进遗传算法重点进行了噪声条件下的结构损伤定位和定量研究,并对遗传算法的参数选取问题进行了深入探讨。遗传算法的主要改进包括:采用浮点编码、采用基于标准化几何分布排名的选择策略、采用最优保存策略、采用算术交叉算子、采用自适应变异算子。最后,本文用一个连续梁桥模型进行了数值模拟,验证了所提出方法的有效性,并对方法应用中存在的一些问题进行了深入探讨。     

Stable growth of penny-shaped crack in viscoelastic composite material under time-dependent loading

Considered is the long-term cracking of the three-dimensional fiber-reinforced viscoelastic composite with a plane penny-shaped crack under time-dependent loading. The composite has a hexagonal structure and consists of elastic isotropic fibers and viscoelastic isotropic matrix. The material is modeled by transversally isotropic homogeneous linearly viscoelastic medium with some averaged characteristics. The crack propagation planecoincides with the plane of isotropy. A ring-shaped yield zone in front of the moving crack is modeled as a Dugdale's zone with time-dependent stresses. Crack growth under deformation of the composite occurs by application of a slowly increasing tensile load; it is normal to the plane of crack propagation. A convolution-type time operator describes the viscoelastic properties of the matrix material. Use is made of the Volterra principle and the theory of long-term cracking of viscoelastic bodies. The irrational function of integral operator associated with the viscoelastic crack opening expression is expanded into a continued fraction of operators. The solution is reduced to the nonlinear integral equations of crack growth. Numerical results are obtained for a specific material. Crack growth kinetics is discussed in connection with the onset of stable crack growth and crack border stress intensity factor.     

Subcritical growth of a crack in a transversally isotropic aging composite under long-term static loading

The problem of long-term fracture of an aging reinforced composite array having hexagonal symmetry and weakened by a flat circular macrocrack is considered based on the Boltzmann-Volterra principle and the theory of long-term fracture of viscoelastic bodies. The array is under the action of stationary tensile forces applied at infinity and normal to the crack plane. The Maslov-Arutyunyan operator is used to describe the aging strain properties of the array. The irrational functions of integral Volterra operators obtained during the solution are determined by expanding them into continued fractions. The crack growth equations derived are numerically solved for a specific material (ferroconcrete). Curves of the rupture life of the composite array, kinetics of crack growth, and safe loading are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 12, pp. 72–79, December, 1999.     

A Multiscale Theory of Swelling Porous Media: II. Dual Porosity Models for Consolidation of Clays Incorporating Physicochemical Effects

A three-scale theory of swelling clay soils is developed which incorporates physico-chemical effects and delayed adsorbed water flow during secondary consolidation. Following earlier work, at the microscale the clay platelets and adsorbed water (water between the platelets) are considered as distinct nonoverlaying continua. At the intermediate (meso) scale the clay platelets and the adsorbed water are homogenized in the spirit of hybrid mixture theory, so that, at the mesoscale they may be thought of as two overlaying continua, each having a well defined mass density. Within this framework the swelling pressure is defined thermodynamically and it is shown to govern the effect of physico-chemical forces in a modified Terzaghi's effective stress principle. A homogenization procedure is used to upscale the mesoscale mixture of clay particles and bulk water (water next to the swelling mesoscale particles) to the macroscale. The resultant model is of dual porosity type where the clay particles act as sources/sinks of water to the macroscale bulk phase flow. The dual porosity model can be reduced to a single porosity model with long term memory by using Green's functions. The resultant theory provides a rational basis for some viscoelastic models of secondary consolidation.     

精细积分法在迷宫密封转子系统非线性动力学特性计算中的应用

基于Muszynska密封力模型,建立了迷宫密封转子系统的非线性动力学模型,将精细积分法推广应用于非线性情况,计算了迷宫密封不平衡转子系统的动力学特性,依据Floquet理论讨论其分岔特性。研究表明：在2^N类算法计算指数矩阵基础上提出的精细积分法和传统的数值计算方法相比,其精度高,在分析中通过取不同步长计算对比,表明该方法在某些情况下可以采取较大时间步长,有效提高了计算速度。     

On the application of branched operator continued fractions for a boundary problem of linear viscoelasticity

In solving linear viscoelastic problems for composite materials, the problem arises of representing a multivariable operator function. To resolve this problem, the method of operator continued fractions is generalized to the case of a multivariable operator function. The method is based on the theory of branched continued fractions. Branched operator continued fractions are considered. Using the convolution theorem, fractions can be represented in terms of operators of basic class. This representation makes it possible to effectively solve boundary problems of linear viscoelasticity Published in Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 133–143, January 2006.     

