Elastic-viscoplastic constitutive equations of K439B superalloy and thermal stress simulation during casting process

K439B nickel-based superalloy is a new type of high-temperature material.There is insufficient research on its constitutive equations and numerical modeling of thermal stress.Isothermal tensile experiments of K439B superalloy at different temperatures(20°C-1,000°C)and strain rates(1.33×10^(-3)s^(-1)-5.33×10^(-3)s^(-1))were performed by using a Gleeble-3800 simulator.The elastic moduli at different temperatures(20°C-650°C)were measured by resonance method.Subsequently,stress-strain curves were measured for K439B superalloy under different conditions.The elastic-viscoplastic constitutive equations were established and the correspongding parameters were solved by employing the Perzyna model.The verification results indicate that the calculated values of the constitutive equations are in good agreement with the experimental values.On this basis,the influence of process parameters on thermal stress was investigated by numerical simulation and orthogonal experimental design.The results of orthogonal experimental design reveal that the cooling mode of casting has a significant influence on the thermal stress,while pouring temperature and preheating temperature of shell mold have minimal impact.The distribution of physical fields under optimal process parameters,determined based on the orthogonal experimental design results,was simulated.The simulation results determine separately the specific positions with maximum values for effective stress,plastic strain,and displacement within the casting.The maximum stress is about 1,000.0 MPa,the plastic strain is about 0.135,and the displacement is about 1.47 mm.Moreover,the distribution states of thermal stress,strain,and displacement are closely related to the distribution of the temperature gradient and cooling rate in the casting.The research would provide a theoretical reference for exploring the stress-strain behavior and numerical modeling of the effective stress of the alloy during the casting process.

Perfect elastic-viscoplastic field at mode I dynamic propagating crack-tip

The viscosity of material is considered at propagating crack-tip. Under the assumption that the artificial viscosity coefficient is in inverse proportion to power law of the plastic strain rate, an elastic-viscoplastic asymptotic analysis is carried out for moving crack-tip fields in power-hardening materials under plane-strain condition. A continuous solution is obtained containing no discontinuities. The variations of numerical solution are discussed for mode Ⅰ crack according to each parameter. It is shown that stress and strain both possess exponential singularity. The elasticity, plasticity and viscosity of material at crack-tip only can be matched reasonably under linear-hardening condition. And the tip field contains no elastic unloading zone for mode I crack. It approaches the limiting case, crack-tip is under ultra-viscose situation and energy accumulates, crack-tip begins to propagate under different compression situations.

Steady-state growing crack-tip field characteristics of mode Ⅲ elastic-viscoplastic/rigid interface cracks

The existence of viscosity effect at the interface of double dissimilar materials has an important impact on the distribution of the interface crack-tip field and the properties variety of the interface itself. The singularity and viscosity are considered in the crack-tip. The elastic-viscoplastic governing equations of double dissimilar materials at the interface crack-tip field are established. The displacement potential function and boundary condition of interface crack-tip are introduced. The numerical analysis of elastic-viscoplastic/rigid interface for mode Ⅲis worked out. The stress-strain fields are obtained at the crack-tip and the variation rules of solutions are discussed according to each parameter. The numerical results show that the viscosity effect is a main factor of the interface propagating in the crack-tip field, and the interface crack-tip is a viscoplastic field governed by the viscosity coefficient, Mach number （Ma）, and singularity exponent.

Elastic-viscoplastic field at mixed-mode interface crack-tip under compression and shear

For a compression-shear mixed mode interface crack, it is difficult to solve the stress and strain fields considering the material viscosity, the crack-tip singularity, the frictional effect, and the mixed loading level. In this paper, a mechanical model of the dynamic propagation interface crack for the compression-shear mixed mode is proposed using an elastic-viscoplastic constitutive model. The governing equations of propagation crack interface at the crack-tip are given. The numerical analysis is performed for the interface crack of the compression-shear mixed mode by introducing a displacement function and some boundary conditions. The distributed regularities of stress field of the interface crack-tip are discussed with several special parameters. The final results show that the viscosity effect and the frictional contact effect on the crack surface and the mixed-load parameter are important factors in studying the mixed mode interface crack- tip fields. These fields are controlled by the viscosity coefficient, the Mach number, and the singularity exponent.

Field structure at mode Ⅲ dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.

