Deformation characteristics and strain-compensated constitutive equation for AA5083 aluminum alloy under hot compression

The hot deformation behavior of AA5083 aluminum alloy was studied using isothermal compression tests with a Gleeble 3500 thermal simulator at strain rate of 0.0110 s 1 and at temperature of 300500°C.The experimental results indicate that dynamic recrystallization(DRX)tends to occur at high strain rates and temperatures,and therefore the flow stress is decreased.To predict the flow behavior under different deformation conditions,a strain-compensated constitutive equation based on Arrhenius-type equation and Zener Hollomon parameters was proposed.The flow stresses obtained from the constitutive equation are consistent with the experimental results.The average absolute relative error is only 4.52%over the entire experimental range,indicating that the proposed constitutive equation exhibits high prediction precision for the hot deformation behavior of AA5083 aluminum alloy.

Simple physically-based constitutive equations for hot deformation of 2024 and 7075 aluminum alloys

The hot working behaviors of 2024 and 7075 aluminum alloys were studied through constitutive analysis based on a physically-based approach which accounts for the dependence of the elastic modulus and the self-diffusion coefficient of aluminum on temperature. It was demonstrated that the lattice self-diffusion activation energy of aluminum（142 k J/mol） can be used in the Zener-Hollomon parameter＇s formula as the deformation activation energy and the theoretical exponent of 5 can be set in the modified hyperbolic sine law to describe the peak flow stresses. By consideration of physically-based material＇s parameters, it was possible to conduct a comparative study on the hot flow stress of 2024 and 7075 aluminum alloys. It was concluded that the used approach in the current work can be considered as a versatile tool in future comparative hot working studies, especially in studies dedicated to alloy development.

Modeling of hot deformation behavior and prediction of flow stress in a magnesium alloy using constitutive equation and artificial neural network(ANN)model

The aim of the present study was to investigate the modeling and prediction of the high temperature flow characteristics of a cast magnesium(Mg-Al-Ca)alloy by both constitutive equation and ANN model.Toward this end,hot compression experiments were performed in 250-450℃and in strain rates of 0.001-1 s^(−1).The true stress of alloy was first and foremost described by the hyperbolic sine function in an Arrhenius-type of constitutive equation taking the effects of strain,strain rate and temperature into account.Predictions indicated that unlike low strain rates and high temperature with dominant DRX activation,in relatively high strain rate and low temperature values,the precision of the models become decreased due to activation of twinning phenomenon.At that moment and for a better evaluation of twinning effect during deformation,a feed-forward back propagation ANN was developed to study the flow behavior of the investigated alloy.Then,the performance of the two suggested models has been assessed using a statistical criterion.The comparative assessment of the gained results specifies that the well-trained ANN is much more precise and accurate than the constitutive equations in predicting the hot flow behavior.

CONSTITUTE EQUATIONS OF 40Cr STEEL UNDER SUPERPLASTIC COMPRESSIVE DEFORMATION

The microstructure of 40Cr steel sample and its surface is ultra-fined through salt-bath cyclic quenching and high frequency hardening, then the superplasticity is studied under isothermal superplastic compressive deformation condition. The experimental results indicate that the stress-strain curves are shown to take place obvious superplastic flow characteristic at the temperature of 750-770℃ and at the initial strain rate of (1.7-5.0)×10-4 s-1. Its strain rate sensitivity is 0.30-0.38, the steady superplastic flow stress is 60-70MPa, the superplastic flow activation energy is 198-217kJ/mol, and it is close to α-Fe grain boundary self-diffusion activation energy. The super-plastic compressive constitute equations of this steel are correspondingly set up. Due to the finer microstructure of high frequency hardening, it appears bigger strain rate sensitivity value, smaller the steady superplastic flow stress and the superplastic flow activation energy, so it has better superplastic deformation capability.

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.

New constitutive equation utilizing grain size for modeling of hot deformation behavior of AA1070 aluminum

A new phenomenological and empirically-based constitutive model was proposed to modify the term in the original Johnson−Cook constitutive model.The new model can be used to describe and predict the flow stress of AA1070 aluminum with different initial grain sizes in the hot working process.This developed model considers thermal softening,strain-rate hardening,strain hardening,initial grain size,and interactions with each other and can correctly model the behavior of AA1070 at elevated temperature with different strains,strain rates,and initial grain sizes.The hot flow behavior of AA1070 was investigated through compression tests over wide ranges of temperature from 623 to 773 K,strain rate from 0.005 to 0.5 s−1 and initial grain size from 50 to 450μm.Results show that the initial grain size has a significant effect on the flow behavior of AA1070.Then,correlation coefficient(R),average absolute relative error(AARE),and relative error were examined for comparative predictability of the model.Results show that flow stresses for different initial grain sizes calculated by the new proposed model perfectly correlate with experimental ones,with a mean relative error of 1.19%,which confirms that the new modified Johnson−Cook relation can give a precise estimation of the hot flow stress of AA1070 aluminum by considering the initial grain size.

