A generalized tube model of rubber elasticity.

In the present paper, a new type of micro-mechanically motivated chain network model for rubber-like materials is proposed. The model captures topological constraints of polymer network chains, in particular, entanglements. The model demonstrates how the local molecular packing constraints modify under deformation and shows the impact of these changes on the macroscopic elasticity of the material. To this end, we combine concepts of a confining tube and a slip-link (reptation) model. In these models, entanglements of polymer chains play an important role. The nature of entanglements is discussed, and relationships governing entanglements are formulated in terms of molecular physics. In the context of nonlinear elasticity, we apply a non-affine concept which captures the liquid-like behavior of polymer networks at smaller scales in a more realistic way. Model predictions show good agreement with experimental results from uniaxial and biaxial tension tests.

Machine learning-based structure-property predictions in silica aerogels.

The structural features in silica aerogels are known to be modelled effectively by the diffusion-limited cluster-cluster aggregation (DLCA) approach. In this paper, an artificial neural network (ANN) is developed for predicting the fractal properties of silica aerogels, given the input parameters for a DLCA algorithm. This approach of machine learning substitutes the necessity of first generating the DLCA structures and then simulating and characterising their fractal properties. The developed ANN demonstrates the capability of predicting the fractal dimension for any given set of DLCA parameters within an accuracy of R2 = 0.973. Furthermore, the same ANN is subsequently inverted for predicting the input parameters for reconstructing a DLCA model network of silica aerogels, for a given desired target fractal dimension. There, it is shown that the fractal dimension is not a unique characteristic defining the network structure of silica aerogels, and the same fractal dimension can be obtained for different sets of DLCA input parameters. However, the problem of non-uniqueness is solved by using a guided gradient descent approach for predictive modelling purposes within certain bounds of the input parameter-space. Model DLCA structures are generated from the constrained and unconstrained inversion, and are compared against several parameters, amongst them, the pore-size distributions. The constrained inversion of the ANN is shown to predict the DLCA model parameters for a desired fractal dimension within an error of 2%.

Stability Studies of Starch Aerogel Formulations for Biomedical Applications.

Starch aerogels are attractive materials for biomedical applications because of their low density and high open porosity coupled with high surface areas. However, the lack of macropores in conventionally manufactured polysaccharide aerogels is a limitation to their use as scaffolds for regenerative medicine. Moreover, the stability under storage of polysaccharide aerogels is critical for biomedical purposes and scarcely studied so far. In this work, the induction of a new macropore population (1-2 µm) well integrated into the starch aerogel backbone was successfully achieved by the incorporation of zein as a porogen. The obtained dual-porous aerogels were evaluated in terms of composition as well as morphological, textural, and mechanical properties. Stability of aerogels upon storage mimicking the zone II (25 °C, 65% relative humidity) according to the International Council for Harmonization guideline of climatic conditions was checked after 1 and 3 months from morphological, physicochemical, and mechanical perspectives. Zein incorporation induced remarkable changes in the mechanical performance of the end aerogel products and showed a preventive effect on the morphological changes during the storage period.

Micro-mechanical modelling of cellulose aerogels from molten salt hydrates.

In this paper, a generalised micro-mechanical model capable of capturing the mechanical behaviour of polysaccharidic aerogels, in particular cellulose aerogels, is proposed. To this end, first the mechanical structure and properties of these highly nanoporous cellulose aerogels prepared from aqueous salt hydrate melts (calcium thiocyanate, Ca(SCN)2·6H2O and zinc chloride, ZnCl2·4H2O) are studied. The cellulose content within these aerogels is found to have a direct relation to the microstructural quantities such as the fibril length and diameter. This, along with porosity, appears to influence the resulting mechanical properties. Furthermore, experimental characterisation of cellulose aerogels was done using scanning electron microscopy (SEM), pore-size data analysis, and compression tests. Cellulose aerogels are of a characteristic cellular microstructures and accordingly a network formed by square shaped cells is considered in the micro-mechanical model proposed in this paper. This model is based on the non-linear bending and collapse of such cells of varying pore sizes. The extended Euler-Bernoulli beam theory for large deflections is used to describe the bending in the cell walls. The proposed model is physically motivated and demonstrates a good agreement with our experimental data of both ZnCl2 and Ca(SCN)2 based cellulose aerogels with different cellulose contents.

Deep reinforcement learning for microstructural optimisation of silica aerogels.

