Microstructure optimization in design of forging processes

International Journal of Machine Tools & Manufacture 40 (2000) 691–711

Microstructure optimization in design of forging processes Zhenyan Gao, Ramana V. Grandhi

*

Department of Mechanical and Materials Engineering, Wright State University, Dayton, OH 45435, USA Received 9 April 1999; accepted 9 September 1999

Abstract A new approach based on sensitivity analysis for optimizing the microstructure development during the forging processes is proposed in this work. The analytical sensitivities of the recrystallization volume fraction and dynamically recrystalized grain size with respect to the design variables are derived. The mean grain size in each finite element is introduced so that the complex recrystallization mechanics, such as no recrystallization, partial recrystallization and complete recrystallization are all considered. The objective is to minimize a function describing the variance of mean grain size and the average value of mean grain size in the whole final product. Two constraints are imposed on die underfill and excessive material waste. Two different kinds of design variables are considered, including state parameter (initial shape of billet) and process parameter (die velocity). The optimization scheme is demonstrated with the design of a turbine disk made of Waspaloy in non-isothermal forging process. The optimal initial shape of billet and the die velocity are obtained.  2000 Elsevier Science Ltd. All rights reserved. Keywords: Microstructure; Optimization; Sensitivity analysis; Forging

1. Introduction Forging is a metal working process to obtain the desired final product shape with sound mechanical properties and fine metallurgical structures. One of the design objectives in the forging processes is to meet the geometrical dimensional accuracy for the final product, i.e. to obtain the prescribed shapes. The other design objective is to achieve superior mechanical properties and microstructure. Traditional build and test methods result in high tooling, set up costs and long lead times before production, which do not adequately meet the demands of the modern industry. With the development of the simulation technique and its application in metal forming processes,

* Corresponding author. Tel.: +1-937-775-5090; fax: +1-937-775-5147. E-mail address: [email protected] (R.V. Grandhi)

0890-6955/00/$ - see front matter  2000 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 9 9 ) 0 0 0 8 3 - 8

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a lot of changes have occurred in design methodologies. The simulation research of metal forming processes can be divided into three levels [1]: global modeling (prediction of deformation load or work using the slab method and upper bound), local modeling (calculation of thermo-mechanical variables using Finite Element Method (FEM)) and microstructure modeling (evaluation of the metallurgical and mechanical properties including microstructure, texture and anisotropy). Till now, a lot of works have been done on the second level to simulate the complex 2D or 3D forming processes and great achievements have been made. However, during the forming process, the material undergoes a series of microstructure changes, such as dynamic/static recovery, recrystallization and grain growth. All these influence the forming process (such as forming force, the distribution of temperature, strain and strain rate) and the quality of the final products (including the strength, ductility and toughness). Therefore, microstructure modeling forms a vital part of the overall modeling of thermo-mechanical processing operations. It is very important to combine microstructure changes and FEM simulation and develop a general model for the microstructural evolution during thermo-mechanical material processing. In recent years, research on the third level has increasingly received a great deal of attention. Some works in simulating the microstructure evolution have been done and the mechanisms of the microstructure and the thermo-mechanics have been studied extensively for a wide range of metals and alloys. Sellars et al. [2] presented microstructure simulation in the hot rolling of steel strip and plate. Semi-empirical mathematical models of microstructure evolution using the thermo-mechanical parameters were established for C-Mn steel. Yada [3] had found the critical strain to be independent of both the initial grain size and the strain rate and another model was given. Devadas [4] made a summary of the earlier works, predicted the grain size and the recrystallization kinetics during the hot rolling of plain C-Mn steel. Kopp [1] first introduced the microsimulating technique into the FEM modeling of forging processes. Xu [5] simulated the evolution of microstructure in ring rolling of hot steel. Shen [6] modeled microstructural development during the forging of Waspaloy. All these works focused on microstructure prediction without considering microstructure optimal design. In recent years, sensitivity analysis and optimization techniques have been introduced into the design of the metal forming. Badrinarayanan [7] developed a sensitivity analysis method for large deformations of hyperelastic viscoplastic solids that can be applied to the preform design problem in metal forming. Fourment and Chenot [8,9] developed sensitivity analysis and preform shape optimization methods to achieve desired shapes in bulking forming. Grandhi et al. [10,11] explored an optimization scheme based on the sensitivity analysis method for preform shape design of billet and die in forging processes. Hwang et al. [12] optimized the die shape using a derivative-based approach in three-dimensional extrusion. Gao et al. [13] developed a sensitivity analysis method in thermo-mechanical coupled FEM simulation of forging processes. Although there are some works on microstructure prediction [1–6] and optimum design [7–13] in metal forming processes, there are two issues that have not been studied yet. One issue is that there has been no reported work on microstructure optimization. Only Malas et al. [14] studied control of microstructure development during hot working processes based on optimal control theory and developed state-space models for describing the material behavior and the mechanics of the process. This method was applied to the optimization of grain size and process parameters such as die geometry and ram velocity during the steady extrusion of plain carbon steel. The other issue is that the process design variables, which are different from state design variables,

