US20180001380A1 - Casting simulation method - Google Patents

Casting simulation method Download PDF

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US20180001380A1
US20180001380A1 US15/544,595 US201615544595A US2018001380A1 US 20180001380 A1 US20180001380 A1 US 20180001380A1 US 201615544595 A US201615544595 A US 201615544595A US 2018001380 A1 US2018001380 A1 US 2018001380A1
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inelastic
strain
effective
stress
equivalent
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Makoto Yoshida
Yuichi MOTOYAMA
Toshimitsu OKANE
Yoya Fukuda
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Waseda University
National Institute of Advanced Industrial Science and Technology AIST
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    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B22—CASTING; POWDER METALLURGY
    • B22D—CASTING OF METALS; CASTING OF OTHER SUBSTANCES BY THE SAME PROCESSES OR DEVICES
    • B22D46/00—Controlling, supervising, not restricted to casting covered by a single main group, e.g. for safety reasons
    • C—CHEMISTRY; METALLURGY
    • C21—METALLURGY OF IRON
    • C21D—MODIFYING THE PHYSICAL STRUCTURE OF FERROUS METALS; GENERAL DEVICES FOR HEAT TREATMENT OF FERROUS OR NON-FERROUS METALS OR ALLOYS; MAKING METAL MALLEABLE, e.g. BY DECARBURISATION OR TEMPERING
    • C21D11/00—Process control or regulation for heat treatments
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N3/00—Investigating strength properties of solid materials by application of mechanical stress
    • G01N3/08—Investigating strength properties of solid materials by application of mechanical stress by applying steady tensile or compressive forces
    • G01N3/18—Performing tests at high or low temperatures
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • G06F30/23—Design optimisation, verification or simulation using finite element methods [FEM] or finite difference methods [FDM]
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2113/00—Details relating to the application field
    • G06F2113/22—Moulding
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F2119/00—Details relating to the type or aim of the analysis or the optimisation
    • G06F2119/08—Thermal analysis or thermal optimisation

Definitions

  • This disclosure relates to a casting simulation method using thermal stress and deformation analysis.
  • JP2007330977A PTL 1
  • Dong Shuxin Yasushi Iwata, Toshio Sugiyama, and Hiroaki Iwahori, “ Cold Crack Criterion for ADC 12 Aluminum Alloy Die Casting”, Casting Engineering, 81(5), 2009, pp. 226-231, ADC12 (NPL 1) are reference examples of Toyota Central R&D Labs. Inc.
  • the amount of equivalent inelastic strain (including the amount of plastic strain and the amount of viscoplastic strain) is used as a measure of hardening, and inelastic strain that occurs at such a high temperature at which recovery occurs simultaneously with deformation is also treated as contributing to work hardening as much as inelastic strain produced at room temperature. This causes an unrealistic increase in yield stress at room temperature, causing problems in the accuracy of thermal stress analysis.
  • NPL 2 Hallvard et al. proposes a constitutive equation that is expressed as Constitutive Eq. (I) below representing the relationship between stress and inelastic strain as described below, and that considers inelastic strain produced at or above a certain temperature as not contributing to hardening, while the other produced below that temperature as contributing to work hardening. From a metallurgical viewpoint, however, it is clear that recovery does not happen suddenly at a certain temperature. Therefore, this constitutive equation has a problem.
  • Van Haaften et al. proposes a constitutive equation that is expressed as Constitutive Eq. (II) below between stress and inelastic strain, a function a which is 0 at high temperature and 1 at low temperature is used to smoothly consider the contributions of inelastic strain produced at different temperatures to work hardening.
  • Alankar et al. proposes a constitutive equation that is expressed as Constitutive Eq. (III) below representing the relationship between stress and inelastic strain.
  • Constitutive Eq. (III) representing the relationship between stress and inelastic strain.
  • recovery occurs more frequently as the ratio of a work hardening index at high temperature n (T) to a work hardening index at room temperature n RT decreases.
  • n (T) /n RT determines the ratio between plastic strain contributing to work hardening and creep strain not contributing to work hardening (strain making no contribution to work hardening).
  • K (T) , n (T) , and m (T) in the following equation cannot be determined accurately.