Silica aerogels are being extensively studied for aerospace and transportation applications due to their diverse multifunctional properties. While their microstructural features dictate their thermal, mechanical, and acoustic properties, their accurate characterisation remains challenging due to their nanoporous morphology and the stochastic nature of gelation. In this work, a deep reinforcement learning (DRL) framework is presented to optimise silica aerogel microstructures modelled with the diffusion-limited cluster-cluster aggregation (DLCA) algorithm. For faster computations, two environments consisting of DLCA surrogate models are tested with the DRL framework for inverse microstructure design. The DRL framework is shown to effectively optimise the microstructure morphology, wherein the error of the material properties achieved is dependent upon the complexity of the environment. However, in all cases, with adequate training of the DRL agent, material microstructures with desired properties can be achieved by the framework. Thus, the methodology provides a resource-efficient means to design aerogels, offering computational advantages over experimental iterations or direct numerical solutions.

Polymer Mechanochemistry in Microbubbles.

Polymer mechanochemistry is a promising technology to convert mechanical energy into chemical functionality by breaking covalent and supramolecular bonds site-selectively. Yet, the mechanochemical reaction rates of covalent bonds in typically used ultrasonication setups lead to reasonable conversions only after comparably long sonication times. This can be accelerated by either increasing the reactivity of the mechanoresponsive moiety or by modifying the encompassing polymer topology. Here, a microbubble system with a tailored polymer shell consisting of an N2 gas core and a mechanoresponsive disulfide-containing polymer network is presented. It is found that the mechanochemical activation of the disulfides is greatly accelerated using these microbubbles compared to commensurate solid core particles or capsules filled with liquid. Aided by computational simulations, it is found that low shell thickness, low shell stiffness and crosslink density, and a size-dependent eigenfrequency close to the used ultrasound frequency maximize the mechanochemical yield over the course of the sonication process.

Computational design of biopolymer aerogels and predictive modelling of their nanostructure and mechanical behaviour.

To address the challenge of reconstructing or designing the three-dimensional microstructure of nanoporous materials, we develop a computational approach by combining the random closed packing of polydisperse spheres together with the Laguerre-Voronoi tessellation. Open-porous cellular network structures that adhere to the real pore-size distributions of the nanoporous materials are generated. As an example, κ-carrageenan aerogels are considered. The mechanical structure-property relationships are further explored by means of finite elements. Here we show that one can predict the macroscopic stress-strain curve of the bulk porous material if only the pore-size distributions, solid fractions, and Young's modulus of the pore-wall fibres are known a priori. The objective of such reconstruction and predictive modelling is to reverse engineer the parameters of their synthesis process for tailored applications. Structural and mechanical property predictions of the proposed modelling approach are shown to be in good agreement with the available experimental data. The presented approach is free of parameter-fitting and is capable of generating dispersed Voronoi structures.

Modeling and Simulation of the Aggregation and the Structural and Mechanical Properties of Silica Aerogels.

Mechanical properties of aerogels are controlled by the connectivity of their network. In this paper, in order to study these properties, computational models of silica aerogels with different morphological entities have been generated by means of the diffusion-limited cluster-cluster aggregation (DLCA) algorithm. New insights into the influence of the model parameters on the generated aerogel structures and on the finite deformation under mechanical loads are provided. First, the structural and fractal properties of the modeled aerogels are investigated. The dependence of morphological properties such as the particle radius and density on these properties is studied. The results are correlated with experimental small-angle X-ray scattering (SAXS) data of a silica aerogel. The DLCA models of silica aerogels are analyzed for their mechanical properties with finite element simulations. There, the aerogel particles are modeled as nodes and the interparticle bonds as beam elements to account for bond stretching, bending, and torsion. The scaling relation between the elastic moduli E and relative density ρ, E ∝ ρm, is investigated and the exponent m = 3.61 is determined. Backbone paths evidently appear in the 3-d network structure under deformation, while the majority of the bonds in the network do not bear loads. The sensitivity of particle neck-sizes on the mechanical properties is also studied. All the results are shown to be qualitatively as well as quantitatively in agreement with the experimental data or with the available literature.

Temperature-Dependent Stiffening and Inelastic Behavior of Newly Synthesized Fiber-Reinforced Super Flexible Silica Aerogels.

