Effective Elastic Material Properties Research Articles

Micromechanical analysis for effective elastic moduli and thermal expansion coefficient of composite materials containing ellipsoidal fillers oriented randomly

In this study, we examine to derive the solutions of effective elastic moduli and thermal expansion coefficient for composite materials containing ellipsoidal fillers oriented randomly in the material using homogenization theories, which are the self-consistent method and the Mori–Tanaka method. This analysis is carried out by micromechanics combining Eshelby’s equivalent inclusion method for each theory. The solutions for effective elastic moduli and thermal expansion coefficient obtained on each theory are expressed by common coefficients composed of both the physical properties of the constituents of the composite material and geometrical factors depending upon the shape of the fillers. Moreover, these solutions enable us to calculate effective elastic moduli and thermal expansion coefficient for composite materials that contain randomly oriented fillers of various shapes and physical properties. By taking the limit of eliminating the existence of the matrix for these solutions, we can derive effective physical properties of polycrystalline materials. Using the obtained solutions, we investigate the effects of the shape of the fillers on the effective elastic moduli and thermal expansion coefficient. As a result, we confirm that these effective properties fall within the lower and upper bounds, and find that a characteristic result appears when the shape of the fillers is flake or oblate. Through comparisons between the analytical and experimental results, we confirm the practical usability of the solutions obtained in this analysis. Furthermore, we determine originally the shape factor for the filler and can show that this factor has the potential to provide guidelines for the optimal design of filler shape to improve the effective elastic properties of materials.

Accelerated multiscale mechanics modeling in a deep learning framework

Microstructural heterogeneity affects the macro-scale behavior of materials. Conversely, load distribution at the macro-scale changes the microstructural response. These up-scaling and down-scaling relations are often modeled using multiscale finite element (FE) approaches such as FE-squared (FE2). However, FE2 requires numerous calculations at the micro-scale, which often renders this approach intractable. This paper reports an enormously faster machine learning (ML) based approach for multiscale mechanics modeling. The proposed ML-driven multiscale analysis approach uses an ML-model that predicts the local stress tensor fields in a linear elastic fiber-reinforced composite microstructure. This ML-model, specifically a U-Net deep convolutional neural network (CNN), is trained separately to perform the mapping between the spatial arrangement of fibers and the corresponding 2D stress tensor fields. This ML-model provides effective elastic material properties for up-scaling and local stress tensor fields for subsequent down-scaling in a multiscale analysis framework. Several numerical examples demonstrate a substantial reduction in computational cost using the proposed ML-driven approach when compared with the traditional multiscale modeling approaches such as full-scale FE analysis, and homogenization based FE2 analysis. This approach has tremendous potential in efficient multiscale analysis of complex heterogeneous materials, with applications in uncertainty quantification, design, and optimization.

An Analysis of Numerical Homogenisation Methods Applied on Corrugated Paperboard

Corrugated paperboard is a sandwich structure composed of wavy paper (fluting) bonded between two flat paper sheets (liners). The analysis of an entire package using three-dimensional numerical finite element models is computationally expensive due to the waved geometry of the board that requires the use of a relatively large number of elements in a simulation. Because of this, homogenisation approaches are used to evaluate equivalent homogenous models with similar material properties. These techniques have been successfully implemented by various researchers to evaluate the strength of corrugated paperboard. However, studies analysing the various homogenisation techniques and their ranges of applicability are limited. This study analyses the application of three homogenisation techniques: classical laminate plate theory, first-order shear deformation theory and deformation energy equivalence method in the evaluation of effective elastic material properties. In addition, inverse analysis has been applied to determine the effective properties of the board. Finite element models have been used to evaluate the accuracy of the three homogenisation techniques in comparison to the inverse method in modelling four-point bending tests and the results reported.

