Dynamic glass transition dramatically accelerates crack propagation in rubberlike solids

A crack propagating in a strained rubberlike solid exhibits an abrupt change of the propagation velocity by typically more than ${10}^2$ times at a specific threshold strain energy, which is a phenomenon called the “velocity jump.” Despite its scientific and industrial significance, the mechanism of the velocity jump had been unsolved for more than 30 years. This paper gives a clear answer to the mechanism, showing dynamic glass transition at the crack tip is the true origin of the crack velocity jump. We present concerted investigations involving theoretical analysis, numerical calculation, and experiment to establish an integrated understanding of the mechanism. Our findings indicate that the velocity jump can be found in general viscoelastic materials.

Figure 1

Velocity jump phenomenon of crack propagation in rubber materials and hypotheses of mechanism of the jump proposed in recent researches. (a) Procedure of pure shear test. 1. The top and bottom edges of the strip specimen are clamped and stretched up to a given strain ${\varepsilon }_{\mathrm{load}}$. 2. A side edge is cut to introduce an initial crack, and the crack propagation velocity $V$ is measured as a function of ${\varepsilon }_{\mathrm{load}}$. 3. After testing with various ${\varepsilon }_{\mathrm{load}}$, the $V\text{−}\mathrm{\Gamma }$ relationship is obtained via Eqs. (1) and (2). (b) Typical $V\text{−}\mathrm{\Gamma }$ relationship in a previous experiment [11]. $V$ exhibits an abrupt jump at a specific $\mathrm{\Gamma }$ that we call the “tearing energy at velocity jump” ${\mathrm{\Gamma }}_{\mathrm{jump}}$ (indicated by the dashed lines). Three series of data were obtained at different temperatures for an identical compound. (c) The mechanism of the velocity jump proposed by the FEM analysis [20]. At the moment of crack propagation, stress and strain rises rapidly at the crack tip (this stage is shown in blue). After a nonmonotonic behavior (in green-yellow), the strain and stress change gradually with time (in red). The time for the crack-tip element to fracture ($\mathrm{\Delta }t$) depends on at what stage the crack-tip element reaches the fracture criterion. (d) The mechanism of the velocity jump proposed by the MMCP analysis [12]. It is argued that the velocity jump appears due to a dynamic glass transition in the vicinity of the crack tip.

Figure 2

Schematic illustration of a mechanical model at a crack tip (step-loading model). Fracture of the element A induces a rapid loading on the new crack-tip element B. Mechanical behavior of the element B in this process is modeled as a viscoelastic response to a step loading. Note that $m$ has the unit of mass per area and is dependent on the specimen height $L_0$.

Figure 3

Schematic illustration of typical stress-strain relationship of uncracked rubber specimen under loading and unloading conditions. The area of stress-strain curve integrated up to the strain at break ${\varepsilon }_b$ indicates the strain energy density at break [i.e., Eq. (2) with ${\varepsilon }_{\mathrm{load}}={\varepsilon }_b]$; the sum of red and blue areas is the energy density applied during the loading process ($W_b$), the blue area is the energy released during the unloading process, and the red area is the hysteresis loss during the loading cycle ($H_b$).

Figure 4

Mechanical response obtained by SLM and FEM analyses. (a) Step-loading response obtained by SLM analysis: (top) strain response and (bottom) stress response. The normalization factors for strain and stress are defined as ${\varepsilon }_{\infty }:={\sigma }^{\mathrm{ext}}/E_0$ and ${\sigma }_{\infty }:={\sigma }^{\mathrm{ext}}$, respectively. Three representative regimes are indicated in different colors according to the explanation in Ref [20]; the blue/green (I/II) boundary is determined by the local maximum point of strain, and the green/red (II/III) boundary is determined by the time at that the strain reaches again to the level of the local maximum point; i.e., the strain values at the both boundaries are equal. (b) Mechanical response at a crack tip obtained with FEM simulation under the condition identical to that in Ref. [20]: (top) maximum nominal principal strain; (bottom) maximum true principal stress.

