Unit Cell Analysis Research Articles

Elastic-plastic properties of metal matrix composites: Validation of mean-field approaches

Abstract Several micromechanical and numerical approaches to estimating the effective properties of heterogeneous media are analyzed. First, micromechanical mean-field estimates of elastic moduli for selected metal matrix composite systems are compared with the results of finite element calculations performed for two simplified unit cells: spherical and cylindrical. Advantages and deficiencies of such numerical verification of analytical homogenization schemes are indicated. Next, predictions of both approaches are compared with available experimental data for two composite systems for tension and compression tests in the elastic-plastic regime using tangent and secant linearization procedures. In the examined range of strain and ceramic volume content, both the Mori-Tanaka averaging scheme and the generalized self-consistent scheme lead to reliable predictions when combined with the tangent linearization, while the use of secant moduli results in a too stiff response. It is also found that the mean-field predictions for a small ceramic volume content are very close to the results obtained from the finite-element analysis of a spherical unit cell.

Macroscopic damage laws based on analysis of microscopic unit cells

AbstractThe paper deals with the influence of the stress state on the damage and fracture behavior of ductile metals. A continuum damage model is discussed taking into account the effect of the stress triaxiality and the Lode parameter on damage criteria and on evolution equations of damage strains. To get insight in the complex damage and failure mechanisms on the micro‐level and their effect on the macroscopic behavior of material samples series of three‐dimensional micro‐mechanical numerical analyses of void containing unit cells have been performed with special focus on shear dominated loading cases. In the present analysis periodic boundary conditions have been used to model the deformation behavior of the unit cells. Numerical results are compared with those based on symmetry boundary conditions. The results are also used to develop equations for stress‐state‐dependent damage criteria and to propose evolution equations of damage strains also depending on the current stress state. The effect of the respective boundary conditions on damage equations on the macro‐scale is critically discussed.

On the description of ductile fracture in metals by the strain localization theory

Numerical simulations based on the bifurcation and imperfection versions of the strain localization theory are used in this paper to predict the failure loci of metals and applied to an advanced high strength steel subjected to proportional loading paths. The results are evaluated against the 3D unit cell analyses of Dunand and Mohr (J Mech Phys Solids 66(1):133–153, 2014. doi: 10.1016/j.jmps.2014.01.008 ) available in the literature. The Gurson porous plasticity model (Gurson in J Eng Mater Technol 99(1):2–15, 1977. doi: 10.1115/1.344340 ) is used to induce strain softening and drive the localization process. The effects of the void growth, void nucleation and void softening in shear are investigated over a large range of stress triaxialities and Lode parameters. A correlation between the imperfection and bifurcation results is established.

A Bloch-based procedure for dispersion analysis of lattices with periodic time-varying properties

We present a procedure for the systematic estimation of the dispersion properties of linear discrete systems with periodic time-varying coefficients. The approach relies on the analysis of a single unit cell, making use of Bloch theorem along with the application of a harmonic balance methodology over an imposed solution ansatz. The solution of the resulting eigenvalue problem is followed by a procedure that selects the eigen-solutions corresponding to the ansatz, which is a plane wave defined by a frequency-wavenumber pair. Examples on spring-mass superlattices demonstrate the effectiveness of the method at predicting the dispersion behavior of linear elastic media. The matrix formulation of the problem suggests the broad applicability of the proposed technique. Furthermore, it is shown how dispersion can inform about the dynamic behavior of time-modulated finite lattices. The technique can be extended to multiple areas of physics, such as acoustic, elastic and electromagnetic systems, where periodic time-varying material properties may be used to obtain non-reciprocal wave propagation.

Unit cell simulations and porous plasticity modelling for strongly anisotropic FCC metals

