A Note on Parameter Identification of the AISI 304 Stainless Steel Using Micromechanical-Based Phenomenological Approaches

Abstract

Austenitic stainless steels have largely been used in industrial equipment, architectural components and consumer items amongst others. Numerical simulation of manufacturing processes of such components and parts requires adequate hardening descriptions and accurate inelastic parameters. This work addresses these issues for the AISI 304 stainless steels based on phenomenological approaches using a higher order logarithmic yield stress equation and alternative micromechanical-based equations. Identification of the inelastic parameters is performed by either curve-fitting strategies or optimization techniques. The experimental-numerical comparative assessments demonstrate that the micromechanical-based yield stress equation derived from Bergström's dislocation model provides the best hardening description.

Keywords:
AISI 304 stainless steel; parameter identification; inverse problems

1. Introduction

Austenitic stainless steels, in especial the AISI 304 grade, have been used in a wide range of applications, such as industrial equipment (e.g. food, pharmaceutical and mining equipment), architectural components (decorative panels, ornamental work, etc.) and consumer items (sinks, cutlery, pans, etc.) amongst others. The reason lies on its good corrosion/oxidation resistance, weldability, formability, toughness and ductility. The high industrial interest has prompted the present discussion on (i) the suitability of some selected heuristic and micromechanic-based phenomenological equations to model hardening evolution of the AISI 304 stainless steel and (ii) strategies to determine the corresponding parameters.

A successful numerical simulation of metal forming operations to manufacture AISI 304 components requires appropriate inelastic modelling. The key aspects to be observed are the hardening law and corresponding inelastic parameters, and the level of plastic deformation. In general, inelastic parameters can be obtained by using either (a) curve-fitting strategies or (b) inverse problem techniques. The former directly correlates true stresses and true strains under uniform deformation, whereas the latter obtains the inelastic parameter by using optimization methods associated with global measures (e.g. geometry and load information).

The literature shows that hardening parameters of stainless steels are mostly determined by curve-fitting strategies associated with tensile tests. For instance, Kashyap et al.^1,2 described the hardening behaviour of the AISI 316L stainless steel under high temperature based on two and three stages of the Holloman³ relation, σ=Kεⁿ, where σ and ε are the uniaxial stresses and strains, K is the strength coefficient, and n is the strain-hardening exponent. Samuel and co-workers^4,5 addressed the suitability of selected hardening equations to describe plastic deformation of the AISI 316L stainless steel. The authors concluded that, in most cases, Ludwigson⁶ equation, σ = Kεⁿ + exp (K₁ + n₁ε), better describes the hardening behaviour, in which K₁ and n₁ are additional inelastic constants. Similar conclusion was reached by Isaac Samuel and Choudhary⁷ addressing the type 316LN stainless steel. Following the works of Kashyap et al.^1,2, Hertelé et al.⁸ indicate that the DIN 1.4462 duplex stainless steel can be described by two hardening stages based on the Ramberg-Osgood equation⁹, ε = σ/E + K (σ/E)ⁿ, where E is the Young's modulus. El-Magd¹⁰ investigated application of non-logarithmic yield stress equations to austenitic steels. The author adopted Voce's ¹¹, σ = a + b [1 - exp (-cε)], and a modified version of Voce's yield stress equation, σ = C₁ + C₂ ε + C₃ [1 - exp(-C₄ ε)], to describe plastic deformation of the austenitic steels X6CrNi18-11 and X8CrNiMoNb16-16, respectively, where a, b, c, C₁, C₂, C₃ and C₄ are material parameters. Hardening descriptions for the AISI 316LN were discussed by Ashraf et al.¹² for cyclic loading, who also adopted Voce's¹¹ yield stress equation to model isotropic hardening.

Contrasting to curve-fitting strategies, application of optimization-based parameter identification techniques to austenitic stainless steels has been rarely reported in the literature. Notwithstanding, preliminary issues on use of optimization techniques to obtain inelastic parameters of the AISI 304 were discussed by the authors in Vaz Jr. et al.¹³. However, it is important to mention that use of optimization methods is well established and has been used to determine inelastic parameters for both metallic and non-metallic materials. For instance, the authors addressed use of different optimization methods to parameter identification of materials as different as AISI 1045 carbon steel and high-density polyethylene (HDPE)¹⁴.

