In an effort to tackle pollution in seas, fields and waterways, regulatory authorities support a wide-ranging ban on single-use plastics. The trend towards minimizing our dependency on synthetic crude-oil derived plastics is also echoed by the chemical industry, both in Europe and worldwide. One common solution currently in research and development is the use of natural and renewable sources of biopolymers. Examples are marine biopolymers (alginate, agar, chitin and chitosan) and plant-derived biopolymers such as cellulose, starch, lignin pectin and proteins. The biopolymers are generally recognized as the most sustainable starting molecules for beeing converted into functional material alternatives to plastics.
Although some difficulties in their use, such as variable molecular weight, impurities and lability against microorganisms, can successfully be overcome [1
], the area of high-performance materials is traditionally thought to be the least prone to changes. In particular, lightweight thermal and sound insulation (for clothing, buildings, aircrafts, packaging) has been dominated essentially by polyurethane- and polystyrene-based synthetic materials.
In 2014–2015 it was reported that cellulose and alginate can be converted into highly porous nanostructured materials—aerogels—with exceptionally low thermal conductivities below that of air (26 mW/m·K), see Rudaz et al. [2
], Gurikov et al. [3
], Kobayashi et al. [4
]. Shortly after these publications, this fact was confirmed for other biopolymers such as pectin and chitosan [5
]. These achievements showed that even high-performance lightweight insulating materials could be of purely biopolymer nature. However, biopolymer aerogels often possess poor mechanical performance impeding their further development as functional materials. This problem has been tackled from chemical perspectives [6
] but mechanical performance of aerogels in relation to the structural features is still far from being understood.
Many biopolymer aerogels have qualitatively similar fibrillar structures, making them very different from classical oxide-based aerogels with interconnected nanoparticles. For classical oxide-based aerogels a substantial body of experimental data have been reported and a wide range of modelling methods, including all-atom molecular dynamics, coarse-grained and macroscopic models, have been suggested [9
]. Mechanical tests, primarily axial compression tests, for fibrillar biopolymer aerogels only recently started to appear in the literature [15
]. To the best of our knowledge, only one approach towards modelling of mechanical properties of biopolymer aerogels, the so called micromechanical constitutive model, has been proposed by Rege et al. [16
]. This model considers the cellular network of biopolymer aerogels to be made up of continuously and isotropically distributed idealized square shaped microcells. The macroscopic compressive deformation is dictated by the bending of the cell wall fibrils at the microscale. The overall macroscopic predictions of the model have been shown to be in good agreements with the uniaxial compression data for several biopolymers.
Another modelling approach is the bonded-particle method (BPM). It has been developed as an extension of the discrete element method (DEM) for modelling of granular materials [19
]. A material modeled by the BPM is represented as a set of discrete primary particles connected with solid or liquid bonds. Each bond connects two primary particles and depending on the relative motion of the connected particles forces and moments emerge in the bond. These forces and moments are finally acting on primary particles and leading to their translational and rotational motion. In each time step additional breakage criteria are analyzed. If one of the criteria is fulfilled the bond is destroyed and removed from the simulation domain.
In recent years BPM has been widely applied for modelling of different types of materials [20
] ranging from glass agglomerates [21
] to concrete [22
] or sandstone [23
]. It is also applicable for a wide range of scales. By coupling the BPM with a finite-element simulation Wu et al. [24
] successfully applied the BPM even to structural-scale boundary value problems like borehole instabilities. Often, experimental data on internal structure and sample shape is used to generate the structural model in BPM. In many cases data obtained from micro computed tomography (
CT) can be effectively employed. For automated approximation of the material structure from
CT data sets, a bonded-particle extraction technique was developed [25
]. From a rheological point of view in most cases purely elastic materials are modeled with BPM. In some contributions authors have applied this approach for modelling of plastic behavior. Nguyen et al. [26
] has developed a combined damage-plasticity cohesive model for cement bridges in soft rocks, for modelling of asphalt concrete. Kim and Buttlar [27
] have proposed bilinear cohesive model, Tran V.T., Donze F.V., Marin P. [28
] have applied a elastic-hardening-damage law to reproduce irreversible compaction for concretes under high confining pressure. For the simulation of failure behavior of quasi-brittle materials Ma et al. [29
] proposed a displacement-softening contact model.
However, until now the BPM was not applied for the modelling of highly porous plastically deformable materials such as biopolymer aerogels. In this contribution we extend the BPM with a new functional model for solid bonds and apply it for the modelling of the mechanical behavior of alginate aerogels.
In the work at hand, a DEM-based model for the investigation of the mechanical behavior of aerogels has been proposed. The model consists of three main subparts: the structural model to represent the internal material structure, the functional model to describe the mechanical behavior of single components and the model parameters which have to be adjusted to the experimental data. The examined biopolymer aerogels consist of pores in the nanometer or sub-micrometer range. However, this was simplified for the modelling. Instead, the porous material structure was represented on the mesoscale as a set of primary particles connected with cylindrical solid bonds with diameters around 100 m. To describe forces and moments acting in the solid bonds, a new elastic-plastic rheological model was developed. For this purpose, self-similarity of the material behavior on the meso- and macroscales has been assumed. The unknown model parameters were adjusted by using a simplex search optimization method.
To analyze the capabilities of the model, it was applied to cylindrical alginate aerogels samples with varied crosslinking degree controlled by the calcium chloride concentration. The aerogel samples were produced with the following method: cylindrical gels were prepared in dialysis bags immersed in aqueous solution of calcium chloride followed by solvent exchange, polishing and supercritical drying. Uni-axial compression tests were performed to obtain stress-strain curves of the samples. Two different regimes are identifiable: an elastic and a plastic range. The results are in good agreement with general theory of mechanical behavior of foamed structures and reported results for aerogels [29
]. However, due to the high material porosity, no densification regions were observed for samples with CaCl
concentrations of 0.5 and 1 g/L. Only for aerogels crosslinked at 5 g/L a slightly pronounced densification region was observed for strains above 35%. Similar results have been obtained by Karadagli et al. [30
], where 50% strain was defined as transition stage into the densification region. Also for the dependency of the Young’s modulus on the crosslinking degree (or density) similar correlations as the one found (Equation (1
)) have been reported for alginate and other biopolymer aerogels [15
The experimental results were used as input for the simulations. Moreover, to test applicability of the model for other type of mechanical stressing, three-point bending tests have been performed for samples with high calcium chloride concentration. Analysis of the obtained simulation results has shown that, after adjustment of the model parameters for a specific calcium chloride concentration, the elastic-plastic material behavior can be accurately predicted for all other concentrations with different elasticity moduli. Only the unloading shows discrepancies between the experiments and simulations. They result from the construction of the bond model. Since the yield criteria in the bond model depend only on the strain, the recovered strain does not change with the Young’s modulus. By including a dependency on stresses this could be solved. Nevertheless, the developed model can mimic buckling effects occurring for cylindrical samples and can describe material behavior during three-point bending tests. Overall, we can conclude that the derived elasto-plastic model is capable to reproduce the macro-scale mechanical behavior of the studied aerogel with different crosslinking degrees. Since the fitting was performed only for one crosslinking degree, the parameters of the functional model seem to be material specific. The moderate deviations between experimental and simulation results indicate that the developed DEM-based model can be used for the modelling of aerogels with high efficiency.