  • the present disclosure has been developed in view of the above circumstances, and provides a casting simulation method using thermal stress and deformation analysis, in which an amount of equivalent inelastic strain effective for work hardening is determined by multiplying an equivalent inelastic strain rate calculated in the analysis by an effective inelastic strain coefficient ⁇ representing a proportion of inelastic strain contributing to work hardening to obtain an effective inelastic strain rate, and integrating it over a time from 0 second in analysis, and the amount of equivalent inelastic strain thus obtained is used as a measure of work hardening in a constitutive equation.
  • the present disclosure also provides a method of experimentally determining an effective inelastic strain coefficient ⁇ (T) which represents the contributions of inelastic strain produced at different temperatures to work hardening.
  • T denotes a temperature with inelastic strain
  • h (T) denotes an increment of yield strength at room temperature with respect to an amount of inelastic strain at the temperature with inelastic strain
  • h (RT) denotes an increment of yield strength at room temperature with respect to an amount of inelastic strain applied at room temperature
  • h (T) /h (RT) denotes an effective inelastic strain coefficient ⁇ (T)
  • ⁇ inelastic / ⁇ t denotes an equivalent inelastic strain rate
  • t denotes a time from 0 second in analysis.
  • the effective inelastic strain coefficient ⁇ (T) is obtained by: applying different inelastic pre-strains to a test piece at different temperatures; cooling the test piece to room temperature; performing a tensile test or a compression test on the test piece at room temperature; and measuring influence of amounts of the inelastic pre-strains applied at the different temperatures on the increase in yield stress.
  • FIG. 1 a graph conceptually illustrating the temperature history of a test piece in a test for determining an effective inelastic strain coefficient
  • FIG. 2 is a graph conceptually illustrating the influence of amounts of inelastic pre-strains at different temperatures on yield stress rise at room temperature;
  • FIG. 3 is a graph illustrating a temperature history of JIS ADC12 used in tests for determining an effective inelastic strain coefficient in examples
  • FIG. 4 is a graph illustrating the experimental results obtained in examples of examining the influence of inelastic pre-strains at different temperatures on the 0.2% offset yield stress of JIS ADC12 at room temperature;
  • FIG. 5 is a graph illustrating material constants K, m, n, and ⁇ obtained for JIS ADC12 in examples
  • FIG. 6 is a graph illustrating the results obtained in examples of comparing experimental values with calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at room temperature;
  • FIG. 7 is a graph illustrating the results obtained in examples of comparing experimental values with calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at 200° C.;
  • FIG. 8 is a graph illustrating the results obtained in examples of comparing experimental values and calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at 250° C.;
  • FIG. 9 is a graph illustrating the results obtained in examples of comparing experimental values and calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at 300° C.;
  • FIG. 10 is a graph illustrating the results obtained in examples of comparing experimental values and calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at 350° C.;
  • FIG. 11 is a graph illustrating the results obtained in examples of comparing experimental values and calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at 400° C.;
  • FIG. 12 is a graph illustrating the results obtained in examples of comparing experimental values and calculated values according to the present disclosure in stress-strain curves for JIS ADC12 at 450° C.;
  • FIG. 13 is a diagram illustrating “the influence of inelastic pre-strains at room temperature and at 450° C. on the increase in yield stress at room temperature” calculated in examples using the conventional extended Ludwik's law;
  • FIG. 14 is a graph illustrating “the influence of inelastic pre-strains applied at different temperatures on increase in yield stress at room temperature” calculated in examples according to the present disclosure
  • FIG. 15 is a graph illustrating the results obtained in examples of comparing experimental values and calculated values according to the present disclosure to examine the influence of inelastic pre-strains applied at different temperatures on the increase in 0.2% offset yield stress at room temperature;
  • FIG. 16 is a graph illustrating experimental results in examples of examining the influence of inelastic pre-strains applied at different temperatures on the 0.2% offset yield stress of FCD400 at room temperature.
  • an equivalent inelastic strain rate calculated by thermal stress analysis is multiplied by an “effective inelastic strain coefficient ⁇ (T) indicative of temperature dependency”, which is preferably experimentally determined, and the result is used as an effective inelastic strain rate.
  • the effective inelastic strain rate is then integrated over a time from 0 second in analysis to determine the amount of effective equivalent inelastic strain, which in turn is, in place of the amount of equivalent inelastic strain, used as a measure of hardening in a constitutive equation.
