Fracture of liquid crystal elastomers

Significance

Understanding the fracture mechanics of soft materials is critical for enabling their actuation and shape programming, where mechanical integrity under extreme deformation is essential. Among these materials, liquid crystal elastomers (LCEs) stand out for their ability to undergo large, reversible shape changes due to a unique combination of elasticity and liquid crystal ordering, which presents new challenges for analyzing their fracture. We report a previously unobserved fracture phenomenon in LCEs: Cracks dynamically reorient during propagation attributed to deformation-coupled rotation of liquid crystal directors. We further develop a predictive, rate-dependent phase-field fracture model, validated across diverse geometries and loading conditions. These findings reveal microstructure-governed fracture behavior and provide a platform for understanding and engineering failure in soft materials with evolving microstructures.

Abstract

Liquid crystal elastomers (LCEs) are anisotropic, viscoelastic materials integrating polymer networks and liquid crystals. While their mechanical responses have been extensively studied, their fracture behavior remains largely unexplored. Specifically, the effect of the deformation-director coupling on LCE fracture paths is unknown, and fracture criteria for LCEs are not yet established. To address this gap, we combine experimental and theoretical approaches to investigate fracture propagation in LCEs. We stretch edge-cracked monodomain LCE samples, recording their stress-stretch responses and crack paths under varying initial directors and stretching rates. Our findings reveal that cracks can change direction during propagation, which are highly dependent on both the initial director and the stretching rate. To further understand LCE fracture behavior, we develop a rate-dependent phase-field fracture model, which is validated through experiments, and demonstrates the ability to predict complex fracture paths. Our study paves the way for designing LCEs with enhanced fracture properties, imperative for their future applications.

Soft materials, known for their ability to easily deform and mimic the properties of biological tissues, have gained significant interest in fields such as actuation, soft robotics, bioelectronics, and tissue engineering (1–8). These applications often involve large and reversible deformations, under which soft materials fail in ways that differ markedly from those of stiff engineering materials (9, 10). Diverse fracture behaviors have been observed in soft materials, closely linked to their microstructures. For instance, stretch-induced crystallization in silicone elastomers can cause sideways crack propagation, allowing for enormous stretchability (11). In polymer composites reinforced by fibers or nanosheets, cracks usually propagate parallel to the reinforced direction (12, 13). In contrast, blood clots composed of platelets and fibrin exhibit crack propagation perpendicular to the stretched fiber direction (14, 15). Additionally, viscoelasticity of polymer networks can introduce even richer fracture behaviors, such as arrested crack propagation observed in vitrimers (16).

Liquid crystal elastomers (LCEs) have distinctive molecular structures, composed of cross-linked polymer networks and liquid crystal mesogens (17). The mesogens, aligned in a specific orientation (called the director), can rotate in response to stress, resulting in unique mechanical behavior coupling deformation and director (17, 18). Although LCEs are anisotropic materials like fiber-reinforced elastomers, their deformation-director coupling makes them stand apart from fiber-reinforced elastomers. Moreover, LCEs are highly viscoelastic and dissipative (18–20). Consequently, LCEs are expected to exhibit fundamentally different fracture behavior from conventional elastomers. Although the unique mechanical properties of LCEs, such as nonlinear elasticity and viscoelasticity (18–23), have been extensively studied, and promising applications have been widely explored, the fracture behavior of LCEs remains largely unexplored, likely due to the complex director rotation involved (24). Only few studies have examined LCE fracture, primarily focusing on characterizing the crack-tip fields without crack propagation (24), and measuring the fracture energy of LCEs with directors parallel or perpendicular to the stretching direction through pure shear tests (25–27). The crack propagation and failure criteria with respect to the director, stretching rate, and geometry of LCEs remain unresolved. For example, the relationship between liquid crystal alignment and fracture behavior is not fully understood due to insufficient experimental characterization. Additionally, the lack of established fracture criteria for LCEs makes it challenging to predict the crack initiation and propagation under complex loading conditions. Moreover, simulation tools are unavailable for studying the fracture process of LCEs.

To address these issues, we combine experimental and theoretical approaches to systematically investigate fracture in monodomain LCEs. We perform tensile tests on edge-cracked LCE samples, recording their stress–stretch responses and crack trajectories under varying initial directors and stretching rates. Our results demonstrate that cracks in LCEs interact with evolving microstructure in ways fundamentally distinct from those in conventional elastomeric materials. Based on these observations, we establish fracture criteria and develop a rate-dependent phase-field fracture model to predict the initiation and evolution of crack propagation. Validated across diverse geometries and loading conditions, the model accurately captures complex crack paths. These findings uncover a class of microstructure-coupled fracture behavior and provide a general framework for understanding and engineering failure in soft materials with evolving internal structure.

Results

Tilted Fracture Path.

We conducted fracture tests on monodomain LCE samples with length H = 30 mm and width W = 5 mm (Fig. 1A). A sample with a tilted initial director ${\theta }_0={45}^{\circ }$ and a horizontal edged crack of length $l_c=0.4W$ was subjected to axial tension at different stretching rates, $\dot{\lambda }=0.1\%/s$ (Movie S1) and $\dot{\lambda }=10\%/s$ (Movie S2). The stress-stretch curves (the red curve for rate 0.1%/s and the brown curve for rate 10%/s) show that the stress softens under intermediate stretch due to director rotation, and stiffens at large deformation. At the lower stretching rate 0.1%/s, the crack propagation initiates when the applied stretch increases to a certain value (λ ∼ 2.18), and then the nominal stress reaches its maximum as the stretch further increases (λ ∼ 2.43), followed by a decrease to zero when the sample is completely fractured (λ ∼ 2.46). Points (i − v) marked on the curve correspond to different stages of deformation and crack propagation shown in Fig. 1C. In the tests, two blue horizontal lines drawn on the samples serve as visual references. As the stretch increases, the precrack opens asymmetrically, and the blue lines rotate, indicating substantial shear deformation. As images iv and v in Fig. 1C show, the crack propagates almost horizontally in the deformed configuration. When the sample is fully fractured and relaxed, the fracture path is tilted (Fig. 1D). We define the fracture angle ${\theta }_c$ as the crack angle with respect to the horizontal direction averaged over the entire crack length, specifically between region X = 0.4W and X = W in this case. The final crack orientations differ for the two stretching rates, ${\theta }_c={35}^{\circ }$ for $\dot{\lambda }=0.1\%/s$ and ${\theta }_c={30}^{\circ }$ for $\dot{\lambda }=10\%/s$, both of which are different from the initial director.