The present work aims to further discuss hardening descriptions of the AISI 304 stainless steels based on phenomenological approaches using (A) a higher order logarithmic yield stress equation and (B) alternative micromechanical-based equations. The corresponding parameters are determined by using either curve-fitting techniques (logarithmic equation) or optimization methods (micromechanical-based equations). The article is organized as follows: section 2 presents a third-order logarithmic yield stress equation and two hardening equations based on micromechanical concepts. Section 3 introduces parameter identification based on optimization strategies using a composed objective function. Application of the aforementioned hardening equations and parameter identification methods is presented in section 4. The main conclusions and final remarks are presented in section 5.

2. Micromechanical-based and phenomenological yield stress equations

The brief review shows that stainless steels present a peculiar hardening behaviour, frequently described by models based on multiple hardening stages or non-logarithmic equations. Furthermore, the AISI 304 stainless steel is subject to deformation-induced phase transformation of austenite to martensite, causing inflections in the stress-strain curve. However, De et al. ¹⁵ show for the AISI 304 steel that, for low strain rates and temperatures higher than 298 K (as defined in the present experiments), such effects do not take place, making possible to model inelastic deformation using straightforward phenomenological approaches.

2.1 Heuristic phenomenological yield stress equations

The literature shows a wide range of empirical stress-strain relations, such as the classical equations proposed by Hollomon³, Ludwigson⁶ and Swift¹⁶. The material parameters of most such equations are determined by curve-fitting procedures of true stresses and true strains in a logarithmic plot. This technique requires the assumption of uniform deformation, thereby restricting evaluation of strains and stresses up to necking onset (instability). The literature review presented in section 1 highlights the nonlinear dependence of logarithmic stresses on logarithmic strains exhibited by the AISI 304 stainless steel (oftentimes modelled by two^1,7> or three² hardening stages) . Therefore, the present work investigates the suitability of a higher order logarithmic yield stress curve.

Within this framework, Dimatteo et al.¹⁷ proposed a 3^rd order (ln σ)×(ln ε) logarithmic fit to Dual Phase (DP) 450/600/800/1000 and Transformation Induced Plasticity (TRIP) 800 steels attempting to predict the influence of alloying elements and heat treatment conditions in a simple manner. The present work investigates a possible extension to the AISI 304 stainless steel. Complementing the equation introduced by Dimatteo et al.¹⁷, this study includes a provision for elastic deformations, so that

(1) $\ln \left({\sigma }_Y\right)=A{\left[\ln \left({\overline{\epsilon }}_p+{\epsilon }_0\right)\right]}^3+B{\left[\ln \left({\overline{\epsilon }}_p+{\epsilon }_0\right)\right]}^2+C\left[\ln \left({\overline{\epsilon }}_p+{\epsilon }_0\right)\right]+D,$

where A, B, C, and D are the model parameters to be determined by the curve-fitting procedure, ε₀, is the initial strain which is obtained by solving the cubic-logarithmic equation resulted by the intersection between the elastic and plastic curves, A[ln(ε₀)]³ + B[ln(ε₀)]² + (C-1)[ln(ε₀)] + (D-lnE) = 0, in which E is the Young's modulus.

2.2 Micromechanical-based yield stress equations

Phenomenological yield stress models can be approximated from micromechanical-based concepts. This section summarises strategies to obtain two such equations. The approach accounts for physical mechanisms of dislocation theory assuming a Taylor-like correlation¹⁸ between the yield stress, σ_Y, and total dislocation density, ρ, as

(2) ${\sigma }_Y={\sigma }^{*}+M\alpha \mathrm{Gb}\left(\sqrt{\rho }-\sqrt{{\rho }^{*}}\right),$

where M is a factor which correlates the shear flow stress of a single crystal and the uniaxial flow stress of a polycrystal, α is a parameter which depends on the dislocation/dislocation interaction and thermal activation kinetics, G is the shear modulus, b is the magnitude of the Burgers vector, and σ^* and ρ* are reference values. All micromechanical parameters are assumed constant in the present approximation. The following paragraphs summarizes key aspects of two alternative phenomenological yield stress equations obtained from Eq. (2).

Modified Voce's yield stress equation: Voce's original equation was proposed based on empirical considerations¹¹. However, El-Magd¹⁰ demonstrated that this hardening law can also be derived from micromechanical approach. Nevertheless, the author indicated that its application to austenitic steel X6CrNi18-11 at different temperatures provided good approximation only in the range of relatively small plastic strains up to 0.2¹⁰.