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are not considered in the optimum design of non-isothermal forging processes. The method of calculating sensitivities for process variables in thermo-mechanical coupled FEM simulation has not been studied yet. This work attempts to resolve these issues by developing sensitivity analysis for different design variables and optimum design for microstructure. This work proposes a new approach based on sensitivity analysis for optimizing the microstructure during the forging processes. The analytical sensitivities of microstructure with respect to the design variables are described. Not only the recrystallized grain size and recrystallization extent, but no recrystallization parts are also considered, which all affect the microstructure features in the final forging. The mean grain size is introduced to describe the complex recrystallization mechanics. The objective is to minimize the variance of mean grain size and the average value of mean grain size in the whole final product. Two constraints are imposed on the die underfill and excessive material waste. Different kinds of design variables are applied, including state parameters for initial shape of billet and process variables for die velocity. The flowchart of microstructure optimization based on shape design variables and die velocity variables in the thermo-mechanical FEM coupled simulation is described. The optimization scheme is demonstrated with the microstructure optimal design of a turbine disk forging made of Waspaloy in the non-isothermal forging process. 2. Optimal design problem In metal forming processes, the appropriate upon the final mechanical and microstructural to obtain optimal parameters for initial billet microstructure in the final forging with a high

optimality criterion places a significant emphasis states of the material. The goal of this design is shape and die velocity so as to get the desired dimensional accuracy and less flash.

2.1. Design objective Microstructural changes, especially grain sizes, play a very important role in the microstructural properties. In theory, it is best to obtain a uniform/fine grain size and a large recrystallized volume fraction. However, due to the discretization of FEM simulation and nonuniform deformation behavior, there are various grain sizes in every element no matter how fine the meshing used. Therefore, the property of the final forging should be determined by all substructures. In this work, the mean grain size in each finite element is introduced in which not only the recrystallized grain size and recrystallization extent are considered, but the grain sizes in no recrystallization part and their volume fractions are also taken into account. The objective is to minimize the variance of mean grain size of all elements and the average of mean grain size in the final forging, which is expressed as:

min

冘冕

冤

NE

i⫽1

(Dim−Dave)2dVi

冘冕

⫹bi(Dave⫺Ddes)2

NE

i⫽1

dVi

冥

(1)

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where NE is the total number of elements in the billet for this simulation; Dim is the actually obtained mean grain size for every element in the simulation; b1 is the weight factor; Ddes is the desired mean grain size and Dave is the average of mean grain size in the final product, which is expressed as:

冘冕 冘冕 NE

DimdVi

Dave⫽

i⫽1

NE

(2)

dVi

i⫽1

2.2. Constraints The primary shape requirement in forging design is to obtain the final forging which fills the die cavity and has less flash. In this work, two shape constraints are imposed on this optimal design. The first constraint is on filling the die cavity and the second constraint is on the volume constancy, which are expressed by Eqs. (3) and (4), respectively. Vdesire−Vin−desire ⬍y1 Vdesire

(3)

Vactual−Vdesire ⬍y2 Vdesire

(4)

where Vdesire and Vactual are the desired volume and actually realized volume of the final forgings, respectively; Vin-desire is the volume of actually obtained forging inside the desired final forging; y1 and y2 are two small critical values for controlling shape constraints. 2.3. Design variables In forging processes, there are two different kinds of design variables (state variables and process variables) which play important roles in the mechanical and metallurgical properties of the final products. Shape design variables for the initial billet are chosen as state variables and die velocity variables are chosen as process design variables. (1) Shape design variables. Microstructure development has a close relationship with the shape of initial billet. The shape of initial billet affects the forming processes, such as influencing the distribution of strain, strain rate and temperature, then further influencing the microstructural behavior. In practice, there are many products whose geometries are not very complex, which could be obtained by one stage forging processes. In this work, one stage forging process is used for turbine disk and the geometry of initial billet is cylinder, whose radius and height are chosen as two shape design variables. (2) Velocity design variables. Many materials are very sensitive to strain rate, and the distribution of strain rate in the forging process greatly affects the microstructure feature, especially

