A GENERAL CONSTITUTIVE RELATION FOR FATIGUE CRACK GROWTH ANALYSIS OF METAL STRUCTURES

Crack growth rate curves are the fundamental material property for metal structures under fatigue loading. Although there are many crack growth rate curves available in the literature, few of them showed the capability to explain various special phenomena observed in tests. A modified constitutive relation recently proposed by McEvily and his co-workers showed very promising capability. This modified constitutive relation is further generalized by (1) introducing an unstable fracture condition; (2) defining a virtual strength to replace the yield stress; and (3) defining an overload and underload parameter. The performances of this general constitutive relation for fatigue crack growth is extensively studied and it is found that this general constitutive relation is able to explain various phenomena observed with particular strong capability on load sequence effect.

Constitutive relation of an orthorhombic polycrystal with the shape coefficients

An orthorhombic polycrystal is an orthorhombic aggregate of tiny crystallites. In this paper, we study the effect of the crystalline mean shape on the constitutive relation of the orthorhombic polycrystal. The crystalline mean shape and the crystalline orientation arrangement are described by the crystalline shape function （CSF） and the orientation distribution function （ODF）, respectively. The CSF and the ODF are expanded as an infinite series in terms of the Wigner D-functions. The expanded coefficients of the CSF and the ODF are called the shape coefficients s^lm0 and the texture coefficients c^lmn respectively. Assuming that Ceff in the constitutive relation depends on the shape coefficients s^lm0 and the texture coefficients c^lmn by the principle of material frame-indifference we derive an analytical expression for C^eff up to terms linear in s^lmo and c^lmn and the expression would be applicable to the polycrystal whose texture is weak and whose crystalline mean shape has weak anisotropy. C^cff contains six unspecified material constants （λ, μ, c, s1, s2, s3）, five shape coefficients （s^2 00, s^2 20, s^4 00, s^4 20, s^4 40）, and three texture coefficients （c^4 99,c^4 20, c^4 40）, The results based on the perturbation approach are used to determine the five material constants approximately. We also find that the shape coefficients 2 and a s^2mo and s^4m0 are all zero if the crystalline mean shape is a cuboid. Some examples are given to compare our computational results.

Dynamic Response Analysis of Bird Strike on Aircraft Windshield Based on Damage-modified Nonlinear Viscoelastic Constitutive Relation

Damage-modified nonlinear viscoelastic constitutive equation and failure criterion are introduced and the three-dimensional incremental forms are deduced based on the updated Lagrangian approach. A simple tensile test model and a split Hopkinson pressure bar model are built to verify the accuracy of the subroutine implemented within the non-linear finite element program LS-DYNA. A numerical model of bird strike on windshield is established to study the responses of windshield under three different bird velocities at three sites. The bird is represented by a cylinder with a hemisphere at each end and the contact-impact coupling algorithm is used in this study. It is found that the implemented subroutine can properly describe the mechanical behavior of polymethyl methaerylate under low and high strain rates and large deformation, and can be used validly.

QUICK ASSESSMENT METHODOLOGY FOR RELIABILITY OF SOLDER JOINTS IN BALL GRID ARRAY (BGA) ASSEMBLY——PART Ⅰ: CREEP CONSTITUTIVE RELATION AND FATIGUE MODEL

In this study, a new unified creep constitutive relation and a mod- ified energy-based fatigue model have been established respectively to describe the creep flow and predict the fatigue life of Sn-Pb solders. It is found that the relation successfully elucidates the creep mechanism related to current constitutive relations. The model can be used to describe the temperature and frequency dependent low cycle fatigue behavior of the solder. The relation and the model are further employed in part Ⅱ to develop the numerical simulation approach for the long-term reliability assessment of the plastic ball grid array (BGA) assembly.

Particle characteristics and rheological constitutive relations of high concentration red mud

Red mud has relatively small solid particles （d50= 13.02 μm） and will flow in paste form under high pressure during pipeline transport. Red mud belongs to a two-phase flow of materials with high viscosity and a high concentration of non-sedimentation, homogeneous solid-liquids. It is difficult to test its rheological properties under atmospheric pressure. Measurements such as rotational viscometry can not reflect the real state of the material when it is flowing in a pipe. Tested rheological parameters are somewhat higher than the actual values. In our investigation, grain shape, distinctive modality and grain size distribution of red mud were tested. Based on the principle of tube measurement, rheological experiments on red mud at different concentrations were carried out by using our independently developed tube-type pressure theology test facility, and obtained constitutive equations. We conclude that red mud behaves as non-Newtonian pseudo-plastic fluid in pipe flows. Its consistency and power-law indices vary considerably with different concentrations.

