Crack propagation in staggered structures of biological and biomimetic composites
Graphical abstract
Crack Trajectories in staggered biological composites predicted by phase field simulations. High aspect ratios lead to straight crack propagation as opposed to lower aspect ratios.
Introduction
Biocomposites such as bones and nacre are naturally occurring materials that possess remarkable mechanical properties [1], [2], [3], [4], [5], [6], [7]. Interestingly the individual constituents of these composites are often found to be weak and/or brittle in nature [8]. Yet, the combination of these constituents in a very complex hierarchical manner comprising several length scales is believed to be the reason for the extraordinary fracture toughness of biocomposites [9], [10]. How these microstructural arrangements of the constituents affect the fracture toughness and the effective stiffness of the biocomposites has attracted lot of research interest [1], [2], [11], [12], [13], [14], [15], [16], [17], [18], [19]. Such knowledge would immensely help in creating artificial materials with high fracture toughness and strength combined with low densities which can be exploited in several engineering applications [20], [21], [22].
While the actual microstructure of the naturally evolved biocomposites such as bone is very complex comprising of several levels of hierarchies [9], [10], a simplified model such as the one shown in Fig. 1 is often found to be useful for analytical and computer simulations for the study of fracture mechanisms in biocomposites [23], [24], [25], [26], [27], [28].
Wagner and Weiner [2] have studied the relationship between the microstructure of bone and its mechanical stiffness. Baron and Wagner [4] have investigated the elastic modulus of hard biological tissues by considering their staggered platelet microstructure. Zhang et al. [19] showed the existence of an optimum level of structural hierarchies due to the limited selection of structural proteins. In their study, they predict the optimum level of hierarchies for a human bone as six which is close to what is actually observed.
Although, the above mentioned studies shed important light on the structure-mechanical properties correlation of biocomposites, a detailed analysis of how crack paths are influenced by the underlying mineral-organic microstructure is still lacking. For predicting crack paths in staggered composites, a model which can describe the thermodynamics of the fracture process as well the dynamics of cracks is essential. The phase field models [29], [30], [31], [32], [33], [34], [35], [36], [37] for crack propagation are ideally suited for such problems since they can describe crack dynamics, without making any a priori assumptions on the crack path. In this work we use phase field approach for studying crack propagation in composites made up of staggered micro structures in which the elastic modulus mismatch plays an important role in determining the crack trajectories. In particular, we study the effects micro-structural parameters such as aspect ratio and volume fraction f of the mineral platelets and modulus mismatch characterised by the ratio of Young’s modulus of the organic to that of the mineral (), on the crack propagation behaviour in these composites.
This paper is organised in the following way. The next section presents the details of phase field model for crack propagation. Section 3 gives the details of crack simulations, numerical scheme and the boundary conditions of the present study. The results of the crack simulations are presented in Section 4. In Section 5, a simple theory is presented which is used in explaining some of our simulation results and also helpful in generalising the trends predicted by the phase field crack simulations. Section 6 gives a brief summary and concluding remarks of the present work.
Section snippets
Phase field model
Phase field modelling (PFM) of crack propagation in solids is emerging as an important tool for studying fracture behaviour especially due to its simplicity and flexibility in handling dynamic crack growth in complex microstructure [31], [32], [38], [39], [40]. The PFM is a variational approach in which a free energy functional is constructed which depends on a spatially varying order parameter and its gradient representing the interface. The time evolution
Crack simulations
Crack propagation simulations are conducted on staggered composites with a typical configuration as shown in Fig. 1. In this figure, the dark grey regions correspond to mineral platelets and the black region corresponds to the organic matrix while the thin white portion is the initial crack with . The simulation domain is an rectangular area with a central crack of length a. We have applied periodic boundary conditions in both x and y directions and have chosen and . The
Results
Fig. 2, Fig. 3 show the snapshots of the crack trajectories near the initial crack tip for the composites with mineral volume fractions and respectively for different aspect ratios . Both these figures pertain to a modulus ratio . Similar simulations have been performed on composites having a modulus ratio of which resulted in straight crack propagation in all the cases. While these snapshot pictures correspond to the magnified view of the regions close to the
Theory and discussion
In order to predict the crack propagation paths in staggered composites, we analyse the stress distribution near the vicinity of a crack. We calculate the stresses in a representative volume element ‘RVE’ shown in Fig. 1. The RVE under consideration is a rectangular area of dimensions . A uniform macroscopic stress field is assumed to be applied on the RVE, neglecting the higher order effects that arise from crack-tip singularities. If the crack size is much larger than the
Summary and conclusions
The staggered structure of biocomposites can make the cracks in these materials deflect from their initial pathways and may lead to crack branching as well as microcracking. In this work we have used phase field method to simulate crack propagation in these materials and have observed many of these interesting mechanisms that are commonly found in biocomposites and biomimetic composites. We conducted a systematic study to link this crack propagation behaviour with their underlying
Conflict of interest
The authors declare no financial or commercial conflict of interest in the subject matter or materials discussed in this manuscript.
Acknowledgments
We would like to thank Prof. Huajian Gao, Brown University, Dr. Syed Nizamuddin Khadri, Indian Institute of Technology Hyderabad for useful discussions on biomaterials, fracture and phase field method. Palla Murali acknowledges funding from OPERA grant, Birla Institute of Technology and Science, Pilani.
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