CONSTITUTIVE EQUATION OF CO-ROTATIONAL DERIVATIVE TYPE FOR ANISOTROPIC-VISCOELASTIC FLUID

A constitutive equation theory of Oldroyd fluid B type,i.e.the co-rotational derivative type,is developed for the anisotropic-viscoelastic fluid of liquid crystalline(LC)polymer.Analyzing the influence of the orientational motion on the material behavior and neglecting the influence,the constitutive equation is applied to a simple case for the hydrodynamic motion when the orientational contribution is neglected in it and the anisotropic relaxation,retardation times and anisotropic viscosi- ties are introduced to describe the macroscopic behavior of the anisotropic LC polymer fluid.Using the equation for the shear flow of LC polymer fluid,the analytical expressions of the apparent viscosity and the normal stress differences are given which are in a good agreement with the experimental results of Baek et al.For the fiber spinning flow of the fluid,the analytical expression of the extensional viscosity is given.

Cyclic constitutive equations of rock with coupled damage induced by compaction and cracking

In this paper,the cyclic constitutive equations were proposed to describe the constitutive behavior of cyclic loading and unloading.Firstly,a coupled damage variable was derived,which contains two parts,i.e.,the compaction-induced damage and the cracking-induced damage.The compaction-induced damage variable was derived from a nonlinear stress–strain relation of the initial compaction stage,and the cracking-induced damage variable was established based on the statistical damage theory.Secondly,based on the total damage variable,a damage constitutive equation was proposed to describe the constitutive relation of rock under the monotonic uniaxial compression conditions,whereafter,the application of this model is extended to cyclic loading and unloading conditions.To validate the proposed monotonic and cyclic constitutive equations,a series of mechanical tests for marble specimens were carried out,which contained the monotonic uniaxial compression(MUC)experiment,cyclic uniaxial compression experiments under the variable amplitude(CUC-VA)and constant amplitude(CUC-CA)conditions.The results show that the proposed total damage variable comprehensively reflects the damage evolution characteristic,i.e.,the damage variable firstly decreases,then increases no matter under the conditions of MUC,CUC-VA or CUC-CA.Then a reasonable consistency is observed between the experimental and theoretical curves.The proposed cyclic constitutive equations can simulate the whole cyclic loading and unloading behaviors,such as the initial compaction,the strain hardening and the strain softening.Furthermore,the shapes of the theoretical curves are controlled by the modified coefficient,compaction sensitivity coefficient and two Weibull distributed parameters.

A set of microstructure-based constitutive equations in hot forming of a titanium alloy

A physical model of microstructure evolution including dislocation density rate and grain growth rate was established based on the deformation mechanism for the hot forming of a class of two-phase titanium alloys. Further, a set of mechanism-based constitutive equations were proposed, in which the microstructure variables such as grain size and dislocation density were taken as internal state variables for characterizing the current material state. In the set of constitutive equations, the contributions of different mechanisms and individual phase to the deformation behavior were analyzed. The present equations have been applied to describe a correlation of the flow stress with the microstructure evolution of the TC6 alloy in hot forming.

Constitutive equation and hot deformation characteristic of Ti-6.5Al-3.5Mo-1.5Zr-0.3Si alloy with equiaxed microstructure

The hot deformation behavior of Ti-6.5Al-3.5Mo-1.5Zr-0.3Si alloy with equiaxed microstructure was characterized in the temperature range of 900～1060 ℃ and strain rate range of 10-3～10 s-1.The experimental results indicate that the plastic deformation behavior of the titanium alloys is rather sensitive to temperature and strain rate.In the α+β phase temperature region,all of the stress-strain curves exhibit different degrees of the flow softening after a peak stress.In the β phase temperature region,the titanium alloy shows a stress softening at high strain rates and a steady flow stress at low strain rates.On the basis of the peak stress data,the constitutive equations were constructed in the α+β phase temperature region and β phase temperature region,respectively.Activation energy parameters were calculated to be 344.923 kJ·mol-1 in the β phase temperature region and 628.3 kJ·mol-1 in the α+β phase temperature region.Microstructure of the compressed specimens in water-quenched conditions was found to be quite dependent on the conditions of deformation.

