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Since its advent in the 1960s, elastoplastic micromechanics has been confronted by continuous challenges, as the classical incremental elastoplastic tangents are known to deliver unrealistically stiff material responses. As a complement to the various “secant” approximations targeting this challenge, we here develop a theoretical framework based on...
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... estimates for the strain concentration and influence tensors appearing in (22) and (23) are obtained by means of generalized Eshelby matrix-inclusion-type problems (Pichler and Hellmich, 2010;Zaoui, 2002). The pore phase is represented as a spherical inclusion embedded in a matrix with stiffness C hom and eigenstress Π 0 , subjected to homogeneous strains E 0 acting at the infinite boundary of the aforementioned matrix, see Figure 3. The needle-shaped solid phases are represented each as a cylindrical inclusion with stiffness C solid and eigenstresses π θφ = −C solid : ε p θφ , embedded into the very same matrix, and subjected to the very same strains E 0 , see Figure 3. ...
Context 2
... pore phase is represented as a spherical inclusion embedded in a matrix with stiffness C hom and eigenstress Π 0 , subjected to homogeneous strains E 0 acting at the infinite boundary of the aforementioned matrix, see Figure 3. The needle-shaped solid phases are represented each as a cylindrical inclusion with stiffness C solid and eigenstresses π θφ = −C solid : ε p θφ , embedded into the very same matrix, and subjected to the very same strains E 0 , see Figure 3. This results to inclusion/phase strains reading as (Zaoui, 2002) ...
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... Additionally, the integration of constitutive models significantly impacts the accuracy and stability of the numerical simulation. Integration algorithms are traditionally categorized into forward and backward Euler methods [50][51][52]. Furthermore, a semi-implicit return mapping algorithm has been developed for implementing constitutive models, including elasto-plastic models [53,54], plasticdamage coupled models [55][56][57], and viscoelasto-plastic models [58,59], due to its user-friendliness and high accuracy. However, we recently verified that the calculation accuracy and convergence of the semi-implicit algorithm are not always assured, primarily due to the high curvature of the plastic potential in plastic models [60] or plastic-damage models [9]. ...
In this work, a new micromechanics-based model is developed for describing the anisotropic mechanical behavior of quasi-brittle rocks in monotonic compression. The material is conceptualized as an isotropic elastic matrix embedded with randomly distributed initial microcracks. Based on experimental evidences and as an innovative approach, the stress-induced coupled anisotropic damage and plastic distortion are assumed related to the propagation and sliding of the dominant microcracks in the most unstable orientations under given loading state, instead of related to microcracks in all orientations as done in the previous studies. In this frame, by combining the thermodynamics framework and linear homogenization scheme, an energy-release-rate based damage law is developed for describing the induced anisotropic evolution of dominant microcracks. The plastic distortion is related to frictional sliding along the rough dominant microcracks and described by a damage-dependent friction law. The direction of dominant microcracks is identified by using Mohr maximization postulate. Further, the macroscopic failure strength is predicted from the coupled friction-damage analysis. The process for calibration of the microscopic parameters from macroscopic strength is developed. As another novelty of the present work, an enhanced semi-implicit return mapping (ESRM) algorithm is proposed to more efficiently dealing with two dissipation processes of plastic deformation and damage evolution. The effectiveness of the proposed model and the robustness of the ESRM algorithm are finally evaluated.
... The latter are directly related to the lattice spacing of hydroxyapatite, which amounts to 0.34 nm (Zhuang et al. 2012). The crystals found in the cement lines in lamellar bone show deviations of up to 30 from the fibrillar orientation (Grünewald et al. 2023), and this has important consequences for the elastoplastic behavior of bone, both at the scale of the porous polycrystal found in the cement line (Morin et al. 2017), and beyond (Fritsch et al. 2009b). ...
