Two-step Homogenization Procedure Research Articles

Experimental investigation and micromechanics-based constitutive modeling of the transition from brittle to ductile behavior in saturated low-porosity rocks

This paper presents a unified experimental and numerical investigation on the transition from brittle to ductile behavior in a low-porosity sandstone under drained conditions. The experimental results demonstrate a transition in the mechanical behavior from brittle faulting to dilatant ductile flow at room temperature with an increase in effective confining pressure, suggesting that microcracking-controlled local friction is the underlying plastic deformation mechanism. For constitutive modeling, the sandstone is considered as a heterogeneous medium composed of a pores-weakened elastic solid matrix and distributed microcracks. By following a two-step homogenization procedure and irreversible thermodynamics framework, a micromechanics-based elastoplastic damage model incorporating a non-associated local plastic flow rule is formulated, in which the coupling between plasticity, damage and pore pressure is taken into account. In this context, a non-associated macroscopic effective strength criterion as an inherent part of the corresponding model is derived. Originally, a theoretical linear relation between critical state of damage at peak strength and effective confining pressure is established, which is efficient in describing post-peak softening behavior. Comparisons of numerical simulations with experimental data demonstrate that the proposed model effectively reproduces the main features of the sandstone with a brittle-ductile transition.

Two-step homogenized elastic model for in-plane analysis of coated masonry walls

A two-step homogenization procedure is used to identify the (membrane) macroscopic elastic constants of coated masonry walls, solicited in their plane. The two steps deal separately with in- and out-of- wall plane homogenization, based on the modeling of an elementary cell by means of suitable designed assemblies of springs. As a result of the different step combinations, two homogenization strategies are derived. Both of them provide simple closed-form expressions for the linear elastic constants of an “equivalent” homogeneous and orthotropic material. The influence of the order of the homogenization steps is also investigated. Parametric analyses are carried out to examine the reinforcement effectiveness on the global behavior of the system. The accuracy of the theoretical results is assessed by comparisons with finite element (FE) models.

Effects of geometry and loading on the elastic behavior of a single osteon via analytical structural mechanics approaches

In this study, two analytical approaches for lamella texture at micro-scale including disordered mineralized and organized collagen fibers are employed to provide analytical models of an osteon. A two-step homogenization procedure is also applied to characterize the elastic properties of the osteon. Numerical evaluations are accomplished to explore the effects of loading and geometrical parameters on the behavior of the osteon. Results show the diameter of the Haversian canal is the dominant geometrical parameter influencing the elastic stresses in the osteon. The radial and hoop stresses as well as radial displacement are insensitive to the changes in the number of lamellae. The ordered motif osteon model can also be used to explain circumferential micro-cracks along lamellae leading to irregular profiles of fracture in osteon direct tensile tests. The theoretical model of osteon based on the lamination theory generally indicated better agreement with existing experimental results.

Nano-emulsion system consisting of non-ionic surfactant, silicone, and polyol

This study aimed to evaluate the optimum formulation of a niosomal nano-emulsion system consisting of silicone oil, non-ionic surfactant, and polyol. The nano-emulsions were prepared through a two-step homogenization procedure by a homogenizer and microfluidizer. The experiments were carried out by changing the composition ratio of dimethicone (DC200/100cs), polyglyceryl-2 dioleate, and glycerin, respectively. The variation of dimethicone resulted in showing a tendency for the content and particle size to be proportional. The variation of polyglyceryl-2 dioleate and the glycerin variation resulted in showing a similar tendency of the content and particle size to be inversely proportional. In zeta potential measurement, all nano-emulsions marked absolute value over around 30 mV, which is considered to be stable, while no significant correlation was observed with the variation. Based on these verifications, the author proposes the optimal conditions for preparing niosomal nano-emulsions.

