Thermal effects on fracture and the brittle-to-ductile transition

The fracture behavior of brittle and ductile materials can be strongly influenced by thermal fluctuations, especially in micro- and nanodevices as well as in rubberlike and biological materials. However, temperature effects, in particular on the brittle-to-ductile transition, still require a deeper theoretical investigation. As a step in this direction we propose a theory, based on equilibrium statistical mechanics, able to describe the temperature-dependent brittle fracture and brittle-to-ductile transition in prototypical discrete systems consisting in a lattice with breakable elements. Concerning the brittle behavior, we obtain closed form expressions for the temperature-dependent fracture stress and strain, representing a generalized Griffith criterion, ultimately describing the fracture as a genuine phase transition. With regard to the brittle-to-ductile transition, we obtain a complex critical scenario characterized by a threshold temperature between the two fracture regimes (brittle and ductile), an upper and a lower yield strength, and a critical temperature corresponding to the complete breakdown. To show the effectiveness of the proposed models in describing thermal fracture behaviors at small scales, we successfully compare our theoretical results with molecular dynamics simulations of Si and GaN nanowires.

Figure 1

(a) Scheme of a crack propagating within an arbitrary crystal lattice. (b) Reduced scheme of the elasto-fragile model, based on symmetry assumption. The central horizontal chain (colored in black) is composed of $N+1$ linear elastic springs of elastic constant $k$. The nodes of this chain are connected to the top layer of the system (at $y=Y$) with $N$ vertical linear elastic springs of elastic constant $l$ (colored in yellow or light gray). Moreover, the first $\eta$ nodes ($i=1,\cdots ,\eta$) are also linked to the bottom layer (at $y=0$) through $\eta$ vertical breakable springs of elastic constant $h$ (intact in blue or dark gray; broken in orange or intermediate gray). We underline that the first node ($i=0$) and the last one ($i=N+1$) are anchored to the bottom and the top layer, respectively. Hence, the first and the last shear springs fix the direction of the crack propagation from the right to the left of the system.

Figure 2

Potential energy of a single breakable spring with elastic constant $h$ (top panel) and corresponding force (bottom panel). The quantity $Y_M$ is the elongation after which the spring breaks, resulting in a force equal to zero.

Figure 3

Behavior of the brittle-fracture model with a variable number of units ($N=\{50,125,200\}$ as indicated by arrows) and a variable thermal to elastic energy ratio $K_{B}T/\left(h{Y}_M^2\right)=0.5$ (blue or dark gray), and $K_{B}T/\left(h{Y}_M^2\right)=2$ (orange or light gray). The dimensionless quantities $\langle f\rangle /\left(Nh{Y}_M\right)$ (top panel) and $\langle \xi \rangle /N$ (bottom panel) are represented vs the dimensionless parameter $Y/Y_M$, where, in both cases, $l/h=k/h=1$. We also fixed $\eta =N$, which means there are no missing or broken elements in the initial configuration.

Figure 4

Comparison between the approximated quantities given by Eqs. (28) and (31) (dashed lines) and the corresponding exact results in Eqs. (18) and (19) (continuous lines) for $\langle f\rangle /\left(Nh{Y}_M\right)$ (top panel) and $\langle \xi \rangle /N$ (bottom panel) vs the dimensionless extension $Y/Y_M$. The thermal to elastic energy ratio $K_{B}T/\left(h{Y}_M^2\right)$ is set to 0.5 (blue or dark gray curves) and 2 (orange or light gray curves). The total number of units is set to $N=50$ and $N=200$ (see arrows). The dimensionless quantities are set to $l/h=1$ and $k/h=1$. We also considered $\eta =N$.

Figure 5

Comparison between the thermodynamic limit obtained for $N\to \infty$ and the approximations obtained with large values of $N$ (see arrows). We plotted $\langle f\rangle /\left(Nh{Y}_M\right)$ (top panel) and $\langle \xi \rangle /N$ (bottom panel) vs the dimensionless extension $Y/Y_M$. The thermal to elastic energy ratio is set to $K_{B}T/\left(h{Y}_M^2\right)=0.5$ (in blue or dark gray) and to $K_{B}T/\left(h{Y}_M^2\right)=2$ (in orange or light gray), while the total number of units for the large $N$ approximation is set to $N=500$. The dimensionless quantities are set to $l/h=1$ and $k/h=1$. Moreover, we adopted $\eta =N$ corresponding to $\varphi =0$.