High-resolution imaging technologies such as micro-computed tomography (CT) have not only revealed the intricate geometrical properties of biomedical materials and systems such as hard tissue engineering scaffolds, but also provided important data sources for the mechanical integrity of these systems, including implants. For the latter purpose, mechanical properties such as elasticity or strength need to be assigned to the voxels making up the CT scan. This is customarily done from back-analysis of mechanical tests performed on the entire scaffold structure, so as to downscale the overall mechanical response to that of an average solid voxel. Accordingly, local inhomogeneities remain unconsidered, and this motivates the review of a more fundamental, bottom-up-type strategy which unleashes, in an interdisciplinary fashion, important additional information “hidden” inside each and every voxel: Namely, the focus is set on resolving microstructural and nanostructural features located inside each and every voxel of a CT image, and on quantifying their effects on the voxel-specific material properties. For this purpose, the basics of X-ray physics, which allow for the translation of voxel-specific attenuation information into compositional quantities, are combined with micromechanical modeling. This provides a theoretically founded, rigorous link between material composition and mechanical properties. The feasibility and usefulness of this approach are highlighted by application examples concerning porosity and elasticity distributions throughout glass- and ceramic-based tissue engineering scaffolds, together with the results of quasistatic structural simulations performed on the geometrically and mechanically reconstructed objects. Thereafter, generalizations concerning hierarchical systems such as bone are discussed, together with the dynamic loading realm and the consideration of tissue anisotropy, also as far as interaction with implants is
concerned.
... This drawback limits existing approaches to small two-scales problems, even in the simplest case of hierarchical materials with linear elastic constituents. Continuum micromechanics provides a rigorous framework to derive analytical estimates of macroscale elastoplastic properties (Zaoui 2002;Suquet 2014;Morin et al. 2017) and has been successfully applied to describe many multiphase hierarchical systems such as plant, wood, bone, and cementitious materials (Gangwar and Schillinger 2019;Hofstetter et al. 2005;Hellmich et al. 2004;Fritsch et al. 2009;Pichler and Hellmich 2011). In the context of concurrent material and structure optimization, we recently integrated continuum micromechanics based homogenized estimates in end-compliance optimization problems, which rendered our framework computationally tractable for multiphase hierarchical systems with several material length scales . ...
... can be augmented to include hardening contributions by utilizing continuum micromechanicsbased homogenization schemes, such as outlined inFritsch et al. (2009);Morin et al. (2017) for the example of bones. ...
The concept of concurrent material and structure optimization aims at alleviating the computational discovery of optimum microstructure configurations in multiphase hierarchical systems, whose macroscale behavior is governed by their microstructure composition that can evolve over multiple length scales from a few micrometers to centimeters. It is based on the split of the multiscale optimization problem into two nested sub-problems, one at the macroscale (structure) and the other at the microscales (material). In this paper, we establish a novel formulation of concurrent material and structure optimization for multiphase hierarchical systems with elastoplastic constituents at the material scales. Exploiting the thermomechanical foundations of elastoplasticity, we reformulate the material optimization problem based on the maximum plastic dissipation principle such that it assumes the format of an elastoplastic constitutive law and can be efficiently solved via modified return mapping algorithms. We integrate continuum micromechanics based estimates of the stiffness and the yield criterion into the formulation, which opens the door to a computationally feasible treatment of the material optimization problem. To demonstrate the accuracy and robustness of our framework, we define new benchmark tests with several material scales that, for the first time, become computationally feasible. We argue that our formulation naturally extends to multiscale optimization under further path-dependent effects such as viscoplasticity or multiscale fracture and damage.
... Hill's method introduces the elastic homogenization for the elastoplastic upscaling, which gives rise to an overestimation of the overall stiffness. 53 Doghri and Ouaar 54 point out the ''isotropization'' of the overall/homogenized tangent moduli or using the consistent tangent moduli can alleviate the overestimation. ...