Application of statistical functions to the numerical modelling of ceramic foam: From characterisation of CT-data via generation of the virtual microstructure to estimation of effective elastic properties

This paper illustrates a numerical methodology to quantify and recreate the microstructure of a ceramic foam using statistical functions and physical descriptors (together referred to as microstructure descriptors). The microstructure descriptors were employed to characterize a real foam material obtained from X-ray tomography (CT) scans. The statistical functions were further used to determine an appropriate size of a statistical volume element (SVE) within the foam sample to be further used in numerical determination of effective elastic properties. A ranking method was also developed to reduce the number of realizations of SVEs used in this calculation. Further, a novel reconstruction procedure was developed to synthesize artificial microstructure of the foam material that had the same microstructure descriptors as the real foam sample. Effective elastic properties of these artificial microstructures were determined as well. Comparison of these properties with experimental results showed that the statistical functions can be used to minimize the computational effort required for determining effective elastic properties of real microstructures. Further, the reconstruction methodology generates microstructure that matches the real microstructure not only in terms of its microstructural descriptors but also in terms of the effective elastic material properties.

A low-parametric rhombic microstructure family for irregular lattices

New fabrication technologies have significantly decreased the cost of fabrication of shapes with highly complex geometric structure. One important application of complex fine-scale geometric structures is to create variable effective elastic material properties in shapes manufactured from a single material. Modification of material properties has a variety of uses, from aerospace applications to soft robotics and prosthetic devices. Due to its scalability and effectiveness, an increasingly common approach to creating spatially varying materials is to partition a shape into cells and use a parametric family of small-scale geometric structures with known effective properties to fill the cells. We propose a new approach to solving this problem for extruded, planar microstructures. Differently from existing methods for two-scale optimization based on regular grids with square periodic cells, which cannot conform to an arbitrary boundary, we introduce cell decompositions consisting of (nearly) rhombic cells. These meshes have far greater flexibility than those with square cells in terms of approximating arbitrary shapes, and, at the same time, have a number of properties simplifying small-scale structure construction. Our main contributions include a new family of 2D cell geometry structures, explicitly parameterized by their effective Young's moduli E , Poisson's ratios v , and rhombic angle α with the geometry parameters expressed directly as smooth spline functions of E, v , and α. This family leads to smooth transitions between the tiles and can handle a broad range of rhombic cell shapes. We introduce a complete material design pipeline based on this microstructure family, composed of an algorithm to generate rhombic tessellation from quadrilateral meshes and an algorithm to synthesize the microstructure geometry. We fabricated a number of models and experimentally demonstrated how our method, in combination with material optimization, can be used to achieve the desired deformation behavior.

Nonlinear material and geometric analysis of thick functionally graded plates with nonlinear strain hardening using nonlinear finite element method

The present paper deals with three dimensional nonlinear geometrically and elastoplastic analyses of thick functionally graded plates with nonlinear strain hardening. The boundary conditions are assumed to be general. The plate consists of ceramic (SiC) and metal (Al) phases varying smoothly through the thickness. The effective elastic material properties are obtained by Mori-Tanaka scheme where the plastic behavior of the FG plate is described by the von-Mises yield criterion, nonlinear isotropic strain hardening and the Prandtl-Reuss constitutive equations in combination with the TTO model. The governing equations are obtained from the incremental theory of plasticity in conjunction with the Green-Lagrange nonlinear kinematics. In order to investigate the nonlinear elastoplastic response of the FGM, the nonlinear graded finite element method is developed from three-dimensional continuum concepts that admit arbitrarily large displacements and rotations. This formulation has been implemented and the effect of material gradient index, boundary conditions and thickness-to-length ratio on the nonlinear elastoplastic analysis of the FG plate as well as the development of plastic zone are presented and discussed.

Is pallial mucus involved in oyster defense against the parasite Bonamia ostreae?

The present paper deals with three dimensional nonlinear geometrically and elastoplastic analyses of thick functionally graded plates with nonlinear strain hardening. The boundary conditions are assumed to be general. The plate consists of ceramic (SiC) and metal (Al) phases varying smoothly through the thickness. The effective elastic material properties are obtained by Mori-Tanaka scheme where the plastic behavior of the FG plate is described by the von-Mises yield criterion, nonlinear isotropic strain hardening and the Prandtl-Reuss constitutive equations in combination with the TTO model. The governing equations are obtained from the incremental theory of plasticity in conjunction with the Green-Lagrange nonlinear kinematics. In order to investigate the nonlinear elastoplastic response of the FGM, the nonlinear graded finite element method is developed from three-dimensional continuum concepts that admit arbitrarily large displacements and rotations. This formulation has been implemented and the effect of material gradient index, boundary conditions and thickness-to-length ratio on the nonlinear elastoplastic analysis of the FG plate as well as the development of plastic zone are presented and discussed.

