CASTING SIMULATION METHOD

Abstract

Provided is a casting simulation method capable of expressing influence of different inelastic strains produced at different temperatures on strain hardenability at room temperature. The following amount of effective equivalent inelastic strain εeffective inelastic is substituted into a constitutive equation in which an amount of equivalent inelastic strain is used as a degree of work hardening: an amount of effective equivalent inelastic strain εeffective inelastic=∫ot{h(T)/h(RT)}{(Δεinelastic/Δt)}dt , where 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 α(T), Δεinelastic/Δt denotes an equivalent inelastic strain rate, and t denotes a time from 0 second in analysis.

Description

TECHNICAL FIELD

This disclosure relates to a casting simulation method using thermal stress and deformation analysis.

BACKGROUND

Conventionally, elasto-plastic constitutive equation is used in analysis programs in order to predict the residual stress and deformation that occurs as metals processed at high temperature, such as cast products, hot forged products, and hot rolled products, cool back to room temperature. JP2007330977A (PTL 1) and Dong Shuxin, Yasushi Iwata, Toshio Sugiyama, and Hiroaki Iwahori, “Cold Crack Criterion for ADC12 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.

However, in elasto-plastic constitutive equation and elasto-viscoplastic constitutive equation that do not take into account recovery, 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.

To address this issue, there have been proposed a number of constitutive equations that can take into account some recovery. Any of these constitutive equations, however, fail to give physical or experimental grounds for how recovery should be considered, and thus suffer from problems with prediction accuracy.

The problems of conventional findings related to the above-described inventions will be described below.

CITATION LIST

Patent Literature

• • PTL 1: JP2007330977A
  • PTL 2: Japanese Patent Application No. 2014-004578

Non-Patent Literature

• • NPL 1: Dong Shuxin, Yasushi Iwata, Toshio Sugiyama, and Hiroaki Iwahori, “Cold Crack Criterion for ADC12 Aluminum Alloy Die Casting”, Casting Engineering, 81(5), 2009, pp. 226-231, ADC12
  • NPL 2: Hallvard G. Ejar and Asbjorn Mo, “ALSPEN-A mathematical model for thermal stresses in direct chill casting of aluminum billets”, Metallurgical Transactions B, December 1990, Volume 21, Issue 6, Pages 1049-1061
  • NPL 3: W. M. van Haaften, B. Magnin, W. H. Kool, and L. Katgerman, “Constitutive behavior of as-cast AA1050, AA3104, and AA5182”, Metallurgical and Materials Transactions A, July 2002, Volume 33, Issue 7, Pages 1971-1980
  • NPL 4: Alankar Alankar and Mary A. Wells, “Constitutive behavior of as-cast aluminum alloys AA3104, AA5182 and AA6111 at below solidus temperatures”, Materials Science and Engineering A, Volume 527, Issues 29-30, 15 Nov. 2010, Pages 7812-7820

SUMMARY

Technical Problem

Hallvard et al. (NPL 2) 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.

Constitutive Eq. (I) representing the relationship between stress and inelastic strain:

$\stackrel{\_}{\sigma }=f\left(\alpha ,{\stackrel{\_}{\stackrel{.}{\varepsilon }}}_p,T\right)=c\left(T\right){\left(\alpha +{\alpha }_0\right)}^{n\left(T\right)}{\left({\stackrel{\_}{\stackrel{.}{\varepsilon }}}_p\right)}^{m\left(T\right)}$ $d\alpha =\{\begin{array}{cc}d{\stackrel{\_}{\varepsilon }}_p& \mathrm{when}T\le T_0\\ 0& \mathrm{otherwise}\end{array}$

Van Haaften et al. (NPL 3) 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.

However, the amount of cumulative inelastic strain is directly multiplied by a, which is not in incremental form, and inelastic strain produced at high temperature eventually contribute to work hardening at low temperature. Thus, as in Constitutive Eq. (I), the constitutive equation proposed by NPL 3 inevitably involves an unrealistic increase in yield stress.

