Meso-scale analysis of FRC using a two-step homogenization approach

This paper describes the development of a two-step homogenization approach for evaluating the elastic properties of fiber reinforced concrete (FRC). For that purpose, a finite element model of an FRC unit cell was generated. Prior to the generation of the unit cell finite element model, the interface transition zones (ITZ) and the aggregates were homogenized using an analytical approach. In the first step, the properties of a spherical aggregate and its concentric ITZ layer were homogenized via an analytical procedure. Then a numerical homogenization procedure was applied to the homogenized aggregate, the mortar, and the fibers to obtain the macroscopic properties of the FRC. The suggested framework executes the multi-scale analysis of FRC structures by incorporating an original concrete unit cell generator into a commercial finite element software package intended for simulating nonlinear solid mechanical problems. The results, obtained using the presented algorithm, are in very good agreement with experimental results.     

Optimal design of periodic functionally graded composites with prescribed properties

The computational design of a composite where the properties of its constituents change gradually within a unit cell can be successfully achieved by means of a material design method that combines topology optimization with homogenization. This is an iterative numerical method, which leads to changes in the composite material unit cell until desired properties (or performance) are obtained. Such method has been applied to several types of materials in the last few years. In this work, the objective is to extend the material design method to obtain functionally graded material architectures, i.e. materials that are graded at the local level (e.g. microstructural level). Consistent with this goal, a continuum distribution of the design variable inside the finite element domain is considered to represent a fully continuous material variation during the design process. Thus the topology optimization naturally leads to a smoothly graded material system. To illustrate the theoretical and numerical approaches, numerical examples are provided. The homogenization method is verified by considering one-dimensional material gradation profiles for which analytical solutions for the effective elastic properties are available. The verification of the homogenization method is extended to two dimensions considering a trigonometric material gradation, and a material variation with discontinuous derivatives. These are also used as benchmark examples to verify the optimization method for functionally graded material cell design. Finally the influence of material gradation on extreme materials is investigated, which includes materials with near-zero shear modulus, and materials with negative Poisson’s ratio.     

Topology optimization for microstructural design under stress constraints

This work aims at introducing stress responses within a topology optimization framework applied to the design of periodic microstructures. The emergence of novel additive manufacturing techniques fosters research towards new approaches to tailor materials properties. This paper derives a formulation to prevent the occurrence of high stress concentrations, often present in optimized microstructures. Applying macroscopic test strain fields to the material, microstructural layouts, reducing the stress level while exhibiting the best overall stiffness properties, are sought for. Equivalent stiffness properties of the designed material are predicted by numerical homogenization and considering a metallic base material for the microstructure, it is assumed that the classical Von Mises stress criterion remains valid to predict the material elastic allowable stress at the microscale. Stress constraints with arbitrary bounds are considered, assuming that a sizing optimization step could be applied to match the actual stress limits under realistic service loads. Density–based topology optimization, relying on the SIMP model, is used and the qp–approach is exploited to overcome the singularity phenomenon arising from the introduction of stress constraints with vanishing material. Optimization problems are solved using mathematical programming schemes, in particular MMA, so that a sensitivity analysis of stress responses at the microstructural level is required and performed considering the adjoint approach. Finally, the developed method is first validated with classical academic benchmarks and then illustrated with an original application: tailoring metamaterials for a museum anti–seismic stand.     

A generalized macroscopic model for sound-absorbing poroelastic media using the homogenization method

This paper proposes a new macroscopic model for sound-absorbing poroelastic media which is derived by using the homogenization theory based on the method of asymptotic expansions. The derivation of the macroscopic properties and governing equations takes into account the multiphysics occurring in poroelastic media for sound absorption, including elastic motions of the solid phase, compressible viscous fluid flow, and the distributions of pressure and temperature in the fluid phase. The coupled effects between the elastic solid and the fluid pressure, and the temperature and the fluid pressure are also considered. In contrast to the conventional Biot’s model, which includes heuristic formulae, the proposed method yields a rigorous model that is consistent with the principal governing equations on the microscopic scale. Utilizing several models that have simple microscopic geometry and comparing the numerical solutions obtained using the proposed method with corresponding analytical solutions, we demonstrate that the derived macroscopic governing equations can provide accurate and effective predictions.     

Design of periodic elastoplastic energy dissipating microstructures

The design of periodic elastoplastic microstructures for maximum energy dissipation is carried out using topology optimization. While the topology optimization of elastic microstructures has been performed in numerous studies, microstructural design considering inelastic behavior is relatively untouched due to a number of reasons which are addressed in this study. An RVE-based multiscale model is employed for computational homogenization with periodic boundary constraints, satisfying the Hill-Mandel principle. The plastic anisotropy which may be prevalent in materials fabricated through additive manufacturing processes is considered by modeling the constitutive behavior at the microscale with Hoffman plasticity. Discretization is done using enhanced assumed strain elements to avoid locking from incompressible plastic flow under plane strain conditions and a Lagrange multiplier approach is used to enforce periodic boundary constraints in the discrete system. The design problem is formulated using a density-based parameterization in conjunction with a SIMP-like material interpolation scheme. Attention is devoted to issues such as dependence on initial design and enforcement of microstructural connectivity, and a number of optimized microstructural designs are obtained under different prescribed deformation modes.