In recent years, flexible silica aerogels have gained significant attention, owing to their excellent thermal and acoustic insulation properties accompanied by mechanical flexibility. Fiber reinforcement of such aerogels results in a further enhancement of the strength and durability of the composite, while retaining the excellent insulation properties. In this paper, the influence of four different kinds of fibers within a flexible silica aerogel matrix is studied and reported. First, a description of the synthesis procedure and the resulting morphology of the four aerogel composites is presented. Their mechanical behavior under uniaxial quasi-static tension and compression is investigated, particularly their performance under uniaxial compression at different temperature conditions (50 °C, 0 °C, and -50 °C). The reinforcement of the flexible silica aerogels with four different fiber types only marginally influences the thermal conductivity but strongly enhances their mechanical properties.

Towards Patient-Specific Computational Modelling of Articular Cartilage on the Basis of Advanced Multiparametric MRI Techniques.

Cartilage degeneration is associated with tissue softening and represents the hallmark change of osteoarthritis. Advanced quantitative Magnetic Resonance Imaging (qMRI) techniques allow the assessment of subtle tissue changes not only of structure and morphology but also of composition. Yet, the relation between qMRI parameters on the one hand and microstructure, composition and the resulting functional tissue properties on the other hand remain to be defined. To this end, a Finite-Element framework was developed based on an anisotropic constitutive model of cartilage informed by sample-specific multiparametric qMRI maps, obtained for eight osteochondral samples on a clinical 3.0 T MRI scanner. For reference, the same samples were subjected to confined compression tests to evaluate stiffness and compressibility. Moreover, the Mankin score as an indicator of histological tissue degeneration was determined. The constitutive model was optimized against the resulting stress responses and informed solely by the sample-specific qMRI parameter maps. Thereby, the biomechanical properties of individual samples could be captured with good-to-excellent accuracy (mean R2 [square of Pearson's correlation coefficient]: 0.966, range [min, max]: 0.904, 0.993; mean Ω [relative approximated error]: 33%, range [min, max]: 20%, 47%). Thus, advanced qMRI techniques may be complemented by the developed computational model of cartilage to comprehensively evaluate the functional dimension of non-invasively obtained imaging biomarkers. Thereby, cartilage degeneration can be perspectively evaluated in the context of imaging and biomechanics.

Worm-like chain model extensions for highly stretched tropocollagen molecules.

Tropocollagen plays a very important role in the load bearing functionality of soft tissues. In the context of multi-scale modeling the response of tropocollagen molecules to stretch should be very carefully predicted in order to describe the mechanical behavior of soft tissues. To this end, the worm-like chain (WLC) model is often applied, although it is restricted to the entropic force regime which is essential at moderate deformations. To describe molecular forces under larger stretches several extensions of the WLC have been proposed for deoxyribonucleic acid (DNA). This contribution aims to investigate the applicability of these models in the context of tropocollagen and discusses the feasibility of their application. Finally, the models are validated in comparison to experimental data available in the literature.

Fatigue of soft fibrous tissues: Multi-scale mechanics and constitutive modeling.

In recent experimental studies a possible damage mechanism of collagenous tissues mainly caused by fatigue was disclosed. In this contribution, a multi-scale constitutive model ranging from the tropocollagen (TC) molecule level up to bundles of collagen fibers is proposed and utilized to predict the elastic and inelastic long-term tissue response. Material failure of collagen fibrils is elucidated by a permanent opening of the triple helical collagen molecule conformation, triggered either by overstretching or reaction kinetics of non-covalent bonds. This kinetics is described within a probabilistic framework of adhesive detachments of molecular linkages providing collagen fiber integrity. Both intramolecular and interfibrillar linkages are considered. The final constitutive equations are validated against recent experimental data available in literature for both uniaxial tension to failure and the evolution of fatigue in subsequent loading cycles. All material parameters of the proposed model have a clear physical interpretation. STATEMENT OF SIGNIFICANCE: Irreversible changes take place at different length scales of soft fibrous tissues under supra-physiological loading and alter their macroscopic mechanical properties. Understanding the evolution of those histologic pathologies under loading and incorporating them into a continuum mechanical framework appears to be crucial in order to predict long-term evolution of various diseases and to support the development of tissue engineering.

Correlating Synthesis Parameters to Morphological Entities: Predictive Modeling of Biopolymer Aerogels.

