Investigations into Multi-scale Mechanical Characterization of Bamboo- a Natural Material

Abstract

Bamboo is a naturally occurring composite material evolved with soft parenchymatous matrix reinforced by unidirectional fibre bundles. Moreover, it is functionally graded with fibre density varying smoothly over the transverse cross-section and it is characterised by structural hierarchy existing at various length scales. Therefore, mechanical characterisation of the material needs to be carried out using multi-scale modelling. In this paper, mechanical characterization of bamboo is discussed using state-of-the-art hybrid numerical-experimental technique. The techniques followed here are very modern and use the most recent methods. To achieve this, three length scales are identified and specimens are prepared accordingly. The specimens correspond to namely, nano-indentation, brittle failure of fibre bundle and inter-laminar fracture in bulk bamboo. Effective elastic moduli of bulk bamboo obtained through homogenisation are used for numerical analyses. An analytical framework for nano-indentation of anisotropic substrate is presented and results are compared with experimental results. The indentation modulus is observed to lie between 19 and 21 GPa whereas, the results obtained experimentally lie between 17 and 23 GPa. Further, finite element analysis of nano-indentation of anisotropic surface of bamboo is also discussed, where indentation modulus is found to be 20.2 GPa. While discussing brittle failure of fibre bundle, statistical weak-link theory based on progressive failure is presented and comparison is made between statistical and experimental results. In the last part, fracture of bulk specimen is studied using asymptotic crack-tip stress field in anisotropic material that is used for estimation of energy release rate. The fracture energy is seen to vary between 500 and 1800 N/m. The results are compared with fracture experiments conducted on inter-laminar bamboo specimens. It is concluded that mechanical properties of a hierarchical material can be estimated through multi-scale modelling and experiments on specimens selected suitably at different length scales.

Appendices

Appendix A. Progressive Failure Model for Bulk Bamboo

Appendix B. Inter-laminar Cracks in Anisotropic Material

For an anisotropic material as shown in Fig. 6, generalised Hooke’s law is expressed as:

$$\begin{aligned} \varepsilon _{{i}}= & {} \sum _{{j}=1}^6 \mathbbm {S}_{{ij}}\sigma _{{j}},\;\mathbbm {S}_{{ij}}=\mathbbm {S}_{{ji}},\;({i}=1,2\ldots 6), \end{aligned}$$

(B.1)

where,

Fig. 6

Inter-laminar crack of initial length a subjected to displacement field. The stressed element is also shown for elaboration

The compliances $\mathbbm {S}_{{ij}}$ are defined in Sadd [25]. Sih et al. [24] have formulated the stress and displacement are formulated in terms of analytic functions $\phi _{j}(z_j)$, of the complex variable, $z_{j}=x_{j}+iy_{j}$ (${j}=1, 2$), where

$$\begin{aligned} x_{j}=x+\alpha _{j}y,\;y_{j}=y+\beta _{j}y\;({j}=1, 2). \end{aligned}$$

(B.2)

The complex numbers $\mu _{j}$ are formed from the parameters $\alpha _{j}\;\mathrm {and}\;\beta _{j}$ (${i.e.,}\;\mu _{j}=\alpha _{j}+i\beta _{j}$) which are determined from

$$\begin{aligned}&\mathbbm {S}_{11}\;\mu ^4\;-\;2\;\mathbbm {S}_{14}\;\mu ^3\;\;+\;(2\;\mathbbm {S}_{12}+\mathbbm {S}_{44})\;\mu ^2\;\nonumber \\&\quad -\;2\;\mathbbm {S}_{24}\;\mu \;+\;\mathbbm {S}_{22}\;=\;0, \end{aligned}$$

(B.3)

The roots, $\mu _{j}$ turn out to be always complex and will form conjugate pairs. In this work, the roots with positive imaginary parts are used. In addition, the stress-intensity factors $K_{j}$ ($j=1, 2, 3)$are related to the energy release rates, $G_{j}$. For inter-laminar fracture (mode−I), it is found that [24]:

$$\begin{aligned} G_{1}= & {} -\dfrac{\pi {K_1}^2}{2} S_{22}\;\mathrm {Im}\Big [ \dfrac{\mu _1+\mu _2}{\mu _1\mu _2}\Big ]. \end{aligned}$$

(B.4)

Appendix C. Compliance Calibration (ASTM D5528)

• 1. Plot log $(\delta _{{i}}/P_{i})$ versus log $a_{i}$.

• 2. Draw a least squares straight line fit through the data.

• 3. Determine the slope of straight line, $n=\Delta _{{y}}/\Delta _{x}$.

• 4. Find out mode−I inter-laminar fracture toughness as $G_{\mathrm {I}}=\dfrac{nP\delta }{2ba}$, where C is compliance ($C=\delta /P$) of specimen.