J.C. Nadeau

Microstructural optimization of a functionally graded transversely isotropic layer

Abstract Motivated by current problems in coating technology, this paper addresses the microstructural optimization of a layer which is free of tractions, transversely isotropic, infinite and subjected to a prescribed thermal gradient. The layers microstructure is characterized as a bi-constituent composite in the form of a continuous matrix perfectly bonded to embedded spheroidal short fibers. Both constituents are assumed to be isotropic in both mechanical and thermal properties. The microstructural parameters are taken to be the volume fraction, aspect ratio, and orientation distribution of the fibers. The composite layer is made functionally graded by assuming that the microstructural parameters vary through the thickness of the layer. The effective properties of the bi-constituent composite are given by the Mori–Tanaka, Hatta–Taya and Rosen–Hashin homogenization theories. The compositional and microstructural properties are determined such that an objective function defined in terms of strain energy and curvature is minimized. Specific results are presented for an aluminum (Al) layer reinforced with silicon carbide (SiC). Comparisons are made to conventional coating technology.

A multiscale model for effective moduli of concrete incorporating ITZ water–cement ratio gradients, aggregate size distributions, and entrapped voids

Abstract This paper develops a model for the effective elastic properties of concrete, which is a function of the volume fractions, size distributions, and elastic properties of fine aggregate (FA) and coarse aggregate (FA) and entrapped voids. Furthermore, the model is a function of the overall water–cement ratio and specific gravity of cement. Explicitly modeled are the water–cement ratio gradients through the interfacial transition zone (ITZ), which, in turn, affect the variation of the cement paste elastic properties through the ITZ, while maintaining the total fractions of cement and water consistent with the overall water–cement ratio. The ITZ volume is also conserved.

Water–cement ratio gradients in mortars and corresponding effective elastic properties

Abstract Water–cement ratio gradients are modeled through the interfacial transition zone (ITZ) of a mortar with spherical inclusions. The model is a function of the over-all water–cement ratio, volume fraction and radius of sand, specific gravity of cement and thickness of ITZ. Based on experimental data from the literature, the dependence of saturated, homogeneous cement paste is modeled as a function of water–cement ratio. Subsequently, the effective bulk and shear moduli for mortars are determined using a generalized self-consistent method. Finally, application of the model to data in the literature pertaining to elastic wave speeds in saturated mortars composed of 20–30 screened sand with an overall water–cement ratio of 0.3 yielded a mean ITZ thickness of 48.3 μm.

On the use of effective properties for the fracture analysis of microstructured materials

This paper addresses some fundamental theoretical and numerical issues concerning the application of effective properties for the failure analysis of microstructured materials, with a focus on functionally graded materials. An edge-cracked square region is considered which is taken to be on the size of a representative volume element in a larger structural system. The region is made functionally graded by utilizing a non-homogeneous probability density function (PDF) governing the spatial distribution of the constituents. The non-homogeneity of the PDF exists only in the direction of the crack plane. Based upon the type of microstructure considered in the region, an accurate homogenization method is employed to obtain effective properties. For functional forms of the PDF, energy release rates and stress intensity factors (SIFs) are then calculated using both effective properties based on pointwise homogenization and realizations of specific microstructures generated from the PDF. For all calculations, we employ the eXtended finite element method, alleviating the need to remesh the domain between different microstructural realizations. Enrichment strategies are employed by the method to capture the singular stress fields near the crack tip as well as the discontinuous strain fields at fiber–matrix interfaces, even with relatively coarse meshes. SIFs are calculated using a domain form of the J-integral that does not exhibit any domain dependence. The results show excellent agreement between the implicit and the mean of the explicit trials for the energy release rates and strain energy densities. The results also seem to indicate that the implicit SIFs predict the mean SIF of the explicit trials for an arbitrary initial crack geometry. Furthermore, the results indicate that a good approximation of the SIF at specific crack tips may be estimated by re-interpreting the implicit results according to knowledge of the distinct crack-tip location in the microstructure. Quasi-static crack growth studies are performed to investigate the probability of finding crack tips within either of the composite phases.

Invariant Tensor-to-Matrix Mappings for Evaluation of Tensorial Expressions

A class of mappings is presented, parameterized by a variable η, that operates on tensorial expressions to yield equivalent matrical expressions which are then easily evaluated, either numerically or symbolically, using standard matrix operations. The tensorial expressions considered involve scalar, second- and fourth- order tensors, double contractions, inversion and transposition. Also addressed is coordinate transformation and eigenanalysis of fourth- order tensors. The class of mappings considered is invariant, meaning that for a given η the corresponding mapping depends only on the order of the tensor upon which it acts and not, for example, on its physical interpretation (e.g., stress vs. strain, or stiffness vs. compliance). As a result the proposed mappings avoid ad hoc definitions like that of engineering shear strain (i.e., γij:=2∈ij for i ≠ j) which is inconsistent with an invariant mapping. Two convenient choices for the parameter η are presented.

