The fracture toughness of polymer cellulose nanocomposites using the essential work of fracture method

Abstract

This work reinforced both a glassy polymer (high stiffness matrix) and a rubbery polymer (low stiffness matrix) with cellulose nanocrystals (CNC) derived from natural sources. CNC addition always increased stiffness while it increased toughness for a rubbery polymer and caused no loss in toughness for a glassy polymer. These results contradict many claims that when stiffness increases, the toughness decreases. We show that these claims depend on how toughness is measured. Our results were based on toughness measured using the essential work of fracture method. In contrast, toughness determined from area under the stress–strain curve shows a significant decrease, but that method may be a poor measure of toughness. Property enhancements usually require a good fiber/matrix interface. We used modeling of stiffness properties to confirm that CNC has a good interface with the studied polymer matrices.

Appendix

Mean-field modeling for composite properties with randomly oriented or partially aligned fibers has two steps. The first step is to find properties of a unit cell containing aligned fibers [8, 24]. The second step is to use mean-field averaging to find properties of randomly oriented composites.

The first step will result in $E_A$, $E_T$, $\nu _A$, $\nu _T$, and $G_A$ for the five independent mechanical properties of the transversely isotropic, aligned fibers, unit cell (axial and transverse tensile moduli and Poisson ratios and axial shear modulus, respectively). $E_A$ can be found using the recent end-capped shear lag model [24]. This model applied optimal shear lag methods [25–27] to an axisymmetric unit cell where a cylindrical fiber with radius $r_f$ and length $l_f$ is encased in a cylinder of matrix with radius $r_m = r_f+\Delta$ and length $l_m = l_f+2\Delta$. In other words, the distance from the fiber side to the unit cell side ($\Delta$) is set equal to the distance from the fiber end to the ends of the unit cell. All fiber matrix interfaces were modeled using imperfect interface parameters $r_fD_n$ and $r_fD_t,$ where $D_n$ and $D_t$ are interface stiffnesses for normal and tangential sliding (when $D = 0$, the interface is debonded, $D = \infty$ is perfect interface, and all other values are imperfect [8]). The axial modulus from the end-capped model, $E_{EC}$ was derived to be [24]:

$$\begin{aligned} {E_2\over E_{EC}} = 1 +\left( {E_f\over E_m}-1\right) (V_1-V_f )+{E_fV_f\over E_mV_m}\Lambda (\rho ) \end{aligned},$$

(7)

where $E_i$ is modulus and $V_i$ is volume fraction. Subscripts f and m refer to fiber and matrix, but $V_1 = r_f^2/r_m^2$ is fiber volume fraction ignoring the end caps and $E_2 = E_fV_1 + E_m(1-V_1)$. The key function of aspect ratio $(\rho )$ was derived to be:

$$\begin{aligned} \Lambda (\rho ) = {V_m\over V_2}{{E_2\over E_f}{\tanh (\beta _1^*\rho )\over \beta _1\rho }+ \Lambda _1(\rho )\over 1 + {\tanh (\beta _1^*\rho )\over \beta _1\eta } + {E_2\over \eta E_f}{\tanh (\beta _2\rho )\over \beta _2}} \end{aligned},$$

(8)

where $\beta _1^* = (V_1-V_f)\beta _1/(2V_f),$ $\eta =E_m V_2/(r_fD_n),$

$$\begin{aligned} \Lambda _1(\rho )= & {} \left( 1+\left( 1-{E_2\over E_f}\right) ^2{\tanh (\beta _1^*\rho )\over \beta _1\eta } \right) {\tanh (\beta _2\rho )\over \beta _2\rho } \end{aligned}$$

(9)

$$\begin{aligned} \beta _1^2= & {} -{4G_mV_2\over E_m(V_2+\ln V_1)} \end{aligned}$$

(10)

$$\begin{aligned} \beta _2^2= & {} {{4 E_2\over E_fE_m}\over {V_2\over 2G_f} - {1\over G_m}\left( {V_2\over 2}+1 + {\ln V_1\over V_2}\right) + {2V_2\over r_fD_t}} \end{aligned}$$