Figure 5

Viscoelastic behavior of a Zener element and mechanical response of SLM. (a) Dynamic modulus $E$ of a Zener element as a function of frequency. $E_0$ and $\lambda E_0$ correspond to the elastic moduli at the rubbery and glassy states, respectively. (b) Schematic illustration of strain response in SLM (black line). This behavior is characterized by three parameters, i.e., ${\widehat{\tau }}_{\mathrm{slow}}, {\widehat{\tau }}_{\mathrm{fast}}$, and $\omega$. The contributions from ${\widehat{\tau }}_{\mathrm{slow}}$ and ${\widehat{\tau }}_{\mathrm{fast}}$ are represented by a solid red line and dashed blue lines, respectively.

Figure 6

Schematic illustrations for deduction of $T_g$-dependence in SLM. (a) Stress relaxation functions with various $T_g$. (b) Corresponding strain behaviors in SLM (the glass-rubber transition behavior is simplified and exaggerated for clarity). (c) Expected relationships between the tearing energy and crack propagation velocity, where the tearing energy at velocity jump is not affected by $T_g$.

Figure 7

Correlations between $c_x, T_g, H_b$, and $W_b$. [(a)–(c)] Correlations between $c_x, H_b$, and $W_b$ for SBR's with various $c_x$ obtained by experiment; (a) $W_b\text{−}{c}_x$, (b) $H_b\text{−}{c}_x$, and (c) $W_b\text{−}{H}_b$ relationships. The dashed lines indicate the power-law approximations of the correlations. [(d)–(f)] Correlations between $T_g, H_b$, and $W_b$ for NBR, SBR, and BR compounds with equal $c_x$ obtained by experiment; (d) $W_b\text{−}{T}_g$, (e) $H_b\text{−}{T}_g$, and (f) $W_b\text{−}{H}_b$ relationships.

Figure 8

Crack propagation velocity $V$ as a function of tearing energy $\mathrm{\Gamma }$. The relationships were obtained at $T=298$ K with (a) three different rubbers (NBR, SBR, BR; all cross-linked and unfilled) with identical cross-linker concentrations $c_x=1.4$ wt% and (b) SBR with various $c_x$. The data for SBR with $c_x=1.4$ wt% are shown in both (a) and (b), indicated by green squares. The vertical lines indicate the velocity jump (guide for eyes).

Figure 9

Tearing energy at velocity jump ${\mathrm{\Gamma }}_{\mathrm{jump}}$ as a function of energy density at break $W_b$ obtained by experiments. The previous result is adopted from Ref. [11], in which data were obtained at various temperatures for an identical compound [cross-linked SBR with silica filler, shown in Fig. 1 in this paper]. All data in this work were obtained with unfilled compounds at $T=298$ K [Figs. 8 and 8(b)]. The orange circles indicate three different substances (NBR, SBR, and BR) with identical $c_x$ but different $T_g$. The dashed line shows a result of linear fitting to the data. The error bars indicate the standard deviation of $W_b$ examined with 8 specimens for each compound.

Figure 10

Mechanical properties of NBR, SBR and BR with equal $c_x$ obtained by experiment. (a) Nominal stress-strain relationships under quasistatic condition. (b) Dynamic moduli; storage modulus (solid line) and loss modulus (dashed line). The entire curves of dynamic moduli were reconstructed by the time-temperature superposition from the experimental data at various temperature (see text). The data of dynamic moduli were smoothened for clarity.

Figure 11

Mechanical properties of SBR's with various $c_x$ obtained by experiment. (a) Nominal stress-strain relationships under quasistatic condition. (b) Dynamic moduli; storage modulus (solid line) and loss modulus (dashed line). The entire curves of dynamic moduli were reconstructed by the time-temperature superposition from the experimental data at various temperature (see text).

Figure 12

Stress relaxation function of filled NBR by experiment [10] (dashed black line) and its Prony-series approximation (solid red line) employed for SLM analysis.

Figure 13

Effective elastic moduli $E_{\mathrm{eff}}\left(t\right)$ in the cases of $m=0$ and $m>0$, compared with the original stress relaxation function $E\left(t\right)$ obtained by an experiment [10].

Figure 14

Effective elastic modulus $E_{\mathrm{eff}}\left(t\right)$ compared to the original stress relaxation function $E\left(t\right)$ for a massless Zener element.