Fig. 1.

Tilted fracture path of a LCE with a tilted initial director. (A) Schematic of a LCE sample with a tilted initial director and a horizontal edged crack subjected to uniaxial tension. The initial director is illustrated by the orange ellipse. (B) Stress-stretch responses for two different stretching rates, $\dot{\lambda }=0.1\%/s$ and $\dot{\lambda }=10\%/s$. (C) Images showing the deformation and crack propagation of a LCE sample under different stretches at a stretching rate of 0.1%, with the corresponding stretch conditions marked on the stress-stretch curve in Fig. 1B. (D) Fully fractured and relaxed samples show tilted fracture paths at the two stretching rates.

The observed fracture behavior, characterized by a tilted fracture path in a rate-dependent manner, is very different from that of conventional fiber-reinforced elastomers. In a conventional fiber-reinforced elastomer, fibers reorient strictly following the local deformation gradient of the material. Consequently, a fracture path is usually parallel to the fiber direction in the reference configuration (28), and tilted in the deformed configuration. In contrast, the director of a LCE, governed by its own evolution law, is an independent variable from the local deformation gradient. It is reasonable to think that a fracture path of a LCE is associated with the current director in the deformed configuration. On the other hand, the minimization of the total free energy of the system requires the director to couple with the deformation gradient in a way associated with the initial director and stress state. Our previous uniaxial tension test for a LCE with a tilted initial director (similar to the far-field stress state here) shows that the director gradually rotates toward the stretch direction as the external stretch increases, and becomes almost vertical when the stretch reaches around 2.5 (29). From the observation of the horizontal fracture propagation and the expectation of vertical director distribution, we hypothesize that a crack in a LCE prefers to propagate perpendicular to the director. Similarly, a recent study on the fracture of blood clots made up of platelets and fibrin has shown that cracks propagate perpendicular to the direction of stretched fibers (14, 15), which agrees with our assumption. Additionally, rate-dependent fracture paths are also anticipated due to rate-dependent deformation and director of LCEs.

Phase-Field Fracture.

We will next quantitatively understand the observed rate-dependent fracture phenomena and predict crack propagation of LCEs in various conditions. As the crack paths in LCEs can be tilted and complex, we develop a phase-field fracture model, which does not require knowing crack paths a priori. The constitutive model is based on our previous constitutive framework of viscoelastic LCEs (29). To describe damage degrees, we introduce a phase field d ∊ [0, 1], which models crack surfaces diffusively to avoid the treatment of discontinuously sharp cracks, and incorporates dissipation via the fracture process (30–32). Following the recent studies on viscoelastic fracture (33, 34), we assume that fracture propagation is determined by the competition between the equilibrium elastic energy and intrinsic fracture energy. Based on nonequilibrium thermodynamics, we derive the governing equation for the phase field d as

$$-\frac{\partial g\left(d\right)}{\partial d}{\psi }_0=G_c{\delta }_d\gamma \left(d,\nabla d\right),$$ [1]

where g(d) is the degradation function, describing the material deterioration due to damage, ${\psi }_0$ is the intact equilibrium free energy density, G_c is the intrinsic fracture toughness, γ is the crack surface density function, which is a function of the phase field d and its gradient with respect to the reference coordinates ▽d, γ = γ(d, ▽d), and ${\delta }_d\gamma$ simply means the variational derivative of γ. With the assumption that a crack prefers to propagate perpendicular to a director, the specific form of γ is derived in relation to the director (SI Appendix). The left-hand side of Eq. 1 represents the energetic crack driving force, while the right-hand side corresponds to the fracture resistance. More details about the formulation of the phase-field fracture model are presented in Materials and Methods section.

We numerically predicted the deformation and crack propagation in a LCE sample with the same geometry, initial director and boundary conditions as the experiment (Fig. 2 and Movies S3 and S4). Our results show that the stress-stretch response and the stretch at complete fracture at different stretching rates obtained from the simulations (solid curves) agree very well with those from the experiments (dashed curves) (Fig. 2A). The snapshots for the predicted deformation and crack propagation of the LCE sample under $\dot{\lambda }=0.1\%/s$, with the contours of the phase field d (Fig. 2B), are similar to the experimental observations. The horizontal black lines are drawn for visual reference, and their large rotation indicates significant shear deformation. The fracture angles in the fully fractured and relaxed configuration are distinct for the two different stretching rates (Fig. 2C), consistent with the experimental findings. The tilted fracture path can be explained by the intriguing deformation and director rotation of LCEs (Fig. 2 B and D). The contour plot in Fig. 2D shows the x component of the director, cosθ, with θ the angle between the director and the x axis, overlaid with arrows showing the director distribution near the crack tip. In front of the crack tip, we can clearly see that the director is almost vertical, and cosθ is close to 0. The vertical director leads to horizontal crack propagation in the deformed configuration (Fig. 2B), based on our assumption that the crack prefers to propagate perpendicular to the director. After the fractured sample is fully relaxed, the fracture path is tilted. Therefore, the tilted fracture path is an outcome of both the shear deformation and the director-deformation coupling effect.

Fig. 2.

Numerical predictions of LCE fracture. (A) Numerical and experimental comparison of the stress-stretch responses for two different stretching rates. (B) Numerical predictions of the deformation and crack propagation at a stretching rate of 0.1%/s. The color bar represents the phase field, and the material with d > 0.95 has been removed. (C) Samples in the fully fractured and relaxed configuration for the two stretching rates. (D) Director distribution in front of the crack tip, where the color bar represents the x component of the director, cosθ.

Influence of Stretching Rates and Initial Directors.

We further conducted simulations and experiments to investigate the influence of stretching rates and initial directors on fracture of LCEs (Fig. 3). Our experiments show that the stretch at complete fracture increases when the stretching rate increases (Fig. 3A and SI Appendix, Fig. S1). As the initial director increases, the maximum applied stress increases, while the stretch at complete fracture decreases. It is noted that for ${\theta }_0={75}^{\circ }$, a high amount of stretch is needed from the stage of maximum applied stress to complete fracture. The stress-stretch response and the stretch for LCEs to fracture predicted by our model agree with the experimental results very well (Fig. 3A). Nonetheless, there are some discrepancies when ${\theta }_0={30}^{\circ }$ and ${\theta }_0={75}^{\circ }$ due to the inaccuracy of the constitutive model for LCEs.

