Fixed point theorems in probabilistic inner product spaces

The purpose of this paper is to introduce a new maditying detinition for probabilistie inner product space, and to establish seveeral new fixed point theorems for mappings on such kind of spaces, As an example of applications. we utilize the results of this paper to study the existence and uniqueness of solution of Uryson’s integral equation in IV.     

Integral constitutive equations of elastic-plastic materials

In this paper the integral constitutive equations of elastic-plastic materials are studied. The endochronic theory can be deduced from this theory. It is shown that the endochronic should be selected compatible with the yield function of the calssical plasticity and this can be considered as a principle of selecting endochronic. Applying this principle the appropriate endochronics of the plastically compressible materials and the orthotropic materials are derived. The second approximate theory of the integral constitutive equation is also discussed in this paper.This paper was reported by the National Natural Science Foundation of China.     

Exact integration of constitutive equations of kinematic hardening materials and its extended applications

In this paper, an exact formula for the integration of the constitutive equations of kinematic hardening material is presented. Its algorithms are simple and clear. For isotropic hardening or mixed hardening material, the formula is still an exact solution for the case of radial loading, and it is an approximate solution with reasonable accuracy for the case of non-radial loading. The computation results show that the procedure proposed in this paper improves both accuracy and efficiency of numerical integration schemes adopted widely in elastic-plastic finite element analysis.     

On instabilities of single-integral constitutive equations for viscoelastic liquids

Various types of instabilities are exposed in this paper for time-strain separable single-integral viscoelastic constitutive equations (CE's). They were distinguished into two groups and defined as Hadamard and dissipative type of instabilities. As for the Hadamard-type, previously obtained criteria are found to be necessary only. They are necessary and sufficient only for thermodynamic stability. Improved, stricter Hadamard stability criteria are described briefly in this paper, and then applied to study of stability of several CE's. It is shown that the Currie potential with the K-BKZ equation and the model proposed by Papanastasiou et al. are Hadamard unstable. In the case of dissipative stability, the necessary and sufficient condition for stress boundedness in any regular flow with a given history, is proved. Then, this criterion was applied to the neoHookean, Mooney, and Yen and McIntire specifications of the general K-BKZ model, to exhibit unbounded solutions. In addition, Larson-Monroe potential which is later proved to be Hadamard unstable but satisfies the above criterion of boundedness, is shown to have unstable decreasing branch in steady simple shear flow. At present, to the authors' knowledge, there is no viscoelastic single-integral CE of factorable type proposed in the literature which can satisfy all the Hadamard and dissipative stability criteria.     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part II

Summary From the continuum mechanics points of view, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity in a continued work (Part III, forthcoming).The supports from the Alexander von Humboldt Foundation, Germany and the Research Foundation of Tsinghua University, P. R. China are acknowledged by the first author.     

Asymptotic integration of elasto-plastic constitutive relation for isotropic hardening material and its computational precision

In this paper, an asymptotic expansion solution of the constitutive equation of hardening materials is presented. Its 1st asymptotic integration can give an approximate one with good enough accuracy and the second asymptotic one improves the precision of solutions further. The steps of its algorithms are fairly simple and clear, and its computational workload is considerably reduced. It can be easily incorporated into a general purpose finite element program.The Chinese original of this article was published in the Chinese edition ofActa Mechanica Solida Sinica, No. 1, 1986.     

Alternative derivation of differential constitutive equations of the Oldroyd-B type

Now that almost 60 years have passed since the pioneering works of J.G. Oldroyd it seems appropriate as an homage to consider here constitutive equations that can be viewed as generalisations of the by now classical Oldroyd-B model. In this short communication we shall address heuristically the theme of differential constitutive models and will provide an alternative way of deriving a “modified FENE” equation (FENE-M) and inter-relating the PTT and FENE-P-like models.     