     

Design of piezocomposite materials and piezoelectric transducers using topology optimization—Part I

Summary Currently developments of piezocomposite materials and piczoelectric actuators have been based on the use of simple analytical models, test of prototypes, and analysis using the finite element method (FEM), usually limiting the problem to a parametric optimization. By changing the topology of these devices or their components, we may obtain an improvement in their performance characteristics. Based on this idea, this paper discusses the application of topology optimization combined with the homogenization method and FEM for designing piezocomposite materials. The homogenization method allows us to calculate the effective properties of a composite material knowing its unit cell topology. New effective properties that improves the electromechanical efficiency of the piezocomposite material are obtained by designing the piezocomposite unit cell. This method consists of finding the distribution of the material and void phases in a periodic unit cell that optimizes the performance characteristics of the piezocomposite. The optimized solution is obtained using Sequential Linear Programming (SLP). A general homogenization method applied to piczoelectricity was implemented using the finite element method (FEM). This homogenization method has no limitations regarding volume fraction or shape of the composite constituents. The main assumptions are that the unit cell is periodic and that the scale of the composite part is much larger than the microstructure dimensions. Prototypes of the optimized piezocomposites were manufactured and experimental results confirmed the large improvement. Department of Mechanical Engineering and Applied Mechanics Department of Mechanical Engineering and Applied Mechanics     

High-aspect-ratio microstructural posts electroforming modeling and fabrication in LIGA process

This paper describes a theoretical model that predicts the metal ion concentration distribution during electroforming high aspect ratio microstructures (HARM). The applied current density and microstructure aspect ratio were found as two important factors that affect the electroforming outcome. The analytical results are verified using experiments that electroforming microstructural posts with an aspect ratio of 10. Good agreement was obtained between the experimental and analytical solutions. Based on the ion concentration analytical prediction on the cathode surface, one can estimate the electroforming time required for fabricating a microstructure for a given aspect ratio.     

Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level

The aim of this paper is to present a microstructural topology optimization methodology for the structural-acoustic coupled system. In the structural-acoustic system, the structure is considered to be a thin composite plate composed of periodic uniform microstructures. The discrete design variables are used in the microstructural topology optimization, and the constitutive matrix is interpolated by the power-law scheme at the micro scale. The equivalent macro material properties of the microstructure are computed through the homogenization method. The design objective is to minimize the sound pressure level (SPL) in an interior acoustic medium. The sensitivities of the SPL with respect to design variables are derived. The bi-directional evolutionary structural optimization (BESO) method is extended to solve the structural-acoustic coupled optimization problem to find the optimal material distribution of the microstructure. Numerical examples of a hexahedral box and an automobile passenger compartment are given to demonstrate the efficiency of the presented microstructural topology optimization method.     

New solution method for homogenization analysis and its application to the prediction of macroscopic elastic constants of materials with periodic microstructures

A new solution method is proposed for the homogenization analysis of materials with periodic microstructures. A homogeneous integral equation is derived to replace the conventional inhomogeneous integral equation related to the microscopic mechanical behavior in the basic unit cell by introducing a new characteristic function. Based on the new solution method, the computational problem of the characteristic function subject to initial strains and periodic boundary conditions is reduced to a simple displacement boundary value problem without initial strains, which simplifies the computational process. Applications to the predication of macroscopic elastic constants of materials with various two-dimensional and three-dimensional periodic microstructures are presented. The numerical results are compared with previous results obtained from the Hapin-Tsai equations, Mori–Tanaka method and conventional homogenization calculations, which proves that the present method is valid and efficient for prediction of the macroscopic elastic constants of materials with various periodic microstructures.     

Optimal shape design in biomimetics based on homogenization and adaptivity

Optimal shape design of microstructured materials has recently attracted a great deal of attention in materials science. The shape and the topology of the microstructure have a significant impact on the macroscopic properties. The paper is devoted to the shape optimization of new biomorphic microcellular silicon carbide ceramics produced from natural wood by biotemplating. This is a novel technology in the field of biomimetics which features a material synthesis from biologically grown materials into ceramic composites by fast high-temperature processing. We are interested in finding the best material-and-shape combination in order to achieve the optimal prespecified performance of the composite material. The computation of the effective material properties is carried out using the homogenization method. Adaptive mesh-refinement technique based on the computation of recovered stresses is applied in the microstructure to find the homogenized elasticity coefficients. Numerical results show the reliability of the implemented a posteriori error estimators.     