Approximate plastic yield criteria of geomaterials with pores and grains embedded in a porous matrix

Mineral grains and porosity are two common properties of geomaterials. This study focuses on establishing a macroscopic yield criterion for geomaterials having pores and mineral grains at the mesoscale which is embedded in a porous matrix. At the microscale, the solid phase is pressure-sensitive and obeys to the Drucker–Prager criterion which considers the asymmetric property between tension and compression. Based on a two-step homogenization procedure and the modified secant method, a macroscopic yield criterion is firstly derived which takes into account the effects of microstructure on the macroscopic mechanical behaviors of the studied material, such as the meso-porosity and mineral grains content at the mesoscale, the micro-porosity and the pressure-sensitive of the solid phase at the microscale. However, it overestimates the material strength in compressive loading by comparing with the Fast Fourier Transform (FFT) based numerical results. Then, this criterion is improved by considering the exact solution in pure hydrostatic loading of a porous medium with a Drucker–Prager matrix. By comparing to the numerical results with different microstructures and to experimental data, the new macroscopic yield criterion is estimated and validated. The main features of the studied geomaterial are well captured by this criterion.

Barrier Dispersion-Based Coatings Containing Natural and Paraffin Waxes.

Petroleum, synthetic, and natural waxes have been used as hydrophobic bases for dispersions intended for use as barrier coatings for packaging paper. Oil-in-water dispersions with alkaline pH were prepared by a two-step homogenization procedure containing paraffin wax, with various characteristics, the Fischer–Tropsch synthesis product or beeswax. The size of the dispersed particles determined by dynamic light scattering depended on the type of hydrophobic base used and was in the range of 350–440 nm. The ability of dispersion particles in aggregation driven by electrostatic attraction, evaluated by Zeta potential analysis by electrophoretic light scattering, was from −26 to −50 mV. Static multiply light scattering was used for 30 days of stability assessment and helped to select the dispersion with a Sarawax SX70 wax base as the most stable. Dispersions were further used for coating the backing of kraft paper by the Meyer rod method. Coated paper with an applied coating of 6 g/m2 had very good hydrophobic properties (Cobb60 < 4 g/m2), sufficient strength properties, and air permeation, which enabled its application as a packaging material. The dispersions based on Sarawax SX70 wax were evaluated as the best coating for Mondi ProVantage Kraftliner 125 g/m2 backing paper. Good hydrophobic properties and strength properties indicate the possibility of using the SX70-based wax dispersion coating as a replacement for PFAS coatings in some applications.

Friction-damage coupled models and macroscopic strength criteria for ice-saturated frozen silt with crack asperity variation by a micromechanical approach

This paper aims to present physical friction-damage coupled models and derive macroscopic nonlinear strength criteria for ice-saturated frozen silt. Two material scales are considered here. The material at the mesoscale is composed of a frozen silt matrix and embedded mesocracks. At the microscale, the frozen silt matrix is composed of elastic mineral grains and elastic ice crystals. Considering this microstructure, for the ice-saturated frozen silt, the plastic deformation is related to frictional sliding along the mesocracks, and damage is due to the propagation of mesocracks. We obtain the effective elastic properties with a two-step linear Mori-Tanaka homogenization procedure. For the second homogenization step, we deduce the macroscopic stress-strain relations and local thermodynamic forces associated with damage and plasticity. Then, an energy release-based damage criterion is developed to capture the evolution of damage, and a friction sliding criterion with an associated/nonassociated local plastic flow rule is introduced to describe the rate of plasticity. The friction sliding criterion considers the variation of surface asperities of mesocracks as a result of pressure melting and damage evolution. In this context, associated/nonassociated multiscale friction-damage models are established. Furthermore, associated/nonassociated macroscopic nonlinear strength criteria as inherent parts of the corresponding multiscale models are derived under a large range of compressive stresses. For application, a semi-implicit local numerical algorithm of the multiscale models is developed. The accuracy of associated/nonassociated multiscale models is assessed by comparing their numerical simulations with conventional triaxial compression (CTC) tests. Although the associated multiscale model correctly predicts the nonlinear strength behavior and axial deformation of the Lanzhou frozen silt under CTC tests with different temperatures, it fails to quantitatively reproduce volumetric deformation. Nonassociated multiscale model predictions with experimental data of volumetric deformation are significantly improved compared to the associated multiscale model.

