Finite Element Unit Cell Model Research Articles

Simulation of the strain rate sensitive flow behavior of SiC-particulate reinforced aluminum metal matrix composites

Strain rate dependent compression mechanical behavior of an SiC-particulate reinforced Al (2024-O) metal matrix composite (MMC) with different particle volume fractions was numerically investigated at various strain rates. Calculations were performed using axisymmetric finite element unit cell model, in which an elastic SiC particle was embedded inside a strain rate sensitive viscoplastic Al matrix. Stress–strain curves of Al matrix material were derived from Split Hopkinson Pressure Bar experiments at various strain rates and used as inputs in the FEM model. Numerically computed stress–strain curves and strain rate sensitivity were compared with those of experiments for a 15% SiC-particulate reinforced MMC. Computed strain rate sensitivity of the MMC was found to be higher than that of the matrix alloy and increased with increasing strain contrary to the strain independent matrix strain rate sensitivity. The strain rate sensitivity of the MMC was also found to increase with increasing particle volume fraction at the same particle size. Finally, several possible reasons including assumptions used in the model, adiabatic heating, microstructural variations between the composite matrix and matrix alloy, particle shape and distribution and damage accumulation for the small discrepancy found between computed and experimental stress–strain curves and strain rate sensitivity of the composite were discussed.

Micromechanical modelling of cortical bone

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.

Analytical and Computational Models for Investigations of Local Buckling in Honeycomb Sandwiches

This work highlights and solves problems with the prediction of the compressive strength, limited by local instabilities, of sandwich material compounds based on honeycomb cores and very thin facesheets. Analytical methods in conjunction with periodic finite element unit cell models are utilized for this task. The finite element models are found to be well suited for all kinds of buckling predictions. Different uni- and bi-axial loadings are considered as well as influences of core height, core material, core geometry, and facesheet thickness are investigated. Finally, a new analytical approach is introduced for the treatment of the rather unexpected core cell wall buckling under in-plane compression of the sandwich, which predicts the critical load very accurately.

Elastic constants estimation of stitched NCF CFRP laminates based on a finite element unit-cell model

Usually polymer matrix laminates are made of individual plies with unidirectional fiber reinforcement laid upon each other with varying fiber orientations with respect to the thickness coordinate. Due to missing fiber reinforcement in the thickness direction, this results in poor through-thickness (out-of-plane) properties of composite laminates. One method to enhance the out-of-plane properties is the stitching of dry semi-finished fiber products in the thickness direction prior to the infusion of the resin. On the other hand, the insertion of stitching yarns in the thickness direction may affect the mechanical properties in the laminate plane such as stiffness and strength. To quantify the effect of structural through-thickness reinforcement in non-crimp fabric carbon fiber/epoxy laminates under uniaxial tension and compression loading an experimental study was carried out which included a systematic variation of the yarn diameter, the stitching pattern and direction as well as the load direction. The results showed a significant effect of the stitching on the in-plane Young’s modulus which was attributed to local displacements of the in-plane fibers and changes in the fiber volume fraction. A finite element based unit-cell model was developed to estimate the elastic constants of structurally stitched non-crimp fabric laminates taking into consideration the yarn diameter, the stitching pattern and direction as well as the load direction. Depending on these parameters, local changes of the fiber volume fraction as well as regions with undisturbed and disturbed fiber orientations within the laminate plies are taken into account.

The influence of particle distribution and volume fraction on the post-necking behaviour of aluminium alloys

Experimental observations show that the continuous cast (CC) Al–Mg alloy with 0.21 wt% Fe has a considerably lower fracture strain than that of a CC alloy with 0.08 wt% Fe and a direct chill cast (DC) alloy with 0.21% Fe. However, the forming limit strains are almost the same for the three alloys. The similarities and differences are thought to relate with the distribution of hard second-phase particles. In the present paper, the influence of particle distribution characteristics (random vs. stringer pattern) and volume fraction (1.2% vs. 3.5%) is examined in detail with the use of a simple plane stress finite element unit cell model. The model permits extension of the composite stress–strain curve beyond the diffuse necking condition so that the nominally uniform response is followed directly by localized necking. The model results agree very well with tensile tests with respect to the differing behaviour of the three materials in the deformation range between the onset of diffuse necking and that for localized necking.