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY(Ⅰ)-NONLINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The nonlinear constitutive equations and field equations of unsaturated soils were constructed on the basis of mixture theory. The soils were treated as the mixture composed of three constituents. First, from the researches of soil mechanics, some basic assumptions about the unsaturated soil mixture were made, and the entropy inequality of unsaturated soil mixture was derived. Then, with the common method usually used to deal with the constitutive problems in mixture theory, the nonlinear constitutive equations were obtained. Finally, putting the constitutive equations of constituents into the balance equations of momentum, the nonlinear field equations of constituents were set up. The balance equation of energy of unsaturated soil was also given, and thus the complete equations for solving the thermodynamic process of unsaturated soil was formed. Foundation items: the National Natural Science Foundation of China (59678003); Special Research Plan of the Education Department of Shaanxi Province (01JK178) Biographies: HUANG Yi (1936-) ZHANG Yin-ke (1964-)     

On plastic anisotropy of constitutive model for rate-dependent single crystal

An algorithm for single crystals was developed and implemented to simulate plastic anisotropy using a rate-dependent slip model. The proposed procedure was a slightly modified form of single crystal constitutive model of Sarma and Zacharia. Modified Euler method, together with Newton-Raphson method was used to integrate this equation which was stable and efficient. The model together with the developed algorithm was used to study three problems. First, plastic anisotropy was examined by simulating the crystal deformation in tension and plane strain compression, respectively. Secondly, the orientation effect of some material parameters in the model and applied strain rate on plastic anisotropy for single crystal also is investigated. Thirdly, the influence of loading direction on the active slip system was discussed.     

Differential constitutive equations of incompressible media with finite deformations

A method is proposed for constructing a system of constitutive equations of an incompressible medium with nonlinear dissipative properties with finite deformations. A scheme of the mechanical behavior of a material is used, in which the points are connected by horizontally aligned elastic, viscous, plastic, and transmission elements. The properties of each element of the scheme are described with the use of known equations of the nonlinear elasticity theory, the theory of nonlinear viscous fluids, and the theory of plastic flow of the material under conditions of finite deformations of the medium. The system of constitutive equations is closed by equations that express the relation between the deformation rate tensor of the material and the deformation rate tensor of the plastic element. Transmission elements are used to take into account a significant difference between macroscopic deformations of the material and deformations of elements of the medium at the structural level. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 158–170, May–June, 2009.     

Flow effects of blood constitutive equations in 3D models of vascular anomalies

The effects of different blood rheological models are investigated numerically utilizing two three‐ dimensional (3D) models of vascular anomalies, namely a stenosis and an abdominal aortic aneurysm model. The employed CFD code incorporates the SIMPLE scheme in conjunction with the finite‐volume method with collocated arrangement of variables. The approximation of the convection terms is carried out using the QUICK differencing scheme, whereas the code enables also multi‐block computations, which are useful in order to cope with the two‐block grid structure of the current computational domain. Three non‐Newtonian models are employed, namely the Casson, Power‐Law and Quemada models, which have been introduced in the past for modelling the rheological behaviour of blood and cover both the viscous as well as the two‐phase character of blood. In view of the haemodynamical mechanisms related to abnormalities in the vascular network and the role of the wall shear stress in initiating and further developing of arterial diseases, the present study focuses on the 3D flow field and in particular on the distribution as well as on both low and high values of the wall shear stress in the vicinity of the anomaly. Finally, a comparison is made between the effects of each rheological model on the aforementioned parameters. Results show marked differences between simulating blood as Newtonian and non‐Newtonian fluid and furthermore the Power‐Law model exhibits different behaviour in all cases compared to the other models whereas Quemada and Casson models exhibit similar behaviour in the case of the stenosis but different behaviour in the case of the aneurysm. Copyright © 2005 John Wiley & Sons, Ltd.     