On micromechanical modeling of orthotropic solids with parallel cracks

A detailed comparison of the accuracy of several popular micromechanical schemes utilized for prediction of the effective elastic properties of materials with parallel cracks is presented. In particular, the non-interaction, Mori–Tanaka, differential and self-consistent schemes are compared against the direct finite element simulations. The latter are performed on the periodic representative volume elements containing 30 strongly oblate spheroids representing the penny-shaped cracks. This work extends the existent results to a more general class of matrix materials – orthotropic materials, which requires the ability to calculate the Eshelby tensor for an ellipsoid in non-isotropic matrix. In addition to the implementation of the integration procedure used for the Eshelby tensor calculation, this work also presents a variation of the Random Sequential Adsorption algorithm modified for periodic structures.Analysis of the results indicates that in the case of parallel nearly flat cracks (strongly oblate spheroids) the overall out-of-plane moduli are best predicted by the differential scheme. On the other hand, Mori–Tanaka scheme should be used for estimation of the in-plane moduli. It also appears that as the cracks are inflated from strongly oblate spheroids to slightly deformed spheres, the best choice of the micromechanical scheme for the out-of-plane properties gradually shifts towards Mori–Tanaka.

Dynamic mechanical analysis of pure and fiber‐reinforced thermoset‐ and thermoplastic‐based polymers and free volume‐based viscoelastic modeling

Fiber‐reinforced polymer composites have gained significant importance in engineering applications and are widely used as structural components. The optimal choice of combinations of the type of polymer matrix and the fiber content depend on the required applications. In view of this, polypropylene (PP), a thermoplastic, reinforced with 30 wt.% of short glass fibers (PP‐GF30) is considered for this study. Additionally, a rather new composite system based on unsaturated polyester‐polyurethane hybrid resin (UPPH) is introduced. This thermoset‐based material is reinforced with 41 wt.% of long discontinuous glass fibers (UPPH‐GF41). Dynamic mechanical analysis (DMA) is performed on the pure polymer and the composite samples to experimentally characterize the temperature‐ and frequency‐dependent material behavior. It is observed that the stiffness of the materials is highly temperature‐dependent. PP exhibits a nonlinear viscoelastic behavior in the temperature range of possible applications whereas the UPPH material shows a less pronounced behavior. The resulting experimental data not only give general information on the material behavior of the composite subjected to a temperature and frequency load but also provide input as well as validation data for the developed material modeling methods. In an industrial environment, homogenized elastic material properties are desirable for analysis of the structural components at room temperature conditions. A mean field homogenization method is introduced to estimate the effective elastic material properties on a macroscopic scale. This method is formulated explicitly in terms of orientation tensors of second‐ and fourth‐order, describing quantitatively the microstructure of the corresponding composite. Additionally, the thermoviscoelastic material behavior for the considered temperature range is modeled by the free volume concept. In this context, the material parameters for the pure PP and UPPH material are identified. The calculated simulation results are compared with the experimental data entailing good agreements.

Mean and full field homogenization of artificial long fiber reinforced thermoset polymers

AbstractBased on two artificial microstructures representing a long fiber reinforced thermoset material, the effective linear elastic material properties are calculated by both a mean and a full field homogenization method. Concerning the mean field method, the effective elastic material properties are approximated using the homogenization scheme by Mori and Tanaka, formulated explicitly in terms of orientation averages. This allows to use orienation tensors of 2nd and 4th order describing the orientation information on the micro level. The full field method is based on the fast Fourier transformation (FFT), for which the effective material properties are determined by volume averaging. The comparison between both methods show good agreements, the deviations are in the range between 2% and 12%. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Finite element modeling of effective properties of elastic materials with random nanosized porosity

Finite element modeling of effective properties of elastic materials with random nanosized porosity

Mean field homogenization and experimental investigation of short and long fiber reinforced polymers