Comparative Study on Constitutive Modeling of Tantalum and Tantalum Tungsten Alloy

As a model bee metal, tantalum and its alloys have wide applications in defense-related fields. The KHL （Khan, Huang, Liang, 1999） model and the constitutive model proposed by Nemat-Nasser et al （Nemat-Nasser and Kapoor, 2001） for tantalum and its alloys were analyzed and compared with each other. A set of published data recorded during elastic-plastic deformations of tantalum, tantalum alloy containing tungsten of 2.5% （Ta-2.5W）, over a wide range of strains, strain rates, and temperatures were used to correlate the two models. Overall, it can be concluded that KHL model correlates much better with the data than the model used by Nemat-Nasser et al.

VISCOPLASTIC SOLUTION TO FIELD AT STEADILY PROPAGATING CRACK TIP IN LINEARHARDENING MATERIALS

An elastic-viscoplastic constitutive model was adopted to analyze asymptotically the tip-field of moving crack in linear-hardening materials under plane strain condition. Under the assumption that the artificial viscosity coefficient was in inverse proportion to power law of the rate of effective plastic strain, it is obtained that stress and strain both possess power law singularity and the singularity exponent is uniquely determined by the power law exponent of the rate of effective plastic strain. Variations of zoning structure according to each material parameter were discussed by means of numerical computation for the tip-field of mode II dynamic propagating crack, which show that the structure of crack tip field is dominated by hardening coefficient rather than viscosity coefficient. The secondary plastic zone can be ignored for weak hardening materials while the secondary plastic zone and the secondary elastic zone both have important influence on crack tip field for strong hardening materials. The dynamic solution approaches to the corresponding quasi-static solution when the crack moving speed goes to zero, and further approaches to the HR （Hui-Riedel） solution when the hardening coefficient is equal to zero.

DYNAMICALLY ASYMPTOTIC SOLUTIONS NEAR MODE II CRACK-TIPS

Description on dynamic behaviours of crack-tips is one of the important foundations to develop a reasonable dynamic fracture criterion. In order to describe the dynamic behaviours of the crack-tip in a material with low viscosity-number, the displacement potential function is assumed as a mathematical expression with exponential singularity. The asymptotic linear differential equations determining plane crack-tip field are established based on the mechanical constitutive model for elastic-viscoplastic materials. According to the conditions of determining solutions for dynamic cracks of mode II, the crack-tip stress fields are numerically simulated based on the asymptotic linear differential equations. Results show the asymptotic linear equations can well describe the crack-tip fields of plane dynamic cracks in the elastic-viscoplastic material with low viscosity-number.

SHEAR BAND FORMATION IN THERMO-VISCOPLASTIC SOLIDS UNDER PLANE STRAIN COMPRESSION

The plane strain compression of a rectangular block is numericallyinvestigated for the study of dynamic shear band development inthermo-elasto-viscoplastic materials from an internal in-homogeneity. As expected, it plays an important role in triggeringthe onset of shear localization as well as thermal softening. And thecompetition between the strain, strain-rate hardening and thermalsoftening exists throughout the process. It is found that shear banddevelops at a 45-degree angle to the compression axis. In the lightof given patterns of deformation and temperature, shear bandevolution accelerated by thermal softening is retarded by theinertial effects.

Mean-field homogenization of elasto-viscoplastic composites based on a new mapping-tangent linearization approach

In this work, a new homogenization method of elasto-viscoplastic composites is developed. Using the fully implicit backward Euler's integration algorithm, the nonlinear ordinary differential equations in the constitutive laws of the matrix and inclusion phases are discretized. Three classical incremental linearization approaches, i.e., direct, secant and tangent ones are adopted and an affine relationship between the stress and strain increments is deduced. In order to reduce the interaction between the inclusion and matrix phases, a second-ordered mapping tensor is introduced and a new mapping-tangent linearization approach is proposed. Different linearization approaches are implemented by the incremental self-consistent scheme to predict the overall stress-strain response of particle-reinforced composites. It is shown that the predicted stress-strain curves given by the proposed mapping-tangent linearization approach are softer than that by other three classical ones, and are much closer to that from a fullfield finite element simulation. Moreover, the linearized elasto-viscoplastic constitutive equation based on the proposed mapping-tangent approach has the same mathematical structure as the rate-independent elasto-plastic constitutive law. In this sense,the homogenization problems faced in the elasto-plastic and elasto-viscoplastic heterogeneous materials can be unified.