Constitutive Eq. (II) representing the relationship between stress and inelastic strain:

$\stackrel{.}{\varepsilon }={A\left[\sinh \left(\frac{{\sigma }_{\mathrm{ss}}}{{\sigma }_0}\right)\right]}^{\mathrm{nH}}\exp \left(\frac{-Q}{\mathrm{RT}}\right)$ ${\sigma }_H=\left({\sigma }_0+k\sqrt{\alpha \left(T\right)\varepsilon }\right)·f\left(Z\right)$ $f\left(Z\right)=\min \left(1,\arcsin {h\left(\frac{Z}{A}\right)}^{\mathrm{mH}}\right)$ $\alpha =\frac{1}{1+\exp \left(a_0+a_1T\right)}$

Moreover, Alankar et al. (NPL 4) proposes a constitutive equation that is expressed as Constitutive Eq. (III) below representing the relationship between stress and inelastic strain. With this constitutive equation, 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. However, there are no metallurgical grounds for considering that the ratio 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). Additionally, it is not specified how to identify material constants in the constitutive equation when inelastic strain is divided into creep strain and plastic strain. Thus, K_(T), n_(T), and m_(T) in the following equation cannot be determined accurately.

Constitutive Eq. (III) representing the relationship between stress and inelastic strain:

$\sigma =K\left(T\right){{\varepsilon }_p^{n\left(T\right)}\left(\frac{{\stackrel{.}{\varepsilon }}_p}{{\varepsilon }_0}\right)}^{m\left(T\right)}$ $\sigma K\left(T\right){\left({\varepsilon }_p+{\varepsilon }_{p_0}\right)}^{n\left(T\right)}{\left({\stackrel{.}{\varepsilon }}_p+{\stackrel{.}{\varepsilon }}_{p_0}\right)}^{m\left(T\right)}$ ${\varepsilon }_{\mathrm{creep}}={\varepsilon }_{\mathrm{plastic}}\left(1-\frac{n_{\left(T\right)}}{n_{\mathrm{RT}}}\right)$ $\%\mathrm{strain}\mathrm{softening}=\left(1-\frac{n_{\left(T\right)}}{n_{\mathrm{RT}}}\right)\times 100$

Recently, one of the applicants of the present application proposed a constitutive equation to solve at least part of the above problems in Japanese Patent Application No. 2014-004578 (PTL 2, an unpublished earlier application). This is expressed as a constitutive equation expressing the relationship between stress and elastic strain/inelastic strain as explained below. In this constitutive equation, inelastic strain is divided into plastic strain contributing to work hardening and creep strain not contributing to work hardening, and inelastic strain produced at high temperature is expressed mainly by creep strain. In this case, measures are taken to prevent the yield stress from excessively rising at room temperature by causing plastic strain to develop gradually as the temperature decreases.

Constitutive Eq. expressing the relationship between stress and elastic strain/inelastic strain:

$\varepsilon ={\varepsilon }_{\mathrm{elastic}}+{\varepsilon }_{\mathrm{plastic}}+{\varepsilon }_{\mathrm{creep}}+{\varepsilon }_{\mathrm{thermal}}$ $\sigma =E·{\varepsilon }_{\mathrm{elastic}}$ ${\varepsilon }_{\mathrm{creep}}=A·\exp \left(\frac{\sigma }{\mathrm{RT}}\right)$ $\sigma =f\left({\varepsilon }_p,T\right),$

where f denotes a yield function.

With the constitutive equation proposed by PTL 2, however, the proportion of plastic strain and creep strain is determined based on the shape of a stress-equivalent inelastic strain curve obtained at different temperatures, rather than on the metallurgical grounds. In addition, as a steady-state creep law is used, it is inevitable to estimate plastic strain excessively beyond the actual value, while estimating creep strain low in low strain regions. Therefore, problems remain in the prediction accuracy of residual stress and deformation.