Following similar considerations, based on Mecking and Kocks'¹⁹ evolution of the dislocation density, $d\rho /d{\epsilon }_p=k_1^{\mathrm{MK}}\sqrt{\rho +k_2^{\mathrm{MK}}\rho ,}$ a better representation for larger plastic strains can be derived by assuming a constant dynamic recovery rate, $k_2^{\mathrm{MK}}$k₂^MK, and by approximating the effects of the dislocation density and dislocation free path, l, upon the dislocation storage rate¹⁰ as $k_1^{\mathrm{MK}}=1/\left(l\sqrt{\rho }\right)=k^{*}\left(1+c^{*}{\epsilon }_{\rho }\right),$ where k* and c* are constants of the model. The evolution of the dislocation density is, therefore, $d\rho /d{\epsilon }_p=k^{*}\left(1+c^{*}{\epsilon }_p\right)\sqrt{\rho }+k_2^{\mathrm{MK}}\rho .$ The yield stress equation is determined by solving the differential equation, $d{\sigma }_Y/d{\epsilon }_p=\left(d{\sigma }_Y/d\rho \right)\left(d\rho /d{\epsilon }_p\right),$ where dσ_Y/dρ is obtained from Eq. (2), so that

(3) ${\sigma }_Y={\sigma }_0^t+{\varsigma }^t{\overline{\epsilon }}_p+\left({\sigma }_{\infty }^t-{\sigma }_0^t\right)\left[1-\exp \left(-{\delta }^t{\overline{\epsilon }}_p\right)\right],$

in which ${\sigma }_0^t$ and ${\sigma }_{\infty }^t$ are the initial stress and saturation stress, respectively, ς^t is the linear hardening coefficient, and δ^t is the rate of approximation to the linear hardening. The correspondence between the phenomenological and micromechanical parameters obtained from solution of the differential equation is presented in Table 1. Phenomenological hardening laws similar to Eq. (3) have been referred as Modified Voce model. The literature shows increasing use of this hardening equation to describe the yield stress evolution of different materials, such as the DP 780 / TRIP 780 high strength steels²⁰ and X8CrNiMoNb16-16 austenitic steel¹⁰.

Table 1
Phenomenological correspondence of micromechanical parameters.

Equation	Hardening parameters
Modified Voce	${\sigma }_0^t=\sigma i$
	${\sigma }_{\infty }^t={\sigma }^{*}-M\alpha \mathrm{Gb}\left[\sqrt{{\rho }^{*}}-\left(1+2c^{*}/k_2^{\mathrm{MK}}\right)\left(k^{*}/k_2^{\mathrm{MK}}\right)\right]$
	${\varsigma }^t=M\alpha {\mathrm{Gbc}}^{*}\left(k^{*}/k_2^{\mathrm{MK}}\right)$
	${\delta }^t=k_2^{\mathrm{MK}}/2$
Bergström	${\sigma }_0^b={\sigma }^{*}-M\alpha \mathrm{Gb}\left(\sqrt{p^{*}}-\sqrt{p_0}\right)$
	${\sigma }_{\infty }^b={\sigma }^{*}-M\alpha \mathrm{Gb}\left[\sqrt{{\rho }^{*}}-{\left(U^B-A^B\right)}^2/{\Omega }^{B^2}\right]$
	${\sigma }_k^b=M\alpha \mathrm{Gb}\sqrt{{\rho }_0}$
	δb=ΩB

Bergström's yield stress equation: Bergström²¹ proposed a micromechanical stress-strain relation for polycrystalline monophasic steels based on the behaviour of large number of mobile and immobile dislocations. In this case, the evolution of the dislocation density with respect to plastic deformation was derived as dρ/dε_p = U^B - A^B - Ω^B ρ, where Ω^B is associated with mobilization and annihilation of dislocations, A^B and U_B represent parameters which account for mobile dislocations and its corresponding rate of increase, respectively. The phenomenological yield stress equation is derived from dρ/dε_p and Eq. (2) as

(4) ${\sigma }_Y={\sigma }_0^b-{\sigma }_k^b+{\left\{{\left({\sigma }_{\infty }^b-{\sigma }_0^b+{\sigma }_k^b\right)}^2\left[1-\exp \left(-{\delta }^b{\overline{\epsilon }}_p\right)\right]+{\sigma }_k^{b^2}\exp \left(-{\delta }^b{\overline{\epsilon }}_p\right)\right\}}^{1/2},$

where ${\sigma }_0^b$ is the initial stress, ${\sigma }_{\infty }^b$ is the saturation stress, ${\sigma }_k^b$ is the stress correction for the initial dislocation density, ρ₀, and δ^b, is the rate to approximate the saturation stress. Table 1 also presents the correspondence between phenomenological and micromechanical parameters for Bergström's hardening equation obtained from solution of the differential equation.

