Dynamic mechanical properties and comparison of two constitutive models for martensitic stainless steel 0Cr17Ni4Cu4Nb

Under quasi-static conditions, the stress–strain curves under load at different strain rates (0.001, 0.01, and 0.1 s⁻¹) are shown in figure 5. Under dynamic conditions, the stress–strain curves loaded at the same temperature (25 °C) and different strain rates (750, 1500, 2000, 2600, 3500, and 4500 s⁻¹) are illustrated in figure 6. As shown in figures 5 and 6, the dynamic stress–strain curve in the initial stage at a high strain rate presents an increasing slope and enters a plastic stage thereafter. In this case, the materials are subject to plastic deformation. Due to the generation of a dislocation, the stress increases slowly with the strain (albeit to an insignificant extent), indicating that the materials have entered a stage of steady plastic deformation.

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It can be seen from figure 6 that the materials show a certain strain-rate strengthening effect. To quantify the influence of the strain rate on the stress on such materials, a strain-rate sensitivity parameter β is introduced, which is defined as follows [15, 16]:

$$\begin{eqnarray}&&\beta =\displaystyle \frac{\partial \sigma }{\partial \,\mathrm{ln}\,\dot{\varepsilon }}=\displaystyle \frac{{\sigma }_{2}-{\sigma }_{1}}{\mathrm{ln}({\dot{\varepsilon }}_{2}-{\dot{\varepsilon }}_{1})}\end{eqnarray} \tag{ 1 }$$

where, ${\sigma }_{1}$ and ${\sigma }_{2}$ separately refer to the measured stresses in the quasi-static state (${\dot{\varepsilon }}_{1}=0.01{{\rm{s}}}^{-1}$) and high strain rates (${\dot{\varepsilon }}_{2}$ = 2600, 3500, or 4,500 s⁻¹).

The strain-rate sensitivity is calculated based on the test data, as shown in table 2: the strain-rate sensitivity of the materials increases with the growing strain rate while decreases with increasing strain, showing the strain-rate strengthening effect.

Table 2. Strain rate sensitivity parameters of 0Cr17Ni4Cu4Nb stainless steel.

Strain Rate $\dot{\varepsilon }/{{\rm{s}}}^{-1}$	True strain $\varepsilon$	Strain rate sensitivity $\beta$
2600	0.08	56.8
	0.10	44.0
	0.12	36.7
	0.14	32.2
3500	0.08	61.4
	0.10	50.7
	0.12	43.4
	0.14	39.6
4500	0.08	62.3
	0.10	51.0
	0.12	44.0
	0.14	42.1

The stress–strain curve of the 0Cr17Ni4Cu4Nb stainless steel at a strain rate of 4500 s⁻¹ and different temperatures is shown in figure 7: the materials deliver the temperature sensitivity, that is, the stress decreases with increasing temperature. To assess the influence of temperature on the stress on a material, the temperature sensitivity n_t is introduced, which is defined as follows [15, 16]:

$$\begin{eqnarray}&&{n}_{t}=\left|\displaystyle \frac{\partial \sigma }{\partial \,\mathrm{ln}\,T}=\displaystyle \frac{\mathrm{ln}({\sigma }_{2}/{\sigma }_{1})}{\mathrm{ln}({T}_{2}/{T}_{1})}\right|\end{eqnarray} \tag{ 2 }$$

where, ${\sigma }_{1}$ and ${\sigma }_{2}$ represent the corresponding stresses under a certain strain at the same strain rate and test temperatures of ${T}_{1}$ and ${T}_{2},$ respectively.