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for recrystallized grain size and recrystallized volume fraction. The control of strain rate is usually applied by the control of die velocity. Therefore, optimal design of die velocity profile plays an important role in the design of forming process. In the optimal design of die velocity, the desired function for die velocity should be differentiable and be able to describe die velocity using fewer parameters. Typically, the velocity of top die is given as a constant, exponential form or tabular data while the bottom die is fixed. In this work, the velocity of top die is chosen as the exponential form, which is described by three design variables pA, pB and pC: Vd⫽pAe−pBW⫹pC

(5)

where: Strokecur W⫽ Stroketotal Strokecur and Stroketotal are the strokes for the current simulation step and the final step, respectively. It should be noted that shape design variable (state parameter) and velocity design variable (process parameter) are two different kinds of design variables. Special consideration for the sensitivity information with respect to process parameters must be made in calculation of their gradients as shown in the following section. 3. Sensitivity analysis In sensitivity analysis based optimum design of forging process, the gradients of objective and constraints are needed to determine the new search direction. The procedure can be divided into three parts: sensitivity analysis for microstructure, sensitivity analysis for flow behavior and sensitivity analysis for temperature. 3.1. Microstructure behavior and its sensitivity analysis In order to optimize the microstructure in forging processes, especially to calculate the microstructure parameters and the sensitivity information of microstructure, the relationship of the microstructure behavior and the thermo-mechanical variables needs to be made available. In this optimization, Waspaloy, a nickel-base superalloy, is used which has received increasing attention, particularly for aircraft engine disks and some critical parts in high-temperature environments. The microstructure greatly affects the mechanical life of turbine engine disks, which are critical in the aircraft design. Therefore, the prediction and optimization of microstructure for Waspaloy have great importance. A model of dynamic recrystallization (DRX) kinetic for Waspaloy is described in Ref. 6. The kinetic of dynamic recrystallization (DRX) is described using critical strain, recrystallized volume fraction and dynamically recrystallized grain size. The mean grain size is introduced to describe the complex recrystallization mechanics. These microstructure changes and their gradients can be calculated as follows. (1) Critical strain. The onset of dynamic recrystallization usually occurs at a stain e¯ c (the critical strain for dynamic recrystallization), which is expressed as:

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5 0.106 e¯ c⫽ ⫻5.375⫻10−4d 0.54 (Tⱕ1027°C) 0 Z 6

(6)

5 0.106 (T⬎1027°C) e¯ c⫽ ⫻1.685⫻10−4d 0.54 0 Z 6

(7)

where d0 is the preheated grain size (prior to hot working) and Z is the Zener-Hollomon parameter (temperature-compensated strain rate), which is expressed as:

冉 冊

Q · Z⫽e¯ exp RT

Q is the activation energy for hot working (468 000 J/mole for Waspaloy) and R is the gas constant (8.314 J/mole·K). When the microstructure evolution is performed, it is necessary to determine whether or not the strain for the current point is greater than the critical strain. Before it reaches the critical amount of strain e¯ c, the recrystallization behavior cannot happen. (2) Recrystallized volume fraction and its gradients. The recrystallized volume fraction c is defined as the ratio of recrystallized grain volume to total volume. For Waspaloy, it is a function of strain, strain rate and temperature, which can be calculated as:

冉 冉 冊冊

c⫽1⫺exp ⫺ln(2)

e¯ e¯ 0.5

n

(8)

where e¯ 0.5 is the plastic strain for 50% recrystallized volume fraction, given as: e¯ 0.5⫽Ad B0 ZC

(9)

there are three regimes: A=0.145, B=0.32, C=0.03, n=3.0 (T⬍1010°C) A=0.056, B=0.32, C=0.03, n=2.0 (1010°CⱕTⱕ1027°C) A=0.035, B=0.29, C=0.04, n=1.8 (T⬎1027°C) It should be noted that the recrystallized volume fraction c is equal to zero before the critical amount of strain e¯ c is reached. Differentiating Eq. (8) with respect to each design variable, the gradient of the recrystallized volume fraction c can be calculated as:

冋 冉 冊册

e¯ ∂c ⫽⫺exp ⫺ln2 ∂pl e¯ 0.5 where:

冉 冊冋

n

(⫺nln2)

e¯ e¯ 0.5

冋 冉 冊册 冉 冊冋

Q ∂e¯ 0.5 · ⫽Ad B0 C e¯ exp ∂pl RT

C−1

exp

n−1

e¯ ∂e¯ 0.5 ∂e¯ 1 ⫺ ∂pl e¯ 0.5 (e¯ 0.5)2 ∂pl

册

· Q ∂e¯ · Q ∂T ⫺e¯ RT ∂pl RT 2 ∂pl

册

(10)

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(3) Recrystallized grain size and its gradients. The recrystallized grain size Dr defined as the grain size after dynamic recrystallization, is a function of strain rate and temperature for Waspaloy, which can be calculated as: Dr⫽FZG (mm)

(11)

where: F⫽8103.00, G⫽⫺0.16 (Tⱕ1027°C) F⫽108.85, G⫽⫺0.0456 (T⬎1027°C) Differentiating Eq. (11) with respect to each design variable, the gradient of the average recrystallized grain size can be calculated as: · Q G−1 Q ∂e¯ · Q ∂T ∂Dr ·¯ ⫽F⫻G eexp exp ⫺e¯ (12) ∂pl RT RT ∂pl RT 2 ∂pl

冋 冉 冊册 冉 冊冋

册

(4) Mean grain size and its gradients. The objective of microstructure optimal design is to obtain more uniform/fine grain size and a large recrystallized volume fraction. For Waspaloy, the recrystallization occurs in a small temperature range and a complete recrystallization does not happen in most areas during non-isothermal forging process. Therefore, the following situations may occur in final forgings: no recrystallization, partial recrystallization or complete recrystallization. If the effective strain of a substructure lies below the critical strain for the onset of dynamic recrystallization, or if the recrystallized volume fraction in the partial recrystallization is smaller than the threshold (taken as 0.05), this case is regarded as no recrystallization. If the newly recrystallized and no recrystallized fraction of the structure are larger than the given threshold value, this case is regarded as partial recrystallization. If the fraction of the unrecrystallized structure during an increment is less than the given threshold value, the recrystallization cycle for a substructure is regarded as having been completed, and this case is called complete recrystallization. When the recrystallized volume fraction is not too large, the recrystallized areas are not big enough to play the main role in the microstructure property of the final forging. Thus, the property of the total structure is determined by all substructures. The mean grain size is introduced in which this not only takes into account the recrystallized grain size and recrystallization extent, but also considers no recrystallization parts. The weighted effect of different microstructure states is considered as follows: ec(N)⫽e¯ (N)(1⫺c(N)), Dm(N)⫽D0(N)(1⫺c(N))⫹Dr(N)c(N)

(13)

where N is element number; c(N) is recrystallized volume fraction; Dr(N) is recrystallized grain size; ec(N) is residual strain; e¯ (N) is effective strain; Dm(N) is the mean grain size and D0(N) is initial grain size. Then no recrystallization and complete recrystallization can be treated as two special cases. For no recrystallization: c(N)=0.0, Dr(N)=0.0 ec(N)=e¯ (N), Dm(N)=D0(N)

(14)

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For complete recrystallization: c(N)=1.0, Dr(N)=Dr(N) ec(N)=0.0, Dm(N)=Dr(N)

(15)

Differentiating Eq. (13) with respect to each design variable, the gradient of the mean grain size can be expressed by gradients of average recrystallized grain size and recrystallized volume fraction: ∂c ∂Dr ∂Dm ⫽(Dr⫺D0) ⫹ c ∂pl ∂pl ∂pl

(16)