3. Identification of the material parameters

Material parameters of yield stress equations can be determined either by curve-fitting strategies or by inverse problem techniques. The former obtains the inelastic parameters directly from fitted true stress - true strain curves, whereas the latter uses optimization methods to approximate computed and measured global/local quantities.

Curve-fitting strategies: The present work makes use of tensile tests to obtain experimental data to determine the inelastic parameters. Tensile tests of ductile materials involve five general stages/events: (i) elastic deformation; (ii) plastic deformation under uniform stress states; (iii) necking onset; (iv) inelastic deformation under triaxial stress states; and (v) catastrophic/macroscopic failure. The technique fits the yield stress equation in a true stress/true strain logarithmic plot. However, curve fitting must be performed under uniaxial conditions and, therefore, comprises stage (ii) up to necking onset (i.e. instability) in order to avoid triaxial stress/strain states.

Inverse problem techniques: The inverse problem techniques use optimization methods to determine material parameters of the yield stress equations. The method minimizes the differences between experimental measures and corresponding computed responses with respect to a given norm²². In order to better capture the effects of plastic deformations and improve assessment of the parameters obtained by the optimization method, this work adopts evolution of both tensile load, R_L, and specimen central radius, R_D, measured at selected deformation stages. In the present work no uncertainties were accounted for.

Unconstrained nonlinear optimization is used in the present parameter identification problem, which can be generally stated as

(5) $\begin{array}{c}\mathrm{Minimise}g\left(\mathit{p}\right)={\lambda }_L{g}_L\left(\mathit{p}\right)+{\lambda }_D{g}_D\left(\mathit{p}\right)\mathit{p}∊R^{n_p}\hfill \\ \mathrm{Such}\mathrm{that}{p}_t^{\inf }\le p_i\le p_i^{\sup }i=1,\cdots ,n_p\hfill \end{array}$

where

(6) $g_L\left(\mathit{p}\right)=\sqrt{\frac{1}{N_L}\sum _{j=1}^{N_L}{\left[\frac{R_{L,j}^{\mathrm{Exp}}-R{\left(\mathit{p}\right)}_{L,j}^{\mathrm{Num}}}{R_{L,j}^{\mathrm{Exp}}}\right]}^2}\mathrm{and}{g}_D\left(\mathit{p}\right)=\sqrt{\frac{1}{N_D}\sum _{j=1}^{N_D}{\left[\frac{R_{D,j}^{\mathrm{Exp}}-R{\left(\mathit{p}\right)}_{D,j}^{\mathrm{Num}}}{R_{D,j}^{\mathrm{Exp}}}\right]}^2}$

are individual objective functions for the load (subscript L) and specimen central radius (subscript D), N_L and N_D are corresponding number of experimental points, $\mathit{p}={\left[p_1{p}_2\cdots p_i\cdots p_{n_p}\right]}^T$is the design vector containing n_p design variables p_i (material parameters), and $p_i^{\inf }$ and $p_i^{\sup }$ are minimum and maximum values of each material parameter. The weights λ_L and λ_D are the contribution of each measure to the global objective function which, in the present study, were assumed 0 ≤ λ_L, λ_D ≤ 1 and λ_L + λ_D = 1.

The optimization technique adopted in this work uses the gradient-free downhill simplex method, also known as Nelder-Mead (NM) algorithm²³. It is important to mention that, in principle, any optimization method can be used in conjunction with the proposed strategy. The NM algorithm was selected owing to its high success rate and computational cost-wise competitiveness in similar problems when compared against heuristic and gradient-based schemes¹⁴. The method defines a regular polytope of n_p + 1 vertices for a n_p number of material parameters, which contracts or moves towards the minimum by replacing the worst vertex. Selection of the new vertex is performed after the following possible operations: reflection, expansion, internal/external contraction or shrinkage. Convergence is assumed when the normalized global fitness with respect to the best and worst vertices reaches a pre-defined value. The reader is referred to Nelder and Mead²³ and Lagarias et al.²⁴ for additional details on the NM optimization scheme, and Vaz Jr. et al.²⁵ for the implementation used in this work.