各向异性本构关系在板料成形数值模拟中的应用

对几种能表达面内各向异性的屈服准则Hill、Barlat-Lian、Barlat进行了比较。以弹性变形服从各向同性广义虎克定律的情况下，给出了基于张量算法推导的弹塑性本构关系的一般表达式，并由此导出了相应屈服准则的弹塑性本构关系的显式表达。借助ABAQUS软件本构模块用户子程序接口，分别实现了这些屈服准则在ABAQUS的嵌入。以模拟方形盒的拉延过程为例，分析了不同的屈服准则在板料成形过程数值模拟中的应用。模拟结果表明，基于弹塑性本构关系一般表达所列出的相应屈服准则的显式表达式是正确的；在采用壳元来模拟板料成形时，采用Barlat准则的模拟结果和采用Barlat-Lian准则的结果差别不大。     

Two exact micromechanics-based nonlocal constitutive equations for random linear elastic composite materials

A Hashin-Shtrikman-Willis variational principle is employed to derive two exact micromechanics-based nonlocal constitutive equations relating ensemble averages of stress and strain for two-phase, and also many types of multi-phase, random linear elastic composite materials. By exact is meant that the constitutive equations employ the complete spatially-varying ensemble-average strain field, not gradient approximations to it as were employed in the previous, related work of Drugan and Willis (J. Mech. Phys. Solids 44 (1996) 497) and Drugan (J. Mech. Phys. Solids 48 (2000) 1359) (and in other, more phenomenological works). Thus, the nonlocal constitutive equations obtained here are valid for arbitrary ensemble-average strain fields, not restricted to slowly-varying ones as is the case for gradient-approximate nonlocal constitutive equations. One approach presented shows how to solve the integral equations arising from the variational principle directly and exactly, for a special, physically reasonable choice of the homogeneous comparison material. The resulting nonlocal constitutive equation is applicable to composites of arbitrary anisotropy, and arbitrary phase contrast and volume fraction. One exact nonlocal constitutive equation derived using this approach is valid for two-phase composites having any statistically uniform distribution of phases, accounting for up through two-point statistics and arbitrary phase shape. It is also shown that the same approach can be used to derive exact nonlocal constitutive equations for a large class of composites comprised of more than two phases, still permitting arbitrary elastic anisotropy. The second approach presented employs three-dimensional Fourier transforms, resulting in a nonlocal constitutive equation valid for arbitrary choices of the comparison modulus for isotropic composites. This approach is based on use of the general representation of an isotropic fourth-rank tensor function of a vector variable, and its inverse. The exact nonlocal constitutive equations derived from these two approaches are applied to some example cases, directly rationalizing some recently-obtained numerical simulation results and assessing the accuracy of previous results based on gradient-approximate nonlocal constitutive equations.     

Accurate numerical integration of Drucker-Prager's constitutive equations

The paper illustrates the application of a general two-step integration scheme for the rate plasticity equations to the case of Drucker-Prager's model with linear mixed hardening and associated flow rule. The integration scheme coincides, for the case of the Mises equations, with a tangent predictor-radial return method with automatic sub-incrementation, the sub-increment size being governed by a predefined tolerance on the value of the yield stress. However, in contrast with the integration methods commonly adopted in several codes, the return step is here based on a precisely formulated rate problem, and it is not necessarily a radial one, in general. Thus, the accuracy characteristics of the method should carry over to every constitutive model tackled, depending only on the choice of the tolerance parameter value. The application of the integration scheme to Drucker-Prager's equations shows that the accuracy is in fact comparable to that obtained in the Mises case, for similar values of the tolerance parameter; at the same time some peculiarities of Drucker-Prager's yield condition, most notably the presence of a singular point in the stress space, highlight the flexibility and generality of the proposed method. Its theoretical basis, in fact, holds for vector-valued yield functions, thus automatically incorporating the treatment of the cases in which the stress points reach a corner in the yield surface.