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The temperature sensitivity is calculated according to the test data (table 3): the temperature sensitivity of the materials increases with the temperature, evincing the thermal softening effect. The effect is caused by the synergistic effect of the test temperature and an adiabatic temperature rise at a high strain rate. The adiabatic temperature rise during material deformation at a high strain rate is generally calculated according to the following equation [17–20]:

$$\begin{eqnarray}&&{\rm{\Delta }}T(\varepsilon )=\displaystyle \frac{\eta }{\rho {C}_{{\rm{v}}}}\displaystyle {\int }_{0}^{\varepsilon }\sigma {\rm{d}}\varepsilon \end{eqnarray} \tag{ 3 }$$

where, $\sigma ,$ $\varepsilon ,$ $\rho ,$ and ${C}_{{\rm{v}}}$ denote the true stress, true strain, material density, and specific heat capacity of materials, respectively; $\eta$ represents the work to heat conversion factor and $\eta$ = 0.9 [21, 22]. As for the specimens, $\rho =7.8\times {10}^{3}\,{\rm{kg}}\,{{\rm{m}}}^{-3}$ and ${C}_{{\rm{v}}}=0.5\,\,{\rm{kJ}}/\left({\rm{kg}}\cdot ^\circ C\right).$

Table 3. Temperature sensitivity parameters of 0Cr17Ni4Cu4Nb stainless steel ($\dot{\varepsilon }=4500\,{{\rm{s}}}^{-1}$).

Temperature range $T/^\circ {\rm{C}}$	True strain $\varepsilon$	Temperature sensitivity ${n}_{t}$
25–350	0.06	0.032
	0.10	0.029
	0.14	0.039
	0.18	0.043
25–500	0.06	0.047
	0.10	0.049
	0.14	0.053
	0.18	0.058
25–650	0.06	0.081
	0.10	0.078
	0.14	0.083
	0.18	0.083

Based on the data in figure 7, the adiabatic temperature rise is calculated according to equation (3). Figure 8 displays the adiabatic temperature rise of materials at a strain rate of 4500 s⁻¹ and different temperatures. It can be seen from the figure that an increase of strain corresponds to the growth of the adiabatic temperature rise of materials, which is attributed to the accumulation of heat generated during the plastic deformation of the materials. The adiabatic temperature rise of materials decreases with increasing test temperature. The increase in temperature generally causes material softening. At a high strain rate, the strain hardening of materials competes with that induced by adiabatic temperature.

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The J-C model [23–27] can reflect the work hardening effect, strain-rate effect, and thermal softening effect of materials, which is expressed as follows:

$$\begin{eqnarray}&&\overline{\sigma }=(A+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n})(1+C\,\mathrm{ln}\,\dot{\varepsilon }* )(1-{(T* )}^{m})\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&T* =(T-{T}_{{\rm{m}}})({T}_{{\rm{m}}}-{T}_{{\rm{r}}})\end{eqnarray} \tag{ 5 }$$

where, $\overline{\sigma },$ $A,$ $B,$ $n,$ $C,$ and $m$ refer to the yield stress on materials, the yield strength at the reference temperature and reference strain rate (quasi-static state), strain hardening coefficient, strain hardening index, strain-rate hardening coefficient, and thermal softening index, respectively; the equivalent plastic strain is given by ${\overline{\varepsilon }}^{{\rm{p}}}={\varepsilon }^{{\rm{p}}}/{\varepsilon }_{0},$ where ${\varepsilon }^{{\rm{p}}}$ and ${\varepsilon }_{0}$ separately denote the plastic strain and reference strain; $\dot{\varepsilon }* \,=\,\dot{\varepsilon }/{\dot{\varepsilon }}_{0}$ denotes the dimensionless plastic strain rate, where ${\dot{\varepsilon }}_{0}$ and $\dot{\varepsilon }$ refer to the reference strain rate and plastic strain rate; ${T}^{* },$ ${T}_{{\rm{r}}},$ ${T}_{{\rm{m}}}$ and $T$ denote the relative temperature, reference temperature, the melting point of the material, and the instantaneous temperature, respectively.