MOC Doped with Graphene Nanoplatelets: The Influence of the Mixture Preparation Technology on Its Properties.

The ongoing tendency to create environmentally friendly building materials is nowadays connected with the use of reactive magnesia-based composites. The aim of the presented research was to develop an ecologically sustainable composite material based on MOC (magnesium oxychloride cement) with excellent mechanical, chemical, and physical properties. The effect of the preparation procedure of MOC pastes doped with graphene nanoplatelets on their fresh and hardened properties was researched. One-step and two-step homogenization techniques were proposed as prospective tools for the production of MOC-based composites of advanced parameters. The conducted experiments and analyses covered X-ray fluorescence, scanning electron microscopy, energy-dispersive spectroscopy, high-resolution transmission electron microscopy, sorption analysis, X-ray diffraction, and optical microscopy. The viscosity of the fresh mixtures was monitored using a rotational viscometer. For the hardened composites, macro- and micro-structural parameters were measured together with the mechanical parameters. These tests were performed after 7 days and 14 days. The use of a carbon-based nanoadditive led to a significant drop in porosity, thus densifying the MOC matrix. Accordingly, the mechanical resistance was greatly improved by graphene nanoplatelets. The two-step homogenization procedure positively affected all researched functional parameters of the developed composites (e.g., the compressive strength increase of approximately 54% after 7 days, and 37% after 14 days, respectively) and can be recommended for the preparation of advanced functional materials reinforced with graphene.

Effect of agglomeration of the nanotubes on the vibration frequency of the multi-scale hybrid nanocomposite conical shells: a GDQ-based study

In the present manuscript, the generalized differential quadrature method (GDQM) will be employed in order to solve the free vibration problem of three-phase carbon fiber/nanotube/polymer hybrid nanocomposite conical shells, rested on a two-parameter elastic medium, regarding for the influence of the agglomeration of the carbon nanotubes (CNTs) on the dynamic responses of the shell for the first time. The equivalent material properties of the multi-scale hybrid nanocomposite will be derived according to a two-step homogenization procedure including the Eshelby-Mori-Tanaka method incorporated with the modified form of the rule of the mixture. Afterward, the strain-displacement relationships of the continuous system will be obtained using the classical shell theory for thin-walled conical shells. Extending the Hamilton’s principle for the conical shell, the motion equation of the problem will be enriched. At the end, the associated boundary condition will be used in associated with the GDQ discretization for the purpose of extracting the natural frequency of the continuous system. The verification tests reveal that the presented mathematical framework is powerful enough to estimate the dynamic response of the system. The results of this study show that existence of agglomeration phenomenon can dramatically affect the dynamic behaviors of the nanocomposite structure.

Micromechanical Modelling of Dynamic Behavior of Advanced Sheet Molding Compound (A-SMC) Composite

Passive safety, particularly in the transport industry, requires maximizing the dissipation of energy and minimizing the decelerations undergone by a vehicle following a violent impact (crash). This paper proposes a strategy for identifying an anisotropic local damage criterion in a moderate dynamic loading for Advanced Sheet Molding Compound (A-SMC) composite materials. Multi-scale damage modelling based on the Mori-Tanaka approach is put forward. Previously, the results of an experimental campaign carried out on a range of strain rates varying from quasi static to 200 s−1 were used to identify a probabilistic local damage criterion based on Weibull’s formulation and integrate the effect of damage at a fiber-matrix interface scale. Therefore, the progressive local damage occurring under a fast loading may be described. A two-step homogenization procedure allows describing the strain rate effect on the stress-strain curves. The model gives also rise to the prediction of the progressive anisotropic loss of stiffness. Comparing between the experimental and numerical results confirms the ability of the proposed approach to describe the visco-damage effect (delay of damage threshold and decrease in damage kinetics) emphasized in A-SMC composites.

Stiffness Estimates for Composites with Elliptic Cylindrical Voids.