Fig. 3.

Influence of stretching rates and initial directors on LCE fracture. Numerical and experimental (A) stress-stretch responses and (B) fracture angles for various initial directors and stretching rates. Fracture paths from (C) experiments and (D) simulations of samples with different initial directors at a stretching rate of $\dot{\lambda }=1\%/s$.

Both the experiments and simulations show that the crack angle ${\theta }_c$ decreases with the increase of the stretching rate, particularly for a small initial director (θ₀ = 30° and 45°) (Fig. 3B). This can be understood based on our previous viscoelastic study (29), which shows that a higher stretching rate leads to smaller shear deformation, and therefore, smaller crack angle. The simulations underestimate the crack angle when the stretching rate is high, which can be explained by the underestimation of shear stretch at a high stretching rate by our viscoelastic model. Both simulations and experiments show that the crack angle θ_c nonmonotonically increases and then decreases with the increase of θ₀ (Fig. 3 B–D). The typical fracture paths from the experiments and simulations at a stretching rate 1%/s match each other very well (Fig. 3 C and D). The fracture path for larger θ₀ is more curved, correlating to the less steep decrease in stress after reaching the maximum stress shown in Fig. 3A. The overall trends predicted by our model are consistent with the experimental results.

Fracture of LCEs with Different Geometries.

We further explore fracture of LCE samples with different geometries. A LCE sample with an initial director θ₀ = 45°, length H = 10 mm, width W = 15 mm, and a precrack of length l_c = 0.4W, is stretched experimentally and numerically at different stretching rates, $\dot{\lambda }=0.1\%/s$ and $\dot{\lambda }=10\%/s$ (Fig. 4 and Movies S5–S8). Fig. 4 A and D shows the corresponding profiles of the deformation and crack propagation. The horizontal blue and black guiding lines were drawn in the crack front for visual reference for the experiments and simulations, respectively. At the beginning of the crack propagation, both guiding lines remain almost horizontal in front of the crack tip, and the cracks propagate upward. It is worth mentioning that the deformation in front of the crack tip tilts the guiding line downward before the crack propagation, which is clearly shown in the simulation result (SI Appendix, Fig. S2). The crack tip under the lower stretching rate is more rounded due to the material relaxation. After the fractured sample fully relaxes, the final fracture path exhibits an initial upward trajectory followed by a downward trend (Fig. 4B). The fracture paths from the experiments and simulations agree very well (Fig. 4C). The initial slope at the higher stretching rate is smaller. Fig. 4E shows the director distribution in the early stage of the crack propagation, where the director in front of the crack tip is tilted, leading to the upward fracture path in the deformed configuration. Further, due to low shear component ${\lambda }_{21}$, indicated by the almost horizontal black line, the initial fracture path in the fractured and relaxed configuration exhibits an upward trajectory. The subsequent downward trend can be explained in a way similar to the previous case in Fig. 2. As we can see, the complex trajectory of a crack path is highly correlated with the director distribution and deformation state near the crack tip. Comparing the numerical and experimental stress-stretch curves for the two stretching rates (Fig. 4F), we see that a higher stretch is required to completely break the sample for a higher stretching rate. While the theoretical prediction overestimates the stretches at the initiation of crack propagation and at the maximum stress for the higher stretching rate compared to the experiment, the predicted stress response at the lower stretching rate agrees very well with the experiment. Our model shows good overall predictions of the deformation, stress-stretch response, and fracture path of LCEs with different directors and geometries at different loading rates.

Fig. 4.

Influence of geometries on LCE fracture. (A) Experimental snapshots showing the deformation and crack propagation of a LCE under different stretches at two stretching rates, $\dot{\lambda }=0.1\%/s$ and $\dot{\lambda }=10\%/s$. The sample near the boundary is obscured by the gripper, as illustrated in the schematic (λ = 1). (B) Fractured samples and (C) the corresponding fracture paths from the experiments and simulations at the two stretching rates. (D) Numerical snapshots showing the deformation and crack propagation of a LCE under the same conditions as the experiments in (A). (E) Director distribution at stretch λ = 1.82 and a stretching rate of 0.1%/s. (F) Comparison of the numerical and experimental stress-stretch responses for the two stretching rates.

It is important to note that adopting the fracture criterion in Eq. 1 does not imply that bulk dissipation is unimportant in crack propagation. On the contrary, the equilibrium elastic energy used as the fracture driving force depends on the entire deformation field, which is influenced by the material’s full viscoelastic response. Thus, crack propagation remains dependent on the overall viscoelastic behavior, even when the intrinsic toughness is separated from bulk dissipation. As demonstrated by our simulations, the fracture behavior exhibits strong rate dependence, which arises not only from the rate-sensitive nature of intrinsic chain scission but also from bulk viscoelastic dissipation.

Fracture of LCEs with Different Precracks.

We further explore the influence of precracks on the fracture path using two examples, including changing the precrack length and introducing two precracks. We first stretch a sample of the same geometry and director as that in Fig. 4 but a longer precrack $l_c=0.6W$ at a stretching rate $\dot{\lambda }=1\%/s$ (Fig. 5A and Movies S9 and S10). Fig. 5A presents the experimental and numerical profiles of the LCE sample under different stretches, indicating consistent deformation, crack propagation, and final fracture path from the experiments and simulations. The crack initially propagates upward, with the guiding lines tilted upward in the deformed configuration. Compared to the case with a shorter precrack (Fig. 4), the slope of the initial upward trajectory is smaller in the fully fractured and relaxed configuration because of the higher shear deformation ${\lambda }_{21}$ (illustrated by the guiding lines in Figs. 4E and 5B). The combined effect of deformation and director distribution leads to the fracture path. With further increase of the precrack length, an initial downward trajectory is expected.

Fig. 5.

Influence of precracks on fracture of LCEs. (A) Experimental and numerical snapshots showing the deformation, crack propagation, and final fracture path of a LCE with a long precrack of length $l_c=0.6W$. (B) Director distribution of the LCE at stretch λ = 1.95. (C) Schematic of a LCE sample with two horizontal edge-cracks subjected to uniaxial tension. (D) Experimental and (E) numerical snapshots showing the deformation and crack propagation of the LCE with two edge-cracks under different stretches. (F) Comparison of the fractured samples after full relaxation from the experiment and simulation. E and F use the same contour of the phase field d.

