A thermomechanical constitutive model for super-stretchable multi-walled carbon nanotube-enforced polyimide aerogel

Abstract

Multi-walled carbon nanotube (MCNT)-enforced cross-linked polyimide aerogels (MPIAs) show wide design scenarios and engineering applications due to their excellent physical and chemical properties, such as high flexibility, efficient heat insulation, ultra-low density, thermal stability, and strong mechanical properties. A comprehensive micromechanism-based inelastic constitutive model is developed for MPIAs under finite deformation framework in this work, aiming to provide a theoretical foundation for their designs and engineering applications. The overall thermomechanical properties of MPIAs compose of two superimposed chain networks of the physically cross-linked polyimide aerogel network between nanotubes and the cross-linked chain network of polyimide aerogels. Considering uniformly distributed chain lengths of each nanotube pair, a total energy potential is integrated by applying classical microsphere full network model to capture mechanisms of nanotubes interacting with polyimide aerogel networks, where a continuous damage kinetic model is developed to characterize gradual detachments of chain networks. The significantly enhanced material modulus and toughness of MPIAs are captured by applying two flexible beams to model the two-end connected chains and dangling chains. An equivalent viscoplastic deformation mechanism is applied to model the pores breaking, the porosity variation as well as the chains slipping, which are found to affect greatly the material toughening. The linear elastic behavior of cross-linked neat polyimide aerogel chemical network is considered, and the overall energy function density is integrated for MPIA by combining two superimposed networks. The overall model is calibrated by a series of experiments and then applied for simulations of isothermal uniaxial tension of MPIAs. The underlying mechanisms are further discussed for significantly enhanced modulus and toughness of MPIAs by adding different contents of MCNTs. This work provides theoretical understandings and design guidelines for MCNTs-enforced polyimide aerogel.

Appendices

Appended material A: Energy function of two flexible beams

In this appendix, the specific calculation process of energy density when two types of flexible beams (fixed-end beam and cantilever beam) are subjected to coupled deformation of tension and bending by $\lambda$ and $\delta$ is solved.

1.1 Energy function of CBs

Under the action of the force, the CB produces coupled deformation, which can be decomposed into the concentrated force in the y direction and the tension in the x direction, thus resulting in flexure $\delta$ in the y direction and stretching $\lambda$ in the x direction. Therefore, the flexural strain of the CB is.

The CB will stretch and bend under the axial force $F_{N}$ and downward concentration force $F_{P}$, which causes the beam displacements $\lambda$ and $\delta$ in the $x$ and $y$ directions. The tensile strain energy of the beam on the $x$ axis can be obtained by

$$\psi_{{1}} = \frac{1}{2}F_{N} \cdot \lambda$$

(A.1)

where $F_{N} = EA\lambda$ and $\lambda = \sqrt {I_{4} } - 1$, Combine the Eq. (A.1)

$$\psi_{1} = \frac{1}{2}EA\left( {\sqrt {I_{4} } - 1} \right)^{2}$$

(A.2)

Bending strain energy for CBs, as shown in Fig. 

Fig. 19

The deformation diagram of CB under uniaxial loading (a). Only the bending strain energy of the downward displacement beam is considered (b)

19b, the bending strain energy

$$\psi_{b} = \frac{1}{2}F_{P} \cdot \delta$$

(A.3)

According to the mechanics of materials, the deflection at point B should be equal to

$$\delta = \frac{{Fl^{3} }}{3EI}$$

(A.4)

The angle of B is going to be equal to

$$\varphi = \frac{{Fl^{2} }}{2EI}$$

(A.5)

Simultaneous Eqs. (A.3)–(A.5) can be obtained

$$\psi_{b} = \frac{2EI}{{3l}}\varphi^{2}$$

(A.6)

Therefore, the total strain energy of CB can be expressed as

$$\psi_{p}^{CB} = \psi_{s} + \psi_{b} = \frac{2EI}{{3l}}\varphi^{2} + \frac{1}{2}EA\left( {\sqrt {I_{4} } - 1} \right)^{2}$$

(A.7)

1.2 Energy function of FBs

As shown in (a), the right side of CB displaces a downward $\delta$, and then the corresponding support reaction force will be generated (see Fig. 