From Eqs. (10, 12) and (16), it can be seen that these gradients can all be expressed by sensitivities of strain, strain rate and temperature with respect to each design variable — which can be obtained in the sensitivity analysis for flow behavior and temperature. 3.2. Sensitivity analysis of flow behavior The method of calculating the sensitivities of effective strain and effective strain-rate with respect to each design variable in non-isothermal FEM simulations has additional complexities over the formulation in isothermal simulation. There are some differences in sensitivity analysis for shape design variables and die velocity variables. For rigid-viscoplastic materials, s¯ is the flow stress, which can be expressed as a function of effective strain, effective strain-rate and temperature: · s¯ ⫽s¯ (e¯ ,e¯ ,T) (17) Thus, the sensitivity of flow-stress needs the derivatives of flow stress with respect to each variable, given as: · ∂s¯ ∂s¯ ∂e¯ ∂s¯ ∂e¯ ∂s¯ ∂T ⫽ ⫹ · ⫹ (18) ∂pl ∂e¯ ∂pl ∂e¯ ∂pl ∂T ∂pl Using finite element simulations, the nodal coordinates, effective strains and effective strainrates are updated at the incremental simulation steps using the following equations: X(t+⌬t)=X(t)+V(t)⌬t · e¯ (t+⌬t)=e¯ (t)+e¯ ⌬t · e¯ = VTBTDBV

冑

(19)

· where X(t), V(t), e¯ (t) and e¯ are the nodal coordinate vector, nodal velocity vector, effective strain and effective strain-rate at time t, respectively; X(t+⌬t) and e¯ (t+⌬t) are the nodal coordinate vector and effective strain at time t+⌬t, respectively; B is the elemental strain-rate matrix and D is the effective strain-rate coefficient matrix.

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Then the sensitivities of nodal coordinates, effective strains and effective strain-rates with respect to each design variable pl can be calculated as follows: ∂(⌬t) ∂X(t+⌬t) ∂X(t) ∂V(t) ⫽ ⫹ ⌬t⫹V(t) ∂pl ∂pl ∂pl ∂pl · ∂e¯ (t+⌬t) ∂e¯ (t) ∂e¯ · ∂(⌬t) ⫽ ⫹ ⌬t⫹e¯ ∂pl ∂pl ∂pl ∂pl · ∂V ∂e¯ 1 T T ∂B ⫽ · V B D V⫹VTBTDB ∂p1 e¯ ∂pl ∂pl

冉

(20) (21)

冊

(22)

It should be noted that when the shape parameters are chosen as design variables, there is ∂(⌬t) /∂pl =0. When the die velocity parameters are chosen as design variables, the die velocity is considered as a constant over each time step, therefore: ⌬t⫽

⌬s Vd

(23)

where ⌬t and ⌬s are the time step and stroke increment for each time step, respectively. Differentiating Eq. (23) with respect to each design variable for the die velocity, yields: 1 ∂Vd 1 ∂Vd ∂(⌬t) ⫽⫺⌬s 2 ⫽⫺⌬t ∂pl Vd ∂pl Vd ∂pl

(24)

From Eqs. (20)–(22), it can be seen that the sensitivities of nodal coordinates, effective strains and effective strain-rates are all described as the functions of the nodal velocity sensitivities, which can be provided by direct differentiation of the element stiffness equations (given in Refs. 10 and 13). Therefore, the sensitivity for flow behavior with respect to each design variable, including shape design variables and die velocity variables can be obtained. 3.3. Sensitivity analysis of temperature THE thermal equilibrium during metal forming process can be expressed in the following form [15]: CT˙⫹KcT⫽Q

(25)

where T and T˙ are the vectors of nodal temperatures and nodal temperature-rates, respectively; C, Kc and Q are the heat capacity matrix, heat conduction matrix and heat flux vector, respectively. The following equation is employed for the integration of the heat evolution equation: Tt+⌬t⫽Tt⫹⌬t[(1⫺b)T˙t⫹bT˙t+⌬t]

(26)

where b is a parameter varying between 0 and 1 (usually chosen as 0.75) and t denotes time; Tt

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and T˙t are the nodal temperature and nodal temperature rate at time t, respectively and Tt+⌬t, T˙t+⌬t are the nodal temperature and nodal temperature rate at time t+⌬t, respectively. Substituting Eq. (26) into Eq. (25), results in:

冉

Kc⫹

冊

C · Tt+⌬t⫽Qt+⌬t⫺CTˆ b⌬t

(27)

where Qt+⌬t is the heat flux vector at time t+⌬t and the circumflex over T˙ denotes:

冉 冊

Tt 1−b · T˙t Tˆ⫽⫺ ⫺ b⌬t b

(28)