A phenomenological constitutive equation for Rene 95 PM alloy and its application to isothermal forging process of turbine disk

The flow behavior of Rene 95 PM alloy was studied from 1050 to 1150 deg Cwith strain rate of 1 X 10^(-3), 1 X 10^(-2), 1 X 10^(-1) and 1 s^(-1). At a given temperature andstrain rate, flow curves exhibit a peak followed by flow softening up to a steady state. Moreover,at constant strain, flow stress increases with increasing strain rate and decreasing temperature. Anequation relating hyperbolic sine of flow stress to hot working parameters, such as strain, strainrate and temperature, was established by using multiple nonlinear regression method. A very goodagreement was found between predicted and experimental flow stress in all the strain rangeinvestigated. Application of the constitutive equation in predicting forming loads and flow behaviorand temperature distribution in both upper and lower dies in an isothermal forging process ofturbine disk of large dimension (about 630 mm) by means of a finite element code was systematicallyanalyzed.

GENERAL EXPRESSIONS OF CONSTITUTIVE EQUATIONS FOR ISOTROPIC ELASTIC DAMAGED MATERIALS

The general expressions of constitutive equations for isotropic elastic damaged materials were derived directly from the basic law of irreversible thermodynamics. The limitations of the classical damage constitutive equation based on the well-known strain equivalence hypothesis were overcome. The relationships between the two elastic isotropic damage models (i.e. single and double scalar damage models) were revealed. When a single scalar damage variable defined according to the microscopic geometry of a damaged material is used to describle the isotropic damage state, the constitutive equations contain two 'damage effect functions', which describe the different influences of damage on the two independent elastic, constants. The classical damage constitutive equation based on the strain equivalence hypothesis is only the first-order approximation of the general expression. It may be unduly simplified and may fail to describe satisfactorily the damage phenomena of practical materials.

CONSTITUTIVE EQUATION OF A NEW AVIATION LUBRICATING OIL

The traction of a new aviation lubricating oil was measured on a self-made test rig. The calculating formulae of the rheological parameters of the oil such as Erying stress, limiting shear stress and shear elastic modulus were obtained under the condition of the high shear strain rate in elastohydrodynamic lubrication（EHL）. The constitutive equation of this oil was determined and verified by test. The results of experiments show that the behavior of the new aviation lubricating oil behaves as visco-elastic fluid and the theoretical value agrees fairly well with the measured data, which implies that the constitutive equation of this oil is correct and feasible.

THE CONSTITUTIVE EQUATION RESEMBLANCE OF ELASTIC-PLASTIC MULTIPHASE SOLID

Based on composite materials, the equivalent elastic-plastic constitutive equations of multiphase solid are researched. According to the suggested definition of. constitutive equivalence it is demonstrated that the multiphase solid, composed of several kinds of homogeneous elastic-plastic media that conform to the generalized normality rule, has the same type of constitutive equations as its constituents have that also conform to the generalized normality rule.

CONSTITUTIVE EQUATION WITH ORDERED PARAMETERS FOR VISCOELASTIC MATERIAL CONTAINING DEFECTS

Based on the experimental results of local temperature field formed in the evolution period of defects, the defect field is defined by this internal temperature field. The evolutionary processes of statistically correlative meso-scopic defects are analysed with the growth rate and nucleation rate of the meso-defect, and it is deemed that the dynamic failure process of the viscoelastic material with defects can be quantitatively described by the normal method of the procedure of heat wave transmission. The defect field is regarded as a complex system of the union of the real sets and null sets, its main characteristics depending on the stratum hypothesis about the activities of the subsystems of the highest stratum described by ordered parameters. The fluctuation of ordered parameters is demonstrated by means of the projection operator method. The constitutive equation with ordered parameters for the meso-defect evolutionary state of viscoelastic material with defects is deduced from the nonlinear rheological dynamic approach, and its solution is obtained.

