Macroscopic Elastic Constants Research Articles

Two-step homogenized elastic model for in-plane analysis of coated masonry walls

A two-step homogenization procedure is used to identify the (membrane) macroscopic elastic constants of coated masonry walls, solicited in their plane. The two steps deal separately with in- and out-of- wall plane homogenization, based on the modeling of an elementary cell by means of suitable designed assemblies of springs. As a result of the different step combinations, two homogenization strategies are derived. Both of them provide simple closed-form expressions for the linear elastic constants of an “equivalent” homogeneous and orthotropic material. The influence of the order of the homogenization steps is also investigated. Parametric analyses are carried out to examine the reinforcement effectiveness on the global behavior of the system. The accuracy of the theoretical results is assessed by comparisons with finite element (FE) models.

Bounds on the Poisson's ratios of diamond-like structures

Abstract Poisson's ratios of diamond-like structures, such as cubic C, Si, and Ge, have been widely explored because of their potential applications in solid-state devices. However, the theoretical bounds on the Poisson's ratios of diamond-like structures remain unknown. In this paper, we have derived analytical expressions for the minimum and maximum Poisson's ratios, as well as the Poisson's ratios averaged by three different schemes (i.e., Voigt, Reuss, and Hill averaging schemes). These expressions are based on the correlation between macroscopic elastic constants and microscopic force constants of diamond-like structures, and are solely a function of a dimensionless quantity (λ) that characterizes the ratio of mechanical resistances between angle bending and bond stretching. Based on these expressions, we have determined the bounds on the Poisson's ratios, as well as the minimum and maximum Poisson's ratios, and the Poisson's ratios averaged by the three schemes mentioned above. Specifically, these bounds are (−1, 4/5), (−1, 1/5), (0, 4/5), (−1, 1/2), (−1/3, 1/2), and (−2/3, 1/2), respectively. These results were well supported by atomistic simulations. Mechanism analyses demonstrated that the diverse Poisson's behaviors of diamond-like structures result from the interplay between two deformation modes (i.e., bond stretching and angle bending). This work provides the roadmap for finding interesting Poisson's behaviors of diamond-like structures.

Fatigue life prediction method for central hole of needled ceramic matrix composite

Based on the consideration of matrix porosity and representative volume element of needled structure, the macroscopic elastic constants of needled C/SiC ceramic matrix composite and the stress concentration factor of the hole-edge were calculated by the finite element method first. Then, based on the fatigue test data of the smooth and central hole specimens, the fatigue notch factors corresponding to different stress levels were obtained. In view of the different characteristics of the fatigue notch factor obtained under different limit lives, the relationship between the fatigue notch and stress concentration factors was established, and the influence of nominal stress on the fatigue notch factor was also considered. Based on this, a fatigue life prediction method for central hole of needled C/SiC composite was proposed. The proposed method can predict the fatigue life of central hole by using the S–N curve of smooth specimen and the stress concentration factor of central hole specimen. The comparison between predicted and experimental lives showed that the proposed method can better predict the fatigue life of central hole specimen for needled C/SiC composite.

Micromechanical Analysis for Macroscopic Elastic Constants and Thermal Expansion Coefficients of Composite Materials Including Double Inhomogeneous Inclusions (2nd Report, Numerical Calculation of Macroscopic Elastic Constants)

Micromechanical Analysis for Macroscopic Elastic Constants and Thermal Expansion Coefficients of Composite Materials Including Double Inhomogeneous Inclusions (2nd Report, Numerical Calculation of Macroscopic Elastic Constants)

Micromechanical Analysis for Macroscopic Elastic Constants and Thermal Expansion Coefficients of Composite Materials Including Double Inhomogeneous Inclusions (1st Report, Derivation of Solutions for Spheroidal Shapes)

Micromechanical Analysis for Macroscopic Elastic Constants and Thermal Expansion Coefficients of Composite Materials Including Double Inhomogeneous Inclusions (1st Report, Derivation of Solutions for Spheroidal Shapes)

Elastic constant estimation for uni-directional CFRP using laser ultrasonic waves

In general, CFRP is known as an anisotropic material. Several nondestructive testing methods for estimating the elastic constants of anisotropic materials have been proposed. However, previous methods have required cutting off a portion of the specimen or immersion of the specimen in water. In this study, we investigate a method for estimating the macroscopic elastic constants of CFRP using laser ultrasonic visualization testing and machine learning. The finite element method simulations are performed to verify the validity of the proposed method.