On optimal zeroth-order bounds with application to Hashin–Shtrikman bounds and anisotropy parameters

Abstract A new result presented in this paper is the evaluation of the Hashin–Shtrikman bounds for composites composed of arbitrarily anisotropic constituents. To date, evaluation of the Hashin–Shtrikman bounds are limited to composites with isotropic constituents or to polycrystalline composites with specific crystal symmetries. The generality of the exact result presented herein is achieved through a reinterpretation of Kroners (J. Mech. Phys. Solids 25 (1977) 137) recursive relations for n th-order bounds and the optimal zeroth-order ( n =0) bound. The definitions of optimal zeroth-order bounds are extended to all even-ordered tensors and procedures are presented to evaluate these bounds for all second- and fourth-order tensors. While optimal zeroth-order bounds are not new, the ability to calculate them for fourth-order tensors of arbitrary symmetry is new. Utilizing the zeroth-order bounds, material anisotropy parameters are defined which quantify the extent of anisotropy for even-ordered tensors.

Effective thermal expansion of heterogeneous materials with application to low temperature environments

This paper presents direct formulations of the effective thermal linear expansion (TLE) and the effective coefficient of thermal linear expansion (CTLE) of heterogeneous materials, or multi-constituent composites, with temperature dependent constituent properties and an arbitrary initial incompatible eigenstrain field. The effective properties are expressed in terms of the stress and strain concentrators. For bi-phase composites results are expressed in terms of the effective elastic properties rather than the concentrators. These developments are based on the linear theory of uncoupled thermoelasticity. An example is presented for niobium (Nb) fibers embedded in a copper (Cu) matrix at cryogenic temperatures. It is shown that this composite achieves negative CTLE despite the CTLE of both Nb and Cu are strictly greater than zero. In addition, it is shown that the presence of an initial field of incompatible eigenstrains is capable of causing anisotropic thermal expansion coefficients in an otherwise macroscopically isotropic material of isotropic constituents. Due to the form equivalence of the governing equations the developments which are presented are also applicable to the area of moisture swelling.

Transcending the traditional: Using tablet PCs to enhance engineering and computer science instruction

Traditional instructional methods present many obstacles to effective teaching and learning in engineering and computer science courses. These include a reliance on text-based or static mediums to convey equation- and graphics-heavy concepts, a disconnect between theoretical lecture presentations and applied laboratory or homework exercises, and a difficulty in promoting collaborative activities that more accurately reflect an engineering approach to problem solving. Additionally, technical courses can suffer, like any other course, when students are not actively engaged in the learning and when instructors cannot gauge student understanding. This project has explored the utility of Tablet PCs for overcoming these challenges within a sample of courses in engineering and computer science. There were three primary questions: which knowledge domains benefit from the use of Tablet PCs; whether observed benefits are derived from Tablet PC-specific activities; and what problems limit the effectiveness of Tablet PCs in educational settings? The evaluation of assessment data using regression approaches demonstrated that Tablet-PC-specific activities had a consistent, meaningful, and positive impact upon engineering and computer science courses.

Second-rank equilibrium and transport properties of fibrous composites: Effective predictions and bounds

Abstract A direct approach to the evaluation of the effective permeability, permittivity and transport properties of an arbitrary composite is presented in terms of gradient and flux-density concentrators. A set of requirements are presented, which are imposed on the effective properties and ultimately result in conditions of admissibility for the concentrators. In the scope of bi-constituent, poly-phase composites, two approximate choices for the concentrators are discussed in detail: the Hatta-Taya theory and the poly-inclusion theory. The Hatta-Taya formulation is shown, in general, to yield an effective property which is unsymmetric and which depends on the matrix properties at unitary volume fraction of the embedded material. The poly-inclusion theory is here applied for the first time to second-rank properties. Regardless of the constitution, morphology and texture of the inhomogeneities, the poly-inclusion approach is shown to satisfy all admissibility requirements with the exception of consistency here defined to indicate form identity of mutually inverse properties. A proof is presented which infers relationships between the symmetry group of the orientation distribution function and the symmetry group of the effective properties for special classes of composites. Bounds of order n are presented for macroscopically homogeneous and isotropic composites comprised of an arbitrary number of anisotropic constituents. The case of n = 2 corresponds to the Hashin-Shtrikman bounds which to date appear to have only been calculated for composites with isotropic constituents. Application of the effective properties to the analysis of functionally graded materials (FGMs) is addressed.

On the response sensitivity of an optimally designed functionally graded layer

Sensitivity of the response of an optimally designed functionally graded (FG) thermomechanical system is investigated. The system is a transversely isotropic layer subjected to a thermal gradient. The layer is modeled as a biconstituent composite with a microstructure characterized by the design parameters: fiber volume fraction, shape, and orientation. Performance is quantified by a linear combination of the layers curvature and strain energy density. Of particular interest is the comparison of variability of response to variations in the design parameters versus variability in such things as constituent material properties, and fiber misalignment. These are issues of significance for the manufacturing of functionally graded materials (FGMs). It is concluded that while not all FG systems show a significant sensitivity to texture (i.e. fiber shape and orientation) there do exist some systems for which response is markedly affected by texture even in comparison to significant variabilities in constituent material properties and fiber misalignment.