Research on constitutive equation of DIWA353 steel used in hemisphere head of power station nearby recrystallization temperature

To simulate the DIWA353 steel used in boiler nearby recrystallization temperature accurately by using finite element,the high temperature constitutive model of this material must be researched firstly.Thermal simulation experimental machine is used to do the high temperature hot compression experiments for the DIWA353 steel used in boiler under different strain rates and temperatures nearby recrystallization temperature.Multi-bank data of true stress-true strain is gotten,and the hot deformation activation energy of DIWA353 is obtained,and the constitutive equation of steady-state flow stress under the circumstance of high temperature is constructed.The results show that DIWA353 steel is positive strain rate sensitized material when it is at temperature of 850-1050 ℃,and the steady-state flow stress decreases with the increase of deformation temperature,but increases with the increase of strain rate.

HIGH TEMPERATURE DEFORMATION MECHANISM AND CONSTITUTIVE EQUATION OF Ti-40 BUREN RESISTANT TITANIUM ALLOY

Ti - 40 alloy is a single β phase burn resistant titanium alloy.Its high temperature deformation mech- anism is studied and its stress - strain curves are examined he use of Gleeble - 1500 thermal -simulator. The results reveal that there are abrupt flow stress drops followed b steady state.The magnitude of the flow stress drop increases with strain rote and decreases with temperature.Deformation activation energy, Q, is 247. 5 KJ/mol. The deformation mechanism of Ti - 40 alloy is controlled by the lattice diffusion Its constitutive equation is set up, i. e.

An unsymmetric constitutive equation for anisotropic viscoelastic fluid

A continuum constitutive theory of corotational derivative type is developed for the anisotropic viscoelastic fluid-liquid crystalline （LC） polymers. A concept of anisotropic viscoelastic simple fluid is introduced. The stress tensor instead of the velocity gradient tensor D in the classic Leslie-Ericksen theory is described by the first Rivlin-Ericksen tensor A and a spin tensor W measured with respect to a co-rotational coordinate system. A model LCP-H on this theory is proposed and the characteristic unsymmetric behaviour of the shear stress is predicted for LC polymer liquids. Two shear stresses thereby in shear flow of LC polymer liquids lead to internal vortex flow and rotational flow. The conclusion could be of theoretical meaning for the modern liquid crystalline display technology. By using the equation, extrusion-extensional flows of the fluid are studied for fiber spinning of LC polymer melts, the elongational viscosity vs. extension rate with variation of shear rate is given in figures. A considerable increase of elongational viscosity and bifurcation behaviour are observed when the orientational motion of the director vector is considered. The contraction of extru- date of LC polymer melts is caused by the high elongational viscosity. For anisotropic viscoelastic fluids, an important advance has been made in the investigation on the constitutive equation on the basis of which a seriesof new anisotropic non-Newtonian fluid problems can be addressed.

CONSTITUTIVE RELATION OF UNSATURATED SOIL BY USE OF THE MIXTURE THEORY (Ⅱ)—LINEAR CONSTITUTIVE EQUATIONS AND FIELD EQUATIONS

The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot's equations for saturated porous media. The Darcy's laws of unsaturated soil were proved. It is shown that Biot's equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.

THE CONSTITUTIVE EQUATIONS FOR MIXED HARDENING ORTHOTROPIC MATERIAL

A dimensionless stress yield criterion is proposed to describe the mixed hardening of orthotropic material,including kinematic hardening and proportional hardening, and the associated plastic flow law is derived. The generalized effective stress_strain formulae can be obtained correspondingly based on the experimental stress_strain curves in various simple stress states. The initial plastic anisotropy is influenced by the elastic anisotropy. The yield criterion can be reduced to Huber_Mises Criterion for isotropic materials and associated constitutive equations can be degenerated into Prandtl_Reuss equations.

Constitutive Equation of Superalloy In718 in Hammer Forging Process

A constitutive equation reflecting the flow behavior of Superalloy In718 during the counter-blow hammer forging process was developed in terms of the relationship of flow stress and hot-deformation parameters,such as strain,strain rate,and deformation temperature.A new simplified approach for the complex multi-pass stress-strain curves has been attempted.The simulation curves calculated by constitutive equation are consistent with the experimental data.