Recent metallurgical findings revealed that the extent to which inelastic strain at high temperature contributes to work hardening at room temperature depends on the composition of the alloy, thermal history such as in heat treatment, and solidified structure.

To date, however, there has been no such constitutive equation that takes into account all the factors listed above in order to predict the residual stress and deformation that occurs as metals processed at high temperature, such as cast products, hot forged products, and hot rolled products, cool back to room temperature.

It is thus desirable at present in construction of a constitutive equation to experimentally clarify how inelastic strain at high temperature contributes to work hardening at room temperature, and reflect it in the constitutive equation.

However, none of the conventional constitutive equations can reflect “the influence of inelastic strain produced at different temperatures on work hardening at room temperature” that is determined on an experimental or theoretical basis. There has also been no finding that shows how to determine material constants in a constitutive equation that can reflect this effect.

Solution to Problem

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 α 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.

To solve the above-described problems, the present disclosure also provides a method of experimentally determining an effective inelastic strain coefficient α(T) which represents the contributions of inelastic strain produced at different temperatures to work hardening.

Specifically, the primary features of this disclosure are as described below.

1. A casting simulation method capable of expressing influence of different inelastic strains produced at different temperatures on work hardening, namely on increase in yield stress, at room temperature, the influence varying with differences in recovery at the different temperatures, by introducing an amount of effective equivalent inelastic strain to an elasto-plastic constitutive equation and/or an elasto-viscoplastic constitutive equation in which an amount of equivalent inelastic strain is used as a degree of work hardening, namely an amount of increase in yield stress, such as an elasto-plastic constitutive equation in which a yield function is expressed as f=f(σ_eff,ε_eff^p,T) or an elasto-viscoplastic constitutive equation in which a relation between equivalent stress, equivalent viscoplastic strain, viscoplastic strain rate, and temperature is expressed as σ_eff.=F(ε_eff.^vp,{dot over (ε)}_eff.^vp,T), wherein an amount of effective equivalent inelastic strain ε_{effective inelastic} obtained by Eq. (1) below is used:

the amount of effective equivalent inelastic strain ε_{effective inelastic}=∫_o^t{t_(T)/h_(RT)}{(Δε_inelastic/Δt)}dt   (1)

, where 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 α(T), Δε_inelastic/Δt denotes an equivalent inelastic strain rate, and t denotes a time from 0 second in analysis.

2. The casting simulation method according to 1., the effective inelastic strain coefficient α(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.

3. The casting simulation method according to 1. or 2., wherein a stress-equivalent inelastic strain curve is transformed into a stress-effective equivalent inelastic strain curve using α(T), and based on the stress-effective equivalent inelastic strain curve, a determination is made of a material constant in a constitutive equation to which the amount of effective equivalent inelastic strain ε_{effective inelastic} is introduced.

4. The casting simulation method according to any one of 1. to 3., wherein when α(T) is substituted into Eq. (1) in a temperature range in which α(T) is 0 or a negative value and a stress-equivalent inelastic strain curve at the temperature T indicates work hardening, α(T) is corrected from 0 or the negative value to a small positive value.

Advantageous Effect

According to the present disclosure, it is possible to eliminate the physical and metallurgical irrationality of the conventional methods, and to express, based on the experimental fact, the influence of different inelastic strains produced at different temperatures on work hardening at room temperature, which influence varies with differences in recovery at the different temperatures. As a result, it is possible to more accurately simulate the residual stress and deformation occurring as metals processed at high temperature cool back to room temperature.

BRIEF DESCRIPTION OF THE DRAWING

In the accompanying drawings:

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 α 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; and

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.

DETAILED DESCRIPTION

The following describes the present disclosure in detail.

As used herein, an equivalent inelastic strain rate calculated by thermal stress analysis is multiplied by an “effective inelastic strain coefficient α(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.