Abstract The macroscopic behaviour of anisotropic porous solids made from an aggregate of spherical voids embedded in a plastically anisotropic matrix material is investigated by means of unit cell simulations. Plastic yielding of the polycrystalline matrix is governed by the anisotropic yield criterion Yld2004-18p. Generic texture components for face-centred cubic crystals resembling those that typically emerge during rolling and annealing processes are applied in the study. A numerical method for systematic prescription of external stress states is presented and employed in the unit cell calculations. To preclude shear effects in the unit cell model, the material symmetry axes are restricted to coincide with the principal stress directions. This excludes the possibility to properly study the ductile failure mechanism and the current work is thus mainly concerned with the void growth phase. Various stress states ranging from biaxial tension to highly constrained regions in the vicinity of crack tips are employed in the study. The numerical results demonstrate that the matrix anisotropy has a marked effect on the unit cell response, both in terms of void growth and stress-strain curves. Furthermore, the void shape evolves quite differently depending upon the direction of the major principal stress relative to the material axes. A heuristic extension of the Gurson model that incorporates matrix plastic anisotropy is presented and subsequently used to describe the constitutive behaviour of the porous ductile solid. Numerical data from the unit cell analyses are used as target curves in the calibration process of the porous plasticity model. A sequential least-square optimization procedure is invoked to minimize the overall discrepancy between the unit cell calculations and the homogenized response of the plastically anisotropic porous solid for all the imposed stress states. The anisotropic porous plasticity model demonstrates predictive capabilities for the range of stress states covered in this study.

A study of reactivity biases and their dependence on spatial discretization in depleted TRIGA fuel

The impact of spatial discretization on reactivity biases in TRIGA fuel models was analyzed. In particular, unit-cell analyses in 2-D and 3-D were performed using Serpent to understand how spatial discretization affects the accuracy with which the effects of material evolution and temperature feedback are resolved. For temperature-dependent cases, a simple, single-channel model was employed. Analysis of 2-D models showed that essentially no radial discretization resolution is needed to eliminate biases (i.e., to reduce biases to the level of stochastic uncertainties) due to material evolution but that more than eight, equal-area, radial regions are needed to resolve temperature-feedback effects. Analysis of 3-D models showed that at least seven, equal-volume, axial regions may be required to resolve material evolution with temperature feedback leading to insignificant additional bias. Because of memory constraints, a full-core model with radially- and axially-resolved fuel elements may be impractical, especially for production-level analyses. Consequently, an “effective Doppler temperature” was determined empirically as a function of the radially-averaged temperature and may be used for future, full-core analyses.

분산 CRLH 전송선로 4중 모드 공진기를 이용한 이중-광대역 대역통과 여파기 설계

본 논문은 CRLH (composite right/left handed) 전송선로 구조의 4중 모드 공진기를 이용하여 높은 대역간 격리도를 갖는 이중-광대역 대역통과 여파기를 설계하는 방법을 제안한다. CRLH 전송선로 구조를 해석하여 4중 모드 공진기의 s-패러미터와 공진 주파수를 유도하였다. 유도된 s-패러미터를 이용하여 새로운 구조의 이중-광대역 대역통과 여파기를 설계, 제작하였다. 측정 결과는 2.8-5.52 GHz, 9.68-12.26 GHz의 이중 통과대역에서 각각 1.08 dB, 2.01 dB의 삽입손실을 갖고, 6.34-8.42 GHz의 중간대역에서 38dB의 대역 격리도를 가져 설계 결과와 잘 일치하는 특성을 보여 주었다. This paper presents a dual-wideband bandpass filter (BPF) with high band-to-band isolation and skirt selectivity using distributed composite right/left-handed (CRLH) transmission line (TL) quad-mode resonators (QMRs). The results of the proposed distributed CRLH TL unit cell analysis are used to establish the scattering parameters and the resonance frequencies of the QMR constituting the dual-wideband BPF. A novel dual-wideband bandpass filter is designed and fabricated, using the derived scattering characteristics. The measured results show that the fabricated dual-wideband bandpass filter has an insertion loss of less than 1.08dB in the lower band, and of 2.01dB in the upper band, a bandwidth of 2.8-5.52GHz and 9.68-12.26GHz, and a band-to-band isolation of more than 38dB, from 6.34-8.42GHz.

A model for creep of porous crystals with cubic symmetry

A model for description of the creep response of porous cubic single crystal is presented. The plastic potential is obtained by specializing the orthotropic potential of Stewart and Cazacu (Int. J. Solids Struct., 48, 357, 2011) to cubic symmetry. The crystal matrix material response is characterized by power law creep. The predictions of this porous plastic constitutive relation are presented for various values of stress triaxiality (mean normal stress divided by Mises effective stress) and various values of the Lode parameter L (a measure of the influence of the third invariant of the stress deviator). A strong influence of crystal orientation on the evolution of the creep strain and the porosity is predicted. For loadings along the < 100 > directions of the cubic crystal, void growth is not influenced by the value of the Lode parameter. However, for loadings such that the maximum principal stress is aligned with the [110] direction there is a strong influence of the values of the Lode parameter and the fastest rate of void growth occurs for shear loadings (one of the principal values of the applied stress deviator is zero). For loadings such that the maximum applied stress is along the [111] crystal direction the fastest rate of void growth corresponds to L=-1, while the slowest rate corresponds to L=1. These predictions are compared with corresponding predictions of the three dimensional finite deformation unit cell analysis of Srivastava and Needleman (Mech. Mater., 90, 10, 2015). It is found that the phenomenological model predicts the same trends as the cell model calculations and, in some cases, gives good quantitative agreement.