Minimum energy and iteration estimates for longitudinal Young’s modulus and Poisson’s ratio of transversely-isotropic unidirectional composites

Some formal inequalities restricting the macroscopic (effective) longitudinal elastic constants of the n-component transversely-isotropic unidirectional composites, based on the minimum energy and complementary energy principles with the help of appropriate constant strain and piece-wise constant stress trial fields and optimization techniques, have been established. Exploring further the possible longitudinal stress–strain modes, additional starting original bounds are derived leading to the new set of iteration bounds on both effective longitudinal constants, which broaden and improve upon the ones of the first set constructed earlier (Pham, 2020) for the composites with positive longitudinal Poisson’s ratio. Another new set of iteration bounds is constructed for the composites having negative longitudinal Poisson’s ratio. Illustrating numerical examples are presented for comparisons.

Elastic constants obtained analytically from microscopic features for regularly arranged elliptical particle assembly

By the micro-structural mechanics approach, this study establishes the quantitative stress–strain relationships at the elastic stage with respect to the microscopic features of the particles for anisotropic granular materials that are composed of regularly arranged elliptical particles with the same size. Firstly, the elliptical particle assembly is equated with a lattice network described by beam elements attached to the center of particles. Then, the elastic stress–strain relationships, which exactly show the features of orthotropic micropolar continuum, are established through analyzing the triangular and hexagonal cells based on the principle of energy balance. Finally, the analytical expressions of eight independent parameters in stress–strain relationships are obtained as the functions of particle shape, size, and microscopic contact stiffnesses. Further, the relationship between microscopic parameters and macroscopic elastic constants for anisotropic granular materials is proposed. The analytical expressions are verified by comparison between theoretical and discrete element method (DEM) results for elliptical particle assembly, and the influences of microscopic parameters on the macroscopic elastic constants are investigated in detail according to the proposed analytical expressions. Our work provides some useful insights for the microscopic explanation and the influences of microscopic parameters on the macroscopic mechanical behavior of granular materials.

In-situ measurement of α formation kinetics in a metastable β Ti-5553 alloy using laser ultrasonics

In this study, we explored the potential of laser ultrasonics for metallurgy (LUMet) as an advanced in-situ technique to measure the α formation kinetics during high temperature isothermal holdings of a Ti-5553 (Ti-5Al-5Mo-5V-3Cr, wt%) alloy. The measured changes in ultrasound velocity were seen to increase linearly with the α phase volume fraction in the investigated temperature range. Using the law of mixture an analytical model was developed to rationalize this experimental observation based on the temperature dependent macroscopic elastic constants and densities of the α and β phases. The transformation kinetics were modeled in the frame of the Johnson-Mehl-Avrami-Kolmogorov (JMAK) theory and the JMAK exponents coincide with the sequence of precipitation observed. Further, a partial time-temperature-transformation (TTT) diagram, constructed from the LUMet data, was found to be consistent with results from in-situ electrical resistivity measurements reported for the same alloy. Considering the advantages of the real-time, non-contact and non-destructive nature of LUMet, this sensor technique provides an excellent monitoring capacity for tracking microstructure evolution in Ti-alloys.

Minimum energy bounds on longitudinal elastic constants of transversely isotropic unidirectional composites.

We consider the n-component transversely isotropic unidirectional elastic composites, the longitudinal axis of which is parallel to those of the transversely isotropic components as well as the generators of the cylindrical phase boundaries between them. From the minimum energy and complementary energy principles, with appropriate constant strain and piece-wise constant stress trial fields, optimization and iteration techniques, a set of bounds for the macroscopic (effective) longitudinal elastic constants of the composites (including the simple lower arithmetic average estimate for longitudinal Young modulus E eff ≥ E V ) are constructed. Numerical examples are provided to illustrate the obtained results.

Determination of Temperature-Dependent Elastic Constants of Steel AISI 4140 by Use of In Situ X-ray Dilatometry Experiments.

In situ dilatometry experiments using high energy synchrotron X-ray diffraction in transmission mode were carried out at the high energy material science beamline P07@PETRAIII at DESY (Deutsches Elektronen Synchrotron) for the tempering steel AISI 4140 at defined mechanical loading. The focus of this study was on the initial tempering state () and the hardened state (). Lattice strains were calculated from the 2D diffraction data for different planes and from those temperature-dependent lattice plane specific diffraction elastic constants () were determined. The resulting coupling terms allow for precise stress analysis for typical hypoeutectoid steels using diffraction data during heat treatment processes, that is, for in situ diffraction studies during thermal exposure. In addition, by averaging specific and macroscopic temperature-dependent elastic constants were determined. In conclusion a novel approach for the determination of phase-specific temperature-dependent DECs was suggested using diffraction based dilatometry that provides more reliable data in comparison to conventional experimental procedures. Moreover, the averaging of lattice plane specific results from in situ diffraction analysis supply robust temperature-dependent macroscopic elastic constants for martensite and ferrite as input data for heat treatment process simulations.