In another demonstration, two precracks of length $l_c=0.25W$ are introduced into a LCE sample with length H = 10 mm and width W = 20 mm to observe the interaction of cracks (Fig. 5C and Movies S11 and S12). Fig. 5 D and E shows that the deformation and crack path obtained from the experiment and simulation, respectively, agree well with each other. When the stretch is small, the guiding lines in both the experiment and simulation are slightly tilted downward; correspondingly, due to the steeply downward-tilted director in front of the crack tip as shown in SI Appendix, Fig. S3, the left precrack propagates upward, and the right precrack propagates downward, as observed in both the deformed and fully fractured, relaxed configurations (Fig. 5 D–F). When the two cracks propagate toward the vertical centerline, the guiding lines become almost vertical. In the experiment, the two crack tips do not meet; the right crack (enclosed by the black dashed circle) propagates and leads to the final fracture, while the left one is arrested. In contrast, in the simulation, finite damage bands cause the two cracks to intersect, resulting in breaking of the sample at its center. Fig. 5F shows the corresponding experimental and numerical fracture paths in the fully fracture and relaxed configuration. Our model can well capture the overall fracture of LCEs that involve complex crack interactions.

Discussion

By combining experiments and theory, we have thoroughly investigated fracture of LCEs. Attributed to their unique molecular alignment and reorientation in response to stress, LCEs exhibit unusual and varying fracture behavior, such as cracks path deflection during propagation, dictated by various factors, including the initial director, stretching rate, and geometry. Grounded in a hypothesis that cracks prefer to propagate perpendicular to the director of a LCE, our developed phase-field fracture model successfully predicts the crack propagation in LCEs of different geometries with different initial directors and under varying stretching rates. Based on the combined experimental and computational results, we unravel that a fracture path is governed by both the director distribution and deformation state at a crack tip.

The numerical tool developed in this paper opens up some interesting directions for future work. Since the director of a LCE always evolves with loading, which can lead to complex fracture paths, it is challenging to measure the fracture toughness of LCEs with a certain director. Our numerical tool may provide a standard or protocol to measure it. Additionally, to improve the fracture toughness while maintaining high stiffness of LCEs for future applications, our numerical tool allows us to design the director distribution of LCEs for outstanding performance. We envision our study serves as a cornerstone for future research on LCE fracture.

Beyond LCE-specific applications, our results underscore the broader significance of microstructural dynamics in fracture. The proposed modeling framework can potentially be extended to other anisotropic and reconfigurable materials whose internal architecture evolves under mechanical loading. For example, recent studies on blood clots composed of platelets and fibrin have shown that cracks propagate perpendicular to the direction of aligned fibers—a phenomenon analogous to the deformation–director coupling observed in LCEs. Our framework may thus be applicable to biomaterials in which evolving microstructure plays a central role in fracture behavior. This work aims to bridge a critical gap between microscale structural reorganization and macroscale failure—a topic of growing interest in soft matter physics, biomechanics, and materials science. We anticipate that our findings will support future investigations into microstructure-coupled fracture and inform the design of robust, high-performance soft materials for emerging applications.

Materials and Methods

Material and Specimen Preparation.

Main-chain monodomain LCEs were synthesized via a two-stage thiol-acrylate Michael addition-photopolymerization reaction (SI Appendix, Standard Operating Procedure) (35). The chemicals used include diacrylate mesogen, 1,4-Bis-[4-(3-arcyloyoxypropyloxy) benzoyloxy]-2-methylbenzene (RM257, Wilshire Technologies, 95%), crosslinker, pentaerythritol tetrakis(3-mercaptopropionate) (PETMP, Sigma-Aldrich, 95%), chain extender, 2,2-(ethylenedioxy) diethanethiol (EDDET, Sigma-Aldrich, 95%), catalyst, dipropylamine (DPA, Sigma-Aldrich, 98%) and photoinitiator, 2-Hydroxy-4′-(2-hydroxyethoxy)-2-methylpropiophenone (HHMP, Sigma-Aldrich, 98%). RM257 (6.4 g) was first dissolved in toluene (4 g) at 80 °C and then cooled to room temperature. EDDET (1.465 g), PETMP (0.347 g), HHMP (0.041 g), and DPA solution (DPA:toluene = 1:50, 0.768 g in total) were subsequently added to the solution, and thoroughly mixed using a vortex mixer at 3,200 rpm for 15 min. The solution was then degassed, poured into a mold, and cured at room temperature for 24 h in absence of light. Afterward the cured film was placed in an oven at 80 °C for 24 h to remove the solvent. The film was then stretched to twice its original length and exposed to UV light for 1 h to complete the second-stage crosslinking process. The fixed stretch is around 1.88 after the photocrosslinking. Specimens were cut into rectangular shapes with a precrack, each with a certain initial director (Fig. 1A). All samples used in our experiments have a thickness of approximately 0.45 mm, which is much smaller than the other characteristic dimensions of the specimens. As a result, the fracture specimens should be considered under plane stress conditions.

Fracture Testing.

We conducted uniaxial tension tests to LCEs with one precrack or two precracks using an Instron universal testing machine (Model 5944). Samples with various geometries and initial directors were clamped to two pneumatic grips. Then they were stretched under displacement-controlled loading at various stretching rates. Experimental data were collected until samples completely fractured. The deformation and fracture process of samples were recorded by a Canon EOS 6D digital single-lens reflex camera. Fractured samples in the fully fractured and relaxed configuration were also photographed.

Image Analysis.

The fracture angle of fully fractured and relaxed samples with a large length-to-width ratio was analyzed using ImageJ software. The crack trajectory of samples with a small length-to-width ratio was extracted using MATLAB.

Theoretical Approach.