Edge waves in plates with resonators: an elastic analogue of the quantum valley Hall effect

We investigate elastic periodic structures characterized by topologically nontrivial bandgaps supporting backscattering suppressed edge waves. These edge waves are topologically protected and are obtained by breaking inversion symmetry within the unit cell. Examples for discrete one and two-dimensional lattices elucidate the concept and illustrate parallels with the quantum valley Hall effect. The concept is implemented on an elastic plate featuring an array of resonators arranged according to a hexagonal topology. The resulting continuous structures have non-trivial bandgaps supporting edge waves at the interface between two media with different topological invariants. The topological properties of the considered configurations are predicted by unit cell and finite strip dispersion analyses. Numerical simulations demonstrate edge wave propagation for excitation at frequencies belonging to the bulk bandgaps. The considered plate configurations define a framework for the implementation of topological concepts on continuous elastic structures of potential engineering relevance.

An improved parametric analysis of CSRR loaded with microstrip line for ultra wide band frequency range

ABSTRACTIn this letter, an improved electrical parameter extraction scheme for the microstrip line loaded with complementary split ring resonator (CSSR) is proposed. A CSRR loaded with microstrip line behaves as an effective negative permittivity media. The equivalent circuit of CSRR contains four electrical quantities and it needs four different characteristic frequencies for their accurate solution which was neglected in previous studies. In this work, a four variable problem is formulated and solved which accurately gives all the electrical parameters of CSRR equivalent circuit for ultra wide frequency range. The proposed technique is used for a parametric analysis of a rectangular CSRR unit cell. © 2017 Wiley Periodicals, Inc. Microwave Opt Technol Lett 59:661–664, 2017

Buckling and free vibration of shear-flexible sandwich beams using a couple-stress-based finite element

In this work, we create a framework for linear buckling and free vibration analyses of sandwich beams using a microstructure-dependent Timoshenko beam model founded on the modified couple-stress theory. The stiffness parameters of a structural web-core sandwich panel are determined by unit cell analysis. An extension to homogeneous cores is also carried out. By employing the exact general solution to the governing equations of the beam, an accurate approximate finite element stiffness matrix is formulated. Furthermore, the static shape functions are used to derive consistent linear geometric stiffness and mass matrices. A convergence study shows that the approximate finite element has good accuracy although the hyperbolic terms of the exact general solution have been expanded into only relatively low-order polynomial series. Results from examples show that the microstructure-dependent beam can predict critical buckling loads and natural frequencies with very good accuracy when compared to more sophisticated finite element models. Unlike the classical Timoshenko beam model, the microstructure-dependent model yields accurate results also when the sandwich assembly is transversely flexible and the bending stiffness of the faces non-negligible.

熱力学的考察に基づく多孔質材料の異方性弾塑性構成式

Considering the effect of microscopic voids to macroscopic mechanical behavior is important because the fracture of ductile materials occurs through the initiation, the growth and the coalescence of voids. Anisotropy and softening are affected by the voids arrangement, and the growth bahavior of voids depends on not only stress and strain but their histories. The elasto-plastic constitutive equation which can represent anisotropy and compressivility of porous materials at once is presented in order to analyse deformation and damage evolution of the materials including voids to predict these phenomena. Damage tensor is employed to express decrease of effective cross-section area subjected to stress and its evolution equation is also derived to predict three dimensional growth of void. Thermomechanical consideration based on Clausius-Duhem's inequality is used to derive the model. Functional forms of dissipation inequality and dissipation potential are derived so that the anisotropic mechanical responce caused by the arrangement of voids is incorporated. The adequacy of presented model is evaluated through finite element analysis of unit cell assuming grid alignment of spherical voids. Initial and subsequent yield behavior are analysed with triaxial stress conditions. The presented model shows good agreement with finite element model for initial yield condition and relation between stress, strain and damage tensor.