At first, the first term in the right side of the J-C constitutive equation corresponds to the normal temperature and strain rate. The J-C constitutive model is expressed as follows:

$$\begin{eqnarray}&&\overline{\sigma }=A+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n}\end{eqnarray} \tag{ 6 }$$

The constants in the machining process of materials are determined according to the quasi-static compression test of the material at normal temperature. As shown in figure 5 (corresponding to a strain rate of 0.01 s⁻¹), the yield strength of the materials is 1050.95 MPa ($A$) and the reference strain is 0.01. Through transposition according to equation (6), it can be found that:

$$\begin{eqnarray}&&\overline{\sigma }-A=+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n}\end{eqnarray} \tag{ 7 }$$

By taking the logarithm of both sides of equation (7), it can be found that:

$$\begin{eqnarray}&&\mathrm{ln}(\overline{\sigma }-A)=+\,\mathrm{ln}\,B+n\,\mathrm{ln}\,{\overline{\varepsilon }}^{{\rm{p}}}\end{eqnarray} \tag{ 8 }$$

Therefore, the slope $n$ ($n=0.5302$) of the curve and $B={\rm{17.677}}$ are attained through fitting (figure 9).

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According to the equation,

$$\begin{eqnarray}&&\overline{\sigma }=(A+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n})(1+C\,\mathrm{ln}\,\dot{\varepsilon }* )\end{eqnarray} \tag{ 9 }$$

It is found that:

$$\begin{eqnarray}&&\displaystyle \frac{\overline{\sigma }}{A+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n}}=1+C\,\mathrm{ln}\,\dot{\varepsilon }* \end{eqnarray} \tag{ 10 }$$

By taking the reference strain rate as 100 s⁻¹, linear fitting is applied to obtain $C={\rm{0.1018}},$ as illustrated in figure 10.

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According to the equation,

$$\begin{eqnarray}&&\displaystyle \frac{\overline{\sigma }}{(A+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n})(1+C\,\mathrm{ln}\,\dot{\varepsilon }* )}=1-{(T* )}^{m}\end{eqnarray} \tag{ 11 }$$

By taking the logarithm of both sides of equation (11), it can be found that:

$$\begin{eqnarray}&&\mathrm{ln}\left(1-\displaystyle \frac{\overline{\sigma }}{(A+B{({\overline{\varepsilon }}^{{\rm{p}}})}^{n})(1+C\,\mathrm{ln}\,\dot{\varepsilon }* )}\right)=m\,\mathrm{ln}\,T* \end{eqnarray} \tag{ 12 }$$

Finally, the value of the thermal softening term $m$ of materials is determined according to the experimental stress–strain curve at a high strain rate (4500 s⁻¹) and different temperatures (the melting point of the material is taken as ${T}_{{\rm{m}}}=1693\,{\rm{K}}$). On this basis, linear fitting (figure 11) gives $m={\rm{1.40909}}.$ The parameters of the J-C model are summarized in table 4.

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Eventually, the J-C constitutive equation for martensitic stainless steel 0Cr17Ni4Cu4Nb at a high strain rate is expressed as follows:

$$\begin{eqnarray}&&\overline{\sigma }=\left({\rm{1050.95}}+{\rm{17.677}}{\left(\displaystyle \frac{{\varepsilon }^{{\rm{p}}}}{0.01}\right)}^{0.5302}\right)\left(1+{\rm{0.1018}}\,\mathrm{ln}\left(\displaystyle \frac{\dot{\varepsilon }}{100}\right)\right)\left(1-{\left(\displaystyle \frac{T-25}{1395}\right)}^{1.40909}\right)\end{eqnarray} \tag{ 13 }$$