Constitutive equation and processing map of equiatomic NiTi shape memory alloy under hot plastic deformation

In order to describe the deformation behavior and the hot workability of equiatomic NiTi shape memory alloy (SMA) during hot deformation, Arrhenius-type constitutive equation and hot processing map of the alloy were developed by hot compression tests at temperatures ranging from 500 to 1100 °C and strain rates ranging from 0.0005 to 0.5 s?1. The results show that the instability region of the hot processing map increases with the increase of deformation extent. The instability occurs in the low and high temperature regions. The instability region presents the adiabatic shear bands at low temperatures, but it exhibits the abnormal growth of the grains at high temperatures. Consequently, it is necessary to avoid processing the equiatomic NiTi SMA in these regions. It is preferable to process the NiTi SMA at the temperatures ranging from 750 to 900 °C.

Constitutive equations and finite element implementation of strain localization in sand deformation

A recently proposed model coupling with the solid-fluid of the saturated sand was utilized to study the deformation band. Based on the critical state plasticity model by Borja and Andrade, the hydraulic conductivity tensor was naturally treated as a function of the spatial discretization matrix about the displacement and the stress field, allowing a more realistic representation of the physical phenomenon. The fully Lagrangian form of the Darcy law was resolved by Piola algorithm, and then the flow law was gained, leading to the implementation of a modified model of the saturated sand. Then the criterion for the onset of localization was derived and utilized to detect instability. The constitutive model was implemented in a finite element program coded by FORTRAN, which was used to predict the formation and development of shear bands in plane strain compression of saturated sand. At last, the formation mechanism of the shear band was discussed. It is shown that the model works well, and the simulation sample bifurcates at 1.18% axial strain, which is in a good qualitative agreement with the experiment. The pore pressure greatly affects the onset and development of the deformation band, and it obviously increases around the localization-prone regions with the direction toward the outer side of the normal of the shear band, while the pore stress flows nearly horizontally and is distributed equally far away the shear band region.

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.

DEFORMATION OF STRUCTURE AND SPECTRUM OF EVOLUTION EQUATIONS

In this paper, we study the general structure of evolution equations of the AKNS eigenvalue problem q(x,t), r(x,t) with the spectrum varying asand AV BV CV are all positive or negative power polynomials of where q, r are not limited with any additional conditions at infinity.

ANALYSIS OF SIMPLE SHEAR ENDOCHRONIC EQUATIONSFOR FINITE PLASTIC DEFORMATION

Jaumann rate, generalized Jaumann rate,Fu rate and Wu rate were incorporated into endochronic equations for finite plastic deformation to analyze simple shear finite deformation. The results show that an oscillatory shear stress and normal stress response to a monotonically increasing shear strain occurs when Jaumann rate objective model is adopted for hypoelastic or endochronic materials. The oscillatory response is dependent on objective rate adopted,independent on elastoplastic models. Normal stress is unequal to zero during simple shear finite deformation.

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.

Simple shear endochronic equations for finite plastic deformation

The endochronic equations proposed by Valanis (1980) were extended to a finite deformation range. Jaumanns rate, Fus rate and Wus rate were incorporated into the endochronic equations to analyze simple shear finite deformation. Incremental equations and numerical solutions are deduced for three endochronic objective models. The results show that an oscillatory shear stress response to a monotonically increasing shear strain occurs when the Jaumanns rate objective model is employed for endochronic materials. The oscillatory response is dependent on the adopted objective rate. Compared with the Jaumanns rate, the Fus rate and the Wus rate satisfy the restrictions to elastic-plastic constitutive relations and are in agreement with the experimental results.

ON LARGE DEFORMATION UNILATERAL CONTACT PROBLEM WITH FRICTION(Ⅰ)—INCREMENTAL VARIATIONAL EQUATIONS

Based on the theory and technique of nonlinear geometric field theory of continuum, a more general incremental variational equation for elastic and plastic large deformation in co-moving coordinate is established in this paper. An expression for two and three-ditnensicnal continua is derived, and the incremental variational equation for large deformation of changing boundary contact and the variational inequality in rate form tire obtained, which provides the theoretical basis for the computation of elastic-plastic large deformation contact problem with friction.

EQUATION IN COMPLEX VARIABLE OF AXISYMMETRICAL DEFORMATION PROBLEMS FOR A GENERAL SHELL OF REVOLUTION

In this paper, the equation of axisymmetrical deformation problems for a general shell of revolution is derived in one complex variable under the usual Love-Kirchhoff assumption. In the case of circular ring shells, this equation may be simplified into the equation given by F.Tdlke(1938)[3]. R.A. Clark(1950 )[4] and V. V.Novozhilov(1951)[5]. When the horizontal radius of the shell of revolution is much larger than the average radius of curvature of meridian curve, this equation in complex variable may be simplified into the equation for slander ring shells. If the ring shell is circular in shape, then this equation can be reduced into the equation in complex variable for slander circular ring shells given by this author (1979)[6]. If the form of elliptic cross-section is near a circle, then the equation of slander ring shell with near-circle ellipitic cross-section may be reduced to the complex variable equation similar in form for circular slander ring shells.