  • the amount of effective equivalent inelastic strain is applicable to any constitutive equation, whether an elasto-plastic constitutive equation or an elasto-viscoplastic constitutive equation, as long as it is a constitutive equation using the amount of equivalent inelastic strain conventionally as a measure of work hardening as described in paragraph 0008.
  • the determination of the effective inelastic strain coefficient ⁇ (T) is carried out by applying inelastic pre-strains of different magnitudes at the corresponding temperatures at which ⁇ (T) is to be obtained, then cooling to room temperature, conducting a tensile test or a compression test, and measuring the increase in yield stress.
  • a tensile test piece is heated in a way as presented in the temperature history in FIG. 1 .
  • T 3 be the temperature at which ⁇ (T) is to be determined
  • a single inelastic pre-strain is applied at T 3 in T 1 , T 2 , T 3 , . . . , T n per test.
  • the test piece is cooled to room temperature and subjected to a tensile test or a compression test at room temperature.
  • measurement is made of the increase in yield stress.
  • This procedure is repeated for inelastic pre-strains at different temperatures and at different magnitudes. In this way, the influence of the amounts of the inelastic pre-strains applied at the different temperatures on the increase in yield stress at room temperature is measured.
  • the same temperature history is set for all test conditions.
  • the reason for this is to eliminate the influence of the temperature history of the test piece on the measured values.
  • the above-described test is not limited to a particular test, and may be a tensile test or a compression test as long as it can provide a stress-strain curve and enables measurement of yield stress.
  • a tensile test in this disclosure for example, a publicly-known and widely-used tensile test may be used, such as JIS Z 2241:2011.
  • a publicly-known and widely-used compression test may be used, such as JIS K 7181:2011.
  • FIG. 2 illustrates the influence of inelastic pre-strains applied at different temperatures on the increase in yield stress at room temperature.
  • h a gradient of yield stress with respect to the amount of inelastic pre-strains applied at room temperature
  • T h
  • the value of h (T) /h (RT) is used as an effective inelastic strain coefficient ⁇ (T), which is indicative of how much the inelastic strain produced at the temperature T contribute to work hardening at room temperature with respect to the inelastic strain produced at room temperature at which all the inelastic strains produced contribute to work hardening.
  • an effective inelastic strain coefficient ⁇ (T) is experimentally determined to be 0 or a negative value, i.e., in which inelastic strain applied at the temperature T should not contribute to work hardening at room temperature
  • a constitutive equation to which the amount of effective equivalent inelastic strain is introduced involves no effective inelastic strain, and thus is not able to express work hardening in principle.
  • the stress-strain curve displays elasto-perfectly plastic behavior or elasto-perfectly viscoplastic behavior. In that case, the reproducibility of the stress-strain curve deteriorates, resulting in lower accuracy of predictions on thermal stress and deformation.
  • An effective inelastic strain is produced as long as the effective inelastic strain coefficient ⁇ (T) is not 0, making it possible to express work hardening in the stress-inelastic strain curve at the temperature T. Accordingly, even in a temperature range with the inelastic strain coefficient ⁇ (T) being 0 or a negative value, if a positive small value, rather than 0 or a negative value, is corrected appropriately for use as an effective inelastic strain coefficient ⁇ (T), it is possible to express work hardening in a stress-equivalent inelastic strain curve with a slight ineffective inelastic strain.
  • ⁇ (T) at the time of correction is in a range of 0 ⁇ 0.5, and desirably 0 ⁇ 0.1, although it depends on the alloy type.
  • ⁇ (T) is corrected by the following linear interpolation using an effective inelastic strain coefficient at maximum temperature ⁇ min in a temperature range in which ⁇ (T) is experimentally determined to be non-zero and a maximum temperature T max (which may alternatively be a solidus temperature) in a temperature range in which ⁇ is experimentally determined to be 0:
  • ⁇ extrapolation (T) denotes a value of a as corrected in a temperature range in which ⁇ is experimentally determined to be 0;
  • T max denotes the maximum temperature (which may alternatively be a solidus temperature) in a temperature range where a is experimentally determined to be 0,
  • T min denotes a maximum temperature in a temperature range in which ⁇ is experimentally determined to be non-zero
  • ⁇ min denotes a value of ⁇ in T min ;
  • T denotes a temperature above T min
  • material constants in a constitutive equation to which the amount of effective inelastic strain is introduced are determined as explained below in the case of introducing the amount of effective inelastic strain to the constants (K(T), n(T), and m(T)) of the extended Ludwik's law.