The sensitivities of nodal temperatures with respect to the design variables can be obtained by differentiating the heat equilibrium equation (the details of differentiating each term of the heat equilibrium equation are shown in Ref. 13). The new equation of heat equilibrium can be expressed as: · ¯ ∂Tt+⌬t C ∂(⌬t) ∂Tˆt T −T ¯ ¯ ¯ t t+⌬t ¯ Kc⫹ ⫽Qt+⌬t⫺C ⫺C (29) 2 b⌬t ∂pl ∂pl b⌬t ∂pl

冉

冊

where the new heat conduction matrix K¯c includes all terms with ∂T/∂pl; the new heat capacity ¯ includes all terms without ¯ includes all terms with ∂T˙/∂pl and the new heat flux vector Q matrix C ˙ ∂T/∂pl and ∂T/∂pl. Sensitivities of nodal temperature and nodal temperature rate with respect to the design variable pl have the following relationship: · ∂Tt+⌬t 1 ∂T⌬t ∂Tˆt Tt−Tt+⌬t ∂(⌬t) ⫽ ⫹ ⫹ (30) ∂pl b⌬t ∂pl ∂pl b⌬t2 ∂pl After evaluating the sensitivities of the thermal equation, they are assembled for the whole workpiece. The sensitivity of temperature with respect to the design variable pl can be obtained by solving the set of simultaneous nonlinear equations. Then the sensitivities of objective and constraints with respect to each design variable, including shape design variable and die velocity design variable can be obtained. 3.4. Optimal design procedure The procedure for microstructure shape optimal design in thermo-mechanical coupled FEM analysis is shown in Fig. 1. From the figure, it can be seen that for a given initial guess of the billet shape and die velocity, the thermo-mechanical FEM analysis is performed after the initial velocity field is generated. First, the thermal analysis for billet and die is performed in each incremental simulation step. The thermal analysis for billet is performed assuming the temperatures of die are not changed. Similarly, the temperatures of billet are not changed during the thermal analysis of die. Then the flow analysis is performed. When the thermal analysis (for billet and die) and the flow analysis are converged, the sensitivities of nodal velocity and nodal tempera-

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Fig. 1. Flow chart of microstructure optimal design in thermo-mechanical coupled FEM analysis.

ture with respect to shape design variables and die velocity design variables are calculated if they are necessary. The sensitivities of the nodal coordinates, effective strain and temperature to the design variables are updated iteratively because of history dependent parameters. When the final stroke is reached, the microstructure parameters, such as recrystallized volume fraction, dynamically recrystallized grain size, residual strain, mean grain size and their gradients are calculated. Then, the objective, constraints and their gradients are obtained. The optimization program DOT (Design Optimization Tool) [16] is used to provide the new design variables for the initial billet shape and die velocity. SQP (Sequential Quadratic Programming) in the program DOT is used to perform the optimization. Sensitivity information is not necessary for every simulation cycle, and is only provided if DOT needs this information.

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4. Design study The forging process of an axisymmetric turbine disk is optimized in this work. The materials of billet and die are Waspaloy and H-13 steel, respectively. Flow stress values for Waspaloy at different temperatures and strain rates at a strain of 3.0 are given in Ref. 17. The initial shape of billet is a cylinder with 58.0 mm height and 70.0 mm radius. The initial grain size of billet is 65 µm (ASTM =5.0). The initial parameters for die velocity pA, pB and pC are 1.0, 2.0 and 1.0, respectively. The friction coefficient is 0.25. The thermal parameters are given as follows: the heat conductivity for the billet and die is 25 N/s°C and 19 N/s°C, respectively; the heat capacities for the billet and die are all 3.77 N/mm2°C; the coefficients of radiation and heat convection are 8.5×10⫺12 N/mm.s(°C)4 and 0.02 N/mm.s°C for billet and die, respectively; the interface heat transfer coefficient for lubricant is 2.00 N/mm.s°C; the initial temperatures of the billet and die are 1100°C and 300°C, respectively, and the environment temperature is 20°C. When the initial billet is heated to 1100°C, it is directly put on the bottom die. After the deformation process, the forging is quenched into cold water immediately. The transfer time before or after deformation process is ignored. The initial shape of billet and geometry of die are shown in Fig. 2. A quarter model of billet and die are used for this simulation due to axisymmetry. Four points are used to define the initial shape of the billet. Point 3 has two degrees of freedom, which allow it to move in x direction and y direction. Point 2 and 4 are on the symmetric axis which are allowed to move in the x direction and y direction, respectively. Point 1 is fixed because it is symmetric to x and y axis. In theory, only one design variable could be used to define the initial shape of billet if the volume of workpiece is a constant. However, due to the complexity in FEM simulation, especially a lot of remeshing and the complex processing of contact boundary, it is hard to avoid volume loss and keep the volume constant during the simulation. In this design, two independent shape design variables are used to describe the shape of billet and a small volume loss of the billet is permitted.