Specific examples are an elasto-plastic constitutive equation and/or an elasto-viscoplastic constitutive equation in which the amount of equivalent inelastic strain is used as a degree of work hardening (namely an amount of increase in yield stress) such as an elasto-plastic constitutive equation in which a yield function is expressed as f=f(σ_eff,ε_eff^p,T) or an elasto-viscoplastic constitutive equation in which a relation between equivalent stress, equivalent viscoplastic strain, viscoplastic strain rate, and temperature is expressed as σ_eff.=F(ε_eff.^vp,{dot over (ε)}_eff.^vp,T).

The determination of the effective inelastic strain coefficient α(T) is carried out by applying inelastic pre-strains of different magnitudes at the corresponding temperatures at which α(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.

The following describes how to determine α(T) in detail.

A tensile test piece is heated in a way as presented in the temperature history in FIG. 1. For example, let T₃ be the temperature at which α(T) is to be determined, a single inelastic pre-strain is applied at T₃ in T₁, T₂, T₃, . . . , T_n per test. Then, the test piece is cooled to room temperature and subjected to a tensile test or a compression test at room temperature. Then, 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.

Desirably, 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.

In the context of the present disclosure, 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. For the tensile test in this disclosure, for example, a publicly-known and widely-used tensile test may be used, such as JIS Z 2241:2011. For the compression test, for example, a publicly-known and widely-used compression test may be used, such as JIS K 7181:2011.

The results obtained in the above test are conceptually illustrated in FIG. 2. FIG. 2 illustrates the influence of inelastic pre-strains applied at different temperatures on the increase in yield stress at room temperature. As can be seen from FIG. 2, if a gradient of yield stress with respect to the amount of inelastic pre-strains applied at room temperature is expressed as h_(RT), a gradient of yield stress with respect to the amount of inelastic pre-strains applied at different temperatures can be similarly expressed as h_(T). Thus, at each temperature T, the value of h_(T)/h_(RT) is used as an effective inelastic strain coefficient α(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.

In a temperature range in which an effective inelastic strain coefficient α(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, if the stress-equivalent inelastic strain curve at the temperature T indicates work hardening and if the effective inelastic strain coefficient α(T) is 0, then 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. In other words, 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 α(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 α(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 α(T), it is possible to express work hardening in a stress-equivalent inelastic strain curve with a slight ineffective inelastic strain. The value of α(T) at the time of correction is in a range of 0<α<0.5, and desirably 0<α<0.1, although it depends on the alloy type. As an example, α(T) is corrected by the following linear interpolation using an effective inelastic strain coefficient at maximum temperature α_min in a temperature range in which α(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 α is experimentally determined to be 0:

$\begin{array}{cc}{\alpha }_{\mathrm{extrapolation}}\left(T\right)=\frac{0-{\alpha }_{\min }}{T_{\max }-T_{\min }}\left(T-T_{\min }\right)+{\alpha }_{\min }& \left(T>T_{\min }\right)\end{array},$

where

α_{extrapolation}(T) denotes a value of a as corrected in a temperature range in which α 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 α is experimentally determined to be non-zero;

α_min denotes a value of α in T_min; and

T denotes a temperature above T_min

As an example, 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.

The extended Ludwik's law is as follows. ε₀ is a constant necessary for calculation and usually a small value of 1×10⁻⁶.

σ=K(T)(ε_inelastic+ε₀)^n(T)({dot over (ε)}_inelastic)^m(T)

When introducing the amount of effective inelastic strain, it is expressed as:

σ=K(T)(ε_{effective inelastic}+ε₀)^n(T)({dot over (ε)}_inelastic)^m(T)

The term with an index n_(T) representing the degree of work hardening includes the amount of effective equivalent inelastic strain as a variable. In addition, 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.

Following the above procedure, the amount of effective equivalent inelastic strain ε_{effective inelastic} is determined. In the disclosure, 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. Although details of the procedures for casting simulation will be described in the Examples, the points are summarized as follows.