THE CONSTITUTIVE RELATION OF CRACK-WEAKENED ROCK MASSES UNDER AXIAL-DIMENSIONAL UNLOADING

An accurate and efficient numerical method for solving the crack-crack interaction problem is presented. The method is mainly by means of the dislocation model, stress superposition principle and Chebyshev polynomial expansion of the pseudo-traction. This method can be applied to compute the stress intensity factors of multiple kinked cracks and multiple rows of periodic cracks as well as the overall strains of rock masses containing multiple kinked cracks under complex loads. Many complex computational examples are given. The dependence of the crack-crack interaction on the crack configuration, the geometrical and physical parameters, and loads pattern, is investigated. By comparison with numerical results under confining pressure unloading, it is shown that the crack-crack interaction under axial-dimensional unloading is weaker than those under confining pressure unloading. Numerical results for single faults and crossed faults show that the single faults are more unstable than the crossed faults. It is found from numerical results for different crack lengths and different crack spacing that the interaction among kinked cracks decreases with an increase in length of the kinked cracks and the crack spacing under axial-dimensional unloading.

ELASTIC-PLASTIC CONSTITUTIVE RELATION OF PARTICLE REINFORCED COMPOSITES

In this paper, a systematic approach is proposed to obtain the macroscopic elastic-plastic constitutive relation of particle reinforced composites (PRC). The strain energy density of PRC is analyzed based on the cell model, and the analytical formula for the macro-constitutive relation of PRC is obtained. The strength effects of volume fraction of the particle and the strain hardening exponent of matrix material on the macro-constitutive relation are investigated, the relation curve of strain versus stress of PRC is calculated in detail. The present results are consistent with the results given in the existing references.

Neural Network Model for the Constitutive Relations of Soil

The soil constitutive relation is one of the important issues in soil mechanics. It is very difficult to establish mathematical models because of the complexity of soil mechanical behavior. We propose a new method of neural network analysis for establishing soil constitutive models. Based on triaxial experiments of sand in the laboratory, the nonlinear constitutive models of sand expressed by the neural network were set up. In comparison with Duncan\|Chang's model, the neural network method for sand modeling has been proved to be more convenient, accurate and it has a strong fault\|tolerance function.

Method of Numerical Modeling for Constitutive Relations of Clay

In order to study the method of numerical modeling for constitutive relations of clay, on the basis of the principle of interaction between plastic volumetric strain and plastic generalized shear strain, the two constitutive functionals that include the function of stress path were used as the basic framework of the constitutive model, which are able to demonstrate the dependence of stress path. The two partial differential cross terms appear in the expression of stress-strain increment relation, which are used to demonstrate the interaction between plastic volumetric strain and plastic generalized shear strain. The elasoplastic constitutive models of clay under two kinds of stress paths, CTC and TC, have been constracted using the triaxtal test results. The three basic characteristics of deformation of soils, pressure sensitivity, dilatancy, and dependence of stress path, are well explained using these two models. Using visualization, the three-dimensional surfaces of shear and volume strains in the whole stress field under stress paths of CTC and TC are given. In addition, the two families of shear and volmetric yield Ioei under CTC and TC paths are plotted respectively. By comparing the results of deformation under these two stress paths, it has been found that, there are obvious differences in the strain peaks, the shapes of strain surfaces, and the trends of variation of volumetric yield loci, however both families of shear yield loci are similar. These results demonstrate that the influences of stress path on the constitutive relations of clay are considerably large and not negligible. The nmericul modeling method that can sufficiently reflect the dependence of stress path is superior to the traditional one.

Research on Bond-Slip Constitutive Relation for Steel Reinforced Concrete Members

The constitutive relation of bond-slip on steel and concrete interface is proposed for short steel reinforced concrete （SRC） column. Based on the experimental research on bond-slip performance, a mechanical model of short SRC column in pulling or pushing test is established. By means of the elasto-plasticity theory the explicit formulation of bond-slip constitutive relation τ-s in different anchor-hold place of push and pull member is investigated under the conditions of balance and boundary. The study shows that the constitutive relation is relevant to the embedment length and the thickness of concrete cover. The results are continuous descriptions of bond-slip constitutive relation and can be used to analyze the non-linear performance of SRC members. Results indicate that the principle of the method is correct and it performs well for short SRC column.

QUASI-STATIC AND DYNAMICAL ANALYSIS FOR VISCOELASTICTIMOSHENKO BEAM WITH FRACTIONAL DERIVATIVECONSTITUTIVE RELATION

The equations of motion governing the quasi-static and dynamical behavior of a viscoelastic Timoshenko beam are derived. The viscoelastic material is assumed to obey a three-dimensional fractional derivative constitutive relation. ne quasi-static behavior of the viscoelastic Timoshenko beam under step loading is analyzed and the analytical solution is obtained. The influence of material parameters on the deflection is investigated. The dynamical response of the viscoelastic Timoshenko beam subjected to a periodic excitation is studied by means of mode shape functions. And the effect of both transverse shear and rotational inertia on the vibration of the beam is discussed.