In this work, a binary-medium-based constitutive model for porous rocks is proposed under the framework of micromechanics. The porous rock at mesolevel is considered a two-phase composite composed of the matrix of the bonded medium and inclusions of the frictional medium. The bonded medium is defined by a Drucker-Prager solid phase where the pores are embedded, for which an upscaled model is introduced. Once the bonded medium breaks due to cracking, it transforms into the weakened frictional medium which is described by an elastic-perfectly plastic Mohr-Coulomb model. The breakage evolution is assumed to obey a Weibull-type function. The Mori-Tanaka method is extended to homogenize the mesoscopic binary-medium morphology. Numerical implementations of the binary-medium homogenization procedure and an implicit integration method for the bonded medium are conducted. Furthermore, the consistent tangent moduli in the local integration are also formulated. Finally, the indoor experimental investigations of Maldivian coral reef limestone (a type of porous rock) are presented, and the proposed model is applied to describe the macroscopic behavior of Maldivian coral reef limestone. The practicality of replacing the initial porosity in the proposed model with the P-wave velocity is verified as well.
... These simulations can relate the stem strength behaviour with plant traits quantitatively and, therefore, lead to important insights to improve lodging resistance of cereals. We note that, in the past decade, the multi-scale material-model-based finite-element simulations have contributed significantly in understanding and developing patient-specific diagnostic tools for bone diseases [23][24][25][26][27]. ...
Lodging impedes the successful cultivation of cereal crops. Complex anatomy, morphology and environmental interactions make identifying reliable and measurable traits for breeding challenging. Therefore, we present a unique collaboration among disciplines for plant science, modelling and simulations, and experimental fluid dynamics in a broader context of breeding lodging resilient wheat and oat. We ran comprehensive wind tunnel experiments to quantify the stem bending behaviour of both cereals under controlled aerodynamic conditions. Measured phenotypes from experiments concluded that the wheat stems response is stiffer than the oat. However, these observations did not in themselves establish causal relationships of this observed behaviour with the physical traits of the plants. To further investigate we created an independent finite-element simulation framework integrating our recently developed multi-scale material modelling approach to predict the mechanical response of wheat and oat stems. All the input parameters including chemical composition, tissue characteristics and plant morphology have a strong physiological meaning in the hierarchical organization of plants, and the framework is free from empirical parameter tuning. This feature of our simulation framework reveals the multi-scale origin of the observed wide differences in the stem strength of both cereals that would not have been possible with purely experimental approach.
... However, similar to the situation with poromechanical theories, RVE approaches hardly entered the biomedical field before the 1990s, and it was not before the 2000s until they started to flourish. They did particularly well for the intricate hierarchical organization of bone tissue where classical anatomical terms could be well assigned to RVEs of different sizes: trabecular bone to an RVE with a size of several millimeters [22][23][24][25][26][27][28][29][30], cortical bone to an RVE with a size of several hundred micrometers [20,23,[31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46], extravascular bone material to an RVE with a size of hundred micrometers [35,46,47], extracellular material to an RVE with a size of tens of micrometers [32,35,39,41,[47][48][49][50][51], and finally both mineralized collagen fibrils [34,35,47,48,[52][53][54] and extrafibrillar polycrystals [34,35,47,52,55] were assigned to RVEs with a size of about hundred nanometers. The latter two RVE types contain the elementary building blocks (or "major components" [7]) of bone: i.e., nanometer-sized hydroxyapatite crystals and crosslinked type I collagen molecules [52]. ...
... Homogenization of eigenstressed media was particularly helpful for explaining the strength of bone, arising from ductile sliding between hydrated crystals followed by brittle failure of molecular collagen [35]: Therefore, elementary plastic strains fulfilling the flow and consistency rules of classical plasticity [135] were introduced at the level of the hydroxyapatite crystals at Level 2B of Fig. 1, and thereafter upscaled up to the level of macroscopic bone material. More recent studies based on the representation of Level 2B of Fig. 1 have revealed the elastoplastic failure behavior of the cement lines surrounding secondary osteons in Haversian bone from the surrounding bone matrix [55]. ...