Prediction of microelectronic substrate warpage using homogenized finite element models

The focus of this study was to numerically predict effective thermo-mechanical properties and substrate warpage of high-density microelectronic substrates used in organic CPU packages. Microelectronic substrates are typically composed of several polymer, fiber-weave, and copper layers and are filled with a variety of complex features, such as electric traces, plated-through-holes, micro-vias, and adhesion holes. When subjected to temperature changes, these substrates may warp, driven by the mismatch in coefficients of thermal expansion (CTE) of the constituent materials. This study focused on predicting substrate warpage in an isothermal condition. The numerical approach consisted of three major tasks: estimating homogenized (effective) thermo-mechanical properties of the features; calculating effective properties of discretized layers using the effective properties of the features; and assembling the layers to create two-dimensional (2D) finite element (FE) plate models and to calculate warpage of the substrates. The effective properties of the features were extracted from three-dimensional (3D) unit cell FE models, and closed-form approximate expressions were developed using the numerical results, curve fitting, and some simple bounds. The numerical approach was applied to predict warpage of production substrates, analyzed, and validated against experimentally measured stiffness and CTEs. In this paper, the homogenization approach, numerical predictions, and experimental validation are discussed.

Prediction for debonding damage process and effective elastic properties of glass-bead-filled modified polyphenylene oxide

A three-dimensional three-phase finite element unit cell model has been applied to predict the debonding damage process of particulate polymer composites (PPC) during tensile deformation. The model consists of a particle, an interface and a polymer matrix. The particle-matrix debonding process has been simulated by using a particle-matrix debonding criterion and a vanishing finite element technique. For verification of the predicted results, the results obtained in the in situ SEM experiments of our previous study on glass beads reinforced modified polyphenylene oxide (GB/PPO) were used The predicted results are in good agreement with the experimental micro-scale tensile debonding-damage process. Anisotropic damage in materials like PPC is very difficult to measure by conventional experimental methods. In order to provide anisotropic damage information of the damaged composites, this was amongst the first time to predict the effective elastic properties of PPC in different principal directions by performing a virtual load–unload test on the partially debonded cell model. Some predicted results would be compared with those of experimental load–unload test.

Thickness effects on the compressive stiffness and strength of stitched composite laminates

A finite element unit cell model for predicting the 3-D mechanical properties of laminates with through-thickness reinforcement has been developed using a unique voxel mesh approach. As part of its development, the model is being verified to other models and validated with experimental data. The model has been validated to tensile experimental data with much success. For further validation of the model and its assumptions, an experimental program was conducted to investigate the compressive properties of stitched laminates for three different thicknesses. It was found that the unstitched modulus remained independent of thickness, while the stitched modulus increased with thickness due to spreading and kinking of the outer surface plies during the stitch formation process. The model predicted the change in modulus well, however did not capture the thickness effect observed in the experimental results. While it was found that stitching did have an influence on the in-plane compressive properties of the test specimens, the known damage tolerance and manufacturing advantages of stitched laminates are considered to outweigh any penalties for certain critical applications.

Estimation of elastic properties of a particulate polymer composite using a face-centered cubic FE model

A three-dimensional (3D) finite element (FE) unit cell model is applied to studies on the elastic properties of ceramic spherical particle–polymer composite. In this model, hydroxyapatite particles (HAp) are assumed to be spherical and are arranged on a face-centered cubic (FCC) array in poly- l-lactide (PLLA) matrix surrounding. The three dimensionality of the proposed model enables simulation of elastic properties as well as developed stress states. The FE calculations provide estimates of compressive elastic modulus, shear modulus, Poisson's ratio and stress state of the composite for a range of particle volume fractions. The FCC unit cell FE models are evaluated by comparison to available experimental results, simple cubic (SC) unit cell FE calculations and Halpin–Tsai equations. In applying unit cell models for predicting elastic properties of particle-filled composites, the FCC arrangement can be observed to be more accurate compared with the SC arrangement.

Influence of interphase layer on the overall elasto-plastic behaviors of HA/PEEK biocomposite

A three-dimensional finite element unit cell model has been designed and constructed for studying mechanical properties of hydroxyapatite (HA) reinforced polyetheretherketone (PEEK) biocomposite. The model consists of an elastic-brittle HA spherical particle, an elasto-plastic matrix and an interphase layer between the particle and the matrix. The interphase layers with four different kinds of material behaviors have been taken into consideration to examine their effects on the overall properties of the composite. The damage evolution in the matrix and the interphase layer, and the interface failure, were also taken into account. Some other factors, such as mesh sensitivity, loading velocity and mass scale scheme, were also discussed in this investigation. A general-purpose finite element software package, ABAQUS, incorporated with a user-defined material subroutine, was used to perform the analysis. The predicted results were compared with the experimental data obtained from existing literatures. The results predicted by using the cell model with consideration of the matrix degradation and the effects of the damage and failure on the interphase layer are in good agreement with the experimental ones. Hence, the suitability of our proposed cell model incorporated with an appropriate type of the interphase layer for modeling the mechanical properties of the particulate biocomposite could be verified.