The P-L model [28–30] can also reflect the work-hardening, strain-rate effect, and thermal softening of materials, which is expressed as follows:

$$\begin{eqnarray}&&\sigma ({\varepsilon }_{s},{\dot{\varepsilon }}_{s},T)=g({\varepsilon }_{s}){\rm{\Gamma }}({\dot{\varepsilon }}_{s}){\rm{\Theta }}(T)\end{eqnarray} \tag{ 14 }$$

$$\begin{eqnarray}&&g({\varepsilon }_{s})={\sigma }_{0}{\left(1+\displaystyle \frac{{\varepsilon }_{s}}{{\sigma }_{0}}\right)}^{1/n}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{\rm{\Gamma }}({\dot{\varepsilon }}_{s})={\left(1+\displaystyle \frac{{\dot{\varepsilon }}_{s}}{{\dot{\varepsilon }}_{0}}\right)}^{1/m}\end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray}&&{\rm{\Theta }}(T)=C{}_{0}+C{}_{1}T+C{}_{2}T^{2}+C{}_{3}T^{3}+C{}_{4}T^{4}+C{}_{5}T^{5}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&T={T}_{{\rm{in}}}+{\rm{\Delta }}T\end{eqnarray} \tag{ 18 }$$

where, $n,$ $m,$ ${T}_{{\rm{in}}},$ ${\rm{\Delta }}T,$ ${\sigma }_{0},$ ${\varepsilon }_{s},$ ${\varepsilon }_{0},$ ${\dot{\varepsilon }}_{s},$ and ${\dot{\varepsilon }}_{0}$ denote the strain hardening index, strain-rate strengthening index, the initial test temperature, adiabatic temperature rise, yield strength at the reference temperature and reference strain rate, plastic strain, reference strain, plastic strain rate, and reference strain rate, respectively; C₀, C₁, ... and C₅, separately represent the polynomial coefficients of temperature.

At first, ${\sigma }_{0}$ and $n$ of materials are deduced. By using the experimental stress–strain relationship of materials in a quasi-static state (that at a strain rate of 0.01 s⁻¹) and supposing that both the strain-rate strengthening term ${\rm{\Gamma }}({\dot{\varepsilon }}_{s})$ and thermal softening term ${\rm{\Theta }}(T)$ are equal to 1, equation (14) is simplified as follows:

$$\begin{eqnarray}&&\sigma ({\varepsilon }_{s})=g({\varepsilon }_{s})={\sigma }_{0}{\left(1+\displaystyle \frac{{\varepsilon }_{s}}{{\sigma }_{0}}\right)}^{1/n}\end{eqnarray} \tag{ 19 }$$

By separately fitting the straight lines in the elastic and plastic stages of the experimental stress–strain curve, the point of intersection of two straight lines corresponds to the yield strength, that is, ${\sigma }_{0}={\rm{1050}}{\rm{.95MPa}};$ moreover, equation (19) is solved by taking the reference strain as 0.001 and the logarithms of both sides of the equation are simultaneously calculated, giving:

$$\begin{eqnarray}&&\mathrm{ln}\left(\displaystyle \frac{\sigma ({\varepsilon }_{s})}{{\sigma }_{0}}\right)=\displaystyle \frac{1}{n}\,\mathrm{ln}\left(1+\displaystyle \frac{{\varepsilon }_{s}}{{\varepsilon }_{0}}\right)\end{eqnarray} \tag{ 20 }$$

The slope ($1/n$) of the straight line is determined as 0.038 52 by substituting the corresponding true stress of the strain within 0.25 ∼ 0.41 and corresponding strains into equation (20) and then the linear fitting is performed. Furthermore, $n=25.9605$ is obtained, as shown in figure 12.

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The value of $m$ materials is determined according to the true stress–strain curve at normal temperature and strain rates between 750 ∼ 4500 s⁻¹. It is supposed that the thermal softening term ${\rm{\Theta }}(T)$ is equal to 1 and thus equation (14) can be simplified as follows:

$$\begin{eqnarray}&&\sigma ({\varepsilon }_{s},{\dot{\varepsilon }}_{s})=g({\varepsilon }_{s}){\left(1+\displaystyle \frac{{\dot{\varepsilon }}_{s}}{{\dot{\varepsilon }}_{0}}\right)}^{1/m}\end{eqnarray} \tag{ 21 }$$