AbstractThe effective elastic material properties of short fiber reinforced polypropylen are determined by means of the self‐consistent (SC) method and the interaction direct derivative (IDD) method. In order to account for thermoelastic effective material properties, a Hashin‐Shtrikman (HS) based two‐step homogenization method with variable reference stiffness is used. The influence of the reference stiffness, dependent on a scalar parameter is investigated. Information on the microstructure are derived by computed tomography scans (µCT) and considered within the homogenization schemes. Thermomechanical properties of a long fiber reinforced polymer (LFRP) and a short fiber reinforced polymer (SFRP) are obtained by means of dynamic mechanical analysis (DMA). Simulation results for SFRP are compared to experimental results. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiscale modeling of functionalized interface effects on the effective elastic material properties of CNT–polyethylene nanocomposites

The effects of functionalization of the interface between carbon nanotubes (CNTs) and the polymer matrix on the macroscale elastic mechanical material properties have been investigated using a multiscale model. Molecular dynamic (MD) simulations are used to investigate the influence of functionalization on the nanoscale load transfer ability of the interface between the CNT and the polymer matrix material in CNT–polyethylene (PE) nanocomposites. Graphene–polymer interface models with variable numbers of functional groups grafted between the graphene sheet and the polymer chains are adopted to represent the interface between the CNTs and the polymer matrix. Normal mode and sliding mode separations are performed using MD simulations in order to characterize the load transfer of the functionalized graphene–polymer interface in terms of force-separation responses. The influence of the functionalization density on the load transfer at the interface is studied parametrically. Two force field potentials, the consistent valence force field (CVFF) and adaptive intermolecular reactive empirical bond order (AIREBO) potential, are used to simulate the interactions between the atoms in the polymer matrix and the functional groups. An energy based bond breaking criteria is introduced into the CVFF potential such that both potentials allow for bond breaking. Comparison of the two potentials is conducted by comparing the MD simulation results where it is found that two potentials give different force-separation responses, but that in both cases, there is significant increase in load transfer capability and toughness of the interface with increasing degree of functionalization of the interface. With cohesive zone laws derived from MD results, a finite element implementation of the cohesive zone models is used to calculate the effective elastic mechanical properties of the CNT–PE nanocomposites in order to study the influence of functionalization at the nanoscale on their macroscale mechanical performance.

Multiscale modeling of the effects of nanoscale load transfer on the effective elastic properties of unfunctionalized carbon nanotube–polyethylene nanocomposites

A multiscale model is proposed to study the macroscale bulk elastic material properties under the influence of interfacial load transfer at the nanoscale in carbon nanotube–polyethylene (CNT–PE) nanocomposites. Molecular dynamic (MD) simulations are performed to characterize the nanoscale load transfer through the identification of representative nanoscale interface elements which are studied parametrically in terms of the length of the polymer chains, the number of the polymer chains and the ‘grip’ position. Once appropriate scales of these parameters are deemed to yield sufficiently converged results, the representative interface elements are subjected to normal and sliding mode simulations in order to obtain the force–separation responses at 100 and 300 K for unfunctionalized CNT–PE interfaces. Cohesive zone traction–displacement laws are developed based on the force–separation responses obtained from the MD simulations and are used in continuum level models to determine the influence of the interface on the effective elastic material properties of the nanocomposites using analytic and computational micromechanics approaches. It is found that the inclusion of the nanoscale interface in place of the perfectly bonded interface results in effective elastic properties which are dependent on the applied strain and temperature in accordance with the interface sensitivity to those effects, and which are significantly diminished from those obtained under the perfect interface assumption.

Modeling the effective elastic properties of materials with parallel pore channels with Y-shaped cross-sections

The paper determines the effective elastic properties of a model material containing pore channels with cross-sections as Y-shaped cracks centered at the nodes of a regular hexagonal network and having arms aligned with the links of the network. A unit cell of relative size $ 1 \times \sqrt {3} $ consisting of two halves of adjacent cracks is considered. Effective Young's modulus and Poisson's ratio of the unit cell are determined for all combinations of relative arm lengths in 0.05 increments. The limits of variation in the effective elastic properties depending on the relative total projection of cracks onto the transverse deformation direction are established.

