Erratum: “Material Strength: A Rational Nonequilibrium Energy Model for Complex Loadings” [ASME J. Appl. Mech., 88(2), p. 021008; DOI: 10.1115/1.4048988]

Zhou, Jie; Wang, Biao

doi:10.1115/1.4051333

Some errors in the derivation and printing process in the original article were corrected. Although it was found that the errors have not created significant change in the main conclusions, the details were still given as follows:

There were two main errors. One error was that Eq. was incorrectly derived, and it should be replaced by

\begin{aligned} Π_{i j α β} & = \frac{\partial^{2} G}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} = \frac{\partial^{2} Δ}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} = A \frac{\partial^{2} I_{1}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} \\ + 2 B (\frac{\partial I_{1}}{\partial ε_{i j}^{T}} \frac{\partial I_{1}}{\partial ε_{α β}^{T}} + I_{1} \frac{\partial^{2} I_{1}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}}) \\ + C \frac{\partial^{2} I_{2}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} + 3 D (2 I_{1} \frac{\partial I_{1}}{\partial ε_{i j}^{T}} \frac{\partial I_{1}}{\partial ε_{α β}^{T}} + I_{1}^{2} \frac{\partial^{2} I_{1}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}}) \\ + E (I_{1} \frac{\partial^{2} I_{2}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} + I_{2} \frac{\partial^{2} I_{1}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} + \frac{\partial I_{2}}{\partial ε_{i j}^{T}} \frac{\partial I_{1}}{\partial ε_{α β}^{T}} + \frac{\partial I_{1}}{\partial ε_{i j}^{T}} \frac{\partial I_{2}}{\partial ε_{α β}^{T}}) \\ + F \frac{\partial^{2} I_{3}}{\partial ε_{i j}^{T} \partial ε_{α β}^{T}} \end{aligned}

(25)

Thus, the components of the free energy for the biaxial loading derived from Eq. (25) should be replaced by

\begin{aligned} Π_{1111} = \frac{\partial^{2} G}{\partial {(ε_{11}^{T})}^{2}} = 2 B + 2 C + 4 E ε_{11}^{T} \\ Π_{2222} = \frac{\partial^{2} G}{\partial {(ε_{22}^{T})}^{2}} = 2 B + 2 C + 4 E ε_{22}^{T} \\ Π_{1122} = \frac{\partial^{2} G}{\partial ε_{11}^{T} \partial ε_{22}^{T}} = 2 B + 2 E (ε_{11}^{T} + ε_{22}^{T}) \\ Π_{2211} = \frac{\partial^{2} G}{\partial ε_{22}^{T} \partial ε_{11}^{T}} = 2 B + 2 E (ε_{11}^{T} + ε_{22}^{T}) \end{aligned}

(31)

To make the matrix given by Eq. positive, the critical conditions of the material failure for the biaxial loading can be determined by

\begin{aligned} Π_{1111} & = 2 C + 4 E ε_{11}^{T} > 0 \\ Π_{1111} Π_{2222} - Π_{1122} Π_{2211} & = (2 C + 4 E ε_{11}^{T}) (2 C + 4 E ε_{22}^{T}) - [2 E (ε_{11}^{T} + ε_{22}^{T})]^{2} \\ = C^{2} + 2 C E (ε_{11}^{T} + ε_{22}^{T}) - E^{2} (ε_{11}^{T} - ε_{22}^{T})^{2} > 0 \end{aligned}

(32)

Another error was that the data in Ref. 17 were incorrectly used, i.e., experimental nominal stress (strain) was confused with true stress (strain). As a result, given Table 1 should be replaced by new Table 1:

Table 1

Fitting parameters for the four types of AZ31 alloy specimens

	A	C	E	−2C/3E
Specimen A	250.0	582.0	−1825	0.2126
Specimen B	182.9	720.9	−2208	0.2177
Specimen C	211.5	719.1	−3032	0.1581
Specimen D	165.8	925.8	−4403	0.1402

Because of the aforementioned changes, Figs. 1–3 should be replaced by the following figures:

As shown in Figs. 1–3, no significant change compared with the results in the original article and the conclusions in the original paper still match with the revised figures.

Fig. 1

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The strength contour of the four types of specimens in plastic strain space. The solid (dashed) curves are given by Eq. (31) with the parameters A, C, and E obtained for the uniaxial tensile (compressive) loading. The dots represent the experimental data.

Fig. 2

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The strength contours for the four types of specimens in stress space. The solid (dashed) curves are given by Eq. (30) with the parameters A, C, and E obtained for the uniaxial tensile (compressive) loading.

Fig. 3

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The strength contours of the four types of specimens. Inside the contour, the specimens were safe. The solid (dash) curves were given by Eq. (30) with the parameters A, C, and E obtained for the uniaxial tensile (compressive) loading.

Besides these two main errors, there were still several printing mistakes in some equations and the corrected forms are as follows:

\begin{aligned} d G & = d U - \int \int_{Γ} δ (Π_{i} u_{i}) d s - d (S T) = d U - \int \int_{Γ} Π_{i} δ u_{i} d s \\ - \int \int_{Γ} u_{i} δ Π_{i} d s - T d S - S d T \end{aligned}

(5)

d S_{i} = - \frac{1}{T} (d G + \int \int_{Γ} u_{i} δ Π_{i} d s + S d T) = - \frac{1}{T} d G; δ Π_{i} = 0, d T = 0

(6)

\int \int \int_{D} σ_{i j}^{0} (ε_{i j} - ε_{i j}^{T}) d v = \int \int \int_{D} σ_{i j} (ε_{i j}^{0} + ε_{i j}) d v = 0

(15)

Compared with the equations in the original article, some integral symbols should be added in Eqs. (5) and (6), the sign before SdT in Eq. (6) should be plus and the first part of Eq. (15) should have a superscript T instead p. All errors above have no effect on the subsequent derivation.

Acknowledgments for this Erratum

Dr. Wan X. J helped the authors to obtain the data and check the results. This work was financially supported by the National Natural Science Foundation of China through the key research project (Grant Nos. 11832019, 11472313, and 13572355).

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Erratum: “Material Strength: A Rational Nonequilibrium Energy Model for Complex Loadings” [ASME J. Appl. Mech., 88(2), p. 021008; DOI: 10.1115/1.4048988]

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Erratum: “Material Strength: A Rational Nonequilibrium Energy Model for Complex Loadings” [ASME J. Appl. Mech., 88(2), p. 021008; DOI: 10.1115/1.4048988]

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