We have developed a rate-dependent phase-field fracture model to simulate the fracture process of viscoelastic LCEs. Unlike other fracture models, such as the extended finite element method or cohesive zone model, the phase-field method uses a phase-field variable to implicitly trace interfaces without dealing with sharp boundaries, which is ideal for the fracture of LCEs with complex fracture paths (36). The phase field d ∊ [0, 1] is introduced to describe the damage degree of the material. We consider a homogeneous LCE body ${\mathrm{\Omega }}_0$ in the reference configuration and denote ${\partial \mathrm{\Omega }}_0$ as its boundary. Considering director rotation in the plane, we define the director in the current state as n = (cosθ, sinθ, 0) to describe the alignment of liquid crystal mesogens, where θ is the angle between the director and the x axis in the deformed configuration. Since the relaxation timescale of the director rotation is very small (18, 19), we only consider the viscosity from the network without contributions from the director. Using our developed viscoelastic model for LCEs (29), we employ a rheological model by connecting the Kelvin-Voigt model and Maxwell model with N branches in parallel. The deformation gradient F in the Maxwell branches is decomposed to elastic and viscous parts, $\mathbf{F}={\mathbf{F}}_i^e{\mathbf{F}}_i^v$ (i = 1, 2, …, N). The free energy density of the material is formulated as a function of the state variables F, ${\mathbf{F}}_i^v$, n, and d,

$$\psi (\mathit{F},{\mathit{F}}_i^v,\mathit{n},d)=g\left(d\right)\left\{{\psi }_0(\mathit{F},\mathit{n})+{\sum }_{i=1}^N{\psi }_i\left[\mathit{F}{\left({\mathit{F}}_i^v\right)}^{-1},\mathit{n}\right]\right\},$$ [2]

where g(d) is the degradation function describing the material deterioration due to damage, ${\psi }_0$ and ${\psi }_i$ are the intact free energy densities of the equilibrium branch and nonequilibrium branches, respectively. We use the phase-field regularized cohesive zone model to describe the quasi-brittle fracture (30, 31), where the degradation function is assumed to be

$$g\left(d\right)=\frac{{\left(1-d\right)}^2}{{\left(1-d\right)}^2+a_{1}d\left(1-0.5d\right)},$$ [3]

where $a_1=\frac{2}{\pi }\frac{b_{\mathit{ch}}}{b_0}$ with the phase field length scale $b_0$ for numerical regularization and the fractocohesive length $b_{\mathit{ch}}$. The form of g(d) leads to a linear softening law (30).

The free energy density of the equilibrium branch is

$$\begin{array}{c}{\psi }_0=\frac{1}{2}{\mu }_0\left[tr\left({\mathit{l}}^{-1}\mathit{F}{\mathit{l}}_0{\mathit{F}}^T\right)-3-2\mathit{log}\left(J\right)\right]+{\mu }_{\alpha }{\left[tr\left({\mathit{l}}^{-1}\mathit{F}{\mathit{l}}_0{\mathit{F}}^T\right)-3\right]}^2\\ +\frac{1}{2}{\mu }_{0}a{\left|\left|{\mathit{n}}_{\mathit{F}}\right|\right|}^2+\frac{1}{2}K{\left(J-1\right)}^2,\end{array}$$ [4]

where ${\mu }_0$ is the shear modulus, ${\mu }_{\alpha }$ is the coefficient of nonlinear elasticity, K is the bulk modulus, $J=det\mathbf{F}$, $a=\frac{a_0}{m\left[tr\left({\mathbf{F}}^T\mathbf{F}\right)-3\right]+1}$ with $a_0$ the semisoft parameter and m a positive scaling factor, ${\left|\left|{\mathbf{n}}_{\mathbf{F}}\right|\right|}^2=tr\left[\mathbf{n}\mathbf{\otimes }\mathbf{n}\bullet \mathbf{F}\left(\mathbf{I}-{\mathbf{n}}_0\otimes {\mathbf{n}}_0\right){\mathbf{F}}^T\right]$, $\mathbf{l}=\left(r-1\right)\mathbf{n}\mathbf{\otimes }\mathbf{n}+\mathbf{I}$ is the dimensionless shape tensor, and ${\mathbf{l}}_0=(r-1){\mathbf{n}}_0\otimes {\mathbf{n}}_0+\mathbf{I}$ is the dimensionless initial shape tensor, where r represents the shape anisotropy of the network distribution, n₀ = (cosθ₀, −sinθ₀, 0) is the initial director with θ₀ the angle between the director and the x axis in the initial state (Fig. 1A), and I is the identity tensor.

The free energy density for the nonequilibrium branch is

$$\begin{array}{c}{\psi }_i=\begin{array}{c}\frac{1}{2}{\mu }_i\left\{\left[tr\left({\mathit{l}}^{-1}{\mathit{F}}_i^e{\mathit{l}}_0{\left({\mathit{F}}_i^e\right)}^T\right)-3-2\mathit{log}\left(J_i^e\right)\right]+2\frac{{\mu }_{\alpha }}{{\mu }_0}{\left[tr\left({\mathit{l}}^{-1}{\mathit{F}}_i^e{\mathit{l}}_0{\left({\mathit{F}}_i^e\right)}^T\right)-3\right]}^2\right\}\hfill \\ +\frac{1}{2}{\mu }_i{a}_i{\left|\left|{\mathit{n}}_{{\mathit{F}}_i^e}\right|\right|}^2+\frac{1}{2}{K}_i{\left(J_i^e-1\right)}^2\hfill \end{array},\hfill \end{array}$$ [5]

where ${\mu }_i$ is the shear modulus of the i-th branch, K_i is the parameter controlling the incompressibility of ${\mathbf{F}}_i^e, J_i^e=det {\mathit{F}}_i^e, a_i=\frac{a_0}{m\left[tr\left({\left({\mathbf{F}}_{\mathrm{i}}^{\mathrm{e}}\right)}^T{\mathbf{F}}_{\mathrm{i}}^{\mathrm{e}}\right)-3\right]+1}$, and ${\left|\left|{\mathit{n}}_{{\mathbf{F}}_{\mathit{i}}^{\mathit{e}}}\right|\right|}^2=tr\left[\mathbf{n}\mathbf{\otimes }\mathbf{n}\bullet {\mathbf{F}}_i^e\left(\mathbf{I}-{\mathbf{n}}_0\otimes {\mathbf{n}}_0\right){\left({\mathbf{F}}_i^e\right)}^T\right]$.

