Mixed-dimensional coupling in finite element models

In many finite element analysis models it would be desirable to combine reduced- or lower-dimensional element types with higher-dimensional element types in a single model. In order to achieve compatibility of displacements and stress equilibrium at the junction or interface between the differing element types, it is important in such cases to integrate into the analysis some scheme for coupling the element types. A novel and effective scheme for establishing compatibility and equilibrium at the dimensional interface is described and its merits and capabilities are demonstrated. Copyright (C) 2000 John Wiley & Sons, Ltd.

Variable mixed-mode delamination in composite laminates under fatigue conditions: testing & analysis

La majoria de les fallades en elements estructurals són degudes a càrrega per fatiga. En conseqüència, la fatiga mecànica és un factor clau per al disseny d'elements mecànics. En el cas de materials compòsits laminats, el procés de fallada per fatiga inclou diferents mecanismes de dany que resulten en la degradació del material. Un dels mecanismes de dany més importants és la delaminació entre capes del laminat. En el cas de components aeronàutics, les plaques de composit estan exposades a impactes i les delaminacions apareixen facilment en un laminat després d'un impacte. Molts components fets de compòsit tenen formes corbes, superposició de capes i capes amb diferents orientacions que fan que la delaminació es propagui en un mode mixt que depen de la grandària de la delaminació. És a dir, les delaminacions generalment es propaguen en mode mixt variable. És per això que és important desenvolupar nous mètodes per caracteritzar el creixement subcrític en mode mixt per fatiga de les delaminacions. El principal objectiu d'aquest treball és la caracterització del creixement en mode mixt variable de les delaminacions en compòsits laminats per efecte de càrregues a fatiga. Amb aquest fi, es proposa un nou model per al creixement per fatiga de la delaminació en mode mixt. Contràriament als models ja existents, el model que es proposa es formula d'acord a la variació no-monotònica dels paràmetres de propagació amb el mode mixt observada en diferents resultats experimentals. A més, es du a terme un anàlisi de l'assaig mixed-mode end load split (MMELS), la característica més important del qual és la variació del mode mixt a mesura que la delaminació creix. Per a aquest anàlisi, es tenen em compte dos mètodes teòrics presents en la literatura. No obstant, les expressions resultants per l'assaig MMELS no són equivalents i les diferències entre els dos mètodes poden ser importants, fins a 50 vegades. Per aquest motiu, en aquest treball es porta a terme un anàlisi alternatiu més acurat del MMELS per tal d'establir una comparació. Aquest anàlisi alternatiu es basa en el mètode dels elements finits i virtual crack closure technique (VCCT). D'aquest anàlisi en resulten importants aspectes a considerar per a la bona caracterització de materials utilitzant l'assaig MMELS. Durant l'estudi s'ha dissenyat i construït un utillatge per l'assaig MMELS. Per a la caracterització experimental de la propagació per fatiga de delaminacions en mode mixt variable s'utilitzen diferents provetes de laminats carboni/epoxy essencialment unidireccionals. També es du a terme un anàlisi fractogràfic d'algunes de les superfícies de fractura per delaminació. Els resultats experimentals són comparats amb les prediccions del model proposat per la propagació per fatiga d'esquerdes interlaminars.

Predicting random effects with an expanded finite population mixed model

Prediction of random effects is an important problem with expanding applications. In the simplest context, the problem corresponds to prediction of the latent value (the mean) of a realized cluster selected via two-stage sampling. Recently, Stanek and Singer [Predicting random effects from finite population clustered samples with response error. J. Amer. Statist. Assoc. 99, 119-130] developed best linear unbiased predictors (BLUP) under a finite population mixed model that outperform BLUPs from mixed models and superpopulation models. Their setup, however, does not allow for unequally sized clusters. To overcome this drawback, we consider an expanded finite population mixed model based on a larger set of random variables that span a higher dimensional space than those typically applied to such problems. We show that BLUPs for linear combinations of the realized cluster means derived under such a model have considerably smaller mean squared error (MSE) than those obtained from mixed models, superpopulation models, and finite population mixed models. We motivate our general approach by an example developed for two-stage cluster sampling and show that it faithfully captures the stochastic aspects of sampling in the problem. We also consider simulation studies to illustrate the increased accuracy of the BLUP obtained under the expanded finite population mixed model. (C) 2007 Elsevier B.V. All rights reserved.