The analysis of composite beams using standard finite element programs

The paper describes how a model using only standard finite elements can be made equivalent to a connecting system in composite construction. Using standard elements and not special slip elements enables composite construction to be analysed by the standard finite element packages now widely available. The method is applied to both the elastic analysis and ultimate load analysis of composite beams and gives results in close agreement to either experimental or other established analytical results.     

Mapping method for sensitivity analysis of composite material property

Composite properties are dependent on the microstructure of materials, which is depicted with a base cell. The parameters for representing the microstructure should include the shape parameters of the base cell and those used to describe the distribution of materials in the base cell. The goal of material design optimization is to find appropriate values of these parameters to make the materials have specific properties. Design optimization needs the sensitivity information of the material properties with respect to the shape parameter of the base cell and the material distribution parameters. Moreover, sensitivity calculation is often expensive. Thus, it is very important to develop an efficient sensitivity analysis method. In this paper, a mapping method is proposed for predicting the material properties and computing their sensitivities with respect to the shape parameters of the base cell. Through mapping transformation, solutions to the micro-scale homogenization problem defined on the domain of a base cell can be obtained by solving a homogenization problem defined on an initial given domain. The composite properties and their sensitivities with respect to the shape parameters of the base cell are explicitly expressed in terms of the properties and their sensitivities of a virtual material with respect to the distribution parameters. This virtual material has an initially given base cell domain. Thus re-meshing for discretizing the problem is avoided and computing cost savings are realized. Numerical examples show that the proposed method is accurate and efficient in both the prediction of material properties and sensitivity calculation.     

Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials

In this paper, two different methods for modelling of functionally graded material (FGM) beam with continuous spatially varying material properties will be presented and compared, namely the multilayering method and the direct integration method. Both the methods are related to homogenization of spatially varying material properties of real FGM beam and to calculation of the secondary variables of the FGM beams. The multilayering method is based on the laminate theory, which is very often used by modelling of the multilayer composite beams. The direct integration method transform spatial continuous varying material properties to the effective ones by direct integration of derived homogenization rules. In next part of the paper, new multiphysical beam finite element will be presented, which in conjunction with the proposed homogenization methods can be used for very effective analysis of the FGM beam structures. The numerical experiment will be presented concerning the multiphysical (electro–thermal–structural) analysis of the chosen FGM beams with spatial continuous variation of material properties.     

Micro- and macro-failure models of heterogeneous media with micro-structure

In this paper, the effectiveness of homogenization techniques for media with micro-structure subject to large deformations has been studied by comparing their micro- and macro-failure mechanisms. The material has been studied by considering its representative volume element (RVE) which entails all the geometric and constitutive information of the micro-structure. First, the formulation of the elastostatic problem governing the non-linear (large deformation and non-linear elastic) behavior of the structure of the RVE is presented. The RVE is subject to loading paths that produce uniform macroscopic strains. In this way it has been possible to use an homogenization procedure in order to simulate the overall behavior of the material, i.e. its constitutive tensor, at each point of the equilibrium path. Then, a macro failure surface has been defined as the locus of the points, in the macroscopic stretches space, corresponding to the loss of positivity of the macroscopic fourth order constitutive tensor in terms of the Biot stress [Encyclopedia of Physics, vol. 3(3), Springer-Verlag, Berlin]. Further, a micro-failure surface is defined as the locus of the points, in the overall stretches space, corresponding to the first critical point detected along the equilibrium path which can be characterized by an eigenmode compatible with the boundary conditions. Finally, a representative volume element, schematized by plane rods with strongly non-linear elastic constitutive behavior, is considered and the corresponding micro- and macro-failure surfaces are obtained in order to validate the proposed methodology.     

Influence of micro-cracking and contact on the effective properties of composite materials

In the present work a novel micro-mechanical approach to analyze the influence of micro-crack evolution and contact on the effective properties of elastic composite materials is proposed, based on homogenization techniques, interface models and fracture mechanics concepts. By means of the finite element method, enhanced non-linear macroscopic constitutive laws are developed by taking into account changes in micro-structural configuration associated with the growth of micro-cracks and with contact between crack faces. Numerical simulations are carried out for the cases of a porous composite with edge cracks and of a debonded fibre reinforced composite, loaded along extension/compression uniaxial macro-strain paths. Micro-crack propagation is modelled by using an original methodology based on the J-integral technique in conjunction with an interface model taking into account the unilateral contact of crack faces. In the context of a micro-to-macro transition obtained by controlling the macro-deformation of the micro-structure, the effects of adopting three types of boundary conditions on the macroscopic constitutive law, namely linear deformation, uniform tractions and periodic deformations and anti-periodic tractions, are studied. As a consequence, the proposed method can be applied to a large class of problems including periodic, locally periodic and irregular composite materials. Micro-crack and contact evolution result in a progressive loss of stiffness and can lead to failure for homogeneous macro-deformations associated with unstable crack propagation.     