Considering an isothermal process and neglecting the body force, the energy balance of the system is

$${\int }_{{\mathrm{\Omega }}_0}\dot{\psi }dV+{\int }_{{\mathrm{\Omega }}_0}\dot{\varphi }dV={\int }_{\partial {\mathrm{\Omega }}_0}\mathit{t}·\dot{\mathit{x}}dA,$$ [6]

where $\dot{\varphi }$ is the dissipated energy per unit time and unit volume through viscous deformation and fracture, t is the traction acting on the surface. Thermodynamics requires that the dissipated energy should not decrease, namely, $\dot{\varphi }\ge 0$. The dissipation rate is contributed from viscous deformation and fracture process,

$$\dot{\varphi }\left(\dot{\mathit{F}},{\dot{\mathit{F}}}_i^v,d\right)={\dot{\varphi }}_0\left(\dot{\mathit{F}},d\right)+{\sum }_{i=1}^N{\dot{\varphi }}_i\left({\dot{\mathit{F}}}_i^v,d\right)+{\dot{\varphi }}_f,$$ [7]

where ${\dot{\varphi }}_0$ is the dissipation rate from the dashpot of the Kelvin-Voigt model, ${\dot{\varphi }}_i$ is from the dashpot of the i-th branch of the Maxwell model, and ${\dot{\varphi }}_f$ is due to fracture.

We assume

$${\dot{\varphi }}_0\left(\dot{\mathit{F}},d\right)=g\left(d\right)\frac{\partial {\phi }_0\left(\dot{\mathit{F}}\right)}{\partial \dot{\mathit{F}}}:\dot{\mathit{F}},$$ [8]

and

$${\dot{\varphi }}_i\left({\dot{\mathit{F}}}_i^v,d\right)=g\left(d\right)\frac{\partial {\phi }_i\left({\dot{\mathit{F}}}_i^v\right)}{\partial {\dot{\mathit{F}}}_i^v}:{\dot{\mathit{F}}}_i^v.$$ [9]

The term ${\phi }_i(i=0,1,\cdots ,N)$ is called the dissipation potential (37), with the assumption of the following form

$${\phi }_0=\frac{1}{8}{\eta }_{0}J\left(\dot{\mathit{F}}{\mathit{F}}^{-1}+{\mathit{F}}^{-T}{\dot{\mathit{F}}}^T\right):\left(\dot{\mathit{F}}{\mathit{F}}^{-1}+{\mathit{F}}^{-T}{\dot{\mathit{F}}}^T\right),$$ [10]

and

$${\phi }_i=\frac{1}{2}{\eta }_i\left({\dot{\mathit{F}}}_i^v{\left({\mathit{F}}_i^v\right)}^{-1}\right):\left({\dot{\mathit{F}}}_i^v{\left({\mathit{F}}_i^v\right)}^{-1}\right),$$ [11]

where ${\eta }_0$ is the viscosity constant and ${\eta }_i$ is the viscosity that changes nonlinearly with deformation.

Organizing Eqs. 2–11, we obtain

$$\begin{array}{c}{\int }_{{\mathrm{\Omega }}_0}\left[-\nabla ·\left(\frac{\partial \psi }{\partial \mathit{F}}+g\left(d\right)\frac{\partial {\phi }_0}{\partial \dot{\mathit{F}}}\right)·\dot{\mathit{x}}+g\left(d\right){\sum }_{i=1}^N\left(\frac{\partial {\psi }_i}{\partial {\mathit{F}}_i^v}+\frac{\partial {\phi }_i}{\partial {\dot{\mathit{F}}}_i^v}\right):{\dot{\mathit{F}}}_i^v+\frac{\partial \psi }{\partial \mathit{n}}·\dot{\mathit{n}}+\left(\frac{\partial \psi }{\partial d}\dot{d}+{\dot{\varphi }}_f\right)\right]dV\hfill \\ +{\int }_{\partial {\mathrm{\Omega }}_0}\left[\left(\frac{\partial \psi }{\partial \mathit{F}}+g\left(d\right)\frac{\partial {\phi }_0}{\partial \dot{\mathit{F}}}\right)·\mathit{N}-\mathit{t}\right]·\dot{\mathit{x}}dA=0\end{array},$$ [12]

where ▽ denotes the gradient with respect to X, and N is the outward unit normal vector. The mechanical equilibrium equation, the evolution equations for ${\mathbf{F}}_i^v$ and d, and the equilibrium equation for the director n can be obtained as the governing equations for the state variables F, ${\mathbf{F}}_i^v$, d, and n.

The mechanical equilibrium and the corresponding boundary condition are

$$\nabla ·\mathit{P}=0,$$ [13]

and

$$\mathit{P}·\mathit{N}=\mathit{t},$$ [14]

with the nominal stress P defined as

$$\mathit{P}=\frac{\partial \psi }{\partial \mathit{F}}+g\left(d\right)\frac{\partial {\phi }_0}{\partial \dot{\mathit{F}}}.$$ [15]

The evolution of ${\mathbf{F}}_i^v$ is

$$\frac{\partial \psi }{\partial {\mathit{F}}_i^v}+\frac{\partial {\phi }_i}{\partial {\dot{\mathit{F}}}_i^v}=0.$$ [16]

Enforcing the constraint $\mathbf{n}\mathbf{·}\mathbf{n}=1$ through introducing a Lagrange multiplier, the equilibrium equation for the director is obtained as (29)

$$\frac{\partial \psi }{\partial \mathit{n}}=\left(\frac{\partial \psi }{\partial \mathit{n}}·\mathit{n}\right)\mathit{n}.$$ [17]

The fracture process is governed by

$${\int }_{{\mathrm{\Omega }}_0}\left(\frac{\partial \psi }{\partial d}\dot{d}+{\dot{\varphi }}_f\right)dV=0,$$ [18]

where the thermodynamics requirement ${\dot{\varphi }}_f\ge 0$ is satisfied when $\frac{\partial \psi }{\partial d}\dot{d}\le 0$, which necessitates $\dot{d}\ge 0$ since $\frac{\partial \psi }{\partial d}=\frac{\partial g\left(d\right)}{\partial d}{\sum }_{i=0}^N{\psi }_i<0$.

