Variability response functions for stochastic systems under dynamic excitations

The concept of variability response functions (VRFs) is extended in this work to linear stochastic systems under dynamic excitations. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of linear stochastic systems under static loads, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is assumed. This assumption is here validated with brute-force Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). The same integral expression can be used to calculate the mean response of a dynamic system using a Dynamic Mean Response Function (DMRF) which is a function similar to the DVRF. These integral forms can be used to efficiently compute the mean and variance of the transient system response together with time dependent spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability system response.     

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.     

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.     

Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poisson's ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random field case

This paper is the second of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. The concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in the first paper for the special case of material property variations modeled by random variables to more general problems involving random fields. Specifically, a hierarchy of spectral- and probability-distribution-free upper bounds on the mean, variance, and exceedance values of the response of stochastic systems is established when only the coefficient of variation and lower limit of the stochastic material properties are known. Furthermore, a hierarchy of probability-distribution-free upper bounds on these quantities is established when the spectral density function describing the stochastic material properties is known in addition to the coefficient of variation and the lower limit.     

Analysis of Bernoulli beams with 3D stochastic heterogeneity

The behavior of stochastically heterogeneous beams, composed of isotropic sub-elements of randomly distributed stiffness is studied. Cross sectional as well as longitudinal heterogeneity are included. Average displacements, reaction forces and their statistical variance are found analytically by a functional perturbation method. Ratio of sub-element to beam characteristic size is not negligible and the use of an equivalent homogeneous structure with the classical effective material properties is not sufficient. The major aim is to study the relation between various microstructure properties (grain size, shape, modulus, statistical correlation lengths etc.) and the overall behavior of linear elastic Bernoulli beams. For the statically determinate case, only cross sectional 2D microstructure statistics is found to affect the elastic response, so that an equal average displacement can be achieved by an equivalent, non-isotropic homogeneous beam. For the indeterminate case, the average values of macro properties are affected by the 3D morphological features. Therefore, the proper equivalent homogeneous beam has to include non-local elastic properties. A simple reciprocal relation, connecting two separate loading systems is found, relating their external forces and displacement statistical variances. Morphological parameters, like two point probability moments, used in the final results are derived analytically, and their physical interpretations are discussed.     

Alternative methods to determine the elastoplastic properties of sintered hydroxyapatite from nanoindentation testing

This study introduces alternative methods to determine the elastoplastic properties of bovine-derived Hydroxyapatite (HA) porous bone graft through a set of nanoindentation tests with a Berkovich indenter. Generally, experimental data obtained from nanoindentation tests are force displacement, hardness and elastic modulus. However, to determine plastic properties such as strength coefficient and work hardening exponent of bovine HA, analytical or inverse finite element models are required. In this paper, the effect of sintering temperature on these properties of HA is studied for the range of 1000–1400 °C. The direct and inverse Finite Element (FE) simulation models for nanoindentation tests were written in MSC, MARC^® software. A special algorithm for the inverse technique was developed to infer the most suitable elastoplastic material model for HA. A semi-empirical method was adapted to calculate the elastoplastic material properties of HA. The numerical results of harder hydroxyapatite showed better agreement with the experiments while the work hardening exponent, or n-value, and strength coefficient k of hard HA were found to be 0.23 and 8.05 GPa respectively. A comparison between the experimental and predicted load–displacement curves showed that the proposed inverse technique is effective in predicting the elastoplastic material properties from the nanoindentation test with error below 4% at maximum load.     

Group characterization of impact-induced,in vivo human brain kinematics

Brain movement during an impact can elicit a traumatic brain injury, but tissue kinematics vary from person to person and knowledge regarding this variability is limited. This study examines spatio-temporal brain–skull displacement and brain tissue deformation across groups of subjects during a mild impact in vivo. The heads of two groups of participants were imaged while subjected to a mild (less than 350 rad s⁻²) impact during neck extension (NE, n = 10) and neck rotation (NR, n = 9). A kinematic atlas of displacement and strain fields averaged across all participants was constructed and compared against individual participant data. The atlas-derived mean displacement magnitude was 0.26 ± 0.13 mm for NE and 0.40 ± 0.26 mm for NR, which is comparable to the displacement magnitudes from individual participants. The strain tensor from the atlas displacement field exhibited maximum shear strain (MSS) of 0.011 ± 0.006 for NE and 0.017 ± 0.009 for NR and was lower than the individual MSS averaged across participants. The atlas illustrates common patterns, containing some blurring but visible relationships between anatomy and kinematics. Conversely, the direction of the impact, brain size, and fluid motion appear to underlie kinematic variability. These findings demonstrate the biomechanical roles of key anatomical features and illustrate common features of brain response for model evaluation.     

