Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors \BC_1 and \BC_2=\Gd\BC_0 comprising the set U=\{\BC_1,\Gd\BC_0\} and mixed in proportions $f$ and $1-f$ is partly characterized in the limit \Gd\to\infty. The material with tensor \BC_2 corresponds to a material which (for technical reasons) is almost rigid in the limit \Gd\to \infty. The paper, and the underlying microgeometries, have many aspects in common with the companion paper "On the possible effective elasticity tensors of 2-dimensional printed materials". The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is almost rigid to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" that is a hierarchical laminate which minimizes an appropriate sum of elastic energies.Comment: 13 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:1606.0330

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Coupling solid and fluid stresses with brain tumour growth and white matter tract deformations in a neuroimaging-informed model

Brain tumours are among the deadliest types of cancer, since they display a strong ability to invade the surrounding tissues and an extensive resistance to common therapeutic treatments. It is therefore important to reproduce the heterogeneity of brain microstructure through mathematical and computational models, that can provide powerful instruments to investigate cancer progression. However, only a few models include a proper mechanical and constitutive description of brain tissue, which instead may be relevant to predict the progression of the pathology and to analyse the reorganization of healthy tissues occurring during tumour growth and, possibly, after surgical resection. Motivated by the need to enrich the description of brain cancer growth through mechanics, in this paper we present a mathematical multiphase model that explicitly includes brain hyperelasticity. We find that our mechanical description allows to evaluate the impact of the growing tumour mass on the surrounding healthy tissue, quantifying the displacements, deformations, and stresses induced by its proliferation. At the same time, the knowledge of the mechanical variables may be used to model the stress-induced inhibition of growth, as well as to properly modify the preferential directions of white matter tracts as a consequence of deformations caused by the tumour. Finally, the simulations of our model are implemented in a personalized framework, which allows to incorporate the realistic brain geometry, the patient-specific diffusion and permeability tensors reconstructed from imaging data and to modify them as a consequence of the mechanical deformation due to cancer growth

Mechanics of Materials: Towards Predictive Methods for Kinetics in Plasticity, Fracture, and Damage

The workshop dealt with current advances of computational methods, mathematics and continuum mechanics directed at thermodynamically consistent forms of constitutive equations for complex evolutionary phenomena in modern materials such as plasticity, fracture and damage. The main aspects addressed in presentations and discussions were multiphysical description of new materials, (visco)plasticity, fracture, damage, structural mechanics, mechanics of materials and dislocation dynamics

Homogenization of Damaged Concrete Mesostructures using Representative Volume Elements: Implementation and Application to SLang

This master thesis explores an important and under-researched topic on the so-called bridging of length scales (from >mesomacrobridge< the representations of events that occur at two different scales. The underlying objective here is to efficiently incorporate material length scales in the classical continuum plasticity/damage theories through the concept of homogenization theory. The present thesis is devoted to computational modeling of heterogeneous materials, primarily to matrix-inclusion type of materials. Considerations are focused predominantly on the elastic and damage behavior as a response to quasistatic mechanical loading. Mainly this thesis focuses to elaborate a sound numerical homogenization model which accounts for the prediction of overall properties with the application of different types of boundary conditions namely: periodic, homogeneous and mixed type of boundary conditions over two-dimensional periodic and non-periodic RVEs and three-dimensional non-periodic RVEs. Identification of the governing mechanisms and assessing their effect on the material behavior leads one step further. Bringing together this knowledge with service requirements allows for functional oriented materials design. First, this thesis gives attention on providing the theoretical basic mechanisms involved in homogenization techniques and a survey will be made on existing analytical methods available in literature. Second, the proposed frameworks are implemented in the well known finite element software programs ANSYS and SLang. Simple and efficient algorithms in FORTRAN are developed for automated microstructure generation using RSA algorithm in order to perform a systematic numerical testing of microstructures of composites. Algorithms are developed to generate constraint equations in periodic boundary conditions and different displacements applied spatially over the boundaries of the RVE in homogeneous boundary conditions. Finally, nonlinear simulations are performed at mesolevel, by considering continuum scalar damage behavior of matrix material with the linear elastic behavior of aggregates with the assumption of rigid bond between constituents

Effective equations governing an active poroelastic medium

In this work we consider the spatial homogenization of a coupled transport and fluid-structure interaction model, to the end of deriving a system of effective equations describing the flow, elastic deformation, and transport in an active poroelastic medium. The `active' nature of the material results from a morphoelastic response to a chemical stimulant, in which the growth timescale is strongly separated from other elastic timescales. The resulting effective model is broadly relevant to the study of biological tissue growth, geophysical flows (e.g. swelling in coals and clays) and a wide range of industrial applications (e.g. absorbant hygiene products). The key contribution of this work is the derivation of a system of homogenized partial differential equations describing macroscale growth, coupled to transport of solute, that explicitly incorporates details of the structure and dynamics of the microscopic system, and, moreover, admits finite growth and deformation at the pore-scale. The resulting macroscale model comprises a Biot-type system, augmented with additional terms pertaining to growth, coupled to an advection-reaction-diffusion equation. The resultant system of effective equations is then compared to other recent models under a selection of appropriate simplifying asymptotic limits

Multiscale mechano-morphology of soft tissues : a computational study with applications to cancer diagnosis and treatment