Fig. 2. Initial geometry of billet and die.

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The coordinates of 3-x and 3-y are chosen as independent shape design variables while the coordinates of 2-x and 4-y are chosen as dependent shape design variables. Physical linkings are used on points 2, 3 and 4, such as the coordinate of 2-x is equal to the one of 3-x and the coordinate of 4-y is equal to the one of 3-y in order to keep the cylinder shape of initial billet. Side bounds are placed on independent design variables to obtain realistic shapes. The final forging using the non-optimized design for initial billet and die velocity is as shown in Fig. 3, in which the material could not fill the die cavity and the volume of flash is very large. The distribution of microstructure parameters in the final forging, such as dynamic recrystallized volume fraction, recrystallized grain size, mean grain size and residual strain, are shown in Figs. 4(a), (b), (c) and (d) respectively. The variance of mean grain size is 87.30 µm2 and the average value of mean grain size is 45.01 µm. From Fig. 4(a), it can be seen that the recrystallized volume fraction is large at the center area of the final forging. It can be explained that due to larger effective strain and higher temperature, there is a higher dislocation density and the recrystallization behavior takes place more adequately. In the center area contacting with the die, the recrystallized volume fraction is not very large because of smaller effective strain and lower temperature on the interface of die and billet. The recrystallized volume fraction for the material under the die deep cavity is very small because of small effective strain. From Fig. 4(b), it can be seen that the recrystallized grain sizes in the flash area are not too large due to lack of enough time for recrystallization grain growth. From Fig. 4(c), it can be seen that all areas which have higher recrystallized volume fraction shown in Fig. 4(a) have smaller mean grain size because recrystallization leads to significant grain refinement from the orignal size. From Fig. 4(d), it can be seen that there is smaller residual strain in the area where the recrystallization behavior happens adequately and most structures are softened to some extent. All these prove that the established method of microstructure simulation is feasible and effective. Two optimal designs are done in this work. The same constraints and the same design variables are used in case 1 and case 2, which are described in Section 2. y1 and y2 in Eqs. (3) and (4) are given as 0.002 and 0.01, respectively. The initial non-optimized design described above is used in both cases. The only difference of these two cases is the objective function. In case 1, the design objective is to minimize the variance of effective strain in order to obtain more uniform distribution of effective strain in the final forging. In case 2, the design objective is to minimize the combination of variance of mean grain size and the average value of mean grain size in the

Fig. 3. Final forging shape for the non-optimized design.

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Fig. 4. Microstructure in the final forging shape for the non-optimized design. (a) Recrystallized volume fraction; (b) Recrystallized grain size; (c) Mean grain size; (d) Residual strain.

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Fig. 4. (continued)

final forging. The weight factor b1 is chosen as 1.0 and the desired mean grain size Ddes is given as 30.0 µm. In case 1, the optimal design is converged after five iterations. The objective function, the variance of effective strain in the workpiece at the final forging is reduced to 0.056 from 0.096. The variance of mean grain size is reduced to 80.14 µm2 from 87.30 µm2 and the average value of mean grain size is reduced to 42.03 µm from 45.01 µm. The other microstructure parameters, such as the variances and the average values of recrystallized volume fraction, recrys-

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tallized grain size and residual strain are also given in Table 1. Comparing the optimal results with the baseline, it can been seen that these microstructure parameters are not changed significantly because they are not optimized. In case 2, the optimal design is converged after seven iterations. The die velocity profiles for non-optimized and optimized cases are shown in Fig. 5. It can be seen that the die velocity is increased during the forging process, which is of benefit to reduce the heat loss on the interface of billet and die, and causes adequate recrystallization. The non-optimized and optimized results are compared in Table 1. The optimal shape of initial billet is shown in Fig. 6. The radius of the initial billet is decreased and it benefits a reduction in flash volume. The optimal height is increased which is beneficial to fill the deep die cavity. The final forging using the optimized billet and die velocity is shown in Fig. 7, in which the die cavity is filled completely with less flash. The shape constraint on cavity volume is reduced to 0.0016 from 0.0045 and the shape constraint on flash volume is also reduced to 0.006 from 0.013. The distribution of microstructure parameters in this final forging is shown in Fig. 8. The dynamic recrystallized volume fractions in most areas are larger than the corresponding areas in Fig. 4(a) and the recrystallized volume fraction in the center area in Fig. 8(a) has reached 1.0, which means a complete recrystallization occurs. The recrystallized volume fraction around the interface of die and billet in the center area is not too large because of lower temperature. The variance of mean grain size is reduced to 58.31 µm2 from 87.30 µm2 and the average value of mean grain size is also reduced to 32.64 µm (ASTM ⬇7.0) from 45.01 µm (ASTM ⬇6.0). This shows that the mean grain sizes in the Table 1 Comparison of the baseline and optimized designs for different objectivesa