In the simulation according to the disclosure, it is possible to adopt a constitutional expression that uses the amount of equivalent inelastic strain as a measure of work hardening conventionally used for casting simulation. An exemplary elasto-plastic constitutive equation is:

ε_ij=ε^e_ij+ε^p_ij

σ_ij=D_ijkl(T)ε^e_kl

f=f(σ_eff,ε_eff^p,T)

, where T is the temperature, σ_ij is the stress, σ_eff. is the equivalent stress, ε_ij is the total strain, ε^e_ij is the elastic strain, ε^p_ij is the plastic strain, f is the yield function, and D_ijkl is the fourth-order constitutive tensor.

Alternatively, an exemplary elasto-viscoplastic constitutive equation is:

ε_ij=ε^e_ij+ε^vp_ij

σ_ij=D_ijkl(T)ε^e_kl

σ_eff.=F(ε_eff.^vp,{dot over (ε)}_eff.^vp,T)

, where σ_eff. is the equivalent stress, ε_eff.^vp is the equivalent viscoplastic strain, {dot over (ε)}_eff.^vp is the equivalent viscoplastic strain rate.

To these constitutive equations, the amount of effective equivalent inelastic strain ε_{effective inelastic} defined by Eq. (1) according to the present disclosure, instead of the amount of equivalent inelastic strain conventionally used, may be introduced or substituted.

The well-known and widely-used procedures for casting simulation are:

(I) element creation step;

(II) element definition step;

(III) heat transfer analysis step;

(IV) thermal stress analysis step; and

(V) analysis result evaluation step.

In the present disclosure, 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.

EXAMPLES

The following describes how the present disclosure enables prediction with high accuracy of the influence of amounts of inelastic pre-strains produced at different temperatures on the work hardening behavior at room temperature in a typical aluminum die-casting alloy, JIS ADC12, using an extended Ludwik equation, which is a typical elasto-viscoplastic constitutive equation.

A specific form of the equation before and after the introduction of the amount of effective inelastic strain is as presented in paragraph 0032.

Firstly, a typical aluminum die-casting alloy, JIS ADC12, is analyzed for an effective inelastic strain coefficient α(T) and material constants (K(T), n(T), and m(T)) at each temperature according to the procedures described in paragraphs 0026 to 0033.

Firstly, 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.

In these tests, all the 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.

In addition, tests for determining an effective inelastic strain coefficient α(T) were carried out at RT, 200° C., 250° C., 300° C., 350° C., 400° C., and 450° C. After solution treatment at 450° C. for 1 hour, the temperature was lowered to a temperature at which the target inelastic pre-strain was to be applied to the test piece following the temperature history presented in FIG. 3, and after the target temperature was reached, an inelastic pre-strain was applied in tension while retaining the test piece at the target temperature. At each temperature, two or three different pre-strains were applied.

After application of pre-strains, 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. For a test piece not imparted with inelastic pre-strain, the 0.2% offset yield stress was 106 MPa. Based on the results presented in the figure, determinations of an effective inelastic strain coefficient α(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 results are presented in FIG. 5. In the temperature range from 350° C. to 400° C., 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. At 450° C., the effective inelastic strain coefficient becomes 0 and any inelastic strain produced does not contribute to work hardening at room temperature. From this experimental fact, as explained in paragraph 0031, α 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.

By using the effective inelastic strain coefficient α_(T) thus obtained, 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. In each case, 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. Except for 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. as T_max described in paragraph 0031. Thus, as described in paragraphs 0030 and 0031, the stress-inelastic strain curve displays elasto-perfectly viscoplastic behavior at 450° C. Therefore, there is a divergence between the experimental values and the calculated values.

It is conceivable, however, that if α is corrected as described in paragraph 0031 with the temperature of α=0, T_max in paragraph 0031, being temporarily set as a liquidus temperature, the reproducibility of the stress-inelastic strain curve also improves.

For extended Ludwik equations incorporating or not incorporating (in the case of a conventional example) the amount of effective equivalent inelastic strain according to the present disclosure, calculation was made to determine the effect of inelastic pre-strains applied at different temperatures on the yield stress at room temperature, and the calculation results were compared as presented in FIG. 13 (a conventional example) and FIG. 14 (an example of the present disclosure).