GENERALIZED FINITE ELEMENT METHOD ON NONCONVENTIONAL HYBRID-MIXED FORMULATION

The generalized finite element method (GFEM) is applied to a nonconventional hybrid-mixed stress formulation (HMSF) for plane analysis. In the HMSF, three approximation fields are involved: stresses and displacements in the domain and displacement fields on the static boundary. The GFEM-HMSF shape functions are then generated by the product of a partition of unity associated to each field and the polynomials enrichment functions. In principle, the enrichment can be conducted independently over each of the HMSF approximation fields. However, stability and convergence features of the resulting numerical method can be affected mainly by spurious modes generated when enrichment is arbitrarily applied to the displacement fields. With the aim to efficiently explore the enrichment possibilities, an extension to GFEM-HMSF of the conventional Zienkiewicz-Patch-Test is proposed as a necessary condition to ensure numerical stability. Finally, once the extended Patch-Test is satisfied, some numerical analyses focusing on the selective enrichment over distorted meshes formed by bilinear quadrilateral finite elements are presented, thus showing the performance of the GFEM-HMSF combination.

Comparison of Mixed and Kinematic Uniform Boundary Conditions in Homogenized Elasticity of Femoral Trabecular Bone Using Microfinite Element Analyses

Mechanical properties of human trabecular bone play an important role in age-related bone fragility and implant stability. Micro-finite element (microFE) analysis allows computing the apparent elastic properties of trabecular bone biopsies, but the results depend on the type of applied boundary conditions (BCs). In this study, 167 femoral trabecular cubic biopsies with a side length of 5.3 mm were analyzed using microFE analysis to compare their stiffness systematically with kinematic uniform boundary conditions (KUBCs) and periodicity-compatible mixed uniform boundary conditions (PMUBCs). The obtained elastic constants were then used in the volume fraction and fabric-based orthotropic Zysset-Curnier model to identify their respective model parameters. As expected, PMUBCs lead to more compliant apparent elastic properties than KUBCs, especially in shear. The differences in stiffness decreased with bone volume fraction and mean intercept length. Unlike KUBCs, PMUBCs were sensitive to heterogeneity of the biopsies. The Zysset-Curnier model predicted apparent elastic constants successfully in both cases with adjusted coefficients of determination of 0.986 for KUBCs and 0.975 for PMUBCs. The role of these boundary conditions in finite element analyses of whole bones and bone-implant systems will need to be investigated in future work.

Finite element simulation of sandwich panels of laminated plaster and rockwool under mixed mode fracture

Sandwich panels of laminated gypsum and rock wool have shown large pathology of cracking due to excessive slabs deflection. Currently the most widespread use of this material is as vertical elements of division or partition, with no structural function, what justifies that there are no studies on the mechanism of fracture and mechanical properties related to it. Therefore, and in order to reduce the cracking problem, it is necessary to progress in the simulation and prediction of the behaviour under tensile and shear load of such panels, although in typical applications have no structural responsability.

Finite element simulation of sandwich panels of plasterboard and rock wool under mixed mode fracture

This paper presents the results of research on mixed mode fracture of sandwich panels of plasterboard and rock wool. The experimental data of the performed tests are supplied. The specimens were made from commercial panels. Asymmetrical three-point bending tests were performed on notched specimens. Three sizes of geometrically similar specimens were tested for studying the size effect. The paper also includes the numerical simulation of the experimental results by using an embedded cohesive crack model.The involved parameters for modelling are previously measured by standardised tests.