Design parameters Billet radius (mm): pW Billet height (mm): pH Die velocity 1: pA Die velocity 2: pB Die velocity 3: pC Constraints and objective Constraint 1 (die cavity under fill) Constraint 2 (flash volume) Variance of effective strain Variance of mean grain size (µm2) Average of mean grain size (µm) Performance Variance of recrystallized volume fraction Average of recrystallized volume fraction Variance of recrystallized grain size (µm2) Average of recrystallized grain size (µm) Variance of residual strain Average of residual strain a

Baseline

Optimum (Case 1)

Optimum (Case 2)

70.00 29.00 1.00 2.00 1.00

61.04 39.04 0.59 1.56 1.10

59.07 41.32 0.75 1.93 1.20

0.0045 0.013 0.096 87.30 45.01

0.0018 0.008 0.056 (obj) 80.14 42.03

0.0016 0.006 0.089 58.31(obj) 32.64(obj)

0.052 0.677 9.30 26.06 0.006 0.152

0.045 0.701 10.53 25.11 0.006 0.144

0.048 0.797 10.92 25.27 0.005 0.139

Case 1: Minimize effective strain variance in the workpiece at the final forging Case 2: Minimize the combination of mean grain size variance and the difference of average and desired mean grain size in the workpiece.

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Fig. 5. Comparison of non-optimized die velocity and optimal die velocity.

Fig. 6.

Optimal initial shape of billet.

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Fig. 7.

Final forging shape using the optimal design for initial billet shape and die velocity.

final forging are more uniform and finer using the optimized design than the ones using the nonoptimized design. Due to the larger recrystallized volume fraction, the residual strains in most areas for optimized design (Fig. 8(d)) are smaller than the residual strains for non-optimized design (Fig. 4(d)). Comparing the optimal results of case 1 and case 2, it can be seen that different design objectives have great influence on the realized optimal results. In case 1, the variance of effective strain is changed greatly (reduced to 0.056 from 0.096), but the combined function of the variance of mean grain size and the average value of mean grain size is not changed too much (reduced to 224.86 µm2 from 312.60 µm2). In case 2, the objective function for the variance of mean grain size and the average value of mean grain size is changed significantly (reduced to 65.28 µm2 from 312.60 µm2), but the variance of effective strain is changed very small (reduced to 0.089 from 0.096). This can be explained that the microstructure change is affected not only by strain, but also by strain rate, temperature and other factors. This proves that the microstructure optimization is necessary if the design goal is to get the desired microstructure. 5. Summary In this work, a sensitivity analysis method for microstructure optimization in the forming processes is developed. The forming process design of a turbine disk made of Waspaloy is performed. Microstructure prediction is combined with the thermo-mechanical FEM simulation, and a sensitivity analysis method for flow behavior, temperature and microstructure is developed. Two shape constraints are imposed on die underfill and excessive material waste. Two different kinds of design variables, including the shape design variables and the die velocity variables are used. By comparing the optimized and non-optimized results, it can be seen that the microstructure properties of the final forging using the optimal design are much better than the non-optimized design. More uniform and fine grain sizes with less flash and complete die cavity fill were obtained. This demonstrates that the microstructure simulation and optimization scheme based on sensitivity analysis is effective. In the future work, the microstructure optimization for the multi-stage forging processes will be investigated.

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Fig. 8. Microstructure in the final forging shape for the optimal design. (a) Recrystallized volume fraction; (b) Recrystallized grain size; (c) Mean grain size; (d) Residual strain.

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Fig. 8. (continued)

Acknowledgements This research work has been sponsored by the National Science Foundation grant DMI9424649.

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