A new method for the evaluation of the effective properties of composites containing unidirectional periodic nanofibers

We propose an innovative numerical scheme (based on complex variable techniques) for the calculation of the effective properties of a composite containing unidirectional periodic fibers in which we additionally incorporate the separate contribution of the ‘interface effect’ between the fibers and the surrounding material. The incorporation of interface effect into the model of deformation allows our model to accommodate the general class of nanocomposite materials, a fast growing area of research and our primary focus in this paper. The composite is loaded by a constant normal strain along the direction parallel to the fibers and by a uniform remote loading in the plane perpendicular to the fibers. Our method is based on the analysis of a representative unit cell with periodic boundary conditions imposed on its edge. Several examples are presented to study the influence of the interface and the volume fraction of the fibers on the effective properties of the composite and the interfacial stress field. We show that when the volume fraction falls below roughly 9%, the interfacial stress distribution recovers effectively to that corresponding to a single fiber with the same interface parameters embedded within an infinite matrix. We find also that if the shear modulus of the fibers exceeds approximately two and a half times that of the matrix, the interface effect is negligible in the determination of the effective properties of the corresponding nanocomposites. Finally, we show that the use of traditional effective medium theories may induce significant errors in the determination of transverse effective properties (in the plane perpendicular to the fibers) of the composite, in particular when the fibers are significantly softer than the matrix.

Design and validation of metamaterials for multiple structural stop bands in waveguides

Environmental and economic requirements lead to a constantly increasing application of lightweight designs. Fulfilling the conflicting goals of favorable noise and vibrational behavior and lightweight design is a challenging tasks, requiring novel low mass and compact volume solutions. Vibro-acoustic metamaterials come to the fore as possible lightweight solutions with superior noise and vibration insulation properties in targeted and tunable frequency ranges, referred to as stop bands. They typically consist of (often periodic) assemblies of unit cells of non-homogeneous material composition and/or topology, adding local resonant behavior to the host structure. Metamaterials targeting different stop bands can be combined to suppress vibrations over a larger frequency zone. This paper investigates various metamaterial layouts for reducing vibrations along a known transmission path, being a duct with a square section. The efficiency of the considered metamaterial designs for vibration attenuation are evaluated and compared both numerically and experimentally. Due to the typical periodic nature of this class of materials, the numerical prediction of the stop band behavior is modeled using unit cell models for the duct; both a one dimensional (1D) waveguide as well as a two dimensional (2D) infinite periodic structure. The efficiency and accuracy of both approaches are evaluated and compared to measurements. It is demonstrated that different resonant structures can be combined to merge stop bands, either using a mixed, checkered pattern, or a sequential pattern. Depending on the transmission path and the targeted frequency zone, either approach or a combination of both can be used. Furthermore it is shown that the 2D unit cell analyses gives an accurate prediction of the stop band locations.

Using unit cell simulations to investigate fracture due to compression–tension loading

Experiments regarding impact against X65 steel pipes show that fracture typically arises in areas subjected to large compressive strains before tension. Fracture surfaces from these areas are brittle in character despite the material exhibiting ductile behaviour elsewhere. Smooth and notched tensile material tests always produced ductile fracture through nucleation, growth and coalescence of voids. The ductile-to-brittle transition seen in the component tests was however recreated in notched axisymmetric material tests, where the specimens were compressed to various levels of plastic strain before being stretched to failure. Increasing compression before tension showed a decrease in strain to fracture as hypothesised, and an increase in the cleavage surface fraction. In an attempt to gain a better understanding of this behaviour, unit cell simulations subjected to tension only and compression–tension loading were carried out. As well as exploring different chosen stress triaxialities, global analyses of the material tests were used to provide an average stress triaxiality for the axisymmetric unit cell simulations. These global simulations were able to represent the material tests with good accuracy. In tension tests where the stress triaxiality was fairly constant (notched tests), the unit cell analyses were able to predict a strain to coalescence within reasonable margin compared with the experimental values. Unit cell simulations including the compressive phase show that the higher the magnitude of the stress triaxiality is during compression, the higher the local stress in the cell, which in turn may trigger cleavage fracture.