Gender Differences in Early Reading Strategies: a Comparison of Synthetic Phonics Only with a Mixed Approach to Teaching Reading to 4-5 Year-Old Children

A survey of primary schools in England found that girls outperform boys in English across all phases (Ofsted in Moving English forward. Ofsted, Manchester, 2012). The gender gap remains an on-going issue in England, especially for reading attainment. This paper presents evidence of gender differences in learning to read that emerged during the development of a reading scheme for 4- and 5-year-old children in which 372 children from Reception classes in sixteen schools participated in 12-month trials. There were three arms per trial: Intervention non-PD (non-phonically decodable text with mixed methods teaching); Intervention PD (phonically decodable text with mixed methods teaching); and a ‘business as usual’ control condition SP (synthetic phonics and decodable text). Assignment to Intervention condition was randomised. Standardised measures of word reading and comprehension were used. The research provides statistically significant evidence suggesting that boys learn more easily using a mix of whole-word and synthetic phonics approaches. In addition, the evidence indicates that boys learn to read more easily using the natural-style language of ‘real’ books including vocabulary which goes beyond their assumed decoding ability. At post-test, boys using the nonphonically decodable text with mixed methods (Intervention A) were 8 months ahead in reading comprehension compared to boys using a wholly synthetic phonics approach.

Previous land use and climate influence differences in soil organic carbon following reforestation of agricultural land with mixed-species plantings

Reforestation of agricultural land with mixed-species environmental plantings (native trees and shrubs) can contribute to mitigation of climate change through sequestration of carbon. Although soil carbon sequestration following reforestation has been investigated at site- and regional-scales, there are few studies across regions where the impact of a broad range of site conditions and management practices can be assessed. We collated new and existing data on soil organic carbon (SOC, 0–30 cm depth, N = 117 sites) and litter (N = 106 sites) under mixed-species plantings and an agricultural pair or baseline across southern and eastern Australia. Sites covered a range of previous land uses, initial SOC stocks, climatic conditions and management types. Differences in total SOC stocks following reforestation were significant at 52% of sites, with a mean rate of increase of 0.57 ± 0.06 Mg C ha−1 y−1. Increases were largely in the particulate fraction, which increased significantly at 46% of sites compared with increases at 27% of sites for the humus fraction. Although relative increase was highest in the particulate fraction, the humus fraction was the largest proportion of total SOC and so absolute differences in both fractions were similar. Accumulation rates of carbon in litter were 0.39 ± 0.02 Mg C ha−1 y−1, increasing the total (soil + litter) annual rate of carbon sequestration by 68%. Previously-cropped sites accumulated more SOC than previously-grazed sites. The explained variance differed widely among empirical models of differences in SOC stocks following reforestation according to SOC fraction and depth for previously-grazed (R2 = 0.18–0.51) and previously-cropped (R2 = 0.14–0.60) sites. For previously-grazed sites, differences in SOC following reforestation were negatively related to total SOC in the pasture. By comparison, for previously-cropped sites, differences in SOC were positively related to mean annual rainfall. This improved broad-scale understanding of the magnitude and predictors of changes in stocks of soil and litter C following reforestation is valuable for the development of policy on carbon markets and the establishment of future mixed-species environmental plantings.

Differences in pelvic floor morphology between continent, stress urinary incontinent, and mixed urinary incontinent elderly women : an MRI study

Aims To compare magnetic resonance imaging (MRI) of the pelvic floor musculature (PFM), bladder neck and urethral sphincter morphology under three conditions (rest, PFM maximal voluntary contraction (MVC), and straining) in older women with symptoms of stress (SUI) or mixed urinary incontinence (MUI) or without incontinence. Methods This 2008–2012 exploratory observational cohort study was conducted with community-dwelling women aged 60 and over. Sixty six women (22 per group), mean age of 67.7 ± 5.2 years, participated in the study. A 3 T MRI examination was conducted under three conditions: rest, PFM MVC, and straining. ANOVA or Kruskal–Wallis tests (data not normally distributed) were conducted, with Bonferroni correction, to compare anatomical measurements between groups. Results Women with MUI symptoms had a lower PFM resting position (M-Line P = 0.010 and PC/H-line angle P = 0.026) and lower pelvic organ support (urethrovesical junction height P = 0.013) than both continent and SUI women. Women with SUI symptoms were more likely to exhibit bladder neck funneling and a larger posterior urethrovesical angle at rest than both continent and MUI women (P = 0.026 and P = 0.008, respectively). There were no significant differences between groups on PFM MVC or straining. Conclusions Women with SUI and MUI symptoms present different morphological defects at rest. These observations emphasize the need to tailor UI interventions to specific pelvic floor defects and UI type in older women.

