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The structural properties of mechanical metamaterials are typically studied with two-scale methods based on computational homogenization. Because such materials have a complex microstructure, enriched schemes such as second-order computational homogenization are required to fully capture their non-linear behavior, which arises from non-local interactions due to the buckling or patterning of the microstructure. In the two-scale formulation, the effective behavior of the microstructure is captured with a representative volume element (RVE), and a homogenized effective continuum is considered on the macroscale. Although an effective continuum formulation is introduced, solving such two-scale models concurrently is still computationally demanding due to the many repeated solutions for each RVE at the microscale level. In this work, we propose a reduced-order model for the microscopic problem arising in second-order computational homogenization, using proper orthogonal decomposition and a novel hyperreduction method that is specifically tailored for this problem and inspired by the empirical cubature method. Two numerical examples are considered, in which the performance of the reduced-order model is carefully assessed by comparing its solutions with direct numerical simulations (entirely resolving the underlying microstructure) and the full second-order computational homogenization model. The reduced-order model is able to approximate the result of the full computational homogenization well, provided that the training data is representative for the problem at hand. Any remaining errors, when compared with the direct numerical simulation, can be attributed to the inherent approximation errors in the computational homogenization scheme. Regarding run times for one thread, speed-ups on the order of 100 are achieved with the reduced-order model as compared to direct numerical simulations.

The lifetime of tungsten (W) monoblocks under fusion conditions is ambivalent. In this work, the microstructure dependent mechanical behaviour of pulsed high heat flux (HHF) exposed W monoblock is investigated. Two different microstructural states, i.e. initial (deformed) and recrystallized, both machined from HHF exposed monoblocks are tested using tensile and small punch tests. The initial microstructural state reveals a higher fraction of low angle boundaries along with a preferred orientation of crystals. Following HHF exposure, the recrystallized state exhibits weakening of initial texture along with a higher fraction of high angle boundaries. Irrespective of the testing methodology, both the microstructural states display brittle failure for temperatures lower than 400∘C. For higher temperatures (>400∘C), the recrystallized microstructure exhibits more ductile behaviour as compared to the initial state. The observed microstructural state-dependent mechanical behaviour is further discussed in terms of different microstructural features. The estimated brittle-to-ductile transition temperature (BDTT) range is noticed to be lower for the recrystallized state as compared to the initial state. The lower BDTT in the recrystallized state is attributed to the high purity of the W in combination with its low defect density, thereby preventing segregation of impurities at the recrystallized boundaries and the related premature failure. Based on this observation, it is concluded that the common opinion of the aggravation of BDTT in W due to recrystallization is not unerring, and as a matter of fact, recrystallization in W could be instrumental for preventing the self-castellation of the monoblocks.

Materials like paper, consisting of a network of natural fibres, exposed to variations in moisture, undergo changes in geometrical and mechanical properties. This behaviour is particularly important for understanding the hygro-mechanical response of sheets of paper in applications like digital printing. A two-dimensional microstructural model of a fibrous network is therefore developed to upscale the hygro-expansion of individual fibres, through their interaction, to the resulting overall expansion of the network. The fibres are modelled with rectangular shapes and are assumed to be perfectly bonded where they overlap. For realistic networks the number of bonds is large and the network is geometrically so complex that discretizing it by conventional, geometry-conforming, finite elements is cumbersome. The combination of a level-set and XFEM formalism enables the use of regular, structured grids in order to model the complex microstructural geometry. In this approach, the fibres are described implicitly by a level-set function. In order to represent the fibre boundaries in the fibrous network, an XFEM discretization is used together with a Heaviside enrichment function. Numerical results demonstrate that the proposed approach successfully captures the hygro-expansive properties of the network with fewer degrees of freedom compared to classical FEM, preserving desired accuracy., Comment: 27 pages, 22 figures, 4 tables, J. Appl. Mech. June 19, 2020