Using waviness to reduce thermal warpage in printed circuit boards

Thermal deformations in printed circuit boards (PCB) were reduced by using wavy rather than straight electric lines or traces. The traces had either in-plane or out-of-plane waviness. Thermal deformations of a few different models were calculated using the finite element (FE) method. Periodic unit cell FE models were developed, and subjected to temperature changes. In-plane trace waviness was seen to reduce the thermal warpage of the PCB by ∼40%. Out-of-plane waviness also reduced the warpage, but to a lesser extent.

The applicability of the generalized method of cells for analyzing discontinuously reinforced composites

The paper begins with a short overview of the recent work done in the field of discontinuous reinforced composites, focusing on the different parameters which influence the material behavior of discontinuous reinforced composites, as well as the various analysis approaches undertaken. Based on this overview it became evident, that in order to investigate the enumerated effects in an efficient and comprehensive manner, an alternative approach to the computationally intensive finite-element based micromechanics approach is required. Therefore, an investigation is conducted to demonstrate the utility of utilizing the generalized method of cells (GMC), a semi-analytical micromechanics-based approach, to simulate the elastic and elastoplastic material behavior of aligned short fiber composites. The results are compared with (1) simulations using other micromechanical based mean field models and finite element (FE) unit cell models found in the literature given elastic material behavior, as well as (2) finite element unit cell and a new semianalytical elastoplastic shear lag model in the inelastic range. GMC is shown to definitely have a window of applicability when simulating discontinuously reinforced composite material behavior.

Effective properties of composites utilising fibres with a piezoelectric coating

Composites with piezoelectric fibres are promising new materials, combining the beneficial properties of very different constituents. Recently hybrid fibres with an inactive core and a piezoelectric coating have been developed. For conventional two-phase systems the correlation between component properties and effective composite behaviour is well approximated using effective field or self-consistent models. Since the latter approaches are commonly based on solutions for homogeneous inclusions, they cannot be directly employed for heterogeneous particles as coated fibres. Different methods are employed to estimate the relevant effective electromechanical parameters of composites continuously reinforced with coated piezoelectric fibres: (a) a unit cell finite element model, (b) an effective field approach built on a rigorous solution for a coated fibre in an infinite matrix and (c) a simple homogeneous field approximation. The results of the approaches are compared and discussed.

A comprehensive unit cell model: a study of coupled effects in piezoelectric 1–3 composites

A finite element unit cell model for investigation of arbitrary loading conditions is developed for composites with periodic arrangements of continuous aligned fibers. Special emphasis is placed on the formulation of the boundary conditions to allow for simulation of all modes of overall deformation arising from any arbitrary combination of mechanical and electrical loading. The model is applied to piezoelectric composites whereby the overall elastic, dielectric, as well as piezoelectric moduli are fully extracted. The boundary conditions are validated for elastic in-plane shear loading and checked by recourse to comparisons with analytical bounds as well as semianalytical bounds and experimental investigations for piezoelectric composites from the literature. Aspects of fiber arrangement and differences between piezoelectric ceramic as well as polymer matrix composites reinforced with piezoelectric fibers are discussed.

Micromechanical Modeling of Ferrite-Pearlite Steels Using Finite Element Unit Cell Models.

An axisymmetric unit cell model based on a regular array of second-phase particles arranged on a BCC lattice is used to study deformation mechanisms of ferrite-pearlite structural steels. Microstructural characteristics of the steels were parameterized by the pearlite volume fraction, the aspect ratio of the pearlite particles, and the neighboring factor, which represents the ratio of interparticle spacing in the longitudinal direction to that in the transverse direction. FE analyses were carried out to investigate the macroscopic and microscopic response of unit cells with morphological features based on idealizations of the microstructures of the actual steels. Tensile properties of each constituent phase were obtained experimentally and used in the analyses. As compared to traditional axisymmetric models, the BCC cell model appears to be able to capture more realistically the behavior of the materials, and it accurately estimates the tensile behavior of the ferrite-pearlite steels even with a relatively large volume fraction of the pearlite phase. The effects of volume fraction and morphology of the second-phase particles on deformation behavior were also investigated.