Modeling the effective elastic properties of materials pressed from a unidirectional hexagonal fiber strand

The paper studies the interrelation between the effective elastic properties and the size of contact areas in the unit cell in modeling a unidirectional hexagonal fiber strand under isostatic and uniaxial pressing in a plastic flow. For a range of relative densities (0.907–1), it is shown that effective Young’s modulus and Poisson’s ratio correlate well with the integral projection of the contact areas relative to the corresponding cell size. For pore channels with cross-sectional shapes close to a three-beam hypocycloid with unequal beams, the elastic properties in the plane perpendicular to the fiber axis are anisotropic because the cross-sectional projections of the pore channel have different sizes.

Simulation and Modeling the Mechanical Behavior of Textile Reinforced Composites by Combining the Binary Model and X‐FEM

AbstractThis paper presents a new efficient modeling strategy based on the combination of Binary Model and X–FEM. It is applied to represent the inner architecture of textile reinforced composites where the fabric is characterized by a complex geometry. Homogenization methods are used to compute the effective elastic material properties. Thereby, the discrete formulation of periodic boundary conditions is adapted. Finally, the results in terms of effective material properties reveal a good agreement with parameters obtained by experimental tests. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Earthquake cycle, fault zones, and seismicity patterns in a rheologically layered lithosphere

We study the coupled evolution of earthquakes and faults in a model consisting of a seismogenic upper crust governed by damage rheology over a viscoelastic substrate. The damage rheology has two types of functional coefficients: (1) a “generalized internal friction” separating states associated with material degradation and healing and (2) damage rate coefficients for positive (degradation) and negative (healing) changes. The evolving damage modifies the effective elastic properties of material in the upper crust as a function of the ongoing deformation. This simulates the creation and healing of fault systems in the upper seismogenic zone. In addition to the vertically averaged thin sheet approximation we introduce a Green function for three‐dimensional elastic half‐space for the instantaneous component of deformation. The formulation accounts in an internally consistent manner for evolving deformation fields, evolving fault structures, aseismic energy release, and spatiotemporal seismicity patterns. These developments allow us to simulate long histories of crustal deformation and to study the simultaneous evolution of regional earthquakes and faults for various model realizations. To focus on basic features of a large strike‐slip fault system, we first consider a simplified geometry of the seismogenic crust by prescribing initial conditions consisting of a narrow damage zone in an otherwise damage‐free plate. For this configuration, the model generates an earthquake cycle with distinct interseismic, preseismic, coseismic, and postseismic periods. Model evolution during each period is controlled by a subset of physical properties, which may be constrained by geophysical, geodetic, rock mechanics, and seismological data. In the more generic case with a random initial damage distribution, the model generates large crustal faults and subsidiary branches with complex geometries. The simulated statistics depend on the space‐time window of the observational domain. The results indicate that long healing timescale, τh, describing systems with relatively long memory, leads to the development of geometrically regular fault systems and the characteristic frequency‐size earthquake distribution. Conversely, short τh (relatively short memory) leads to the development of a network of disordered fault systems and the Gutenberg‐Richter earthquake statistics. For intermediate values of τh the results exhibit alternating overall switching of response from periods of intense seismic activity and the characteristic earthquake distribution to periods of low seismic activity and Gutenberg‐Richter statistics.

A relation between multiple-scattering theory and micromechanical models of effective thermoelastic properties

A relation between multiple-scattering theory and micromechanical models of effective elastic material properties of heterogeneous materials has been established within a unified theoretical framework, and exemplified on three important approximations, the average t-matrix (or Mori-Tanaka), symmetric self-consistency (coherent potential), and asymmetric self-consistency approximations.

Variations of the effective elastic material properties of a brittle solid with distributed void propagation

The initial fracture (damage) process in brittle solids subject to uniform uniaxial or biaxial compression may be simulated by a system of non-self-similarly propagating internal (three-dimensional) 3D ellipsoidal voids. The variations of effective elastic material properties with propagating ellipsoidal voids may thus be estimated. Although both assumptions and approximations are introduced, the estimated effective elastic material properties indicate the general trend of the overall non-linear anisotropic response of a cracking solid. This proposal seems to provide a better understanding of the non-local damage process of a brittle solid fracturing in compression.