In the past decade, biopolymer aerogels have gained significant research attention due to their typical properties, such as low density and thermal insulation, which are reinforced with excellent biocompatibility, biodegradability, and ease of functionalization. Mechanical properties of these aerogels play an important role in several applications and should be evaluated based on synthesis parameters. To this end, preparation and characterization of polysaccharide-based aerogels, such as pectin, cellulose and k-carrageenan, is first discussed. An interrelationship between their synthesis parameters and morphological entities is established. Such aerogels are usually characterized by a cellular morphology, and under compression undergo large deformations. Therefore, a nonlinear constitutive model is proposed based on large deflections in microcell walls of the aerogel network. Different sizes of the microcells within the network are identified via nitrogen desorption isotherms. Damage is initiated upon pore collapse, which is shown to result from the failure of the microcell wall fibrils. Finally, the model predictions are validated against experimental data of pectin, cellulose, and k-carrageenan aerogels. Given the micromechanical nature of the model, a clear correlation-qualitative and quantitative-between synthesis parameters and the model parameters is also substantiated. The proposed model is shown to be useful in tailoring the mechanical properties of biopolymer aerogels subject to changes in synthesis parameters.

Mechanics of Nanostructured Porous Silica Aerogel Resulting from Molecular Dynamics Simulations.

Silica aerogels are nanostructured, highly porous solids which have, compared to other soft materials, special mechanical properties, such as extremely low densities. In the present work, the mechanical properties of silica aerogels have been studied with molecular dynamics (MD) simulations. The aerogel model of 192 000 atoms was created with different densities by direct expansion of ß-cristobalite and subjected to series of thermal treatments. Because of the high number of atoms and improved modeling procedure, the proposed model was more stable and showed significant improvement in the smoothness of the resulting stress-strain curves in comparison to previous models. Resulting Poisson's ratio values for silica aerogels lie between 0.18 and 0.21. The elasticity moduli display a power law dependence on the density, with the exponent estimated to be 3.25 ± 0.1. These results are in excellent agreement with reported experimental as well as computational values. Two different deformation scenarios have been discussed. Under tension, the low-density aerogels were more ductile while the denser ones behaved rather brittle. In the compression simulations of low-density aerogels, deformation occurred without significant increase in stress. However, for high densities, atoms offer a higher resistance to the deformation, resulting in a more stiff response and an early densification. The relationship between different mechanical parameters has been found in the cyclic loading simulations of silica aerogels with different densities. The residual strain grows linearly with the applied strain (≥0.16) and can be approximated by a phenomenological relation ϵp = 1.09ϵmax - 0.12. The dissipation energy also varies with the compressive strain according to a power law with an exponent of 2.31 ± 0.07. Moreover, the tangent modulus under cyclic loading varies exponentially with the compressive strain. The results of the study pave the way toward multiscale modeling of silica as well as reinforced silica aerogels.

T2 MR imaging vs. computational modeling of human articular cartilage tissue functionality.

The detection of early stages of cartilage degeneration remains diagnostically challenging. One promising non-invasive approach is to functionally assess the tissue response to loading by serial magnetic resonance (MR) imaging in terms of T2 mapping under simultaneous mechanical loading. As yet, however, it is not clear which cartilage component contributes to the tissue functionality as assessed by quantitative T2 mapping. To this end, quantitative T2 maps of histologically intact cartilage samples (n=8) were generated using a clinical 3.0-T MR imaging system. Using displacement-controlled quasi-static indentation loading, serial T2 mapping was performed at three defined strain levels and loading-induced relative changes were determined in distinct regions-of-interest. Samples underwent conventional biomechanical testing (by unconfined compression) as well as histological assessment (by Mankin scoring) for reference purposes. Moreover, an anisotropic hyperelastic constitutive model of cartilage was implemented into a finite element (FE) code for cross-referencing. In efforts to simulate the evolution of compositional and structural intra-tissue changes under quasi-static loading, the indentation-induced changes in quantitative T2 maps were referenced to underlying changes in cartilage composition and structure. These changes were parameterized as cartilage fluid, proteoglycan and collagen content as well as collagen orientation. On a pixel-wise basis, each individual component correlation with T2 relaxation times was determined by Spearman's ρs and significant correlations were found between T2 relaxation times and all four tissue parameters for all indentation strain levels. Thus, the biological changes in functional MR Imaging parameters such as T2 can further be characterized to strengthen the scientific basis of functional MRI techniques with regards to their perspective clinical applications.

Mechanics of collagen fibrils: A two-scale discrete damage model.