In the conventional extended Ludwik equation not incorporating the amount of effective equivalent inelastic strain, in principle, 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.

On the other hand, as can be seen from FIG. 14, in the expanded Ludwik equation incorporating the amount of effective inelastic strain according to the present disclosure, the influence of the amount of inelastic pre-strain on work hardening gradually increased with decreasing temperature, which fact reproduced the behavior observed in the experimental results.

To examine the influence of inelastic pre-strain at different temperatures on the yield stress at room temperature, a comparison was made between experimental values and calculated values according to the extended Ludwik equation to which the amount of effective equivalent inelastic strain is introduced, and the results are presented in FIG. 15.

It can be seen from the figure that 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. In addition, 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.

It can be seen from the figure that pre-strain applied at 700° C. does not contribute to work hardening at room temperature. In contrast, 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.

Claims

1. A casting simulation method capable of expressing influence of different inelastic strains produced at different temperatures on work hardening, namely on increase in yield stress, at room temperature, the influence varying with differences in recovery at the different temperatures, by introducing an amount of effective equivalent inelastic strain to an elasto-plastic constitutive equation and/or an elasto-viscoplastic constitutive equation in which an amount of equivalent inelastic strain is used as a degree of work hardening, namely an amount of increase in yield stress, such as an elasto-plastic constitutive equation in which a yield function is expressed as f=f(σeff,εeffp,T) or an elasto-viscoplastic constitutive equation in which a relation between equivalent stress, equivalent viscoplastic strain, viscoplastic strain rate, and temperature is expressed as σeff.=F(εeff.vp,{dot over (ε)}eff.vp,T), wherein, where

an amount of effective equivalent inelastic strain εeffective inelastic obtained by Eq. (1) below is used: the amount of effective equivalent inelastic strain εeffective inelastic=∫ot{h(T)/h(RT)}{(Δεinelastic/Δt)}dt   (1)

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 α(T),

Δεinelastic/Δt denotes an equivalent inelastic strain rate, and

t denotes a time from 0 second in analysis.

2. The casting simulation method according to claim 1, the effective inelastic strain coefficient α(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.

3. The casting simulation method according to claim 1, wherein a stress-equivalent inelastic strain curve is transformed into a stress-effective equivalent inelastic strain curve using α(T), and based on the stress-effective equivalent inelastic strain curve, a determination is made of a material constant in a constitutive equation to which the amount of effective equivalent inelastic strain εeffective inelastic is introduced.

4. The casting simulation method according to claim 1, wherein when α(T) is substituted into Eq. (1) in a temperature range in which α(T) is 0 or a negative value and a stress-equivalent inelastic strain curve at the temperature T indicates work hardening, α(T) is corrected from 0 or the negative value to a small positive value.

5. The casting simulation method according to claim 2, wherein a stress-equivalent inelastic strain curve is transformed into a stress-effective equivalent inelastic strain curve using α(T), and based on the stress-effective equivalent inelastic strain curve, a determination is made of a material constant in a constitutive equation to which the amount of effective equivalent inelastic strain εeffective inelastic is introduced.

6. The casting simulation method according to claim 2, wherein when α(T) is substituted into Eq. (1) in a temperature range in which α(T) is 0 or a negative value and a stress-equivalent inelastic strain curve at the temperature T indicates work hardening, α(T) is corrected from 0 or the negative value to a small positive value.

7. The casting simulation method according to claim 3, wherein when α(T) is substituted into Eq. (1) in a temperature range in which α(T) is 0 or a negative value and a stress-equivalent inelastic strain curve at the temperature T indicates work hardening, α(T) is corrected from 0 or the negative value to a small positive value.

8. The casting simulation method according to claim 5, wherein when α(T) is substituted into Eq. (1) in a temperature range in which α(T) is 0 or a negative value and a stress-equivalent inelastic strain curve at the temperature T indicates work hardening, α(T) is corrected from 0 or the negative value to a small positive value.