Figure 6

Fracture extension $Y_s/Y_M$ (in orange or light gray) and fracture strength ${\sigma }_s/\left(h{Y}_M\right)$ (in blue or dark gray) vs the reduced temperature $T/T_c$. All quantities are written in dimensionless form. We can observe that both quantities present a critical behavior corresponding to a phase transition for $T=T_c$. We adopted $l/h=1, k/h=1$, and $\varphi =0$.

Figure 7

Tensile strength of [110]-oriented silicon nanowires as a function of temperature for wires with different diameters: comparison between molecular dynamics simulations results [94] (symbols) and our theory given by Eq. (39) (continuous lines). The parameters used are reported in the main text.

Figure 8

Scheme of the fracture model with the softening mechanism. The central horizontal chain (colored in black) is composed of $N+1$ linear springs with elastic constant $k$. The nodes of this chain are connected to the top layer (at $y=Y$) by $N$ vertical linear springs with elastic constant $l$ (colored in yellow or light gray). The first $\eta$ nodes ($i=1,\cdots ,\eta$) are also linked to the bottom layer (at $y=0$) by $\eta$ vertical softenable and breakable springs with elastic constant $h$ when intact (colored in blue or dark gray), or $p$ when softened (colored in green, springs with fewer coils). The broken elements are represented in orange and identified by a rupture in the springs. We remark that the first node ($i=0$) and the last one ($i=N+1$) are anchored to the bottom and the top layers, respectively.

Figure 9

Potential energy of a single softenable and breakable spring of elastic constants $h$ and $p$ (top panel) and corresponding force (bottom panel). We see that $Y_p$ is the elongation after which the spring is weakened or softened, and $Y_b$ is the elongation after which the spring is broken.

Figure 10

Behavior of the dimensionless extension thresholds $Y_{\xi }/Y_b, Y_{\chi }/Y_b$, and $Y_{\xi \chi }/Y_b$ vs the dimensionless temperature $K_{B}T/\left(h{Y}_b^2\right)$ of the system. From the physical point of view, $Y_{\xi \chi }$ (yellow or light gray curve) describes the brittle fracture below the transition temperature $T^{*}$, and the couple $Y_{\xi }$ (blue or dark gray), $Y_{\chi }$ (orange or intermediate gray) describes the ductile fracture above the temperature $T^{*}$. The three curves $Y_{\xi }/Y_b, Y_{\chi }/Y_b$, and $Y_{\xi \chi }/Y_b$ vs $K_{B}T/\left(h{Y}_b^2\right)$ intersect at the bifurcation black point, characterized by $T^{*}$. We adopted the parameters $\alpha =7/5, \beta =2/5, \gamma =9/10$, and $\mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/10$.

Figure 11

Brittle response of the fracture phenomenon ($0<T<T^{*}$). Left panel: dimensionless force vs dimensionless extension for different values of $N=100,250,500$ (as indicated by the arrow). Right panel: fraction of intact, softened, and broken elements for different values of $N=100,250,500$ (as indicated by arrows). Inset: same plot as Fig. 10, where the temperature used here is indicated by the black point. We adopted the parameters $\alpha =7/5, \beta =2/5, \gamma =9/10, \mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/10, \varphi =0$ (i.e., $\eta =N$), and $K_{B}T/\left(h{Y}_b^2\right)=3/10$.

Figure 12

Ductile response of the fracture phenomenon ($T^{*}<T<T_{\xi }$). Left panel: dimensionless force vs dimensionless extension for different values of $N=500,1000,1500$ (as indicated by the arrow). Right panel: fraction of intact, softened, and broken elements for different values of $N=500,1000,1500$ (as indicated by arrows). Inset: same plot as Fig. 10, where the temperature used here is indicated by the black points. We adopted the parameters $\alpha =7/5, \beta =2/5, \gamma =9/10, \mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/10, \varphi =0$ (i.e., $\eta =N$), and $K_{B}T/\left(h{Y}_b^2\right)=4/5$.

Figure 13

Overductile response of the fracture phenomenon ($T_{\xi }<T<T_{\chi }$). Left panel: dimensionless force vs dimensionless extension for different values of $N=500,1000,1500$ (as indicated by the arrow). Right panel: fraction of intact, softened, and broken elements for different values of $N=500,1000,1500$ (as indicated by arrows). Inset: same plot as Fig. 10, where the temperature used here is indicated by the black point. We adopted the parameters $\alpha =7/5, \beta =2/5, \gamma =9/10, \mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/10, \varphi =0$ (i.e., $\eta =N$), and $K_{B}T/\left(h{Y}_b^2\right)=6/5$.