A two-step homogenization procedure is presented to investigate the stiffness of a unidirectional continuous fiber-reinforced composite material containing voids of different shapes and volume contents. Since the Mori–Tanaka scheme is limited to moderate volume contents of the inhomogeneity phase, fiber and matrix are homogenized with semi-empirical relations with use of the adjusted fiber volume content in a first step. In the second step, the Mori–Tanaka scheme is applied to obtain the homogenized stiffness tensor of a transversely isotropic material containing voids aligned with the fiber direction. The voids are modelled with infinite length, but an elliptic base characterized by the aspect ratio. The tensor components of the Eshelby tensor for this case are presented in closed form for a transversely isotropic material depending on the aspect ratio and matrix material properties. The scheme is solved directly for easy implementation and the use of fast calculations of the effective engineering constants of a composite material containing voids. Experimental results from literature for different void contents and shapes are compared to the predicted moduli with cylindrical voids. From the results it is further concluded that the aspect ratio of the void and the manufacturing process of the composite should be considered.

Numerical evaluation on the influence of void defects and interphase on the thermal expansion coefficients of three-dimensional woven carbon/carbon composites

ABSTRACT In order to evaluate the influence of void defects and interphase on the thermal expansion properties of three-dimensional woven carbon/carbon composites, a multi-scale numerical model was proposed to predict the effective thermal expansion coefficients considering void defects and interphase. On micro-scale, the effective thermal expansion coefficients of fiber bundles with void defects were predicted. On meso-scale, the effective thermal expansion coefficients of three-dimensional woven carbon/carbon composite with voids and interphase were predicted through a two-step homogenization procedure based on the asymptotic homogenization method and finite element method. The numerical model was validated by comparing with experimental measurements. Moreover, the influences of voids fraction and interphase modulus on the effective thermal expansion coefficients were examined quantitatively. The results indicate that the voids fraction has a negative impact on the coefficients of thermal expansion. The thermal expansion coefficients tend to decrease approximately linearly with the increase of voids fraction in fiber reinforcement and matrix, and decrease nonlinearly with the increase of voids fraction in interphase. The effective coefficients of thermal expansion increase as the increase of interphase elastic modulus, and the influence of interphase voids fraction on the coefficients of thermal expansion is becoming weaken as the increase of interphase elastic modulus.

Multiscale modeling of strength and failure behavior of carbon nanostructure reinforced epoxy composite adhesives in bonded systems

In this study, atomistic based continuum (ABC) multiscale modeling approach is presented for strength and failure prediction of carbon nanotube (CNT) reinforced epoxy composite adhesives in bonded systems. Assuming that the CNTs are uniformly dispersed in the epoxy matrix, the mechanical properties of the nanocomposite adhesive are determined utilizing a two-step sequential homogenization procedure. In the first step, effective mechanical properties of a nanoscale representative volume element (RVE) are determined. The RVE comprises a CNT embedded in the epoxy matrix separated by an interphase. The structure-property relationship at the nanoscale is captured through ABC modeling approach wherein a CNT is modeled as a space-frame structure with beam elements and the epoxy is modeled using solid elements. The interphase between the CNT and the epoxy is modeled using non-linear spring elements to account for non-bonded van-der Waals interactions. In the second step, CNT inclusions with random orientations in the matrix are considered to create a microscale RVE of the nanocomposite. In both the steps, different boundary conditions were applied on the RVEs, and finite element (FE) analyses were conducted to estimate the effective mechanical properties through numerical homogenization. Furthermore, the strength of CNT/epoxy composite is estimated using modified Kelly-Tyson theory. Finally, modeling scheme proposed here is utilized to assess the load carrying capacity and failure behavior of a single lap-joint (SLJ) comprising composite adherends and CNT/epoxy nanocomposite adhesive by conducting continuum scale FE analyses. Results show over 30% improvement in strength for the SLJ with CNT/epoxy bondlayer comprising 2 vol% CNT. The findings of the study indicate that both strength and damage tolerance of the bonded joints can be significantly optimized utilizing nanoreinforced bondlayer.