Differences in pelvic floor morphology between continent, stress urinary incontinent, and mixed urinary incontinent elderly women : an MRI study

Aims To compare magnetic resonance imaging (MRI) of the pelvic floor musculature (PFM), bladder neck and urethral sphincter morphology under three conditions (rest, PFM maximal voluntary contraction (MVC), and straining) in older women with symptoms of stress (SUI) or mixed urinary incontinence (MUI) or without incontinence. Methods This 2008–2012 exploratory observational cohort study was conducted with community-dwelling women aged 60 and over. Sixty six women (22 per group), mean age of 67.7 ± 5.2 years, participated in the study. A 3 T MRI examination was conducted under three conditions: rest, PFM MVC, and straining. ANOVA or Kruskal–Wallis tests (data not normally distributed) were conducted, with Bonferroni correction, to compare anatomical measurements between groups. Results Women with MUI symptoms had a lower PFM resting position (M-Line P = 0.010 and PC/H-line angle P = 0.026) and lower pelvic organ support (urethrovesical junction height P = 0.013) than both continent and SUI women. Women with SUI symptoms were more likely to exhibit bladder neck funneling and a larger posterior urethrovesical angle at rest than both continent and MUI women (P = 0.026 and P = 0.008, respectively). There were no significant differences between groups on PFM MVC or straining. Conclusions Women with SUI and MUI symptoms present different morphological defects at rest. These observations emphasize the need to tailor UI interventions to specific pelvic floor defects and UI type in older women.

Semi-implicit finite strain constitutive integration and mixed strain/stress control based on intermediate configurations

A new semi-implicit stress integration algorithm for finite strain plasticity (compatible with hyperelas- ticity) is introduced. Its most distinctive feature is the use of different parameterizations of equilibrium and reference configurations. Rotation terms (nonlinear trigonometric functions) are integrated explicitly and correspond to a change in the reference configuration. In contrast, relative Green–Lagrange strains (which are quadratic in terms of displacements) represent the equilibrium configuration implicitly. In addition, the adequacy of several objective stress rates in the semi-implicit context is studied. We para- metrize both reference and equilibrium configurations, in contrast with the so-called objective stress integration algorithms which use coinciding configurations. A single constitutive framework provides quantities needed by common discretization schemes. This is computationally convenient and robust, as all elements only need to provide pre-established quantities irrespectively of the constitutive model. In this work, mixed strain/stress control is used, as well as our smoothing algorithm for the complemen- tarity condition. Exceptional time-step robustness is achieved in elasto-plastic problems: often fewer than one-tenth of the typical number of time increments can be used with a quantifiable effect in accuracy. The proposed algorithm is general: all hyperelastic models and all classical elasto-plastic models can be employed. Plane-stress, Shell and 3D examples are used to illustrate the new algorithm. Both isotropic and anisotropic behavior is presented in elasto-plastic and hyperelastic examples.

An experimental and finite element investigation of the biomechanics of vertebral compression fractures