4. Results and Discussions

This section discusses parameter identification of the AISI 304 stainless steel using curve fitting and optimization methods. Tensile tests of cylindrical specimens prepared according to the Brazilian ABNT NBR ISO 6892²⁶ were adopted to determine the inelastic parameters of Eq. (1) and Eqs. (3)-(4). The chemical composition of the austenitic AISI 304 steel used in this study is presented in Table 2.

Table 2
Chemical composition of the AISI 304 steel (in wt.%)^(*).

	Cr	Ni	C	Mn	P	S
Nominal	18-20	8-10.5	max 0.08	max 2.0	max 0.045	max 0.03
This work	18.677	8.780	0.07612	1.9856	0.01984	0.02767

• (*) Mass spectrometer – Shimadzu OES 5500II

The specimens were manufactured with a gauge section diameter d₀ = 10 mm, initial gauge length l₀ = 50 mm (measurements performed by a clip gauge transducer), total gauge section L = l₀ + 2d₀ = 70 mm, shoulder diameter d_s = 20 mm and a transitional radius r_t = 15 mm between the gauge section and shoulder. The tests were performed under displacement control with maximum crosshead speed v_c = 3 mm/min (low strain rate). Evolution of the specimen central radius is estimated by assuming plastic deformation at constant volume up to necking onset, so that R_D = R₀/(e + 1)^1/2, where R₀ is the initial central radius and e = ∆l/l₀ is the specimen elongation (engineering strain). The reference radius data are complemented by measurement of the final specimsen central radius.

The geometrical model used in the simulations consider axisymmetry around the rotation axis Z - Z' and symmetry about the R - R' axis, making possible to model only 1/4 of the specimen. A structured, eight-noded quadrilateral finite element mesh with 200 elements and 661 nodes with progressive refinement towards the specimen R - R' axis was used.

4.1 Curve-fitting of the third-order logarithmic yield stress equations

The curve-fitting strategy accounts for load evolution from plastic onset up to necking. The elastic limit is the lower bound imposed by the elastic curve, whereas the maximum load is assumed as the instability limit (necking onset). The true strains and true stresses are determined as ε=ln (1 + e) and σ = s (1 + e), respectively, where s = R_L/A₀ is the engineering stress, R_L is the axial load, and A₀ is the initial cross-sectional area. The reader is referred to Davis²⁷ for a detailed description of the curve-fitting method, including experimental considerations and requirements.

Figure 1(a) shows the experimental data and fitted equation in a (lnσ) × (lnε) plot, whereas Table 3 presents the corresponding material parameters. For the sake of clarity, not all experimental points are represented in Figure 1(a) and forthcoming figures. The experimental data show that the yield stress of the AISI 304 stainless steel at room temperature prompts a nonlinear correlation between stresses and strains in the (lnσ) × (lnε) space, thereby precluding using of first-order logarithmic equations such as Hollomon's³ and Swift's¹⁶. On the other hand, the 3^rd order logarithmic curve achieved an excellent fit with a coefficient of determination R² = 0.999187.

Table 3
Curve-fitting parameters for the third-order (ln)x(lnε) logarithmic curve.

	Symbol	Value
3rd order polynomial equation	A	3.54744x10-3
	B	8.12467x10-3
	C	0.607523
	D	21.3091
	ε0	1.97659x10-3
Coefficient of determination	R2	0.999187

Figure 1
Third-order (lnσ) × (lnε) logarithmic curve: (a) Fitting of the third-order equation and (b) evolution of the tensile load and specimen central radius.

Verification of the 3^rd order logarithmic equation is performed by comparing load and central radius computed by finite elements (using the aforementioned geometrical model) against available reference data. In this case, the yield stress is defined by Eq. (1) with hardening parameters given in Table 3. Figure 1(b) indicates a very accurate prediction of tensile loads and specimen central radius up to necking onset. However, the simulations show that both tensile load and specimen central radius are grossly overestimated after the maximum experimental load. In this case, no necking was observed, causing the computed load to further increase towards the final stages and macroscopic failure. Therefore, use of the 3^rd order logarithmic equation is restricted to relatively small plastic strains, represented by the equivalent plastic strain limit${\overline{\epsilon }}_p\le 0.317.$