Construction of models of dispersive elastodynamic behavior of periodic composites: A computational approach

A finite element method is described for the construction of continuum models of elastodynamic behavior of periodic composites. The method yields a hierarchy of models including ones which can be used to predict the phenomenon of dispersion of elastic waves in bulk composites; prediction of this effect of small-scale heterogeneity on the macroscopic behavior of composite materials is beyond the scope of the classical effective modulus theories.     

Multiscale simulation of particle-reinforced elastic–plastic adhesives at small strains

Many aerospace, aircraft or automotive mechanical components are joined together by using a structural adhesive. Adherend-to-adherend joint performance is usually carried out by a thin adhesive layer such that loads are transferred through this region, being then a critical point in the design. In order to ensure a proper behaviour of the adhesive under dynamical, mechanical, thermal or rheological loads, they are typically reinforced with a second phase stiffer material in addition to the adhesive matrix. Due to the intrinsic nature of the matrix, it may be approached using an elastic–plastic behaviour. Under these circumstances the adhesive inherently shows a heterogeneous microstructure whereas the loads are applied at the macroscopic adherend scale. In this work, a multiscale formulation is developed to analyze particle-reinforced adhesive joints. The adherend and the adhesive region, which is modelled using cohesive elements, stand macroscopically. On the other hand, the macroscopic adhesive behaviour is obtained by a direct analysis of the two-distinguished phases interaction at the microscopic level, using micromechanics and homogenization. The presented approach provides macroscopic as well as microscopic information about load distribution avoiding phenomenological lab fitting, case to case, of the overall macroscopic behaviour of the adhesive.     

Routes for Efficient Computational Homogenization of Nonlinear Materials Using the Proper Generalized Decompositions

Computational homogenization is nowadays one of the most active research topics in computational mechanics. Different strategies have been proposed, the main challenge being the computing cost induced by complex microstructures exhibiting nonlinear behaviors. Two quite tricky scenarios lie in (i) the necessity of applying the homogenization procedure for many microstructures (e.g. material microstructure evolving at the macroscopic level or stochastic microstructure); the second situation concerns the homogenization of nonlinear behaviors implying the necessity of solving microscopic problems for each macroscopic state (history independent nonlinear models) or for each macroscopic history (history dependant nonlinear models). In this paper we present some preliminary results concerning the application of Proper Generalized Decompositions—PGD—for addressing the efficient solution of homogenization problems. This numerical technique could allow to compute the homogenized properties for any microstructure or for any macroscopic loading history by solving a single but highly multidimensional model. The PGD allows circumventing the so called curse of dimensionality that mesh based representations suffer. Even if this work only describes the first steps in a very ambitious objective, many original ideas are launched that could be at the origin of impressive progresses.     

纳米压痕法对304不锈钢残余应力的研究

利用纳米压痕法研究了304不锈钢的残余应力,采用Suresh理论模型恒定载荷时的公式计算残余应力,最大加载载荷依次为500μN、1 000μN、1 500μN、2 000μN、2500μN。结果表明,不锈钢硬度和弹性模量为定值,退火前后的硬度分别为5.3GPa和4.0 GPa,弹性模量分别为110 GPa和100 GPa。利用Ansys分析软件模拟了压痕过程,发现不锈钢在受压过程中有Sink-in现象发生。纳米压痕法测得了未退火不锈钢存在残余压应力,大小为381 MPa;用XRD测得了未退火不锈钢中有350 MPa±23 MPa的残余压应力,两种测量结果吻合良好,说明了纳米压痕法在残余应力测试时的准确性与可靠性。     

Phenomenological nanoindentation technique in quality control of optoelectronics devices

Multiphase and porous ceramic materials are being used as passive semiconductors for building modern optoelectronics devices. Prescreening of the critical parts for subsurface defects before assembly can prevent device failures in the field. A new nanoindentation based non-destructive method has been proposed where shapes of loading–unloading curves can fingerprint subsurface cracking and material porosity induced inelastic behavior in defected multiphase, rough and porous ceramic parts. In addition to the differences in nanoindentation loading-unloading curve shapes, nanohardness values associated with the cracked contact surfaces were 40% lower. The non-destructive nanoindentation results agree well with the destructive Knoop’s microhardness results. A commercially available nanoindentation instrument integrated into the tribometer together with high-resolution optical microscope was utilized for automated and qualitative/quantitative surface materials properties characterization of optoelectronics parts. The proposed instrument and nondestructive testing method can be used for quality control of optoelectronics parts before assembly, significantly improving the yields.