Fig. 20

FB is displaced $\delta$ downward (a); the force diagram of FB when it displaces $\delta$ downward on the right (b)

20b). Obviously, the beam structure is a cubic statically indeterminate structure. In the following, the "force method" in structural mechanics is used to solve the support reaction force at point B. Firstly, remove the B-end constraint and replace with three generalized unknow forces $X_{1}$, $X_{2}$ and $X_{3}$, assuming directions as shown in Fig. 

Fig. 21

The direction of the generalized unknow force (a); the fixed beam (FB) is antisymmetric with respect to the midpoint C (b)

21:

The regular equations are listed as follows:

$$\left\{ \begin{gathered} \delta_{11} X_{1} { + }\delta_{12} X_{2} { + }\delta_{13} X_{3} { + }\Delta_{1P} { = }\delta \hfill \\ \delta_{21} X_{1} { + }\delta_{22} X_{2} { + }\delta_{23} X_{3} { + }\Delta_{2P} { = 0} \hfill \\ \delta_{31} X_{1} { + }\delta_{32} X_{2} { + }\delta_{33} X_{3} { + }\Delta_{3P} { = }0 \hfill \\ \end{gathered} \right.$$

(A.8)

where $\delta_{ij}$ is the generalized displacement corresponding to $X_{i}$ caused when $X = 1$, $\Delta_{iP}$ is the generalized displacement corresponding to $X_{i}$ caused by load $P$. They can be calculated by a systematic method.

Since there is no external force applied on the beam, $P = 0$, then:

$$\Delta_{1P} = \Delta_{2P} = \Delta_{3P} = 0$$

(A.9)

When $X_{i}$ is a unit force, we call it $\overline{{X_{i} }}$. Then we can write the bending moment equation $\overline{{M_{i} }}$ of the beam under the three generalized unit forces separately:

$$\left\{ \begin{gathered} \overline{{M_{1} }} = l - x \hfill \\ \overline{{M_{2} }} = - 1, \hfill \\ \overline{{M_{3} }} = 0 \hfill \\ \end{gathered} \right.\qquad \left( {0 \le x \le l} \right)$$

(A.10)

According to the compliance calculation formula:

$$\delta_{ij} = \int {\frac{{\overline{{M_{i} }} \overline{{M_{j} }} }}{EI}ds}$$

(A.11)

Calculate $\delta_{ij}$ in turn:

$$\left\{ \begin{gathered} \delta_{33} = \delta_{13} = \delta_{23} = \delta_{31} = \delta_{32} = 0 \hfill \\ \delta_{11} = \frac{{l^{3} }}{3EI},\delta_{12} = \delta_{21} = - \frac{{l^{2} }}{2EI},\delta_{22} = \frac{l}{EI} \hfill \\ \end{gathered} \right.$$

(A.12)

Substitute in Eq. (A.8) and obtain:

$$X_{1} = - \frac{12EI\delta }{{l^{3} }},X_{2} = - \frac{6EI\delta }{{l^{3} }}$$

(A.13)

That's $F_{RAy} = F_{RBy} = - \frac{12EI\delta }{{l^{3} }},M_{A} = M_{B} = - \frac{6EI\delta }{{l^{3} }}$, and the minus sign is the opposite of the initial assumption. Thus, the bending moment equation of the beam can be written as follows:

$$M\left( x \right) = F_{RAy} x - M_{A} = \frac{6EI\delta }{{l^{2} }}\left( {\frac{2}{l}x - 1} \right),\quad \left( {0 \le x \le l} \right)$$

(A.14)

The strain energy of the FBs:

$$\psi_{b} \left( \delta \right) = \int_{0}^{l} {\frac{{M\left( x \right)^{2} }}{2EI}} dx = \frac{{6EI\delta^{2} }}{{l^{3} }}$$

(A.15)