4.2 Application of the optimization-based technique to the Modified Voce and Bergström yield stress equations

Use of optimization methods makes possible to determine inelastic parameters taking into account experimental data up to macroscopic failure. In the present study, the tensile load and specimen central radius constitute the reference measures. The global error function is represented in Eq. (5), whereas the individual objective functions are given by Eqs. (6). The global error g(p) constitutes a weighted sum of the individual fitness g_L(p) and g_D(p) by λ_L and λ_D. This strategy allows a straightforward investigation of the effects of each type of reference measure (load / central radius) in the final hardening parameters. In order to quantify such effects a new index G(p,λ) is defined as

(7) $G\left(\mathit{p},\lambda \right)={\sqrt{\frac{1}{2}\left[{\left(\frac{g_L\left(\mathit{p}\right)-g_L^{\min }}{g_L^{\min }}\right)}^2+{\left(\frac{g_D\left(\mathit{p}\right)-g_D^{\min }}{g_D^{\min }}\right)}^2\right]}}^{},$

where $g_L^{\min }\left({\lambda }_L=1;{\lambda }_D=0\right)$ and $g_D^{\min }\left({\lambda }_L=0;{\lambda }_D=1\right)$ are the best individual fitness for tensile load and central radius, respectively. Qualitatively, the index G(p,λ) represents the average differences between computed load and central radius for a given weight set {λ_L, λ_D} and their best individual approximation.

Three cases were studied for both Modified Voce and Bergström yield stress equations. Case A: parameter identification accounts for only tensile loads (λ_L = 1; λ_D = 0) and provides $g_L^{\min }$ (the individual fitness, g_D, for the specimen central radius is calculated but not included in the computation of the global error function, g(p));Case B: parameter identification accounts for only the specimen central radius (λ_L = 0; λ_D = 1) and provides $g_D^{\min }$ the individual objective function, g_L, for tensile load is computed but not included in the computation of g(p)); and Case C: tensile loads and specimen central radius impose equal effect on obtaining the hardening parameters (λ_L = 0.5; λ_D = 0.5). Table 4 presents the individual fitness, g_L (p) and g_D (p), and the global index, G(p,λ), for all cases. The best individual fitness, $g_L^{\min }$ and $g_D^{\min }$, are represented in boldface. The corresponding load and central radius evolution curves for Modified Voce and Bergström yield stress equations are shown in Figures 2 and 3, respectively.

Table 4
Modified Voce and Bergström equations: best individual fitness(*) and global index.

Hardening Equation	Case	Weights		Individual fitness		Global index
		λL	λD	gL( p )x102	gD( p )x102	G( p ,λ)
Modified Voce	A	1	0	0.794469	0.879569	14.316
	B	0	1	15.1765	0.0413996	12.801
	C	0.5	0.5	1.18416	0.193070	2.614
Bergström	A	1	0	0.775876	0.378298	5.106
	B	0	1	14.2041	0.0460175	12.238
	C	0.5	0.5	0.882198	0.163666	1.810

• (*) and gLmin are gDmin represented in boldface.

Figure 2
Modified Voce's yield stress equation: evolution of the tensile load and specimen central radius.

Figure 3
Bergström's yield stress equation: evolution of the tensile load and specimen central radius.

It is relevant to notice from Table 4 and Figures 2 and 3 that, for both yield stress equations, parameter identification based solely on the specimen central radius (Case B:: λ_L = 0; λ_D = 1) does not guarantee acceptable approximations of the loading curve. The tensile loads are clearly underestimated for both Modified Voce (Figure 2a) and Bergström (Figure 3a) equations. It can be observed in Table 4, that the individual objective function for the load, g_L, is substantially larger than their best minimum $g_L^{\min }$ for both yield equations. A similar behaviour, yet in a smaller scale, can be observed when parameter identification is performed based solely on the tensile load (Case A:λ_L = 1; λ_D = 0). In this case, the greatest effect takes place in the final specimen central radius, reaching a maximum error of 6.1 % with respect to the experimental measure.

The best results are obtained when the influence of the tensile load and specimen central radius are accounted for (Case C:λ_L = 0.5; λ_D = 0.5). Table 5 presents the material parameters for Case C for both Modified Voce and Bergström yield equations. The quantitative assessment for Case C is also shown in Table 4, which indicates that the global index, G(p,λ), are markedly smaller than those evaluated for Case A and Case B.

Table 5
Final parameters for weights λ_L= 0.5 and λ_D= 0.5.