From Boltzmann’s Equation to the Incompressible Navier–Stokes–Fourier System with Long-Range Interactions

We establish a rigorous demonstration of the hydrodynamic convergence of the Boltzmann equation towards a Navier–Stokes–Fourier system under the presence of long-range interactions. This convergence is obtained by letting the Knudsen number tend to zero and has been known to hold, at least formally, for decades. It is only more recently that a fully rigorous mathematical derivation of this hydrodynamic limit was discovered. However, these results failed to encompass almost all physically relevant collision kernels due to a cutoff assumption, which requires that the cross sections be integrable. Indeed, as soon as long-range intermolecular forces are present, non-integrable collision kernels have to be considered because of the enormous number of grazing collisions in the gas. In this long-range setting, the Boltzmann operator becomes a singular integral operator and the known rigorous proofs of hydrodynamic convergence simply do not carry over to that case. In fact, the DiPerna–Lions renormalized solutions do not even make sense in this situation and the relevant global solutions to the Boltzmann equation are the so-called renormalized solutions with a defect measure developed by Alexandre and Villani. Our work overcomes the new mathematical difficulties coming from the consideration of long-range interactions by proving the hydrodynamic convergence of the Alexandre–Villani solutions towards the Leray solutions.     

Development and implementation of a beam theory model for shape memory polymers

In this paper we focus on the development of a beam theory for a small strain continuum model of thermoviscoelastic shape memory polymers (SMP). Rather than a history integral model that is common for viscoelastic materials, a thermodynamically based state evolution model developed by Ghosh and Srinivasa (2011a) is used as the basis for the beam model based on the Euler–Bernoulli beam theory. An example of a three-point bend test is simulated using the beam theory model. The numerical solution is implemented by using an operator split technique that utilizes an elastic predictor and dissipative corrector. The key idea is that the elastic predictor is based on the solution to a beam theory boundary value problem while the dissipative corrector is entirely local (and hence can be parallelized) and is applied by considering the beam as a two or three dimensional body. This enables a very rapid solution of the problem yet maintaining fidelity of the distribution of inelastic strains across the cross-section. A displacement based convergence criterion is used in each time step. This algorithm is validated by using a three-point bending experiment for three different material cases: elastic, plastic and thermoplastic response. Time step convergence and mesh density convergence studies are carried out for the thermoviscoelastic FEM model. Finally, we implement and study this model for a SMP beam undergoing three-point bending strain recovery and stress recovery thermomechanical loading.     

Finite element analysis of functionally graded plates for coupling effect of extension and bending

In this paper, the coupling effect of extension and bending in functionally graded plate subjected to transverse loading for Kirchhoff-Love plate theory equations is studied. The material properties of the FG plates are assumed to vary continuously throughout the thickness direction of the layer according to sigmoid distribution of the volume fractions of constituents. The two plate functionals are used which are developed by Gateaux differential and potential operator concept. A layer wise, isoparametric, mixed finite element approach was used and results of two different quadrilateral elements, one considering the membrane forces and the other one not, were compared by an analytical study. Finally, for different composition profiles the effect of variations of the Young’s moduli and of variations volume fraction index to dimensionless displacement, strain and stress values are studied.     

Error estimate for influence of model reduction of nonlinear dissipative autonomous dynamical system on long-term behaviours

Introduction Forsomenonlineardynamicalsystems,especiallyincomplexmechanicalvibrationand fluid structureinteractionsystems,thefiniteelementmethodisoftenusedtoapproachthe solutionofthegoverningequationduetothedifficultyofdescribingsysteminanalyticalform.Asa…     

分离变量法与哈密尔顿体系

数学物理与力学中用分离变量法求解偏微分方程经常导致自共轭算子的sturmLiouville问题,在此基础上而得以展开求解。然而在应用中有大量问题并不能导致自共轭算子。本文通过最小势能变分原理,选用状态变量及其对偶变量,导向一般变分原理。利用结构力学与最优控制的模拟理论,导向哈密尔顿体系。将有限维的理论推广到相应的哈密尔顿算子矩阵及共轭辛矩阵代数的理论。拓广了经典的分离变量法,证明了全状态本征函数向量的共轭辛正交归一性质及按本征函数向量展开的理论。以条形板为例,说明了应用。     

Dynamics of a Periodically Pulsed Bio-Reactor Model With a Hydraulic Storage Zone

In this paper, we investigate a periodically pulsed bio-reactor model of a flowing water habitat with a hydraulic storage zone in which no flow occurs. The full system can be reduced to a limiting system based on a conservation principle. Then we obtain sufficient conditions in terms of principal eigenvalues for the persistence of single population and the coexistence of two competing populations for the limiting system by appealing to the theory of monotone dynamical systems. Finally, we use the theory of chain transitive sets to lift the dynamics of the limiting system to the full system.     