  • ⁇ 0 is a constant necessary for calculation and usually a small value of 1 ⁇ 10 ⁇ 6 .
  • the term with an index n (T) representing the degree of work hardening includes the amount of effective equivalent inelastic strain as a variable.
  • the term with an index m (T) representing the strain rate dependence of the stress-strain curve includes an equivalent inelastic strain rate as a variable. Since the equation as a whole includes the amount of effective equivalent inelastic strain, for each temperature, by substituting the inelastic strain rate into the term with m (T) , K (T) , n (T) , and m (T) are determined by numerical optimization to fit the stress-effective equivalent inelastic strain curve.
  • the amount of effective equivalent inelastic strain ⁇ effective inelastic is determined.
  • the amount of effective equivalent inelastic strain is used to simulate the influence of inelastic strain applied at the temperature in question on work hardening at room temperature.
  • ⁇ ij ⁇ e ij + ⁇ p ij
  • T is the temperature
  • ⁇ ij is the stress
  • ⁇ eff. is the equivalent stress
  • ⁇ ij is the total strain
  • ⁇ e ij is the elastic strain
  • ⁇ p ij is the plastic strain
  • f is the yield function
  • D ijkl is the fourth-order constitutive tensor.
  • an exemplary elasto-viscoplastic constitutive equation is:
  • ⁇ ij ⁇ e ij + ⁇ vp ij
  • ⁇ eff. F ( ⁇ eff. vp , ⁇ dot over ( ⁇ ) ⁇ eff. vp ,T )
  • ⁇ eff. is the equivalent stress
  • ⁇ eff. vp is the equivalent viscoplastic strain
  • ⁇ dot over ( ⁇ ) ⁇ eff. vp is the equivalent viscoplastic strain rate
  • the amount of effective equivalent inelastic strain ⁇ effective inelastic defined by Eq. (1) instead of the amount of equivalent inelastic strain conventionally used, may be introduced or substituted.
  • material constants in a constitutive equation are determined in step (II) using an equivalent stress-effective equivalent inelastic strain curve, and are input to a constitutive equation to which the amount of effective inelastic strain is introduced. Then, in the thermal stress analysis step (IV), the amount of effective equivalent inelastic strain is calculated, and the result, instead of the amount of equivalent inelastic strain conventionally used, is used as a parameter representing the amount of work hardening to calculate thermal stress.
  • JIS ADC12 a typical aluminum die-casting alloy, JIS ADC12, is analyzed for an effective inelastic strain coefficient ⁇ (T) and material constants (K(T), n(T), and m(T)) at each temperature according to the procedures described in paragraphs 0026 to 0033.
  • tensile tests were performed to obtain stress-equivalent inelastic strain curves required to determine material constants K(T), n(T), and m(T).
  • Stress-inelastic strain curves were obtained under a set of conditions including: experimental strain rates of 10 ⁇ 3 /s and 10 ⁇ 4 /s and test temperatures of RT, 200° C., 250° C., 300° C., 350° C., 400° C., and 450° C.
  • Each test pieces was obtained by casting JIS ADC12 in a copper mold and processing it into the shape of a tensile test piece.
  • test pieces were heated from room temperature to 450° C., then subjected to heat treatment at 450° C. for 1 hour to cause precipitates to be re-dissolved, and cooled to the test temperature as soon as possible so that the mechanical properties at the time of cooling can be examined accurately. As soon as the test temperature was reached, the tensile test was carried out.
  • each test piece was cooled to room temperature and quenched with dry ice to eliminate the effect of the increase in yield stress caused by natural aging. Then, the 0.2% offset yield stress was determined by conducting a tensile test on each test piece at room temperature. The results are presented in FIG. 4 .
  • the 0.2% offset yield stress was 106 MPa. Based on the results presented in the figure, determinations of an effective inelastic strain coefficient ⁇ (T) were made of the rate of increase in 0.2% offset yield stress with respect to the increase in inelastic pre-strain at each temperature, that is, h (T) , and of the value of h (T) at room temperature.
