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.

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.

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.

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.

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.

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.

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.

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.

The Constitutive Relation of a Fabric Membrane Composite for a Stratospheric Airship Envelope Based on Invariant Theory

The study of stratospheric airships has become the focus in many countries in recent years,because of its potential applications in many fields.Lightweight and high strength envelopes are the keys to the design of stratospheric airships,as it directly determines the endurance flight performance and loading deformation characteristics of the airship.A typical envelope of any stratospheric airship is a coated-fabric material which is composed of a fiber layer and several functional membrane layers.According to composite structure,nonlinearity and viscoelasticity are the two main characteristics of such envelope.Based on the analysis on the interaction between the different components in the micro-mechanical model of the coated-fabric,several invariant values reflecting the characteristics of the envelope material are obtained according to invariant theory.Furthermore,the constitutive equation that describes the viscoelasticity of the envelope material is derived.The constitutive equation can represent both the individual roles of the warp and weft fibers,and their further coupled interactions.The theoretical computation results were verified by off-axial tension tests.The results can help gain a deeper understanding of the mechanical mechanism and provide a reference for structural design of envelope material.

Alternative constitutive relation for momentum transport of extended Navier–Stokes equations

The classical Navier–Stokes equation(NSE)is the fundamental partial differential equation that describes the flow of fluids,but in certain cases,like high local density and temperature gradient,it is inconsistent with the experimental results.Some extended Navier–Stokes equations with diffusion terms taken into consideration have been proposed.However,a consensus conclusion on the specific expression of the additional diffusion term has not been reached in the academic circle.The models adopt the form of the generalized Newtonian constitutive relation by substituting the convection velocity with a new term,or by using some analogy.In this study,a new constitutive relation for momentum transport and a momentum balance equation are obtained based on the molecular kinetic theory.The new constitutive relation preserves the symmetry of the deviation stress,and the momentum balance equation satisfies Galilean invariance.The results show that for Poiseuille flow in a circular micro-tube,self-diffusion in micro-flow needs considering even if the local density gradient is very low.

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.

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.

Artificial neural network model of constitutive relations for shock-prestrained copper

Data from the deformation on Split Hopkinson Bar were used for constructing an artificial neural network model. When putting the thermodynamic parameters of the metals into the trained network model, the corresponding yielding stress can be predicted. The results show that the systematic error is small when the objective function is 0.5 , the number of the nodes in the hidden layer is 6 and the learning rate is about 0.1 , and the accuracy of the rate error is less than 3%. [

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.

Speculation of fluid dynamics equations based on Liutex theory and constitutive relation of symmetric shearing deformation

The fluid kinematics of Liutex decomposes a velocity gradient tensor(VGT)of∇v into four components,including rotation(R),stretching/compressing(SC),anti-symmetric shear(Santi-sym)and symmetric shear(Ssym),as oppose to the traditional Cauchy-Stokes decomposition where a VGT was decomposed into the strain rate and vorticity tensors.The current study limpidly clarified the physical meanings of these deformations in the newly-proposed decomposition from the perspectives of both fluid kinematics and dynamics.With in-depth understanding the physical connotations of these deformations,the present study further suggests that the Ssym be the only deformation appropriately correlated to the stress tensor,leading to the establishment of a new constitutive relation for Newtonian fluids with the modified model assumptions originated from Stokes in 1845.Moreover,the present research finds that the“principal decomposition”proposed by Liu is not mathematically unique when a VGT has three real eigenvalues(TR).Within the context,a new decomposition method is introduced to avoid the non-uniqueness issue arising from using the principal decomposition to establish fluid dynamics equations.Based on the modified Stokes assumptions and the novel VGT decomposition method,a set of new fluid dynamics momentum equations are obtained for Newtonian fluid.The added stress tensor of Fadd is identified as the key difference between the newly-derived governing equations and the conventional Navier-Stokes(N-S)equations,which is caused by excluding the SC correlation to the stress tensor in the new constitutive equation.Finally,a preliminary analysis of Fadd is conducted using the existing channel turbulence direct numerical simulations(DNS)data based on the traditional N-S equations.The Fadd is found widely existing in turbulence and is of the same order of magnitude with the other force terms.Therefore,the Fadd is expected to have some nonnegligible effects on altering the current DNS data based on the traditional N-S equations,which will be further verified by performing the“DNS”simulation using the newly-derived fluid dynamics equations in near future.