Variation of transverse and shear stiffness properties of wood in a tree

In this work, the application of the unnotched Iosipescu test for the evaluation of the radial variability of longitudinal–radial stiffness parameters of maritime pine wood was investigated. Rectangular specimens with grain at 45° were tested using the Iosipescu fixture. For this configuration both the transverse (Q₂₂) and the shear (Q₆₆) stiffness parameters can be simultaneously identified by the virtual fields method. Displacement fields were measured by the grid method. The strain fields were reconstructed from the raw displacement fields using a polynomial approximation scheme. For the tested wood material, it was found that both parameters decrease from the centre to about the middle radius of the stem and increase afterwards to the outermost positions. Moreover, a relatively good correlation was obtained between the patterns of radial variability of Q₂₂ and Q₆₆.     

Shape effect on weakly nonlinear elliptical composites

The electric field responses of two types of weakly nonlinear dielectric composites consisting of elliptic cylindrical inclusions, one with identical shape and the another with distributed shapes, randomly embedded in the linear host media in the dilute limit are investigated. The dielectric property of the inclusions is that the relation between the displacement (D) and electric (E) fields obey the form D = ?E + χ∣E∣^βE where β is a nonlinear integer exponent and ? ? χ∣E∣^β. By using the decoupling approximation, the effective nonlinear susceptibility (χ_e) is determined and analyzed for varying the aspect ratios and the shape distribution parameters for the composites with identical and distributed inclusion shapes, respectively. In addition, the exact analytic result of χ_e for the elliptical composites with distributed inclusion shapes for the case of β = 2 is derived in this article.     

Effect of stochastically anisotropic morphology on the mechanical behavior of microbeams

In many Micro-Electro-Mechanical-Systems (MEMS) applications such as polysilicon microstructures, the size of the basic element (grain), compared with the structure’s scale, is not negligible. In these cases, the random microstructure causes statistical dispersion of its response to load, experimentally observed by many researchers. The accompanying calculations, which generally treat the material as homogeneous and isotropic, do not model the structures properly. In this work, we analytically study the relations between morphological properties and the generalized displacements of statically determinate beams, frequently used in MEMS. The analysis includes shear deformation effects, material and structural couplings such as shear and bending deformations due to normal forces, and nonisotropic effects. The governing equations are presented in a special tensorial form, which enables finding analytical expressions for the mean and covariance matrix of the generalized deflections in terms of well defined morphological parameters and their statistics.     

Effective in situ material properties of micron-sized SiO2 particles in SiO2 particulate polymer composites

Several analytical models exist for determination of the Young’s modulus and coefficient of thermal expansion (CTE) of particulate composites. However, it is necessary to provide accurate material properties of the particles as input data to such analytical models in order to precisely predict the composite’s properties, particularly at high particle loading fractions. In fact, the constituent’s size scale often presents a technical challenge to accurately measure the particles’ properties such as Young’s modulus or CTE. Moreover, the in situ material properties of particles may not be the same as the corresponding bulk properties when the particles are embedded in a polymer matrix. To have a better understanding of the material properties and provide useful insight and design guidelines for particulate composites, the concept of “effective in situ constituent properties” and an indirect method were employed in this study. This approach allows for the indirect determination of the particle’s in situ material properties by combining the experimentally determined composite and matrix properties and finite element (FE) models for predicting the corresponding composite properties, then backing out the effective in situ particle properties. The proposed approach was demonstrated with micron-size SiO₂ particle reinforced epoxy composites over a range of particle loading fractions up to 35 vol.% by indirectly determining both the effective Young’s modulus and the effective CTE of the particles. To the best of our knowledge, this study is the first published report on the indirect determination of both the Young’s modulus and the CTE of micron size particles in particulate composites. Similar results on Young’s modulus of micron-size SiO₂ particles measured from nano-indentation testing are encouraging.     

Effect of intumescent ammonium polyphosphate (APP) and melamine cyanurate (MC) on the properties of epoxy/glass fiber composites

The effects of halogen-free flame retardants (FR) such as intumescent ammonium polyphosphate (APP) and melamine cyanurate (MC) on the flammability and mechanical properties of epoxy/glass fiber composite systems were studied. Overall, intumescent APP shows better flammability results compared with MC. The composite with 5 vol.% APP performed sufficiently well in the UL-94 test and LOI, whereas 20 vol.% MC is required to achieve similar results. The addition of 1 vol.% MC into 4 vol.% APP had shown some improvement on the composite flame resistance. The composite attains the maximum flexural strength at 15 vol.% while the dynamic mechanical analysis shows that the addition of flame retardants increased the storage modulus but did not change the glass transition temperature T_g.     

