Orthotropic Elastic Moduli Research Articles

Discrete element models of tooth enamel, a complex three-dimensional biological composite

Enamel, the hard surface layer of teeth, is a three-dimensional biological composite made of crisscrossing mineral rods bonded by softer proteins. Structure-property relationships in this complex material have been difficult to capture and usually require computationally expensive models. Here we present more efficient discrete element models (DEM) of tooth enamel that can capture the effects of rod decussation and rod-to-interface stiffness contrast on modulus, hardness, and fracture resistance. Enamel-like microstructures were generated using an idealized biological growth model that captures rod decussation. The orthotropic elastic moduli were modeled with a unit cell, and surface hardness was captured with virtual indentation test. Macroscopic crack growth was also modeled directly through rupture of interfaces and rods in a virtual fracture specimen with an initial notch. We show that the resistance curves increase indefinitely when rod fracture is avoided, with the inelastic region, crack branching, and 3D tortuosity being the main sources of toughness. Increasing the decussation angle simultaneously increases the size of the inelastic region and the crack resistance while decreasing the enamel axial modulus, hardness, and rod stress. In addition, larger contrasts of stiffness between the rods and their interfaces promote high overall stiffness, hardness, and crack resistance. These insights provide better guidelines for reconstructive dental materials, and for development of bioinspired hard materials with unique combinations of mechanical properties. Statement of SignificanceEnamel is the hardest, most mineralized material in the human body with a complex 3D micro-architecture consisting of crisscrossing mineral rods bonded by softer proteins. Like many hard biological composites, enamel displays an attractive combination of toughness, hardness, and stiffness, owing to its unique microstructure. However few numerical models explore the enamel structure-property relations, as modeling large volumes of this complex microstructure presents computational bottlenecks. In this study, we present a computationally efficient Discrete-element method (DEM) based approach that captures the effect of rod crisscrossing and stiffness mismatch on the enamel hardness, stiffness, and toughness. The models offer new insight into the micromechanics of enamel that could improve design guidelines for reconstructive dental materials and bioinspired composites.

Estimation of measurement errors in orthotropic elastic moduli determined from natural frequencies

Orthotropic elastic moduli of composite structures can be identified by nonlinear least squares fit between measured and computed natural frequencies. However, due to measurement errors contained in the measured natural frequencies and mode shapes, the process of elastic moduli identification is error-prone. This paper proposes an efficient method to estimate the errors in the elastic moduli caused by the measurement errors in the natural frequencies. The method utilizes an efficient semi-analytic expression of the sensitivities of the eigenvalues with respect to the orthotropic elastic moduli. First, the first-order approximation of the analytic sensitivity of orthotropic elastic moduli with respect to the measurement errors is introduced. The approximation is then used to estimate the variability of the identified elastic moduli due to the measurement errors. It is shown that the aspect ratio of the test specimen greatly affects the standard deviation of the identified elastic moduli. Second, using the first-order approximation, the effects of aspect ratio of the test specimen as well as the values of the elastic moduli are further investigated. Based on the results of numerical experiments, guidelines for the dimension of the test specimen for the elastic moduli identification are proposed.

Locally Exact Homogenization of Unidirectional Composites With Cylindrically Orthotropic Fibers

The elasticity-based, locally exact homogenization theory for unidirectional composites with hexagonal and tetragonal symmetries and transversely isotropic phases is further extended to accommodate cylindrically orthotropic reinforcement. The theory employs Fourier series representations of the fiber and matrix displacement fields in cylindrical coordinate system that satisfy exactly equilibrium equations and continuity conditions in the interior of the unit cell. Satisfaction of periodicity conditions for the inseparable exterior problem is efficiently accomplished using previously introduced balanced variational principle which ensures rapid displacement solution convergence with relatively few harmonic terms. As demonstrated in this contribution, this also applies to cylindrically orthotropic reinforcement for which the eigenvalues depend on both the orthotropic elastic moduli and harmonic number. The solution's demonstrated stability facilitates rapid identification of cylindrical orthotropy's impact on homogenized moduli and local fields in wide ranges of fiber volume fraction and orthotropy ratios. The developed theory provides a unified approach that accounts for cylindrical orthotropy explicitly in both the homogenization process and local stress field calculations previously treated separately through a fiber replacement scheme. Comparison of the locally exact solution with classical solutions based on an idealized microstructural representation and fiber moduli replacement with equivalent transversely isotropic properties delineates their applicability and limitations.

A method for the orthotropic coating-substrate system: Green׳s function for a normal line force on the surface

As for the increasing usage of orthotropic and composite materials as a coating of conventional materials in structural applications, a method based on the Green׳s function for the orthotropic coating-substrate system under a normal line force is proposed in this paper. The coating and substrate are modeled as an infinite layer and a semi-infinite substrate respectively. they are perfectly bonded or in smooth contact at the interface. Based on the general solution of orthotropic material, the corresponding harmonic functions with undetermined constants are constructed. Full elastic field can be obtained by substituting these harmonic functions into the general solution. The undetermined constants can be determined by the boundary conditions on the surface and the compatibility conditions at the interface between the coating layer and the semi-infinite substrate. Numerical results are given to show the accuracy and efficiency of this method in studies on the interface effect, interface debonding, coating tensile failure and the relationship between the orthotropic elastic moduli with the stress distributions. The proposed method can suggest possible mechanisms by which orthotropic coated structures can be designed for engineering applications.

A general methodology for inverse estimation of the elastic and anelastic properties of anisotropic open-cell porous materials—with application to a melamine foam

This paper proposes an inverse estimation method for the characterisation of the elastic and anelastic properties of the frame of anisotropic open-cell foams used for sound absorption. A model of viscoelasticity based on a fractional differential constitutive equation is used, leading to an augmented Hooke's law in the frequency domain, where the elastic and anelastic phenomena appear as distinctive terms in the stiffness matrix. The parameters of the model are nine orthotropic elastic moduli, three angles of orientation of the material principal directions and three parameters governing the anelastic frequency dependence. The inverse estimation consists in numerically fitting the model on a set of transfer functions extracted from a sample of material. The setup uses a seismic-mass measurement repeated in the three directions of space and is placed in a vacuum chamber in order to remove the air from the pores of the sample. The method allows to reconstruct the full frequency-dependent complex stiffness matrix of the frame of an anisotropic open-cell foam and in particular it provides the frequency of maximum energy dissipation by viscoelastic effects. The characterisation of a melamine foam sample is performed and the relation between the fractional-derivative model and other types of parameterisations of the augmented Hooke's law is discussed.

Micro-Polar and Second Order Homogenization of Periodic Masonry

Micro-polar and second order homogenization procedures for periodic elastic masonry are implemented to include geometric and material length scales in the constitutive equation. By the solution of the RVE equilibrium problems with properly prescribed boundary conditions the orthotropic elastic moduli of the higher order continua are obtained on the basis of an enhanced Hill–Mandel condition. A shear layer problem is analysed and the results from the heterogeneous models are compared with those ones obtained by the homogenization procedures; the second-order homogenization appears to provide better results in comparison to the micro-polar homogenization.