Collagen is one of the most important structural proteins of biological tissues. Recently, a new damage phenomena caused by a mechanical overloading of the tendon has been revealed on the collagen fibril level by means of scanning electron microscopy (Veres et al., 2013). In order to describe this phenomena we propose in the present paper a constitutive damage model for single collagen fibrils. It utilizes a statistical framework in order to incorporate a physically motivated transition from the entropic- to the intrinsic-elasticity range of single tropocollagen molecules. In this two-scale approach a bridging relation between the tropocollagen and fibril level as well as a failure criterion after exceeding a critical stretch limit are introduced. The final constitutive relations are validated against experimental data available in literature for both, the reversible and the irreversible, stress-strain response.

Multi-scale modeling of soft fibrous tissues based on proteoglycan mechanics.

Collagen in the form of fibers or fibrils is an essential source of strength and structural integrity in most organs of the human body. Recently, with the help of complex experimental setups, a paradigm change concerning the mechanical contribution of proteoglycans (PGs) took place. Accordingly, PG connections protect the surrounding collagen fibrils from over-stretching rather than transmitting load between them. In this paper, we describe the reported PG mechanics and incorporate it into a multi-scale model of soft fibrous tissues. To this end, a nano-to-micro model of a single collagen fiber is developed by taking the entropic-energetic transition on the collagen molecule level into account. The microscopic damage occurring inside the collagen fiber is elucidated by sliding of PGs as well as by over-stretched collagen molecules. Predictions of this two-constituent-damage model are compared to experimental data available in the literature.

Micromechanical model for isolated polymer-colloid clusters under tension.

Binary polymer-colloid (PC) composites form the majority of biological load-bearing materials. Due to the abundance of the polymer and particles, and their simple aggregation process, PC clusters are used broadly by nature to create biomaterials with a variety of functions. However, our understanding of the mechanical features of the clusters and their load transfer mechanism is limited. Our main focus in this paper is the elastic behavior of close-packed PC clusters formed in the presence of polymer linkers. Therefore, a micromechanical model is proposed to predict the constitutive behavior of isolated polymer-colloid clusters under tension. The mechanical response of a cluster is considered to be governed by a backbone chain, which is the stress path that transfers most of the applied load. The developed model can reproduce the mean behavior of the clusters and is not dependent on their local geometry. The model utilizes four geometrical parameters for defining six shape descriptor functions which can affect the geometrical change of the clusters in the course of deformation. The predictions of the model are benchmarked against an extensive set of simulations by coarse-grained-Brownian dynamics, where clusters with different shapes and sizes were considered. The model exhibits good agreement with these simulations, which, besides its relative simplicity, makes the model an excellent add-on module for implementation into multiscale models of nanocomposites.

Constitutive modeling of strain-induced crystallization in filled rubbers.

Strain-induced crystallization is a unique crystallization process taking place solely in polymers subjected to large deformations. It plays a major role for reinforcement and improvement of mechanical properties of polymers with a high regularity of the molecular structure. In this paper, we develop a micromechanical model for the strain-induced crystallization in filled rubbers. Accordingly, the strain-induced crystallization is considered as a process triggered by fully stretched and continued by semistretched polymer chains. The model extends the previously proposed network evolution model [Dargazany and Itskov, Int. J. Solids Struct. 46, 2967 (2009)] and can thus, in addition to the stress upturn and evolution of crystallinity, take into account several inelastic features of filled rubbers, such as the Mullins effect, permanent set, and induced anisotropy. Finally, the accuracy of the model is verified against different set of experimental data both with respect to the stress-strain and crystallization-strain relations. The model exhibits good agreement with the experimental results, which, besides its relative simplicity, makes it a good option for finite-element implementations.

Constitutive modeling of the Mullins effect and cyclic stress softening in filled elastomers.

The large strain behavior of filled rubbers is characterized by the strong Mullins effect, permanent set, and induced anisotropy. Strain controlled cyclic tests also exhibit a pronounced hysteresis as a strain rate independent phenomenon. Prediction of these inelastic features in elastomers is an important challenge with immense industrial and technological relevance. In the present paper, a micromechanical model is proposed to describe the inelastic features in the behavior of filled elastomers. To this end, the previously developed network decomposition concept [Dargazany and Itskov, Int. J. Solids Struct. 46, 2967 (2009)] is extended and an additional network (CP network) is added to the classical elastic rubber (CC) and polymer-filler (PP) networks. The new network is considered to account for the damage of filler aggregates in the cyclic deformation as the source of hysteresis energy loss. The accuracy of the resulting model is evaluated in comparison to a new set of experimental data.