  • the effective inelastic strain coefficient is 0 by definition, yet any inelastic strain produced contributes to work hardening at room temperature, although not depending on the quantity, and exhibits work hardening even in a stress-equivalent inelastic strain curve.
  • the effective inelastic strain coefficient becomes 0 and any inelastic strain produced does not contribute to work hardening at room temperature.
  • ⁇ was set to 0.000185 at 350° C., 0.0000927 at 400° C., and 0 at 450° C., while correcting the effective inelastic strain coefficient at 350° C. and 400° C. from 0 to a very small positive value, so that work hardening could be expressed in the stress-inelastic strain curve in the temperature range of 350° C. to 400° C. and almost no effective inelastic strain would be produced.
  • the stress-equivalent inelastic strain curve obtained in paragraph 0040 was transformed into a stress-effective equivalent inelastic strain curve, and the values of K (T) , m (T) , and n (T) were obtained as described in paragraphs 0032 and 0033.
  • the values of K (T) , m (T) , and n (T) are presented in FIG. 5 .
  • FIGS. 6 to 12 each illustrate stress-inelastic strain curves to compare experimental values with calculated values according to the extended Ludwik equation to which the amount of effective equivalent inelastic strain determined by Eq. (1) is introduced.
  • tensile tests were performed at strain rates of 10 ⁇ 3 /s and 10 ⁇ 4 /s at RT, 200° C., 250° C., 300° C., 350° C., 400° C., and 450° C.
  • all the constitutive equations incorporating the amount of effective equivalent inelastic strain accurately reproduced the strain rate dependence and the shape of the corresponding stress-inelastic strain curve. In this case, we faithfully followed the data obtained in the limited experimental temperature range and considered 450° C.
  • inelastic strains produced in different temperature ranges are all considered as equivalent to one another and included as a measure of hardening. Accordingly, as is clear from FIG. 13 , inelastic strains produced at 450° C. as well as those produced at room temperature contributed to an increase in yield stress, and thus to an unrealistic increase in yield stress. It is noted here that inelastic strains applied at other temperatures also have the same results as at 450° C. with overlapping plots, and thus they are omitted in the figure.
  • the analysis program incorporating the amount of effective equivalent inelastic strain could reproduce the behavior at 300° C. or higher at which an increase in yield stress is independent of the amount of inelastic pre-strain.
  • this program could accurately reproduce the behavior even at 300° C. or lower at which an increase in yield stress depends on the amount of inelastic pre-strain.
  • FIG. 16 depicts the influence of experimentally obtained inelastic strain at high temperature on the 0.2% offset yield stress of a typical cast iron, JIS FCD400, at room temperature.
  • pre-strain applied at 700° C. does not contribute to work hardening at room temperature.
  • inelastic strain applied at 350° C. contributes to work hardening at room temperature and the amount of work hardening is proportional to the amount of inelastic strain applied. This behavior is identical to that observed in ADC12, and from this follows that the present disclosure is also applicable to FCD400.

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CN110135026A (zh) * 2019-04-29 2019-08-16 西北工业大学 多尺度物理本构模型嵌入abaqus的方法
CN111272552A (zh) * 2020-02-28 2020-06-12 鞍钢股份有限公司 一种变速率拉伸曲线评价方法
CN114115165A (zh) * 2022-01-28 2022-03-01 深圳市北工实业有限公司 一种铸锻一体成型机的生产控制方法及系统

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JP4670542B2 (ja) * 2005-08-11 2011-04-13 日本電気株式会社 材料定数算出装置および方法
US8214182B2 (en) * 2009-05-12 2012-07-03 GM Global Technology Operations LLC Methods of predicting residual stresses and distortion in quenched aluminum castings
JP6268584B2 (ja) * 2014-01-14 2018-01-31 日産自動車株式会社 熱変形解析方法、熱変形解析プログラム、および熱変形解析装置

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CN110135026A (zh) * 2019-04-29 2019-08-16 西北工业大学 多尺度物理本构模型嵌入abaqus的方法
CN111272552A (zh) * 2020-02-28 2020-06-12 鞍钢股份有限公司 一种变速率拉伸曲线评价方法
CN114115165A (zh) * 2022-01-28 2022-03-01 深圳市北工实业有限公司 一种铸锻一体成型机的生产控制方法及系统

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