Constitutive Equation of Laminated Veneer Lumber (LVL) at Large Deformation Condition

The stress-strain relation of laminated veneer lumber (LVL) was established by the method of regression, and its constitutive equation at large deformation and constant loading conditions was given to predict the static mechanical behaviors of LVL.

Constitutive equation and model validation for a 31 vol.% B_4Cp/6061Al composite during hot compression

An accurate constitutive equation is essential to understanding the flow behavior of B4C/A1 compos-ites during the hot deformation. However, the constitutive equations developed previously in literature are generally for low strain rate deformation. In the present work, we modified the general consti-tutive equation and take the high strain rate correction into account. The constitutive equation for a 31 vol.% B4Cp/6061AI composite was constructed based on the flow stresses measured during isothermal hot compression at temperatures ranging from 375 to 525 ℃ and strain rates from 0.01 to 10 s^-1. The experimental flow stresses were corrected by considering temperature-dependent Arrhenius factor. The modified equation was then verified by using DEFORM-3D finite element analysis to simulate the exper-imental hot compression process. The results show that the modified equation successfully predicts flow stress, load-displacement, and the temperature rise. This helps to optimize the hot deformation process, and to obtain desirable properties, such as reduced porosity and homogenous particle distribution in B4C/AI composites.

Constitutive Equation Models of Hot-Compressed T122 Heat Resistant Steel

Based on dislocation reaction theory and Avrami equation, a constitutive equation model was developed to describe dynamic recovery and dynamic recrystallization during hot deformation of T122 heat resistant steel, which have taken the effect of dynamic strain aging into account. Uniaxial hot compression test had been carried out over a wide range of strain rate （0.01 to 10 s-1 ） and temperature （900 to 1 200 ~C） with the help of Gleeble 3500. Obtained experimental data was applied to determine the material parameters in proposed constitutive equations of T122 steel, by using the non-linear least square regress optimization method. The calculated constitutive equations are quantita- tively in good agreement with experimentally measured curves and microstructure observation. It shows that propose constitutive equation T122 steel is able to be used to predict flow stress of T122 steel during hot deformation in aus- tenite temperature scope.

Inelastic constitutive equation of plasma-sprayed ceramic thermal barrier coatings

Ceramic thermal barrier coatings （TBCs） are a very important technology for protecting the hot parts of gas turbines （GTs） from a high-temperature environment. The coating stress generated in the operation of GTs brings cracking and peeling damage to the TBCs. Thus, it is necessary to evaluate precisely such coating stress in a TBC system. We have obtained a stress-strain curve for a freestanding ceramic coat specimen peeled from a TBC coated substrate by conducting the bending test. The test results have revealed that the ceramic coating deforms nonlinearly with the applied loading. In this study, an inelastic constitutive equation for the ceramic thermal barrier coatings deposited by APS is developed. The obtained results are as follows： （1） the micromechanics-based constitutive equation was formulated with micro crack density formed at splat boundary, and （2） it was shown that the numerical results for a nonlinearly deformed beam simulated by the developed constitutive equation agreed with the experimental results obtained by cantilever bending tests.

Characterization of hot deformation behavior of wear-resistant steel BTWl using processing maps and constitutive equations

In order to predict flow instability of wear-resistant steel BTW1, the hot compressions of wear-resistant steel BTW1 were firstly performed at the temperature of 900-1150 ℃ and at the strain rate of 0.05-15 s-1. Then, the constitutive relation was established based on Arrhenius-type hyperbolic sine equation. The results demonstrated that the flow stress depended on the deformation temperature and strain rate. When the deformation temperature kept constant, the flow stress increased as the strain rate increased. When the strain rate remained constant, the flow stress decreased as the temperature increased. The flow stresses calculated by constitutive equations were in a good agreement with experimental results. The apparent activation energy for deformation in the above processing region was estimated to be 369 kJ tool-1. A processing map could be obtained by the superimposition of an instability map on a power dissipation map. Based on the analysis of processing map and the microstructures, the theological instability regimes of strain rate and temperature for hot deformation of wear-resistant steel BTWl had been identified.