Osteoporosis is a disease characterized by low bone mass and micro-architectural deterioration of bone tissue, with a consequent increase in bone fragility and susceptibility to fracture. Osteoporosis affects over 200 million people worldwide, with an estimated 1.5 million fractures annually in the United States alone, and with attendant costs exceeding $10 billion dollars per annum. Osteoporosis reduces bone density through a series of structural changes to the honeycomb-like trabecular bone structure (micro-structure). The reduced bone density, coupled with the microstructural changes, results in significant loss of bone strength and increased fracture risk. Vertebral compression fractures are the most common type of osteoporotic fracture and are associated with pain, increased thoracic curvature, reduced mobility, and difficulty with self care. Surgical interventions, such as kyphoplasty or vertebroplasty, are used to treat osteoporotic vertebral fractures by restoring vertebral stability and alleviating pain. These minimally invasive procedures involve injecting bone cement into the fractured vertebrae. The techniques are still relatively new and while initial results are promising, with the procedures relieving pain in 70-95% of cases, medium-term investigations are now indicating an increased risk of adjacent level fracture following the procedure. With the aging population, understanding and treatment of osteoporosis is an increasingly important public health issue in developed Western countries. The aim of this study was to investigate the biomechanics of spinal osteoporosis and osteoporotic vertebral compression fractures by developing multi-scale computational, Finite Element (FE) models of both healthy and osteoporotic vertebral bodies. The multi-scale approach included the overall vertebral body anatomy, as well as a detailed representation of the internal trabecular microstructure. This novel, multi-scale approach overcame limitations of previous investigations by allowing simultaneous investigation of the mechanics of the trabecular micro-structure as well as overall vertebral body mechanics. The models were used to simulate the progression of osteoporosis, the effect of different loading conditions on vertebral strength and stiffness, and the effects of vertebroplasty on vertebral and trabecular mechanics. The model development process began with the development of an individual trabecular strut model using 3D beam elements, which was used as the building block for lattice-type, structural trabecular bone models, which were in turn incorporated into the vertebral body models. At each stage of model development, model predictions were compared to analytical solutions and in-vitro data from existing literature. The incremental process provided confidence in the predictions of each model before incorporation into the overall vertebral body model. The trabecular bone model, vertebral body model and vertebroplasty models were validated against in-vitro data from a series of compression tests performed using human cadaveric vertebral bodies. Firstly, trabecular bone samples were acquired and morphological parameters for each sample were measured using high resolution micro-computed tomography (CT). Apparent mechanical properties for each sample were then determined using uni-axial compression tests. Bone tissue properties were inversely determined using voxel-based FE models based on the micro-CT data. Specimen specific trabecular bone models were developed and the predicted apparent stiffness and strength were compared to the experimentally measured apparent stiffness and strength of the corresponding specimen. Following the trabecular specimen tests, a series of 12 whole cadaveric vertebrae were then divided into treated and non-treated groups and vertebroplasty performed on the specimens of the treated group. The vertebrae in both groups underwent clinical-CT scanning and destructive uniaxial compression testing. Specimen specific FE vertebral body models were developed and the predicted mechanical response compared to the experimentally measured responses. The validation process demonstrated that the multi-scale FE models comprising a lattice network of beam elements were able to accurately capture the failure mechanics of trabecular bone; and a trabecular core represented with beam elements enclosed in a layer of shell elements to represent the cortical shell was able to adequately represent the failure mechanics of intact vertebral bodies with varying degrees of osteoporosis. Following model development and validation, the models were used to investigate the effects of progressive osteoporosis on vertebral body mechanics and trabecular bone mechanics. These simulations showed that overall failure of the osteoporotic vertebral body is initiated by failure of the trabecular core, and the failure mechanism of the trabeculae varies with the progression of osteoporosis; from tissue yield in healthy trabecular bone, to failure due to instability (buckling) in osteoporotic bone with its thinner trabecular struts. The mechanical response of the vertebral body under load is highly dependent on the ability of the endplates to deform to transmit the load to the underlying trabecular bone. The ability of the endplate to evenly transfer the load through the core diminishes with osteoporosis. Investigation into the effect of different loading conditions on the vertebral body found that, because the trabecular bone structural changes which occur in osteoporosis result in a structure that is highly aligned with the loading direction, the vertebral body is consequently less able to withstand non-uniform loading states such as occurs in forward flexion. Changes in vertebral body loading due to disc degeneration were simulated, but proved to have little effect on osteoporotic vertebra mechanics. Conversely, differences in vertebral body loading between simulated invivo (uniform endplate pressure) and in-vitro conditions (where the vertebral endplates are rigidly cemented) had a dramatic effect on the predicted vertebral mechanics. This investigation suggested that in-vitro loading using bone cement potting of both endplates has major limitations in its ability to represent vertebral body mechanics in-vivo. And lastly, FE investigation into the biomechanical effect of vertebroplasty was performed. The results of this investigation demonstrated that the effect of vertebroplasty on overall vertebra mechanics is strongly governed by the cement distribution achieved within the trabecular core. In agreement with a recent study, the models predicted that vertebroplasty cement distributions which do not form one continuous mass which contacts both endplates have little effect on vertebral body stiffness or strength. In summary, this work presents the development of a novel, multi-scale Finite Element model of the osteoporotic vertebral body, which provides a powerful new tool for investigating the mechanics of osteoporotic vertebral compression fractures at the trabecular bone micro-structural level, and at the vertebral body level.