On the derivation of some fundamental expressions for the average stress tensor in systems of interacting particles

The so-called generalized Kramers-Kirkwood expression for the average stress tensor of a system of interacting point particles, derived by Bird and Curtiss on using a phase-space-kinetic formalism has been reconsidered from different points of view. First a derivation based upon volume averaging is discussed, and after that a derivation based upon a virtual work principle. The latter approach offers the possibility of distinguishing reversible (including thermodynamic and Brownian) and dissipative forces and stresses by using a projection operator, associated with the constraints of the system.     

基于高斯原理的非理想系统动力学建模

高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则, 它可以使得多体系统动力学问题不需通过求解微分(代数)方程, 而是采用求解最小值的优化方法来解决, 从而提供了一种适用于优化算法的建模思路, 因此, 如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提. 对于理想系统而言, 约束对系统的作用可以通过约束方程来体现, 故高斯拘束可表达为系统质点加速度的函数, 系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题; 当系统中需要考虑干摩擦等非理想因素时, 部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述, 这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性. 基于变分类的高斯原理, 推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理, 针对目前文献中用于非理想系统的高斯原理进行了讨论, 指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式, 当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时, 该形式失效; 同时根据文中的极值原理, 得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型. 例子中分析了优化模型及相应的线性互补性模型的关系, 分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件, 从而证明了文中优化模型的可靠性; 并采用优化计算方法进行了动力学模拟, 模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.      

DYNAMIC MODELING OF NONIDEAL SYSTEM BASED ON GAUSS'S PRINCIPLE$^{\bf 1)}$

高斯原理给出了通过求函数极值、从可能运动中鉴别出真实运动的规则, 它可以使得多体系统动力学问题不需通过求解微分(代数)方程, 而是采用求解最小值的优化方法来解决, 从而提供了一种适用于优化算法的建模思路, 因此, 如何定义恰当的高斯拘束函数是动力学优化方法得以实现的前提. 对于理想系统而言, 约束对系统的作用可以通过约束方程来体现, 故高斯拘束可表达为系统质点加速度的函数, 系统的动力学问题因此可以描述为目标函数为高斯拘束函数、优化变量为质点加速度的约束最优化问题; 当系统中需要考虑干摩擦等非理想因素时, 部分相互作用不能被所定义的约束方程所涵盖而需要采用额外的物理规律来描述, 这种相互作用破坏了原有的针对理想系统的高斯拘束函数的极值特性. 基于变分类的高斯原理, 推导并证明了目标函数以理想约束力所表达的非理想系统的极值原理, 针对目前文献中用于非理想系统的高斯原理进行了讨论, 指出其实际为文中的极值原理在非理想约束力与理想约束力无明显关联时的一种特殊表达形式, 当非理想约束力与理想约束力有明显的函数关系(如库仑摩擦定律中滑动摩擦力与法向约束力间的线性关系)时, 该形式失效; 同时根据文中的极值原理, 得到了考虑库仑摩擦时非理想的多体系统动力学问题的优化模型. 例子中分析了优化模型及相应的线性互补性模型的关系, 分析发现在满足刚体滑动问题的唯一性条件下二者互为充分必要条件, 从而证明了文中优化模型的可靠性; 并采用优化计算方法进行了动力学模拟, 模拟结果显示了将高斯原理与优化算法相结合的可行性及有效性.     

浮式垂直轴风机的动力学建模、仿真与实验研究

考虑气动力和水动力的耦合研究浮式垂直轴风机系统的运动响应,将固定式垂直轴风机的气动载荷计算方法进一步推广到海上浮式垂直轴风机的气动载荷计算.考虑阻尼力、波浪力、风载荷、系泊力等,建立了浮式垂直轴风机系统的纵荡-垂荡-纵摇运动方程.考虑动态失速和浮式基础运动,基于双致动盘多流管理论,推导了风机叶片气动载荷计算公式,编制了数值计算程序.以Sandia 17 m风机为例,验证了气动载荷计算程序的正确性.最后进行了模型实验,其中模型的风机为Φ型达里厄垂直轴风机,支撑基础为桁架式Spar型浮式基础,将模型实验结果与数值计算结果进行了对比,验证了耦合计算程序.结果表明,数值计算得到的风机系统的垂荡、纵摇运动的RAO(幅值响应算子)曲线与模型实验结果吻合较好,验证了耦合程序的正确性.然而,由于数值计算与模型实验在运动自由度、阻尼、风载荷等方面存在差别,数值计算结果与模型实验结果仍有一定的差异.