Rather than using the total free energy, including both equilibrium and nonequilibrium contributions, as the driving force for crack propagation, Lopez-Pamies and collaborators proposed a modified fracture criterion in which only the equilibrium elastic energy competes with the intrinsic fracture energy (33, 34). This treatment enables the intrinsic fracture process at the crack tip to be addressed independently from bulk viscoelastic dissipation. The approach was recently implemented by Lopez-Pamies and collaborators (38), who demonstrated its applicability and predictive accuracy for crack initiation and propagation in viscoelastic elastomers using a phase-field framework. Inspired by their work, we describe the fracture process in Eq. 18 by considering fracture propagation is determined by the competition between the equilibrium elastic energy and intrinsic fracture energy, in the following form,

$${\int }_{{\mathrm{\Omega }}_0}\left(\frac{\partial g\left(d\right)}{\partial d}{\psi }_0\dot{d}+G_c{\delta }_d\gamma \dot{d}\right)dV=0,$$ [19]

where G_c is the intrinsic fracture toughness and the crack surface density function γ is expressed as,

$$\gamma =\frac{1}{c_0}\left[\frac{1}{b_0}w\left(d\right)+b_0\nabla d·\mathit{A}·\nabla d\right],$$ [20]

where w(d) is the geometric crack function, $c_0$ is a constant, and A is the second-order structural tensor related to the director orientation. In the phase-field regularized cohesive zone model (30, 31), the geometric crack function is defined as w(d) = 2d − d² and the constant is set as $c_0=\pi$. The exact form $\mathbf{A}=\mathbf{I}+\alpha {\mathbf{F}}^{-1}{\mathbf{n}}_{\perp }\mathbf{\otimes }{\mathbf{F}}^{-1}{\mathbf{n}}_{\perp }$ is derived in the SI Appendix based on the assumption that cracks prefer to propagate perpendicular to the director, with α representing the coefficient of anisotropy of the fracture energy, and ${\mathbf{n}}_{\perp }$ being the unit vector perpendicular to the director.

Combining Eqs. 4, 5, 10, and 15, the nominal stress is derived as

$$\begin{array}{c}\mathit{P}=g\left(d\right)\left\{\begin{array}{c}{\mu }_0\left[g_0{\mathit{l}}^{-1}\mathit{F}{\mathit{l}}_0-{\mathit{F}}^{-T}+a\mathit{n}\otimes \mathit{n}·\mathit{F}\left(\mathit{I}-{\mathit{n}}_0\otimes {\mathit{n}}_0\right)-m\left(\frac{a^2}{a_0}\right){\left|\left|{\mathit{n}}_{\mathit{F}}\right|\right|}^2\mathit{F}\right]\hfill \\ +KJ\left(J-1\right){\mathit{F}}^{-T}+\frac{1}{2}{\eta }_{0}J\left(\dot{\mathit{F}}{\mathit{F}}^{-1}+{\mathit{F}}^{-T}{\dot{\mathit{F}}}^T\right){\mathit{F}}^{-T}\hfill \\ +{\sum }_{i=1}^N\left[\begin{array}{c}{\mu }_i\left(g_i{\mathit{l}}^{-1}{\mathit{F}}_i^e{\mathit{l}}_0-{\left({\mathit{F}}_i^e\right)}^{-T}+a_i\mathit{n}\otimes \mathit{n}·{\mathit{F}}_i^e\left(\mathit{I}-{\mathit{n}}_0\otimes {\mathit{n}}_0\right)\right)\hfill \\ -m{\mu }_i\left(\frac{a_i^2}{a_0}\right){\left|\left|{\mathit{n}}_{{\mathit{F}}_{\mathit{i}}^{\mathit{e}}}\right|\right|}^2{\mathit{F}}_i^e+K_i{J}_i^e\left(J_i^e-1\right){\left({\mathit{F}}_i^e\right)}^{-T}\end{array}\right]{\left({\mathit{F}}_i^v\right)}^{-T}\end{array}\right\},\hfill \end{array}$$ [21]

where $g_0=1+4\left(\frac{{\mu }_{\alpha }}{{\mu }_0}\right)\left[tr\left({\mathit{l}}^{-1}\mathit{F}{\mathit{l}}_0{\mathit{F}}^T\right)-3\right]$ and $g_i=1+4\left(\frac{{\mu }_{\alpha }}{{\mu }_0}\right)$$\left[tr\left({\mathit{l}}^{-1}{\mathit{F}}_{\mathrm{i}}^{\mathrm{e}}{\mathit{l}}_0{\left({\mathit{F}}_{\mathrm{i}}^{\mathrm{e}}\right)}^T\right)-3\right]$ describe the nonlinear dependence of the stiffness on the deformation.

Utilizing Eqs. 4, 5, 11, and 16, the evolution of ${\mathbf{F}}_i^v$ is

$$\begin{array}{c}{\dot{\mathit{F}}}_i^v=\frac{{\mu }_i}{{\eta }_i}\left[\begin{array}{c}{g}_i{\left({\mathit{F}}_i^e\right)}^T{\mathit{l}}^{-1}{\mathit{F}}_i^e{\mathit{l}}_0{\mathit{F}}_i^v-{\mathit{F}}_i^v+a_i\left({\left({\mathit{F}}_i^e\right)}^T\mathit{n}\otimes {\mathit{F}}^T\mathit{n}-\left({\left({\mathit{F}}_i^e\right)}^T\mathit{n}·{\mathit{n}}_0\right){\left({\mathit{F}}_i^e\right)}^T\mathit{n}\otimes {\left({\mathit{F}}_i^v\right)}^T{\mathit{n}}_0\right)\hfill \\ -m\left(\frac{a_i^2}{a_0}\right){\left|\left|{\mathit{n}}_{{\mathit{F}}_{\mathit{i}}^{\mathit{e}}}\right|\right|}^2{\left({\mathit{F}}_i^e\right)}^T{\mathit{F}}_i^e{\mathit{F}}_i^v+\frac{K_i}{{\mu }_i}{J}_i^e\left(J_i^e-1\right){\mathit{F}}_i^v\hfill \end{array}\right].\hfill \end{array}$$ [22]

We assume that the viscosity ${\eta }_i$ follows the relation ${\eta }_i={\eta }_{i,0}{g}_0$ (i = 0, 1, …, N), where ${\eta }_{i,0}$ is a constant.