The bending energy of the FBs can be better described by using the angle of the beam with respect to the horizontal direction as the independent variable. Since the beam is antisymmetric with respect to its midpoint C, the energy of the beam can be divided into two identical parts, of which one is taken for analysis, as shown in Fig. 21b. According to the relationship between bending moment and deflection, the angle of the beam is:

$$- M\left( x \right) = EI\frac{{d^{2} w}}{{dx^{2} }},\varphi = EI\frac{dw}{{dx}}$$

(A.16)

Therefore, the relation between the Angle and the bending moment is:

$$\varphi = \frac{1}{EI}\int_{0}^{\frac{l}{2}} {M(x)} dx = \frac{3\delta }{{2l}}$$

(A.17)

Therefore, the bending strain energy of FB is expressed by the angle $\varphi$ as:

$$\psi_{b} = \frac{8EI}{{3l}}\varphi^{2}$$

(A.18)

The tensile strain energy of FB in the $x$ direction is the same as that of CB, so the total strain energy of FB can be expressed as

$$\psi_{FB} = \frac{8EI}{{3l}}\varphi^{2} + \frac{1}{2}EA\left( {\sqrt {I_{4} } - 1} \right)^{2}$$

(A.19)

Appended material B: Parameter identification

The three groups of model parameters are determined to develop viscoplastic thermomechanical by calibrating the experimental results and referred to the published works: (1) 2 parameters related to the PI network can be identified by analyzing the engineering stress-engineering strain curves of the MPIAs without the addition of MCNTs; (2) 5 parameters associated with the carbon nanotube network can be discerned from the stress–strain curves of MPIAs with varying concentrations of carbon nanotubes.

2.1 Material parameters for PI network part $(E,K)$

The PI network part is mainly used to analyze the basic elastic parameters of the chain. Dynamic mechanical analysis (DMA) was used to measure the stress–strain curve of PI aerogel at room temperature (25 °C). Elastic modulus ($E$) and bulk modulus of matrix ($K$) of partial chain of PI network were obtained. The fitting curve is shown in Fig. 

Fig. 22

The fitted engineering stress–strain relation for MPIA with 0 wt% carbon nanotubes (a), the fitted engineering stress–strain relation for MPIA with 10 wt% carbon nanotubes (b)

22.

2.2 Material parameters for the carbon nanotubes network ($N_{0} ,\beta_{0} ,\lambda^{*} ,\eta ,\gamma$ )

To reflect the increase in the modulus and toughness of polyimide aerogel (PIA) after the addition of carbon nanotubes, a carbon nanotube network was established. Among them, two parameters related to the damage evolution and three parameters related to the content evolution of the FB need to be determined. On the precondition of determining the number of initial beams ($N_{0}$), anti-detachment parameters ($\beta_{0}$), debonding threshold ($\lambda^{*}$), modulus reduction factor ($\gamma$) and plasticity modulus ($\eta$) fitted by the engineering stress-engineering strain curve of MPIA supplemented with 10wt% carbon nanotubes content. 5 parameters were obtained by fitting the 10 wt% engineering stress–strain curve. As shown in Fig. 22 the simulation results of the model are in good agreement with the experimental results. The stress–strain curves of 20 and 30 wt% were predicted by adjusting different FBs volume fractions. The predicted results are shown in Fig. 

Fig. 23

The volume fraction of FB is adjusted according to the model to predict the engineering stress–strain curve of the material, 20 wt% (a), 30 wt% (b)

23, which are consistent with each other.

Appended material C: Cyclic loading–unloading experiments

To study the elastic–plastic properties of materials, we conducted cyclic loading and unloading experiments as shown in Fig. 

Fig. 24

cyclic loading–unloading experiments at two different temperatures. a 25 °C; b 100 °C

24. In the experiment, we used the same loading rate as uniaxial tension to stretch to 4% strain, and then unloaded to 0 at 2 N/min. The next step is loaded to 8% strain and so on. It was shown that the material had significant residual strain after unloading, which was attributed to dissipative multiple mechanisms of pore breaking, chain detachment, and plastic deformation.