... According to Popper's famous reasoning on the logic of scientific discovery [145,146], a hypothesis which the human mind comes up with needs to survive as many statistically and physically different falsification tests as possible, in order to eventually reach, after extensive discussions and debates between the peers within a scientific community, the status of a reliable scientific theory. In this spirit, various multiscale micromechanics approaches, as described in Sec. 3, have been carefully validated by means of biomechanical, biophysical, and biochemical experiments [23,35,47,49,52,55,147]. Typically, such experiments concern (i) "universal", i.e. tissue-invariant, mechanical properties of the elementary components making up any tissue belonging to a particular class of tissues (e.g., in the case of bone tissues: molecular collagen type I (making up around 90% of all organic matter found in bone [148]), hydroxyapatite, and water with non-collagenous organic, see Secs. ...
Biological materials and systems are hierarchically organized.The main motivation for hierarchical biomechanics is that the wide variability of mechanical properties encountered at the macroscopic scale may be traced back to just a few universal. i.e. tissue-invariant, mechanical properties of elementary components at a sufficiently small scale (such as collagen, elastin, and water in case of soft tissues; complemented by hydroxyapatite in case of hard tissues), and to the nano and microstructures which the latter build up. This challenging task requires a physically rigorous and mathematically sound basis, as provided by Finite Element and Fast Fourier Transform methods, as well as by continuum micromechanics resting on (semi-)analytical solutions for Eshelby-type matrix-inclusion problems. Corresponding numerical and analytical mathematical models have undergone diligent experimental validation, by means of data stemming from a variety of biophysical, biochemical, and biomechanical testing methods, such as light and electron microscopy, ultrasonic testing and scanning acoustic microscopy, as well as physico-chemical tests associated with dehydration, demineralization, decollagenization, ashing, and weighing in air and fluid. While elastic scale transition and homogenization methods have attained a high maturity level, the hierarchical nature of dissipative (i.e. viscous or strength) properties is still a vibrant field of research. This applies even more to hierarchical approaches elucidating the interface between biological cells and extracellular matrices, and to the highly undiscovered mechanics unfolding within biological cells.
... Due to the simple averaging method proposed by the mean-field homogenization technique, it has been adopted to evaluate the overall behaviors of various composite-like materials (Pierard et al., 2004;Deude et al., 2002;Doghri et al., 2016;Fritsch et al., 2013;Morin et al., 2017;Pardoen and Hutchinson, 2003;Pens é e et al., 2002;Pichler et al., 2007;Ulm et al., 2004;Zhu et al., 2009). However, the application of this technique to masonry has received very limited attention. ...
... This study presents the first step towards developing an analytically based multiscale model for masonry. As a simple and efficient upscaling method, the mean-field homogenization technique has been successfully applied to many analytical or semi-analytical multiscale methods by combining it with different damage, plasticity or fracture models at macro and microscale, for the nonlinear analyses of various isotropic composite materials in different fields ranging from metal composites (Doghri et al., 2016), alloys (Pardoen and Hutchinson, 2003), rocks (Deude et al., 2002;Pens é e et al., 2002), cementitious materials (Pichler et al., 2007;Ulm et al., 2004), geomaterials (Zhu et al., 2009) and bones (Fritsch et al., 2013;Morin et al., 2017). These analytical or semi-analytical multiscale procedures achieve a balance between the accuracy and computational cost, and their implementation to simulate structural behavior is considered feasible. ...
... Furthermore, the proposed model has advantages in nonlinear extension. Many multiscale methods have been established over the past decades for the nonlinear analyses of various composite-like materials, such as metal composites (Doghri et al., 2016), alloys (Pardoen and Hutchinson, 2003), rocks (Deude et al., 2002;Pens é e et al., 2002), cementitious materials (Pichler et al., 2007;Ulm et al., 2004), geomaterials (Zhu et al., 2009) and bones (Fritsch et al., 2013;Morin et al., 2017), by combining the mean-field homogenization technique with different damage, plasticity or fracture models at macroscopic and microscopic scales. The proposed model has the potential to be applied to these well-established mean-field homogenization based multiscale methods to achieve the nonlinear analysis of masonry. ...