(11)

Although numerical calculations [24] show that the end-capped shear lag model is very accurate when the fiber to matrix modulus ratio, $R = E_f/E_m$, is less than 100, those calculations also show the model becomes a lower bound for large R. This inaccuracy is not caused by shear lag methods, because all other models, including finite element models, also degenerate to lower bound results for soft matrices [24]. Because our CNCs with PVdF-HFP had $R=190$, we introduce a correction term to model composites with soft matrices. When $R>100$, we propose the axial modulus to be

$$\begin{aligned} E^* = \phi _LE_{EC} + (1 - \phi _L)E_{UB} \quad (0< \phi _L < 1) \end{aligned},$$

(12)

where

$$\begin{aligned} E_{UB} = \eta _fE_fV_f +E_mV_m \end{aligned}$$

(13)

here $E_{UB}$ is an “upper bound” modulus derived using fiber effectiveness methods advocated in several text books [11], where $\eta _f$ is found by comparing average stress in a short fiber to the average stress that fiber would have in a continuous fiber composite:

$$\begin{aligned} \eta _f = {1\over l_f\sigma _{\infty }}\int _{-l_f/2}^{l_f/2} \sigma _f(x) dx \end{aligned},$$

(14)

where $l_f$ is fiber length, $\sigma _\infty$ is stress the fiber would have for infinitely long fibers, and $\sigma _f(x)$ is average stress in the fiber cross section at position x. Evaluating this integral using the end-capped shear lag stress with imperfect interfaces (rather then the simplistic shear lag used in textbooks [11]) gives

$$\begin{aligned} \eta _f = 1 - {1 + \left( 1-{E_2\over E_f}\right) {\tanh (\beta _1^*\rho )\over \beta _1\eta } \over 1 + {\tanh (\beta _1^*\rho )\over \beta _1\eta } + {E_2\over \eta E_f}{\tanh (\beta _2\rho )\over \beta _2}} {\tanh (\beta _2\rho )\over \beta _2\rho } \end{aligned}$$

(15)

For PSF/CNC nanocomposites ($R < 100$), the end-capped shear lag model can be used directly. For PVdF-HFP/CNC nanocomposites ($R\ge 190$), the model including both $E_{EC}$ and $E_{UB}$ was needed. The only remaining issue is to choose $\phi _L$. We choose $\phi _L=0.93$ based on comparing Eq. (12) to numerical results in Nairn and Shir Mohammadi [24]. This value of $\phi _L=0.93$ appears to fit a wide range of nanocomposites when the matrix is much more compliant than the fibers.

The end-capped shear lag model gives $E_A$ as a function of aspect ratio and interface, but mean-field modeling needs $E_T$, $\nu _A$, $\nu _T$, and $G_A$ as well. Fortunately, both numerical [24] and analytical [2] modeling shows that all other properties are only weakly dependent on aspect ratio. Assuming they are independent of aspect ratio, they can be found for any aspect ratio, such as for continuous fiber composites. All remaining unit cell properties therefore used the Hashin’s analysis [8] for properties of a continuous fiber composite including effects of imperfect interfaces.

The final step is to use mean-field methods for averaging unit cell properties. Here we assumed the nanocomposite films are statistically isotropic in the plane of the film (i.e., fibers tend to lie in the plane of the film). For this special case, an upper bound modulus can be found from unit cell properties using [22]:

$$\begin{aligned} E_c\le & {} 4U_2\left( 1-{U_2\over U_1}\right) \end{aligned}$$

(16)

$$\begin{aligned} U_1= & {} {E_A(3+2\nu _A') + 3E_T + 4G_A(1-\nu _A\nu _A') \over 8(1-\nu _A\nu _A') } \end{aligned}$$

(17)

$$\begin{aligned} U_2= & {} {E_A(1-2\nu _A') + E_T + 4G_A(1-\nu _A\nu _A') \over 8(1-\nu _A\nu _A') } \end{aligned},$$

(18)

where $\nu _A' = \nu _A E_T/E_A$ and $\nu _T$ is not needed.