By taking the logarithms of both sides of equation (21), it can be found that:

$$\begin{eqnarray}&&\mathrm{ln}\left(\displaystyle \frac{\sigma ({\varepsilon }_{s},{\dot{\varepsilon }}_{s})}{g({\varepsilon }_{s})}\right)=\displaystyle \frac{1}{m}\,\mathrm{ln}\left(1+\displaystyle \frac{{\dot{\varepsilon }}_{s}}{{\dot{\varepsilon }}_{0}}\right)\end{eqnarray} \tag{ 22 }$$

By setting the reference strain rate as 1900 s⁻¹ and substituting the measured stress at strains of 0.05 and 0.06 and corresponding strain rates into equation (22), linear fitting is conducted. On this basis, the slope $1/m$ of the straight line is determined as 0.166 35 and furthermore $m={\rm{6.0114}},$ as illustrated in figure 13.

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The temperature terms C₀, C₁, ... and C₅ are determined according to the experimental stress–strain curves at temperatures between 25 °C ∼ 650 °C and strain rates of 750 ∼ 4500 s⁻¹. Equation (14) is transformed as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\sigma ({\varepsilon }_{s},{\dot{\varepsilon }}_{s},T)}{g({\varepsilon }_{s}){\rm{\Gamma }}({\dot{\varepsilon }}_{s})}=C{}_{0}+C{}_{1}T+C{}_{2}T^{2}+C{}_{3}T^{3}+C{}_{4}T^{4}+C{}_{5}T^{5}\end{eqnarray} \tag{ 23 }$$

Considering the influence of adiabatic temperature rise (${\rm{\Delta }}T$) on the plastic deformation of materials, by substituting the actual stresses at different temperatures into equation (23) and performing polynomial fitting, it can be found that C₀ = 0.976 14, C₁ = 1.424 45 × 10⁻⁵, and C₂ = −5.1262 × 10⁻⁷. The fitting curves are shown in figure 14. The parameters of the P-L model are summarized in table 5.

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Finally, the P-L constitutive equation for martensitic stainless steel 0Cr17Ni4Cu4Nb at a high strain rate is shown as follows:

$$\begin{eqnarray}&&\begin{array}{l}\sigma ({\varepsilon }_{s},{\dot{\varepsilon }}_{s},T)=g({\varepsilon }_{s}){\rm{\Gamma }}({\dot{\varepsilon }}_{s}){\rm{\Theta }}(T)\\ \,=\,1050.95{\left(1+\displaystyle \frac{{\varepsilon }_{s}}{0.001}\right)}^{0.03852}{\left(1+\displaystyle \frac{{\dot{\varepsilon }}_{s}}{1900}\right)}^{0.16635}\\ (0.97614+{\rm{1.42445}}\times {10}^{-5}T-{\rm{5.1262}}\times {10}^{-7}{T}^{2})\end{array}\end{eqnarray} \tag{ 24 }$$

Figure 15 compares the values of true stress predicted by the J-C and P-L constitutive models with the test values under different conditions. The further to investigate the accuracies of the two models, the error analysis is performed by using the mean absolute error (MAE) (Δσ), R and AARE, which are expressed as follows [31–36]:

$$\begin{eqnarray}&&{\rm{\Delta }}\sigma =\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left|{E}_{i}-{P}_{i}\right|\end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray}&&AARE=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}\left|\displaystyle \frac{{E}_{i}-{P}_{i}}{{E}_{i}}\right|\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}&&R=\displaystyle \frac{\displaystyle {\sum }_{i=1}^{N}({E}_{i}-\bar{E})({P}_{i}-\bar{P})}{\sqrt{\displaystyle {\sum }_{i=1}^{N}{({E}_{i}-\bar{E})}^{2}\displaystyle {\sum }_{i=1}^{N}{({P}_{i}-\bar{P})}^{2}}}\end{eqnarray} \tag{ 27 }$$

where, ${E}_{i}$ and ${P}_{i}$ separately refer to the test and predicted values of the true stress (MPa); $\bar{E}$ and $\bar{P}$ separately denote the means of ${E}_{i}$ and ${P}_{i};$ $N$ represents the total number of data points used here. The MAEs are summarized in table 6 and R values and AAREs are displayed in figure 16.