Hot Compressive Deformation Characteristics of Al-9.3Zn-2.4Mg-1.1Cu Alloy

To understand the hot compression deformation characteristics of the self-developed Al-9.3Zn-2.4Mg^(-1).1Cu alloy,the hot compression tests of Al-9.3Zn-2.4Mg^(-1).1Cu alloy were investigated by Gleeble 1500 thermo-mechanical simulator to determine the best hot processing conditions.The hot deformation temperatures were 300,350,400,and 450℃,and the strain rates were 1,0.1,0.01,and 0.003 s^(-1),respectively.Based on the experimental results,the constitutive equation and hot processing maps are established,and the corresponding strain rate and temperature-sensitive index are analyzed.The results show that Al-9.3Zn-2.4Mg^(-1).1Cu alloy has a dynamic softening trend and high strain rate sensitivity during the isothermal compression process.The hot deformation behavior can be described by an Arrhenius-type equation after strain compensation.The temperature has a negligible effect on the hot processing properties,while a low strain rate is favorable for the hot working of alloy.The processing maps and microstructure show that the optimal processing conditions were in the temperature range of 400-450℃and strain rate range of 0.003-0.005 s^(-1).

Higher-dimensional Chen-Lee-Liu equation and asymmetric peakon soliton

Integrable systems play a crucial role in physics and mathematics.In particular,the traditional(1+1)-dimensional and(2+1)-dimensional integrable systems have received significant attention due to the rarity of integrable systems in higher dimensions.Recent studies have shown that abundant higher-dimensional integrable systems can be constructed from(1+1)-dimensional integrable systems by using a deformation algorithm.Here we establish a new(2+1)-dimensional Chen-Lee-Liu(C-L-L)equation using the deformation algorithm from the(1+1)-dimensional C-L-L equation.The new system is integrable with its Lax pair obtained by applying the deformation algorithm to that of the(1+1)-dimension.It is challenging to obtain the exact solutions for the new integrable system because the new system combines both the original C-L-L equation and its reciprocal transformation.The traveling wave solutions are derived in implicit function expression,and some asymmetry peakon solutions are found.

Prediction of Hot Deformation Behavior of 7Mo Super Austenitic Stainless Steel Based on Back Propagation Neural Network

The hot compression tests of 7Mo super austenitic stainless(SASS)were conducted to obtain flow curves at the temperature of 1000-1200℃and strain rate of 0.001 s^(-1)to 1 s^(-1).To predict the non-linear hot deformation behaviors of the steel,back propagation-artificial neural network(BP-ANN)with 16×8×8 hidden layer neurons was proposed.The predictability of the ANN model is evaluated according to the distribution of mean absolute error(MAE)and relative error.The relative error of 85%data for the BP-ANN model is among±5%while only 42.5%data predicted by the Arrhenius constitutive equation is in this range.Especially,at high strain rate and low temperature,the MAE of the ANN model is 2.49%,which has decreases for 18.78%,compared with conventional Arrhenius constitutive equation.

Hot deformation behavior and processing map of Cu-Ni-Si-P alloy

The high-temperature deformation behavior of Cu-Ni-Si-P alloy was investigated by using the hot compression test in the temperature range of 600-800 ℃ and strain rate of 0.01-5 s-1. The hot deformation activation energy, Q, was calculated and the hot compression constitutive equation was established. The processing maps of the alloy were constructed based on the experiment data and the forging process parameters were then optimized based on the generated maps for forging process determination. The flow behavior and the microstructural mechanism of the alloy were studied. The flow stress of the Cu-Ni-Si-P alloy increases with increasing strain rate and decreasing deformation temperature, and the dynamic recrystallization temperature of alloy is around 700 ℃. The hot deformation activation energy for dynamic recrystallization is determined as 485.6 kJ/mol. The processing maps for the alloy obtained at strains of 0.3 and 0.5 were used to predict the instability regimes occurring at the strain rate more than 1 s-1 and low temperature （〈650 ℃）. The optimum range for the alloy hot deformation processing in the safe domain obtained from the processing map is 750-800 ℃ at the strain rate of 0.01-0.1 s i The characteristic microstructures predicted from the processing map agree well with the results of microstructural observations.

Hot deformation behavior of novel imitation-gold copper alloy

The hot deformation behavior of a novel imitation-gold copper alloy was investigated with Gleeble-1500 thermo-mechanical simulator in the temperature range of 650-770 °C and strain rate range of 0.001-1.0 s-1. The hot deformation constitutive equation was established and the thermal activation energy was obtained to be 249.60 kJ/mol. The processing map at a strain of 1.2 was developed. And there are two optimal regions in processing map, namely 650-680 °C, 0.001-0.01 s-1 and 740-770 °C, 0.01-0.1 s-1. Optical microscopy was employed to investigate the microstructure evolution of the alloy in the process of deformation. Recrystallized grains and twin crystals were found in microstructures of the hot deformed alloy.