To include different behavior of fracture in tension and compression, the free energy ${\psi }_0$ is decomposed into positive and negative parts,

$$\begin{array}{c}{\psi }_0^{+}=\begin{array}{c}\frac{1}{2}{\mu }_0\left[{\sum }_{i=1}^3\left(〈{\lambda }_{\mathrm{i}}^2-1〉-2〈\mathit{log}{\lambda }_i〉\right)\right]\hfill \\ +{\mu }_{\alpha }{\left[{\sum }_{i=1}^3〈{\lambda }_{\mathrm{i}}^2-1〉\right]}^2+\frac{1}{2}K{\left[〈J-1〉\right]}^2+\frac{1}{2}{\mu }_{0}a{\left|\left|{\mathit{n}}_{\mathit{F}}\right|\right|}^2\end{array},\hfill \end{array}$$ [23]

and

$${\psi }_0^{-}={\psi }_0-{\psi }_0^{+},$$ [24]

where ${\lambda }_i^2$ are the eigenvalues of the tensor ${\mathit{l}}^{\mathbf{-}\mathbf{1}}{\mathit{F}}_i^e{\mathit{l}}_0{\left({\mathit{F}}_i^e\right)}^T$, and $<\bullet >$ are Macaulay brackets, defined as $〈x〉=\max \{0,x\}$. The semisoft contribution is added into the positive part of the free energy without decomposition. To avoid fracture due to compression, we only consider the positive part of the free energy that contributes to the evolution of the phase field. Correspondingly, Eq. 19 is equivalent to the differential form

$$\frac{\partial g\left(d\right)}{\partial d}\frac{{\psi }_0^{+}}{G_c}+\frac{1}{\pi }\left[\frac{1}{b_0}\left(2-2d\right)-2b_0\nabla ·\left(\mathit{A}·\nabla d\right)\right]=0,$$ [25]

with the natural boundary condition $\left(\mathit{A}·\mathrm{\nabla }d\right)·\mathit{N}=0$.

To ensure irreversibility of the phase field, the history variable

$$\mathrm{H}=\underset{\tau \in \left[0,t\right]}{\mathit{max}}\left\{\frac{{\psi }_0^{+}\left(\tau \right)}{G_c\left(\tau \right)}\right\},$$ [26]

is introduced to replace the term $\frac{{\psi }_0^{+}}{G_c}$ in Eq. 25 (32, 39), and Eq. 25 is rewritten as

$$\frac{\partial g\left(d\right)}{\partial d}H+\frac{1}{\pi }\left[\frac{1}{b_0}\left(2-2d\right)-2b_0\nabla ·\left(\mathit{A}·\nabla d\right)\right]=0.$$ [27]

To solve Eq. 27, the intrinsic fracture toughness G_c needs to be specified. Experiments have shown that the intrinsic fracture toughness of elastomers is typically not a constant, but coupled with their viscoelastic responses (40, 41). We assume that the intrinsic fracture toughness is rate-dependent, based on the mechanism that bond dissociation in polymer chains is a thermally activated process, in which applied mechanical stress lowers the activation energy barrier for bond breakage. Chaudhury demonstrated that the intrinsic fracture toughness G_c scales with the stretching velocity V of the bridging chains as $G_c\sim {\left(\log V\right)}^2$ (42). We adopt a similar functional form for G_c associated with rate-dependent chain scission (43, 44)

$$G_c={\left[\sqrt{G_{c0}}+\beta \mathit{log}\left(\frac{r_d}{r_{d0}}\right)\right]}^2,$$ [28]

where $\begin{array}{c}{r}_d=\frac{1}{2}\sqrt{\left(\dot{\mathit{F}}{\mathit{F}}^{-1}+{\mathit{F}}^{-T}{\dot{\mathit{F}}}^T\right):\left(\dot{\mathit{F}}{\mathit{F}}^{-1}+{\mathit{F}}^{-T}{\dot{\mathit{F}}}^T\right)}\end{array}$ represents the deformation rate, β is a scaling factor, and $G_{c0}$ is the reference fracture toughness, and $r_{d0}$ is the reference deformation rate.

By solving the mechanical equilibrium (Eq. 13), the evolution of ${\mathbf{F}}_i^v$ (Eq. 22), the equilibrium equation for the director n (Eq. 17), and the evolution of the phase field d (Eq. 27), we can determine the fracture responses of LCEs. The model is implemented in COMSOL under plane stress conditions. The implementation details and material parameters are described in the SI Appendix.

Supplementary Material

Appendix 01 (PDF)

Movie S1.

A LCE sample with an initial director θ = 45°, length H = 30mm, width W = 5 mm, and a pre-crack of length l_c = 0.4W is stretched experimentally at a stretching rate of 0.1%/s.

Movie S2.

A LCE sample with an initial director θ = 45°, length H = 30mm, width W = 5 mm, and a pre-crack of length l_c = 0.4W is stretched experimentally at a stretching rate of 10%/s.

Movie S3.

A LCE sample with an initial director θ = 45°, length H = 30mm, width W = 5 mm, and a pre-crack of length l_c = 0.4W is stretched numerically at a stretching rate of 0.1%/s.

Movie S4.

A LCE sample with an initial director θ = 45°, length H = 30mm, width W = 5 mm, and a pre-crack of length l_c = 0.4W is stretched numerically at a stretching rate of 10%/s.

Movie S5.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 15 mm, and a precrack of length l_c = 0.4W is stretched experimentally at a stretching rate of 0.1%/s.

Movie S6.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 15 mm, and a precrack of length l_c = 0.4W is stretched experimentally at a stretching rate of 10%/s.

Movie S7.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 15 mm, and a precrack of length l_c = 0.4W is stretched numerically at a stretching rate of 0.1%/s.

Movie S8.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 15 mm, and a precrack of length l_c = 0.4W is stretched numerically at a stretching rate of 10%/s.

Movie S9.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 15 mm, and a precrack of length l_c = 0.6W is stretched experimentally at a stretching rate of 1%/s.

Movie S10.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 15 mm, and a precrack of length l_c = 0.6W is stretched numerically at a stretching rate of 1%/s.

Movie S11.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 20 mm, and two precracks of length l_c = 0.25W is stretched experimentally at a stretching rate of 1%/s.

Movie S12.

A LCE sample with an initial director θ = 45°, length H = 10mm, width W = 20 mm, and two precracks of length l_c = 0.25W is stretched numerically at a stretching rate of 1%/s.

Data, Materials, and Software Availability

Raw experimental data and analysis code have been deposited in GitHub (https://github.com/LihuaJinMSM/LCE_fracture) (45). All other data are included in the article and/or supporting information.