Study of experiment method for triaxial frost heaving constitutive relation of frozen soil

A new experiment method is introduced to study the relations between frost heaving strain and stress of soil or fracture rock under different moisture contents, temperatures and stress states. Based on experiments, a new triaxial frost heaving stress-strain relation is presented aiming at tunnel in cold regions. The experimental equipments are strain-controlled and low temperature buildup, and different restrained conditions are controlled by changing the stiffness of a test-force-ring. Then the frost heaving strain and stress of soil or fracture rock can be obtained under different restrained conditions, and the frost heaving stress-strain relation can be got by regression analysis of some feature points. Experiments of saturated sandy soil conducted by this method show that the triaxial frost heaving stress-strain relation could be expressed by logarithmic curves, and the frost heaving stress changes linearly with the logarithm of the frost heaving strain. The stronger the constraint, the smaller the frost heaving strain and the larger the frost heaving stresses. The frost heaving stresses would tend to a limit value with increasing the constraint intensity. The larger the confining pressure, the larger the frost heaving strain and stress.

One-dimensional Incremental Constitutive Relation of SMA Wire Reinforced Smart Composites with Damages

A new Martensitic transformation kinetic model for shape memory alloy （SMA） is proposed based on the phenomenological description of the Martensitic transformation heat flow-temperature curve and on the linear relationship between the partial derivatives of Martensite fraction and of Gbbis free energy with respect to the temperature. A meso-mechanical model is developed to describe the longitudinal stiffness reduction and thermo-dilatation variation of the composites caused by fiber breaking or fiber peeling off the base material. One-dimensional incremental constitutive relation is then established for SMA wire reinforced smart composites with damages by introducing three parameters to respectively describe the extent of fiber breaking, fiber peeling off the base material and interface weakening. The results presented herein may provide a theoretical basis for further studying on SMA smart composites with damages.

A CONSTITUTIVE RELATION FOR PSEUDO-ELASTIC BEHAVIOUR IN SHAPE MEMORY ALLOYS

A constitutive relation to describe pseudo-elastic deformation in shape memory alloys is pres- ented in this paper. It is capable of describing deformation behaviour of polycrystalline materials under triaxial stress state as well as of monocrystalline materials under one-dimensional condition. Total strain rate is sup- posed to be composed of elastic strain rate and transformation strain rate. Deformation behaviour of Cu-Zn-Sn alloy and Ti-Ni alloy is simulated by use of the proposed constitutive relation. It is shown that simulated results are in a good agreement with experimental data.

NEW CONSTITUTIVE RELATIONSHIP OF INCOMPRESSIBLE HYPERELASTICITY

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PROPERTY DEGRADATION OF ANISOTROPIC COMPOSITELAMINATES WITH MATRIX CRACKING(Ⅰ)──CONSTITUTIVE RELATION DEVELOPING FOR (θ_m/90_n)_s CRACKED LAMINATES BY STIFFNESS PARTITION

The study on property degradation of damaged composite laminates is extendedto anisotropic laminates with matrix cracking. In (I) of the paper, an idea of 'stiffnesspatition' is proposed to deal with the puzzle that the in-plane normal response iscoupled with the shear response of the laminates. For (θm/90n.), laminates containingtransversely cracked layers under general in-plane loading, the constitutive relationsare derived and the effective stiffnesses are expressed as the function of crack density.

STUDY OF CONSTITUTIVE RELATIONS FOR PSEUDO-ELASTICITY AND SHAPE MEMORY BEHAVIOR

Constitutive relations are given for the description of the deformation behavior of shape memory materials. The deformation is the superposition of the elastic, the thermal and the phase transformation deformation caused by the transformation from one to the other among the high temperature phase, the low temperature phase and the stress induced phase. The phase transformation is controlled by the driving force, i.e., the Gibbs energy difference between the phases.

DYNAMICAL STABILITY OF VISCOELASTIC COLUMN WITH FRACTIONAL DERIVATIVE CONSTITUTIVE RELATION

The dynamic stability of simple supported viscoelastic column, subjected to a periodic axial force, is investigated. The viscoelastic material was assumed to obey the fractional derivative constitutive relation. The governing equation of motion was derived as a weakly singular Volterra integro-partial-differential equation, and it was simplified into weakly singular Volterra integro-ordinary-differential equation by the Galerkin method. In terms of the averaging method, the dynamical stability was analyzed. A new numerical method is proposed to avoid storing all history data. Numerical examples are presented and the numerical results agree with the analytical ones.

CRITERION FOR JUDGING SATISFACTION OF CONSISTENCY CONDITIONS IN RATE-DEPENDENT CONSTITUTIVE RELATIONS

Based on irreversible thermodynamics, the criterion for judging the satisfaction of consistency conditions in rate-dependent constitutive relationship is deduced by introducing four basic hypotheses. Formulas for solving internal variables are given. It makes the rate-dependent model applicable no matter whether the consistency conditions can be satisfied or not.

Effect of hydrostatic stress on yield function and plastic constitutive relations

A new yield function taking effect of hydrostatic stress into account is presented through establishing and solving the functional equation satisfied by the yield function, and its characteristic is simple in form and strong in generality. In order to reveal its availability, a comparison is made between the results obtained with it and the experimental results of grey cast iron has be done, both seem to be in good agreement. At the same time, taking the yield function obtained here as a potential function, a new associative plastic constitutive equation taking effect of hydrostatic stress into account is built, and the plastic volume change ratio of plastic deformation is given.