A homogenization-based damage model for stiffness loss in ductile metal-matrix composites

The aim of this paper is to develop a homogenized model that permits to describe the progressive loss of stiffness observed experimentally on ductile metal-matrix composites subjected to mechanical loadings. A two-step homogenization procedure is proposed to describe the effect of damage on the elastic properties. Since damage is solely driven by plastic deformation, a Gurson-type model is considered to describe the nucleation and growth of voids. Completed by appropriate evolution equations of the microstructure, this model of plasticity is used to update the elastic properties that are described by a mean-field homogenization scheme. The complete incremental model is implemented numerically and applied to the prediction of the stiffness loss. The model predictions are in very good agreement with experimental results on Al-SiC composites, cast irons and steel composites.

A new micromechanics approach to the application of Eshelby's equivalent inclusion method in three phase composites with shape memory polymer matrix

In the present work Eshelby's equivalent inclusion method is extended to compute the effective properties of two inclusions in a heterogeneous matrix. Generally Eshelby's problem involves a single inclusion under dilute distribution in two phase composites. Moreover, unlike the dual inclusion theory, which is based on the concept of one inclusion embedded in to the second inclusion, alternate problem is now addressed. In this paper Eshelby's single equivalent inclusion concept is extended to the case of composite consisting of two different inclusions embedded separately in a heterogeneous matrix. In the composite, first inclusion is fibre and second inclusion is carbon nanotube embedded in a shape memory polymer heterogeneous matrix. The solution is obtained through individual eigenstrains in both the inclusions. The general closed form relations for the effective moduli and inelastic strain tensors are derived under infinitesimal strains of constituent materials following a two-step homogenization procedure. Analytical solutions of Eshelby's problem with two cylindrical inclusions explicit of each other, having different properties and placed in the same inelastic matrix domain are obtained. The relations are then applied to characterise the shape memory polymer composite. The consistency of the novel relations are established by comparing the results with other micromechanics methods.

Effects of inclusions and pores on plastic and viscoplastic deformation of rock-like materials

The aim of this paper is to study effects of inclusion and pores on plastic and viscoplastic deformation of rock-like materials. We shall consider a class of clayey rocks with two separate scales of microstructure. At the mesoscopic scale, the material is constituted by a continuous matrix and embedded mineral inclusion. At the microscopic scale, the continuous matrix is a porous medium composed of a solid phase and spherical pores. Macroscopic deformation behavior of the material is determined by using a two-step homogenization procedure. An analytical plastic yield criterion is used for the porous matrix in the first step. It is assumed that the porous matrix exhibits both instantaneous plastic deformation and delayed viscoplastic deformation, which are described by a unified formulation. The plastic criterion is extended to serve as the viscoplastic loading function. At the mesoscopic scale, we shall investigate influences of inclusion stiffness, shape, orientation and volume fraction on plastic and viscoplastic deformation. Due to the absence of analytical solutions, we shall propose an extended Fast Fourier Transform method (FFT) to solve the nonlinear homogenization problem of the unit cell exhibiting time-dependent plastic deformation. A series of numerical simulations are performed and the obtained results show that the proposed numerical model is able to bring a finer description of complex microstructure effect than most analytical models. Finally, the efficacy of this numerical model is checked through comparisons between numerical results and experimental data in triaxial compression creep and relaxation tests on claystone.

On the bounds of applicability of two-step homogenization technique for porous materials

We discuss applicability of the two-step homogenization procedure to microstructures formed by spherical pores of two distinct sizes with total porosity of 40%. Results of one- and two-step homogenizations utilizing Non-Interaction Approximation (NIA), Mori–Tanaka–Benveniste Scheme (MTB) and Differential Scheme (DS) are compared with numerical data obtained by finite element simulations. A modified collective rearrangement method powered by computationally efficient hierarchical k-means tree algorithm is developed for generating microstructures containing spherical pores of different sizes with prescribed partial porosities. Two-step procedure turns to be almost commutative with respect to the sequence of homogenization step showing 0.2% as a maximum relative error. Sensitivity of approximated overall elastic properties to the size difference of spherical inhomogeneities is observed with the shear modulus showing stronger dependence than the bulk modulus. The two-step MTB can be used to approximate effective properties of solids with spherical pores of distinct size as long as this size difference between pore families is larger than 10 times.