Cooperation of engineering and biomedical sciences has produced significant advances in healthcare technology. In particular, computational modelling has led to a faster development and improvement of diagnostic and treatment techniques since it allows exploring multiple scenarios without additional complexity and cost associated to the traditional trial-and-error methodologies. The goal of this thesis is to propose computational methodologies to analyse how the changes in the microstructure of soft tissues, caused by different pathological conditions, influence the mechanical properties at higher length scales and, more importantly, to detect such changes for the purpose of quantitative diagnosis and treatment of such pathologies in the scenario of drug delivery. To achieve this objective different techniques based on quasi-static and dynamic probing have been established to perform quantitative tissue diagnosis at the microscopic (tissue) and macroscopic (organ) scales. The effects of pathologies not only affect the mechanical properties of tissue (e.g. elasticity and viscoelasticity), but also the transport properties (e.g. diffusivity) in the case of drug delivery. Such transport properties are further considered for a novel multi-scale, patient-specific framework to predict the efficacy of chemotherapy in soft tissues. It is hoped that this work will pave the road towards non-invasive palpation techniques for early diagnosis and optimised drug delivery strategies to improve the life quality of patientsJames-Watt Scholarship, Heriot-Watt Annual Fund and the Institute of Mechanical, Process and Energy Engineering (IMPEE) Grant

A micromechanics based constitutive model for fibre reinforced cementitious composites

A new constitutive model for fibre reinforced cementitious composites based on micromechanical solutions is proposed. The model employs a two-phase composite based on the Eshelby matrix-inclusion solution and the Mori-Tanaka homogenization scheme and also simulates directional microcracking. An exterior point Eshelby based criterion is employed to model crack-initiation in the matrix-inclusion interface. Microcrack surfaces are assumed to be rough and able to regain contact under both normal and shear displacements. Fibres are included into the formulation in both cracked and uncracked conditions. Once cracks start to develop, the crack-bridging action of fibres is simulated using a local constitutive equation that accounts for the debonding and pull-out of fibre groups with different orientations. It is shown that the combination of the rough microcrack and fibre-bridging sub-models allows microcracking behaviour deriving from both tensile and compressive loads to be modelled in a unified manner. This ability to model tensile and compressive behaviour using the same micromechanical mechanisms is considered to be a particularly attractive feature of the formulation, which removes the need for multi-parameter triaxial yield surfaces and evolution functions that bedevil many competitor models. The model is successfully validated using a series of examples based on experimental test data

Nanoindentation relaxation study and micromechanics of Cement-Based Materials

Ce travail évalue le comportement mécanique des matériaux cimentaires à différentes échelles de distance. Premièrement, les propriétés mécaniques du béton produit avec un bioplastifiant à base de microorganismes efficaces (EM) sont etudiées par nanoindentation statistique, et comparées aux propriétés mécaniques du béton produit avec un superplastifiant ordinaire (SP). Il est trouvé que l’ajout de bioplastifiant à base de produit EM améliore la résistance des C–S–H en augmentant la cohésion et la friction des nanograins solides. L’analyse statistique des résultats d’indentation suggère que le bioplastifiant à base de produit EM inhibe la précipitation des C–S–H avec une plus grande fraction volumique solide. Deuxièmement, un modèle multi-échelles à base micromécanique est dérivé pour le comportement poroélastique de la pâte de ciment au jeune age. L’approche proposée permet d’obtenir les propriétés poroélastiques requises pour la modélisation du comportoment mécanique partiellement saturé des pâtes de ciment viellissantes. Il est montré que ce modèle prédit le seuil de percolation et le module de Young non drainé de façon conforme aux données expérimentales. Un metamodèle stochastique est construit sur la base du chaos polynomial pour propager l’incertitude des paramètres du modèle à travers plusieurs échelles de distance. Une analyse de sensibilité est conduite par post-traitement du metamodèle pour des pâtes de ciment avec ratios d’eau sur ciment entre 0.35 et 0.70. Il est trouvé que l’incertitude sous-jacente des propriétés poroélastiques équivalentes est principalement due à l’énergie d’activation des aluminates de calcium au jeune age et, plus tard, au module élastique des silicates de calcium hydratés de basse densité.This work assesses the mechanical behavior of cement-based materials through different length scales. First, the mechanical properties of concrete produced with effective microorganisms (EM)-based bioplasticizer are investigated by means of statistical nanoindentation, and compared to the nanomechanical properties of concrete produced with ordinary superplasticizer (SP). It is found that the addition of EM-based bioplasticizer improves the strength of C–S–H by enhancing the cohesion and friction of solid nanograins. The statistical analysis of indentation results also suggests that EM-based bioplasticizer inhibits the precipitation of C–S–H of higher density. Second, a multiscale micromechanics-based model is derived for the poroelastic behavior of cement paste at early age. The proposed approach provides poroelastic properties required to model the behavior of partially saturated aging cement pastes. It is shown that the model predicts the percolation threshold and undrained elastic modulus in good agreement with experimental data. A stochastic metamodel is constructed using polynomial chaos expansions to propagate the uncertainty of the model parameters through different length scales. A sensitivity analysis is conducted by post-treatment of the meta-model for water-to-cement ratios between 0.35 and 0.70. It is found that the underlying uncertainty of the effective poroelastic proporties is mostly due to the apparent activation energy of calcium aluminate at early age and, later on, to the elastic modulus of low density calcium-silicate-hydrate