Accurate assessment of the overall mechanical behavior of masonry, composed of bricks and mortar joints, remains challenging due to its inhomogeneous and orthotropic nature. In this study, the feasibility of various mean-field homogenization schemes for the three-dimensional orthotropic elastic properties of masonry is comprehensively investigated. Three kinds of masonry patterns are considered, including the stack bonded pattern, the running bonded pattern and the double-leaf Flemish bonded pattern that has received limited attention so far. Special attention is paid to the homogenization schemes which have not been applied to the masonry case, such as Lielens’ interpolative double inclusions (D-I) and the interaction direct derivative (IDD) schemes. After a comparison between the well-known mean-field homogenization schemes, an improved micro-mechanical model is proposed by combining the advantages of the IDD and D-I models. The validation of the proposed model is conducted through a comparison against experimental data from literature and numerical results obtained via finite element analyses (FEA). The results show that the proposed model can accurately evaluate the orthotropic elastic properties of the three masonry typologies for a wide range of stiffness ratios between brick and mortar, ranging from 1 to 1000. The proposed model also shows better performance than the classical schemes especially when the stiffness ratios between brick and mortar are higher than 10, which is of major importance for the application of mean-field homogenization based multiscale methods to the nonlinear analysis of masonry. Furthermore, the presented homogenization method can be of interest for other anisotropic materials, e.g., laminate materials.
... Lemarchand et al. 2003). Due to a simple averaging method proposed by the mean-field homogenization technique, the microporomechanics theory has been adopted to study the nonlinear behavior of composite-like materials in different fields ranging from metal composites (Doghri et al. 2016;Pierard et al. 2004), alloys (Pardoen & Hutchinson 2003, rocks (Deude et al. 2002;Pensée et al. 2002), cementitious materials (Pichler et al. 2007;Ulm et al. 2004), geomaterials (Zhu et al. 2009) and bones (Fritsch et al. 2013;Morin et al. 2017). The application of such theory is however mainly limited to isotropic quasi-brittle materials and the study of crack propagation in an initially anisotropic materials, as masonry, has received limited attention. ...
The microporomechanics theory, which combines the mean-field homogenization method and linear fracture mechanics theory, has been successfully adopted to study the nonlinear behavior of composite-like materials, such as alloy, rocks and concretes. The application of such theory is however mainly limited to the isotropic quasi-brittle materials and the study of crack propagation in an initially anisotropic materials, as masonry, has received limited attention. This paper aims to derive the nonlinear material behavior of masonry by adopting microporomechanics theory. In this study, masonry is treated as a composite material, made of bricks, mortar joints and microcracks. At constituents’ level, cracks are idealized as three orthotropic families of penny-shaped inclusions, which are then embedded in an undamaged effective masonry matrix formed by bricks and mortar joints. A crack density variable, containing the information of each crack family (e.g., crack radius), is adopted to define the damage state of masonry. The propagation of each crack family is governed by the energy release rate and its critical value. The results shows that the microporomechanics theory can successfully derive the nonlinear behavior of masonry (e.g., the tensile softening). The proposed model allows using limited input parameters mainly related to properties of constituents, and elastic modulus and tensile strength of the composites. However, it should be mentioned that the model developed in this study only considers the cohesive mechanics by modelling the propagation of open cracks, while the friction on the lips of closed microcracks is not taken into consideration and it will be objective of further study.
... However, since the calculation of localization and influence tensors in an exact form is rather impractical, an estimate of them (denoted by( )) is usually sought for via homogenization schemes. Details of the derivation of these approximate values can be found in Eghbalian (12); Morin et al. (29); Pichler and Hellmich (32). ...