Equation	Symbol	Value
Modified Voce	${\sigma }_0^t$	415.223 [MPa]
	${\sigma }_{\infty }^t$	1049.68 [MPa]
	ςt	383.664 [MPa]
	δt	3.75756
g(pt)=λLgL(pt)+λDgD(pt)		6.88615 × 10-3
Bergström	${\sigma }_0^b$	405.110 [MPa]
	${\sigma }_{\infty }^b$	1726.55 [MPa]
	${\sigma }_k^b$	453.044 [MPa]
	δb	1.05480
g(pb)=λLgL(pb)+λDgD(pb)		5.22932 × 10-3

The good approximations obtained for Case C are illustrated in Figure 4, which shows the finite element mesh and actual specimen at final stages of deformation for Bergström and Modified Voce equations. The geometrical differences between the actual specimen and simulations are very small, virtually imperceptible in Figures 4(a) and 4(b). The equivalent plastic strain, ${\overline{\epsilon }}_p$, for both yield stress equations are also presented in Figure 4. The differences of ${\overline{\epsilon }}_p$ computed using Bergström and Modified Voce equations are also very small, reaching a maximum of 0.1 % at the centre of the specimen.

Figure 4
Finite element mesh and actual specimen at final stage of deformation and corresponding distribution of the equivalent plastic strain for (a) Bergström and (b) Modified Voce equations.

Remark: The results discussed in previous paragraphs show that identification of material parameters for both Bergström and Modified Voce equations using optimization techniques provide good results for both load evolution and geometrical configuration. Notwithstanding, Table 4 shows that, for weights λ_L = 0.5 and λ_D = 0.5 (Case C), Bergström equation provides individual errors, g_L and g_D, 34.2 % and 18.0 % smaller for tensile loads and specimen central radius, respectively, when comparing against Modified Voce hardening law. Furthermore, the global errors, g(p), (see Table 5) for Bergström equation is 31.7 % smaller than those evaluated by the Modified Voce equation. Therefore, based on the results discussed in the previous paragraphs, Bergström hardening law is recommended to represent the yield behaviour of the AISI 304 stainless steel.

5. Concluding Remarks

The AISI 304 stainless steel has been largely used in manufacturing equipment and components owing its balance between good mechanical and corrosion characteristics and relative lower cost (when compared to other stainless steels). Therefore, a hardening description able to describe accurately inelastic deformation in a simplified manner is of great industrial interest. Many researchers agree that a purely micromechanical yield stress criterion is the best and the most elegant approach to account for strain induced phase transformation from austenite to martensite. On the other hand, the present phenomenological equations have shown able to approximate load and geometry evolution with acceptable accuracy, making possible its straightforward use in combination with commercial software. The present work is inserted within this framework and discusses use of a heuristic third-order logarithmic equation and two micromechanical-based phenomenological yield stress equations to describe hardening evolution of the AISI 304 stainless steel. The material parameters of the hardening equations were determined by either curve fitting (third-order logarithmic equation) or inverse problem techniques (micromechanical-based yield stress equations).

Curve-fitting of the third order (lnσ) × (lnε) curve indicates good results up to necking onset, corresponding to a maximum equivalent plastic strain ${\overline{\epsilon }}_p$ = 0.317. Contrastingly, the optimization-based parameter identification was able to approximate both load evolution and geometrical configuration (specimen central radius) up to macroscopic failure, with equivalent plastic strains reaching values as high as ${\overline{\epsilon }}_p$ = 1.355. Therefore, based on the fact that metal forming operations involve large plastic deformations, parameter identification using optimization-based strategies are highly recommended.

The present work investigated Modified Voce and Bergström hardening laws in association with an optimization-based strategy. In both cases, the technique required compliance of both load evolution and geometrical configuration (specimen central radius). The simulations indicated that Bergström equation provided smaller approximation errors than the Modified Voce hardening law. Therefore, Bergström equation is recommended to describe the yield stress evolution for the AISI 304 stainless steel.

6. Acknowledgements

The authors acknowledge the financial support provided by the Brazilian funding agency CNPq - (National Council for Scientific and Technological Development), Grant no. 303412/2016-0.

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Publication Dates

• Publication in this collection
  19 Aug 2019
• Date of issue
  2019

History

• Received
  08 Mar 2019
• Reviewed
  30 Apr 2019
• Accepted
  10 June 2019