Finite element modeling of nanoindentation on C–S–H: Effect of pile-up and contact friction

In this paper we computationally study the indentation response of a rigid axisymmetric indenter on a semi-infinite elasto-plastic material of the Mohr–Coulomb type. The finite element method is used to quantify the effect of material properties (E, c, φ) and contact friction (μ) on the indentation response of C–S–H phases. The high E/c-ratio for both C–S–H phases, together with their cohesive-frictional behavior, leads to pile-up phenomena around the penetrated probe. The influence of all these parameters on the actual area of contact and its subsequent effect on commonly extracted quantities of the indentation test, namely indentation modulus (M) and indentation hardness (H), is investigated. It is shown that contact friction affects the contact area between the indenter and indented material and as a consequence interferes, to a certain extent, with the procedure for estimating elastic and plastic material properties. The effect is more pronounced for the hardness measurements.     

Preparation and in vitro evaluation of mesoporous hydroxyapatite coated β-TCP porous scaffolds

A mesoporous hydroxyapatite (HA) coating was prepared on a β-tricalcium phosphate (β-TCP) porous scaffold by a sol-gel dip-coating method using the block copolymer Pluronic F127 (EO₁₀₆PO₇₀EO₁₀₆) as the template. For application as a bone graft, in vitro cell response and bone-related protein expression of mesoporous HA coated β-TCP scaffold were investigated, using the non-mesoporous HA coated scaffold as the control group, to evaluate the influence of the mesoporous structure on the biological properties of HA coating. It was found that the increased surface area of the mesoporous HA coating greatly affected the response of MC3T3-E1 osteoblasts and the expression of proteins. An enzyme-linked immunosorbent assay recorded a significantly higher expression of alkaline phosphatase (ALP) and bone sialoprotein (BSP) in the mesoporous group than those in the control group (*p < 0.05) after different incubation periods. The introduction of mesopores enhanced the expression of ALP and BSP in the cells grown on the mesoporous HA coatings, on the premise of maintaining the protein expression in a sequence to ensure the correct temporo-spatial expression in osteogenesis. These results indicated that the mesoporous HA coating would provide a good environment for cell growth, suggesting that it could be used as the coating material for the surface modification of the tissue engineering scaffolds.     

Low-temperature sintering and microwave dielectric properties of Ba3Ti4Nb4O21 with MnCO3–CuO

Effects of MnCO₃–CuO (for short MC) additives on densification and dielectric properties of Ba₃Ti₄Nb₄O₂₁ ceramics have been investigated. The densification temperature of Ba₃Ti₄Nb₄O₂₁ is greatly reduced from 1280 °C for pure Ba₃Ti₄Nb₄O₂₁ to 950 °C with the presence of MC. This is caused by the liquid phase sintering taking place between MC and Ba₃Ti₄Nb₄O₂₁ during sintering. The dielectric constant and the quality factor decrease with increasing MC additives. At a given amount of sintering additive, the dielectric constant and the quality factor decrease with increasing Mn content in the MC mixture. The Ba₃Ti₄Nb₄O₂₁ ceramics with 1 wt% 0.2MnCO₃–0.8CuO sintered at 950 °C for 2 h shows dielectric properties: ε _r  = 66, Q  ×  f = 13,400 GHz and τ _f  = 60 ppm/°C. Also, the material is compatible with Ag electrodes and, therefore, is suitable for LTCC application.     

A through interface crack between a transversely isotropic pair of materials (+30°/−60°, −30°/+60°)

This study focuses on a delamination between two layers of a fiber-reinforced composite material oriented in the directions θ/(θ − 90°). Two specific interfaces are examined: the +30°/−60° interface and −30°/+60° interface. The delamination in these cases is treated effectively as a crack between two monoclinic materials. The behavior of the stress and displacement fields near the crack tip is studied. The first term of the asymptotic expansion for the stress and displacement fields are found by means of the Stroh and Lekhnitskii formalisms. A general solution is obtained for an interface crack in the x₂ = 0 plane. The crack is between two monoclinic materials with x₂ = 0 a symmetry plane.In order to calculate the stress intensity factors, a three-dimensional interaction energy or conservative M-integral is extended and implemented in conjunction with the finite element method. For the M-integral, the auxiliary fields used are particular cases of the stress and displacement fields obtained earlier. The displacement extrapolation method is also extended for this case. The crack surface displacements obtained from a finite element analysis are employed. The methods are independent of each other; hence, they may be used for validation of the results determined.Three test cases are analyzed to examine the accuracy of the results obtained by means of the M-integral method. In addition, two problems of a central crack in a symmetric composite under different loadings are solved. Those loadings are tension and in-plane shear. Stress intensity factors and the interface energy release rate are obtained along the crack front for all cases.