Elasto-plastic deformation of compositionally graded metal-ceramic composites

The elasto-plastic deformation due to thermal and mechanical loading of layered metal-ceramic composites with compositionally graded interfaces is analyzed using detailed finite element models. The model material considered is a NiAl 2O 3 layered system with a compositionally graded interface. The analyses consider planar geometries with perfectly periodic arrangements of the constituent phases, by recourse to new classes of square-packing and hexagonal-packing unit cell formulations for the graded material. Also considered are graded phase arrangements within which large numbers of microstructural units of the two phases are randomly placed within the unit cell. It is found that square-packing arrangements provide the best possible bounds for the thermal strains and coefficient of thermal expansion (CTE) of the graded multi-layer, among the different unit cell models examined; however, no unique bounds could be identified for mechanical loading. The numerical predictions of thermal and mechanical response are compared with those provided by the mean-field approach involving an incremental Mori-Tanaka analysis and by the simple rule-of-mixture approximations. The former method provides a stiffer mechanical response than the finite-element unit cell models. The finite element predictions of bending CTE due to thermal excursions match the overall trends observed experimentally for the NiAl 2O 3 graded system, and further provide a quantitative prediction of the temperature for the onset of plastic flow in the graded material.

In Situ Monitoring of Thermally Cycled Metal Matrix Composites by Neutron Diffraction and Laser Extensometry

A novel stroboscopic neutron diffraction data collection system has beendeveloped. In combination with scanning laser extensometry this has beenused to investigate the thermal cycling behaviour of SiC short fibrereinforced Al matrix composites. Three-dimensional unit cell finite elementmodels have been produced, incorporating matrix deformation both by creepand plasticity. Comparison of the experimental results with modelpredictions has allowed conclusions to be drawn about the deformationprocesses which dominate at different parts of the thermal cycle.

Modelling void nucleation and growth processes in a particle-reinforced metal matrix composite material

The nucleation and growth of voids in a particle-reinforced metal matrix composite (MMC) material have been modelled using internal state variable equations implemented in a unit-cell finite element model. The coupled elastic-viscoplastic damage constitutive equations enable a good representation of the dependence of the stress-strain behaviour on particle volume fraction, the influence of the damage process on the stress-strain behaviour, and the nature of the damage evolution over a range of volume fraction of particles. In particular, the volume fraction of reinforcement in the MMC material is found to influence the nature of the damage evolution process, and a volume fraction of approximately 20% has been identified as separating two different modes of damage evolution.

Validity of Finite Element Unit Cell Model for Studying Yield Condition of Isotropic Porous Materials.

Assuming a spherical void in an infinite rigid plastic material, Gurson proposed a yield function for isotropic porous solids. It is, however, well known that the Gurson model gives harder response than those predicted by experimentation on actual porous solids. In numerical studies to check the validity of Gurson's model, most of the past researchers have introduced a cubic unit cell model, in which a spherical void is placed at the center of the cube. The cubic model can be a good approximation of a porous solid, if the void volume fraction is very small. The cubic model, however, may be not appropriate for the study of porous solids with high ratio of void volume fraction since the model automatically introduces an orthotropy effect due to the geometrically repetitive distribution of voids in three orthogonal axis directions. This research will propose a new unit cell model which is appropriate for the study of the yield functions for isotropic porous materials. The model is also favorable for the study of the anisotropic effect due to the void shape because the unit cell includes less of the anisotropy based on the distribution than the cubic model does.

Modelling the thermo-mechanical behaviour of an Al alloy-SiC p composite effects of particle shape and microscale fracture

The mechanical behaviour of an Al alloy/SiC particle reinforced composite undergoing thermomechanical fatigue, i.e. cyclic loading and cyclic temperature change, is simulated. The Continuum Micromechanics approach is taken where individual reinforcing particles are represented using an idealised 3D repeating unit cell finite element model. Continuum constitutive theories are used, viz. elasticity for the SiC and elasticity-J 2 flow theory plasticity for the matrix. The results are compared with published experimental macroscale data for thermo-mechanical fatigue of an Al alloy/SiC composite. The effects of particle shape and particle fracture are assessed. Interfacial debonding is simulated using a simple elastic strain energy/element removal method.