Mechanical metamaterials feature engineered microstructures designed to exhibit exotic, and often counter-intuitive, effective behaviour. Such a behaviour is often achieved through instability-induced transformations of the underlying periodic microstructure into one or multiple patterning modes. Due to a strong kinematic coupling of individual repeating microstructural cells, non-local behaviour and size effects emerge, which cannot easily be captured by classical homogenization schemes. In addition, the individual patterning modes can mutually interact in space as well as in time, while at the engineering scale the entire structure can buckle globally. For efficient numerical macroscale predictions, a micromorphic computational homogenization scheme has recently been developed. Although this framework is in principle capable of accounting for spatial and temporal interactions between individual patterning modes, its implementation relied on a gradient-based quasi-Newton solution technique. This solver is suboptimal because (i) it has sub-quadratic convergence, and (ii) the absence of Hessians does not allow for proper bifurcation analyses. Given that mechanical metamaterials often rely on controlled instabilities, these limitations are serious. To address them, a full Newton method is provided in detail in this paper. The construction of the macroscopic tangent operator is not straightforward due to specific model assumptions on the decomposition of the underlying displacement field pertinent to the micromorphic framework, involving orthogonality constraints. Analytical expressions for the first and second variation of the total potential energy are given, and the complete algorithm is listed. The developed methodology is demonstrated with two examples in which a competition between local and global buckling exists and where multiple patterning modes emerge., Comment: 34 pages, 17 figures, 1 table, 1 algorithm, abstract shortened to fulfill 1920 character limit

Porous materials like acoustic foams can be used for acoustic shielding, which is important for high-tech systems and human comfort. In this paper, a homogenization model is proposed to investigate the relation between the microstructure and the resulting macroscopic acoustic properties. The macroscopic absorption ability is due to the microscopic viscous-thermal coupling between an elastic solid skeleton and a gaseous fluid in the associated Representative Volume Element (RVE). The macro-to-micro relation is realized through the boundary conditions of the microscopic RVE, which relies on the macroscopic solid deformation and fluid pressure gradient. By assuming that the variation of the macroscopic energy per unit volume equals the volume average of the variation of the microscopic energy, the macroscopic solid stress and fluid displacement can be calculated from the corresponding microscopic quantities. Making additional assumptions on this approach, Biot's poroelastic theory is recovered. A case study is performed through the simulations of sound absorption in three porous materials, one made from aluminum and two from different polyurethane foams. For simplicity, an idealized partially open cubic microstructure is adopted. The homogenization results are evaluated by comparison with Direct Numerical Simulations (DNS), revealing an adequate performance of the approach for the studied porous material. By comparing the results of different solid materials, it is found that the solid stiffness has a limited effect when resonance does not occur. Nevertheless, due to the absence of the microscopic fluctuation, Biot's model with the parameters obtained from the homogenization approach predicts a higher resonance frequency than the DNS, whereas a full homogenization modification improves the prediction., QC 20160129

Porous materials like acoustic foams can be used for acoustic shielding, which is important for high-tech systems and human comfort. In this paper, a homogenization model is proposed to investigate the relation between the microstructure and the resulting macroscopic acoustic properties. The macroscopic absorption ability is due to the microscopic viscous-thermal coupling between an elastic solid skeleton and a gaseous fluid in the associated Representative Volume Element (RVE). The macro-to-micro relation is realized through the boundary conditions of the microscopic RVE, which relies on the macroscopic solid deformation and fluid pressure gradient. By assuming that the variation of the macroscopic energy per unit volume equals the volume average of the variation of the microscopic energy, the macroscopic solid stress and fluid displacement can be calculated from the corresponding microscopic quantities. Making additional assumptions on this approach, Biot's poroelastic theory is recovered. A case study is performed through the simulations of sound absorption in three porous materials, one made from aluminum and two from different polyurethane foams. For simplicity, an idealized partially open cubic microstructure is adopted. The homogenization results are evaluated by comparison with Direct Numerical Simulations (DNS), revealing an adequate performance of the approach for the studied porous material. By comparing the results of different solid materials, it is found that the solid stiffness has a limited effect when resonance does not occur. Nevertheless, due to the absence of the microscopic fluctuation, Biot's model with the parameters obtained from the homogenization approach predicts a higher resonance frequency than the DNS, whereas a full homogenization modification improves the prediction.

