Correction to: JMEPEG https://doi.org/10.1007/s11665-021-05639-x

As a result of an error during the publication process, a number of equations were displayed incorrectly in the PDF version of this article as originally published.

The following shows the incorrect and correct presentations of the equations in question:


Mistyped equation:

$$d \stackrel{-} {\alpha }=\sum_{i=1}^{M}d{\stackrel{-}{\alpha }}_{i}, d{\stackrel{-}{\alpha }}_{i}=\frac{2}{3}{C}_{i}d{\stackrel{-}{\varepsilon }}_{P}-{\gamma }_{i}^{^{\prime}}{\stackrel{-}{\alpha }}_{i}dp$$
(1)

Correct equation:

$$d\overline{\alpha }=\sum_{i=1}^{M}d{\overline{\alpha }}_{i}, d{\overline{\alpha }}_{i}=\frac{2}{3}{C}_{i}d{\overline{\varepsilon }}_{P}-{\gamma }_{i}^{^{\prime}}{\overline{\alpha }}_{i}dp$$
(1)

Mistyped equation:

$$dp=\sqrt{\frac{2}{3}d{\stackrel{-}{\varepsilon }}_{P}.d{\stackrel{-}{\varepsilon }}_{P}}$$
(2)

Correct equation:

$$dp=\sqrt{\frac{2}{3}d{\overline{\varepsilon }}_{P}.d{\overline{\varepsilon }}_{P}}$$
(2)

Mistyped equation:

$$d\stackrel{-}{\alpha }=Cd{\stackrel{-}{\varepsilon }}_{p}-{\gamma }_{1}\left(\stackrel{-}{\alpha }-\delta \stackrel{-}{b}\right)dp$$
(3a)

Correct equation:

$$d\overline{\alpha }=Cd{\overline{\varepsilon }}_{p}-{\gamma }_{1}\left(\overline{\alpha }-\delta \overline{b}\right)dp$$
(3a)

Mistyped equation:

$$d\stackrel{-}{b}={\gamma }_{2}\left(\stackrel{-}{\alpha }-\stackrel{-}{b}\right)dp$$
(3b)

Correct equation:

$$d\overline{b}={\gamma }_{2}\left(\overline{\alpha }-\overline{b}\right)dp$$
(3b)

The term \({\stackrel{-}{\varepsilon }}_{P}\) should be \({\overline{\varepsilon }}_{P}\)


The term \(\stackrel{-}{b}\) should be \(\overline{b }\)


The term \(\left(\stackrel{-}{\alpha }-\delta \stackrel{-}{b}\right)\) should be \(\left(\overline{\alpha }-\delta \overline{b }\right)\)


The term \(\sum_{i=1}^{M}d{\stackrel{-}{\alpha }}_{i}\) should be \(\sum_{i=1}^{M}d{\overline{\alpha }}_{i}\)


The term \({d\stackrel{-}{\varepsilon }}_{p}\) should be \({d\overline{\varepsilon }}_{p}\)


The term \(\sum_{i=1}^{M}d{\stackrel{-}{\alpha }}_{i}\) should be \(\sum_{i=1}^{M}d{\overline{\alpha }}_{i}\)


The term \(\left(\stackrel{-}{\alpha }-\delta \stackrel{-}{b}\right)\) should be \(\left(\overline{\alpha }-\delta \overline{b}\right)\)

Under section “2.2 The Ahmadzadeh-Varvani (A-V) Kinematic Hardening Rule” of the published manuscript, terms δ and k in the below lines are mistyped as shown in the following text:

\(\delta ={(lex\alpha/-lexk)}^{m}\). Constant k is the ratio of coefficients \(C\) and \({\gamma }_{1}\) as \(k=lexC/-lex\gamma_{1},\) and exponent \(m\) is a material

The corrected terms δ and k within the text should be given as below:

\(\delta ={(\alpha/k)}^{m}\). Constant k is the ratio of coefficients \(C\) and \({\gamma }_{1}\) as \(k=C/\gamma_{1},\) and exponent \(m\) is a material constant.


The original article has been corrected.