This paper presents a micromechanical description of the swelling behavior of partially saturated clays following a two-stage homogenization procedure on a three-dimensional Representative Elementary Volume (REV) of clay. At the nano-scale, the REV of clay particles includes idealized parallel clay platelets and oblate spheroidal pores in between them that are saturated with an electrolyte solution. Swelling forces at all spacing ranges, including the crystalline and osmotic regimes, are considered. At the micro-scale, the REV of clay is comprised of dispersed clay aggregates and partially saturated (variably-sized) spherical pores in between them. At issue, is reconstructing the couplings of mechanical, hydraulic and electrochemical forces at the macroscopic level and their effect on the overall behavior of clay. In Part I, we focus on reversible deformation mechanisms in clay. A generalized nonlinear form of the classical Biot’s poroelasticity relation is obtained with additional terms accounting for capillary and swelling stresses. Capillary stress emerges from interaction of fluid phases at different scales, and takes into account the surface tension effects. The swelling part on the other hand, correlates with the net disjoining hydration and electrochemical forces at the nano-scale. As a result, the stress description of clays is embedded with microstructural information. In addition, a robust localization procedure is utilized to track the microstructural changes associated to micro-porosity and clay platelet spacing. The nonlinearity of the stress–strain description is shown to originate from the dependency of the particles’ elasticity on spacing with electrochemical origins. Finally, base-line features of the model are investigated through material point simulations of a clay swelling test. The discrepancies between the model predictions and experimental measurements are resolved in Part II of this paper by considering also the plastic deformations within the framework developed in Part I.
... This drawback limits existing approaches to small two-scales problems, even in the simplest case of hierarchical materials with linear elastic constituents. Continuum micromechanics provides a rigorous framework to derive analytical estimates of macroscale elastoplastic properties [37][38][39] and has been successfully applied to describe many multiphase hierarchical systems such as plant, wood, bone, and cementitious materials [40][41][42][43][44][45]. In the context of concurrent material and structure optimization, we recently integrated continuum micromechanics based homogenized estimates in end-compliance optimization problems, which rendered our framework computationally tractable for multiphase hierarchical systems with several material length scales [46]. ...
... We observe that the microstructure design variable set m is implicitly accounted for by the residual definitions in each load increment. The macroscale stress Σ n+1 at each Gauss point is evaluated by solving the nonlinear elastoplastic constitutive relations (39) and (41) with known microstructure configurationm x,j n+1 that solves the material optimization problem detailed in the following subsection. Therefore, the global equilibrium equation (43) is nonlinear and requires iterative solution approaches such as the Newton-Raphson incremental procedure [47,58]. ...
... including all constraints defined through the stress admissible set E Σ n+1 and microscale design admissible set E ad . The first two conditions are essentially the elastoplastic constitutive equations relating the macroscale stress with the macroscale strains via (39) and the constraint on the macroscale stress defined through the homogenized yield criterion (41). The third condition is the definition of the Helmholtz free energy in terms of microscale design variable m x,j n+1 . ...
The concept of concurrent material and structure optimization aims at alleviating the computational discovery of optimum microstructure configurations in multiphase hierarchical systems, whose macroscale behavior is governed by their microstructure composition that can evolve over multiple length scales from a few micrometers to centimeters. It is based on the split of the multiscale optimization problem into two nested sub-problems, one at the macroscale (structure) and the other at the microscales (material). In this paper, we establish a novel formulation of concurrent material and structure optimization for multiphase hierarchical systems with elastoplastic constituents at the material scales. Exploiting the thermomechanical foundations of elastoplasticity, we reformulate the material optimization problem based on the maximum plastic dissipation principle such that it assumes the format of an elastoplastic constitutive law and can be efficiently solved via modified return mapping algorithms. We integrate continuum micromechanics based estimates of the stiffness and the yield criterion into the formulation, which opens the door to a computationally feasible treatment of the material optimization problem. To demonstrate the accuracy and robustness of our framework, we define new benchmark tests with several material scales that, for the first time, become computationally feasible. We argue that our formulation naturally extends to multiscale optimization under further path-dependent effects such as viscoplasticity or multiscale fracture and damage.