Figure 14

Dimensionless force vs dimensionless extension for different values of the thermal to elastic energy ratio $K_{B}T/\left(h{Y}_b^2\right)=\{0.3,0.4,0.5,0.6,0.7,0.8,0.9,1\}$ (as indicated by the arrow). We observe that the behavior of the model changes from brittle, at low values of temperature, ductile for intermediate temperatures, to over ductile at high temperatures. We adopted the parameters $N=1000, \alpha =7/5, \beta =2/5, \gamma =9/10, \varphi =0$, and $\mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/10$.

Figure 15

Behavior of the dimensionless extension thresholds $Y_{\xi }/Y_b, Y_{\chi }/Y_b$, and $Y_{\xi \chi }/Y_b$ vs the dimensionless temperature $K_{B}T/\left(h{Y}_b^2\right)$ for a system always brittle (left panel), and for a system always ductile (right panel). In the left panel we have $Y_{\chi }<Y_{\xi \chi }<Y_{\xi }$ and we adopted the parameters $\alpha =13/10, \beta =3/10, \gamma =4/5$, and $\mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/5$. In the right panel we have $Y_{\chi }>Y_{\xi \chi }>Y_{\xi }$ and we adopted the parameters $\alpha =9/5, \beta =4/5, \gamma =13/10$, and $\mathrm{\Delta }E/\left(h{Y}_b^2\right)=1/10$.

Figure 16

Response of the system in the thermodynamic limit within the three temperature regimes $0<T<T^{*}$ (brittle, first column), $T^{*}<T<T_{\xi }$ (ductile, second column), and $T_{\xi }<T<T_{\chi }$ (overductile, third column). We plotted the average number of intact (first row), softened (second row), and broken elements (third row), and the stress $\langle f\rangle /N$ (fourth row), for $N\to \infty$. To compact the notation, we defined $\overline{\varphi }=1-\varphi$, and we introduced the characteristic stresses ${\sigma }_B, {\sigma }_S^{+}, {\sigma }_S^{-}$, and ${\sigma }_F$, as defined in Eqs. (98, 99, 100), and (102).

Figure 17

Behavior of the characteristic stresses ${\sigma }_B, {\sigma }_S^{+}, {\sigma }_S^{-}$, and ${\sigma }_F$ vs the temperature $T$, as defined in Eqs. (98, 99, 100), and (102) with $\varphi =0$. Moreover, some stress-extension curves are plotted in correspondence of the following values of the temperature: (a) $0<T<T^{*}$; (b) $T={\left(T^{*}\right)}^{-}$ (on the left of $T^{*}$); (c) $T={\left(T^{*}\right)}^{+}$ (on the right of $T^{*}$); (d) $T=T^{**}$; (e) $T^{**}<T<T_{\xi }$; (f) $T_{\xi }<T<T_{\chi }$. While the temperature $T^{*}$ indicates the switching between brittle and ductile behavior [see Eq. (84)], the temperature $T^{**}$ corresponds to ${\sigma }_S^{+}={\sigma }_F$; see Eq. (103). In the panels we also show the stress-extension response for $N=1000$ (dim colors or light gray curves).

Figure 18

Tensile stress-strain curves for a GaN nanowire oriented in the direction [0001] (diameter of 1.92 nm, length of 6.12 nm). The lateral facets are oriented along the $\left\{11\overline{2}0\right\}$ side planes. Comparison between molecular dynamics simulations results [93] (dashed lines) and our theory given by Eqs. (95) and (96) (continuous lines). The parameters used are reported in the main text.

Figure 19

The quantities $S_2\left(\xi \right)/N, S_1\left(\xi \right)$ and $S_0\left(\xi \right)$ are obtained using Usmani relations (continuous curves) for different values of $N=\{10,\cdots ,300\}$ (with step of 10). We observe that, as $N$ increases, the quantities approach their relative approximations obtained for large $N$ (dashed lines).

Figure 20

The quantity $(lndet\mathcal{A})/N$ is obtained using Usmani relations (continuous curves) for different values of $N=\{10,\cdots ,300\}$ (with steps of 10). We observe that, as $N$ increases, $(lndet\mathcal{A})/N$ approaches its approximation for large $N$ (dashed line).