A multiscale homogenization model for strength predictions of fully and partially frozen soils

Soil freezing is often used to provide temporary support of soft soils in geotechnical interventions. During the freezing process, the strength properties of the soil–water–ice mixture change from the original properties of the water-saturated soil to the properties of fully frozen soils. In the paper, a multiscale homogenization model for the upscaling of the macroscopic strength of freezing soil based upon information on three individual material phases—the solid particle phase (S), the crystal ice phase (C) and the liquid water phase (L)—is proposed. The homogenization procedure for the partially frozen soil–water–ice composite is based upon an extension of the linear comparison composite (LCC) method for a two-phase matrix–inclusion composite, using a two-step homogenization procedure. In each step, the LCC methodology is implemented by estimating the strength criterion of a two-phase nonlinear matrix–inclusion composite in terms of an optimally chosen linear elastic comparison composite with a similar underlying microstructure. The solid particle phase (S) and the crystal ice phase (C) are assumed to be characterized by two different Drucker–Prager strength criteria, and the liquid water phase (L) is assumed to have zero strength capacity under drained conditions. For the validation of the proposed upscaling strategy, the predicted strength properties for fully and partially frozen fine sands are compared with experimental results, focussing on the investigation of the influence of the porosity and the degree of ice saturation on the predicted failure envelope.

An anisotropic elasto-plastic material model for injection-molded long fiber-reinforced thermoplastics accounting for local fiber orientation distributions

This paper presents an anisotropic elasto-plastic material model for injection-molded long fiber-reinforced thermoplastics. It considers local heterogeneities which are attributed to process-induced variations of fiber orientation distributions and fiber volume fractions. These inhomogeneities have an effect on the mechanical properties and need to be considered in structural computations. In the material model, this is realized through a two-step homogenization procedure. First, an anisotropic stiffness tensor is approximated using mean field homogenization. Second, the plastic behavior is described using Hill’s transversely isotropic yield criterion averaged over the three principal directions of the fiber orientation. The advantage in combining these two approaches is a micro-mechanically based, yet fast numerical calculation of the composite material behavior within an explicit finite element code. The anisotropic material model is calibrated by simulating tensile tests on specimens taken in different directions from an injection-molded plate of fiber reinforced thermoplastic. The spatial variation of fiber orientation distribution and fiber volume fraction throughout the plate is determined from numerical mold filling simulations and is compared with computer tomography scans at different positions. A validation of the model is performed through simulating position-dependent tensile tests on smooth and notched specimens as well as a punch test which is well reproduced.

Simple holonomic homogenization model for the non-linear static analysis of in-plane loaded masonry walls strengthened with FRCM composites

A two-step homogenization procedure for fast non-linear static analyses of FRCM (Fabric Reinforced Cementitious Matrix) composites reinforced masonry walls in-plane loaded is presented and benchmarked on a series of FRCM strengthened tuff panels experimentally tested under diagonal compression at the University of Naples, Italy. The numerical model relies into a first homogenization step where the unreinforced masonry is substituted with an equivalent homogenized non-linear orthotropic material exhibiting softening. The elementary cell is discretized by means of few triangular elastic elements (bricks) and holonomic interfaces (joints) where all the non-linearity is lumped. The standard homogenization model so obtained is characterized by either two or three unknowns under biaxial and shear stress states, respectively. The homogenized behavior of the elementary cell is thus deduced solving small scale non-linear equations systems. The second step relies into the strengthening application to the already homogenized material at a structural level. In such phase, masonry is modeled with rigid quadrilateral elements and homogenized holonomic interfaces, whereas FRCM by means of equivalent trusses with limited tensile strength and fragile behavior, connecting adjoining rigid elements. Equivalent mechanical properties of the trusses can be eventually tuned accounting for FRCM debonding or rupture of the fibers. In order to further assess the results obtained using homogenization, a 3D large scale heterogeneous micro-modeling strategy is used to reproduce experimental results. Pros and cons of the two approaches are discussed with respect to their reliability in fitting experimental force-displacement curves and crack patterns, as well as to the rather different computational effort required by the two strategies.