Porous materials like acoustic foams can be used for acoustic shielding, which is important for high-tech systems and human comfort. In this paper, a homogenization model is proposed to investigate the relation between the microstructure and the resulting macroscopic acoustic properties. The macroscopic absorption ability is due to the microscopic viscous-thermal coupling between an elastic solid skeleton and a gaseous fluid in the associated Representative Volume Element (RVE). The macro-to-micro relation is realized through the boundary conditions of the microscopic RVE, which relies on the macroscopic solid deformation and fluid pressure gradient. By assuming that the variation of the macroscopic energy per unit volume equals the volume average of the variation of the microscopic energy, the macroscopic solid stress and fluid displacement can be calculated from the corresponding microscopic quantities. Making additional assumptions on this approach, Biot's poroelastic theory is recovered. A case study is performed through the simulations of sound absorption in three porous materials, one made from aluminum and two from different polyurethane foams. For simplicity, an idealized partially open cubic microstructure is adopted. The homogenization results are evaluated by comparison with Direct Numerical Simulations (DNS), revealing an adequate performance of the approach for the studied porous material. By comparing the results of different solid materials, it is found that the solid stiffness has a limited effect when resonance does not occur. Nevertheless, due to the absence of the microscopic fluctuation, Biot's model with the parameters obtained from the homogenization approach predicts a higher resonance frequency than the DNS, whereas a full homogenization modification improves the prediction.

Porous materials like acoustic foams can be used for shielding and their absorption abilities depend on the interaction of the acoustic wave and the complex microstructure. In this paper, a homogenization model is proposed to investigate the relation between the microstructure and the macroscopic properties. A numerical experiment is performed in the form of simulations of sound absorption tests on a porous material made from polyurethane. For simplicity, an idealized partially open cubic microstructure is adopted. The homogenization results are evaluated by comparison with Direct Numerical Simulations (DNS), showing a good performance of the approach for the studied porous material. By comparing the results, it is found that Biot's model with the parameters obtained from the homogenization approach predict a higher resonance frequency than the DNS, whereas a full homogenization modification improves the prediction due to the incorporation of the microscopic fluctuation., QC 20200709

In this paper, a homogenization method is proposed to obtain the parameters of Biot's poroelastic theory from a multiscale perspective. It is assumed that the behavior of a macroscopic material point can be captured through the response of a microscopic Representative Volume Element (RVE) consisting of both a solid skeleton and a gaseous fluid. The macroscopic governing equations are assumed to be Biot's poroelastic equations and the RVE is governed by the conservation of linear momentum and the adopted linear constitutive laws under the isothermal condition. With boundary conditions relying on the macroscopic solid displacement and fluid pressure, the homogenized solid stress and fluid displacement are obtained based on energy consistency. This homogenization framework offers an approach to obtain Biot's parameters directly through the response of the RVE in the regime of Darcy's flow where the pressure gradient is dominating. A numerical experiment is performed in the form of a sound absorption test on a porous material with an idealized partially open microstructure that is described by Biot's equations where the parameters are obtained through the proposed homogenization approach. The result is evaluated by comparison with Direct Numerical Simulations (DNS), showing a superior performance of this approach compared to an alternative semiphenomenological model for estimating Biot's parameters of the studied porous material., QC 20150626

In this paper, a numerical homogenization approach is proposed to obtain isotropic Biot's parameters based on the microstructure of an porous material. It is assumed that a macroscopic point can be represented by a microscopic Representative Volume Element (RVE) consisting of the solid and the fluid. The macroscopic properties are controlled by Biot's equations and the RVE is governed by linearized balance equations for momentum and linear constitutive laws. With suitable boundary conditions, the micro-macro relation is formulated based on consistency of energy. Then, Biot's parameters are calculated through the response of the RVE. By following this new homogenization approach, examples with simple microstructures are given and simulations of two sound absorption experiments are conducted by using Biot's equations. The results are compared with Direct Numerical Simulations and it shows a favourable performance of this new approach compared to the alternative Transfer Matrix Method., QC 20150422

QC 20150422. QC 20160212

In this paper, based on a representative volume element (RVE) and Slattery’s averaging theorem, parameters of Biot’s poroelastic equations for homogenous isotropic porous materials are obtained. According to Slattery’s averaging theorem, the coupling terms, which describe the inertial effects and the viscous effects, are represented by an integral of the solid-fluid interaction force. This relation provides a new approach to obtain the parameters required in Biot’s equations through a direct numerical simulation of the RVE. An example of a 2D RVE is given and simulations of sound propagation in an impedance tube with a foam are conducted using Biot’s equations. It is shown that the numerical coupling mass obtained from this new approach behaves qualitatively the same as an associated phenomenological model., QC 20140102