Probabilistic analyses of soil consolidation by prefabricated vertical drains for single‐drain and multi‐drain systems

SummaryNatural soils are one of the most inherently variables in the ground. Although the significance of inherent soil variability in relation to reliable predictions of consolidation rates of soil deposits has long been realized, there have been few studies that addressed the issue of soil variability for the problem of ground improvement by prefabricated vertical drains. Despite showing valuable insights into the impact of soil spatial variability on soil consolidation by prefabricated vertical drains, available stochastic works on this subject are based on a single‐drain (or unit cell) analyses. However, how the idealized unit cell solution can be a supplement to the complex multi‐drain systems for spatially variable soils has never been addressed in the literature. In this study, a rigorous stochastic finite elements modeling approach that allows the true nature of soil spatial variability to be considered in a reliable and quantifiable manner, both for the single‐drain and multi‐drain systems, is presented. The feasibility of performing an analysis based on the unit cell concept as compared with the multi‐drain analysis is assessed in a probabilistic context. It is shown that with proper input statistics representative of a particular domain of interest, both the single‐drain and multi‐drain analyses yield almost identical results. Copyright © 2016 John Wiley &amp; Sons, Ltd.

Closed-form effective elastic constants of frame-like periodic cellular solids by a symbolic object-oriented finite element program

In this study, the effective elastic constants of several 2D and 3D frame-like periodic cellular solids with different unit-cell topologies are analytically derived using the homogenization method based on equivalent strain energy. The analytical expressions of strain energy of a unit cell under different strain modes are determined using a generic symbolic object-oriented finite element (FE) program written in MATLAB. The obtained analytical expressions of the strain energy are then used to symbolically compute the effective elastic constants that include Young’s moduli, Poisson’s ratios, and shear moduli. The obtained analytical effective elastic constants are numerically verified using results from an ordinary numerical FE program. The obtained closed-form effective elastic constants are also compared with some existing solutions from the literature. This study demonstrates that symbolic computation platforms can be properly used to provide efficient methodologies for finding useful analytical solutions of mechanical problems. Without the symbolic object-oriented FE program in this study, elaborate and tedious analytical analysis has to be manually performed for each different unit cell. The symbolic object-oriented FE program provides analytical analysis of unit cells that is accurate and fast. The object-oriented programming technique allows the symbolic FE program in this study to be efficiently implemented.

Stone column effectiveness in soils with creep: a numerical study

ABSTRACTWhile it is well established that vibro stone columns reduce primary settlement and improve bearing capacity, their effect on creep compression has largely been overlooked to date. However, with increasing pressure to develop marginal sites underlain by soft organic soils, the effect of ground treatment on creep is an important emerging issue in geotechnical engineering. In this paper, a series of axisymmetric unit cell analyses have been carried out using the PLAXIS 2D finite element program in conjunction with the Soft Soil Creep (SSC) model. Examination of the evolution of settlement improvement factor with time has indicated that the presence of creep leads to a lower ‘total’ improvement factor than would be obtained for primary consolidation settlement alone. Separate ‘primary’ and ‘creep’ improvement factors have also been derived; the latter are much lower than the former, but are nevertheless greater than unity. Creep results in a stress transfer process; as the soil creeps, vertical stress is transferred from the soil to the stone column. The additional load carried by the column induces additional yielding and shear-plane formation in closely-spaced columns. The additional increment of stress transferred to the already yielded column reduces its efficacy.

Single-Feed Quad-Beam Transmitarray Antenna Design

We present a design methodology for single-feed multibeam transmitarray antennas through case studies of quad-beam designs. Different far-field pattern masks and fitness functions are studied for multibeam designs, and the particle swarm optimization (PSO) technique is implemented for aperture phase synthesis. A quad-layer configuration of double square loops is used for the transmitarray elements, and a quad-beam transmitarray prototype is fabricated and tested. The effects of various approximations in unit-cell analysis are also investigated in detail. The Ku-band prototype generates four symmetric beams with 50° elevation separation between the beams and gains around 23 dB.

Studying the nonlinear performance of an amplifying reflectarray antenna

In this paper, significance of studying the nonlinear performance of amplifying reflectarray antenna is clarified by showing nonlinear behavior of the unit cell. Furthermore, the procedure for nonlinear analysis of amplifying reflectarray antenna is described and nonlinear analysis is verified by ADS software simulation. Moreover, a typical amplifying reflectarray antenna is designed considering the nonlinear behavior. Nonlinear analysis of the active aperture-coupled patch unit cell is performed using harmonic balance method considering nonlinear model of the amplifier. Then, the effect of nonlinear element in radiation pattern of the antenna is studied. Passive unit cell is simulated using full-wave analysis method considering infinite array approach, and scattering parameters of the unit cell is obtained. Next, these scattering parameters are transformed to admittance parameters and used in obtaining nonlinear response and harmonic distortion of the unit cell containing amplifier. Obtained results show up to 3.4 dB error in predicting pattern of the antenna with linear modeling of the active element.