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Sommario Il lavoro illustra l'applicazione di uno schema di integrazione a due passi delle equazioni della plasticità incrementale al caso del modello di Drucker-Prager con incrudimento lineare misto e legge di scorrimento associata. Lo schema di integrazione si riduce, per il modello di Von Mises, a un metodo tangent predictor-radial return con subincrementazione automatica. L'ampiezza dei subincrementi è governata da una tolleranza prefissata sul valore dello sforzo di snervamento. A differenza dei metodi di integrazione comunemente impiegati in diversi codici di calcolo, il passo di ritorno qui è basato su un problema incrementale ben formulato, e non è necessariamente un passo radiale, in generale. In questo modo le caratteristiche di accuratezza del metodo non dovrebbero dipendere dalla legge costitutiva adottata, ma solo dal valore scelto della tolleranza. In effetti, l'applicazione del metodo alle equazioni di Drucker-Prager mostra che l'accuratezza ottenibile è paragonabile a quella ottenuta per il caso di Von Mises. Allo stesso tempo alcune caratteristiche della condizione di Drucker-Prager, in particolare la presenza di un punto singolare nello spazio degli sforzi, evidenziano la flessibilità e la generalità di applicazione del metodo studiato. Esso infatti ha basi teoriche valide anche per superfici di snervamento vettoriali, che incorporano perciò automaticamente il trattamento dei casi in cui lo sforzo si trova in punti angolosi.

     

MICROMECHANICAL MODELLING OF PHASE TRANSFORMATION BEHAVIOUR OF A TRANSITIONAL SOIL

Experiments show that silts and silty soils exhibit contraction followed by dilation during shearing and the slope of failure line decreases at large strains, termed as phase transformation behaviour. This paper is to develop a new micromechanical stress-strain model that accounts for the phase transformation behaviour by explicitly employing the phase transformation line and its related friction angles. The overall strain includes plastic sliding and plastic compression among grains. The internal-friction angle at the phase transformation state and the void state variable are employed to describe the phase transformation behaviour. The model is examined by simulating undrained and drained triaxial compression tests performed on Pitea silts. The local stress-strain behaviour for contact planes is also investigated.     

An endochronic constitutive model for rate-dependent and nonproportional cyclic plasticity

The total stress response of material is decomposed into a sum of an equilibrium stress response and a non-equilibrium overstress response. Correspondingly, the rate-independent intrinsic time and the rate-dependent intrinsic time are defined respectively. Additional hardening functions for describing the isotropic and anisotropic nonproportional effects are assumed to be related to the accumulation of plastic strain component along the normal of equilibrium stress trajectory, in which the effects of geometry of the loading path are included. An endochronic constitutive model for rate-dependent, nonproportional cyclic plasticity is formulated and applied to simulate the stress responses of stainless steel XCrNi18.9 for some typical loading programs at different loading rates. A comparison between predicted results and experimental ones by Haupt and Lion shows that the former are in agrreement with the latter.     

An efficient algorithm for strain history tracking in finite element computations of non-Newtonian fluids with integral constitutive equations

A new finite element technique has been developed for employing integral-type constitutive equations in non-Newtonian flow simulations. The present method uses conventional quadrilateral elements for the interpolation of velocity components, so that it can conveniently handle viscoelastic flows with both open and closed streamlines (recirculating regions). A Picard iteration scheme with either flow rate or elasticity increment is used to treat the non-Newtonian stresses as pseudo-body forces, and an efficient and consistent predictor-corrector scheme is adopted for both the particle-tracking and strain tensor calculations. The new method has been used to simulate entry flows of polymer melts in circular abrupt contractions using the K-BKZ integral constitutive model. Results are in very good agreement with existing numerical data. The important question of mesh refinement and convergence for integral models in complex flow at high flow rate has also been addressed, and satisfactory convergence and mesh-independent results are obtained. In addition, the present method is relatively inexpensive and in the meantime can reach higher elasticity levels without numerical instability, compared with the best available similar calculations in the literature.     

On a theory of micropolar protoelastic material bodies and constitutive equations for nonlocal micropolar elastic continua

In this paper the definition of micropolar protoclastic material bodies is given and with the help of the principle of virtual power, the variational principle of those bodies is derived. In terms of that same idea and the definition of micropolar protopotential presented here, the constitutive equations for nonlocal micropolar elastic continua are naturally derived.