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Table 6. Absolute error mean value in the two constitutive models under different deformation conditions.

$T/^\circ {\rm{C}}$	$\dot{\varepsilon }/{{\rm{s}}}^{-1}$	${\rm{\Delta }}{\sigma }_{{\rm{PL}}}/{\rm{MPa}}$	${\rm{\Delta }}{\sigma }_{{\rm{JC}}}/{\rm{MPa}}$
25	750	27.62	28.26
	1500	24.15	50.54
	2000	23.37	48.60
	2600	15.45	74.33
	3500	34.94	22.39
	4500	22.55	19.79
350	750	32.24	7.89
	1500	17.07	16.85
	2000	26.69	46.84
	2600	47.22	16.36
	3500	17.90	57.23
	4500	36.01	32.49
500	750	58.22	19.79
	1500	24.63	57.20
	2000	20.66	74.87
	2600	32.57	54.25
	3500	17.86	89.96
	4500	35.37	82.03
650	750	17.88	86.16
	1500	10.65	96.21
	2000	27.81	102.54
	2600	50.67	59.97
	3500	18.84	107.39
	4500	19.93	120.29

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According to figure 15 and table 6, it can be found that the values predicted by the J-C and P-L models differ from the test values under the same deformation conditions. Under most deformation conditions, the MAE of the P-L model is smaller than that of the J-C model. The maximum MAE (58.22 MPa) of the P-L model is determined at 500 °C and 750 s⁻¹ while that (120.29 MPa) of the J-C model occurs at 650 °C and 4500 s⁻¹. Overall, as the temperature rises, the MAE of the P-L model varies to an insignificant extent and is stable while that of the J-C model gradually increases, which reaches a maximum at 650 °C. This is ascribed to the difference of the third terms in expressions of the two models. The third term in the P-L model is expressed by using a polynomial while that in the J-C model is shown as the relative exponent. The P-L model is slightly superior to the J-C model and it shows higher accuracy at high temperature.

Figure 16 compares the correlations of the values of the true stress predicted by the two models with the test values: the R values are 0.977 80 and 0.968 33 while AAREs of the P-L and J-C models are 2.25% and 4.77%, respectively. The P-L model is shown to deliver higher prediction accuracy than the J-C model and it can better predict the stress on martensitic stainless steel 0Cr17Ni4Cu4Nb.

In the case that the strain rate and temperature are kept constant, the derivative of the strain at 25 °C and 750 s⁻¹ is found from equation (24), as shown in figure 17. In the figure, the derivative is greater than 0, indicating that the stress increases with the strain (i.e. the strain-induced strengthening effect); however, when the strain reaches a certain value, the derivative tends to 0. This implies that the growth in stress gradually decreases and the stress gradually stabilizes, which is in agreement with the experimental results.

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The derivative of the strain rate under the strain of ${\varepsilon }_{{\rm{s}}}={\rm{0.04}}$ is obtained using equation (24) when the strain and temperature are constants, as shown in figure 18, in which the derivative is greater than 0, which indicates that the actual stress increases with increasing strain rate (i.e. the strain-rate strengthening effect); however, the derivative drops as the strain rate is further increased, suggesting that the strain-rate strengthening effect is gradually weakened, being consistent with the experimental results.

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On condition that the strain and strain rate are kept constant, the temperature at a strain of ${\varepsilon }_{{\rm{s}}}={\rm{0.04}}$ is calculated by using equation (24), as shown in figure 19. It can be seen from the figure that the derivative is less than 0, implying a reduction in stress with increasing temperature (i.e. the thermal softening effect); this matches the experimental results.

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