In this paper, based on a representative volume element (RVE) and Slattery’s averaging theorem, parameters of Biot’s poroelastic equations for homogenous isotropic porous materials are obtained. According to Slattery’s averaging theorem, the coupling terms, which describe the inertial effects and the viscous effects, are represented by an integral of the solid-fluid interaction force. This relation provides a new approach to obtain the parameters required in Biot’s equations through a direct numerical simulation of the RVE. An example of a 2D RVE is given and simulations of sound propagation in an impedance tube with a foam are conducted using Biot’s equations. It is shown that the numerical coupling mass obtained from this new approach behaves qualitatively the same as an associated phenomenological model., QC 20140102

In this paper, based on a representative volume element (RVE) and Slattery’s averaging theorem, parameters of Biot’s poroelastic equations for homogenous isotropic porous materials are obtained. According to Slattery’s averaging theorem, the coupling terms, which describe the inertial effects and the viscous effects, are represented by an integral of the solid-fluid interaction force. This relation provides a new approach to obtain the parameters required in Biot’s equations through a direct numerical simulation of the RVE. An example of a 2D RVE is given and simulations of sound propagation in an impedance tube with a foam are conducted using Biot’s equations. It is shown that the numerical coupling mass obtained from this new approach behaves qualitatively the same as an associated phenomenological model., QC 20140102

In this paper, based on a representative volume element (RVE) and Slattery’s averaging theorem, parameters of Biot’s poroelastic equations for homogenous isotropic porous materials are obtained. According to Slattery’s averaging theorem, the coupling terms, which describe the inertial effects and the viscous effects, are represented by an integral of the solid-fluid interaction force. This relation provides a new approach to obtain the parameters required in Biot’s equations through a direct numerical simulation of the RVE. An example of a 2D RVE is given and simulations of sound propagation in an impedance tube with a foam are conducted using Biot’s equations. It is shown that the numerical coupling mass obtained from this new approach behaves qualitatively the same as an associated phenomenological model., QC 20140102

In this paper, based on a representative volume element (RVE) and Slattery’s averaging theorem, parameters of Biot’s poroelastic equations for homogenous isotropic porous materials are obtained. According to Slattery’s averaging theorem, the coupling terms, which describe the inertial effects and the viscous effects, are represented by an integral of the solid-fluid interaction force. This relation provides a new approach to obtain the parameters required in Biot’s equations through a direct numerical simulation of the RVE. An example of a 2D RVE is given and simulations of sound propagation in an impedance tube with a foam are conducted using Biot’s equations. It is shown that the numerical coupling mass obtained from this new approach behaves qualitatively the same as an associated phenomenological model., QC 20140102

Multiple length scales are involved in the development of traumatic brain injury, where the global mechanics of the head level are responsible for local physiological impairment of brain cells. In this study, a relation between the mechanical state at the tissue level and the cellular level is established. A model has been developed that is based on pathological observations of local axonal injury. The model contains axons surrounding an obstacle (e.g., a blood vessel or a brain soma). The axons, which are described by an anisotropic fiber-reinforced material model, have several physically different orientations. The results of the simulations reveal axonal strains being higher than the applied maximum principal tissue strain. For anisotropic brain tissue with a relatively stiff inclusion, the relative logarithmic strain increase is above 60%. Furthermore, it is concluded that individual axons oriented away from the main axonal direction at a specific site can be subjected to even higher axonal strains in a stress-driven process, e.g., invoked by inertial forces in the brain. These axons can have a logarithmic strain of about 2.5 times the maximum logarithmic strain of the axons in the main axonal direction over the complete range of loading directions. The results indicate that cellular level heterogeneities have an important influence on the axonal strain, leading to an orientation and location-dependent sensitivity of the tissue to mechanical loads. Therefore, these effects should be accounted for in injury assessments relying on finite element head models., QC 20110613

Grain size reduction in single phase alloys is generally accompanied by a loss of ductility related to a decrease in the strain hardening capacity. The flow behaviour of fine-grained dual-phase steels produced by swaging was investigated in order to address the couplings between grain size reduction and incorporation of a second phase towards optimizing microstructures with different objectives. A physically based grain size dependent strain hardening model has been developed for the ferrite matrix, involving specific laws for the accumulation and saturation of dislocations along grain boundaries and for their net back stress contribution. The back stress increases with the dislocation density, reaches a maximum and finally decreases due to screening effects. The overall behaviour of the ferrite-martensite composite is then evaluated using a mean field homogenization method. After identification of the material parameters, a parametric study provides, for a given carbon content and grain size of the ferrite matrix, the optimum martensite volume fraction, leading either to the maximum strength ductility product or to the maximum strength under the constraint of a minimum ductility. (c) 2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.