On the relation between microstructure and elastic constants of tungsten/steel composites fabricated by spark plasma sintering

Tungsten/steel composites might play an important role in plasma-facing components, especially in joining of tungsten armor to structural parts made of steel. In this work, Young's moduli and shear moduli of a set of tungsten/P91 steel composites fabricated by spark plasma sintering were determined by resonant ultrasound spectroscopy. It was observed that the sintering temperature and the volume fractions of the individual phases have strong effects on the macroscopic elastic constants. The results were interpreted by means of a finite elements numerical model, showing that the regions of imperfectly bonded tungsten particles appearing in the microstructure of some of the composites act effectively as inclusions with very low elastic stiffness. A good correlation between the number of these regions in the microstructure and the elastic constants was observed.

Mechanics of mutable hierarchical composite cellular materials

Cellular structures having the internal volumes of the cells filled with fluids, fibres or other bulk materials are very common in nature. A remarkable example of composite solution is the hygroscopic keel tissue of the ice plant Delosperma nakurense. This tissue, specialized in promoting the mechanism for seed dispersal, reveals a cellular structure composed by elongated cells filled with a cellulosic swelling material. Upon hydrating, the filler adsorbs large amounts of water leading to a change in the cells’ shape and effective stiffness.This paper, inspired by the configuration of the aforementioned hygroscopic keel tissue, deals with the analysis of a two-dimensional honeycomb made of elongated hexagonal cells filled with an elastic material. The system is treated as a sequence of Euler–Bernoulli beams on Winkler foundation, whose displacements are derived by introducing the classical shape functions of the Finite Element Method. The assumption of the Born rule, in conjunction with an energy-based approach, provide the constitutive model in the continuum form. It emerges a strong influence of the infill’s stiffness and cell walls’ inclination on the macroscopic elastic constants. In particular, parametric analysis reveals the system isotropy only in the particular case of regular hexagonal microstructure.Even though a rigorous analysis of the keel tissue is well beyond our aim, the application of the theoretical model to estimate the effective stiffness of such biological system leads to results that are in good agreement with the published data, where the keel tissue is represented as an internally pressurized honeycomb. Specifically, an energetic equivalence gives an explicit relation between the inner pressure and the filler’s stiffness. Optimal values of pressure and cell walls’ inclination also emerge.Finally, the theory is extended to the hierarchical configuration and a closed form expression for the macroscopic elastic moduli is provided. It emerges a synergy of hierarchy and material heterogeneity in obtaining a stiffer material, in addition to an optimal number of hierarchical levels.

Analogy between the twist-bend nematic and the smectic A phases and coarse-grained description of the macroscopic NTB properties

ABSTRACTThe twist-bend nematic, NTB, has unusual elastic properties. Contrary to the usual nematic, the NTB director n is spontaneously distorted, resulting in a heliconical structure with small period and doubly degenerate handedness. Different models have been proposed so far, and are still actively debated, to describe the NTB properties on the microscopic scale. On the macroscopic scale, we propose a symmetry-based analogy between the NTB and the SmA* phases. Introducing a complex order parameter, we write the macroscopic NTB energy by analogy with the de Gennes’ SmA* description. By coarse graining, based on the elastic-instability local model, we calculate the macroscopic elastic constants and characteristic lengths in terms of a few Frank-like elastic coefficients. We discuss the inverse elastic anisotropy of the NTB phase, compared with the SmA*, and its impact on the structure of the NTB defects. We extend the coarse-grained (CG) model to investigate the macroscopic response to electric fields, and the surface anchoring alignment and the microscopic distortions of the NTB phase. Finally, we suggest the existence of a NTB analogue of the twist grain boundary phase TGBA, with a penetration of the twist field in the bulk through a network of screw dislocations of the NTB pseudo-layers.

Optimal design of a honeycomb core composite sandwich panel using evolutionary optimization algorithms

The present paper deals with the optimal design of a composite sandwich panel with honeycomb core structure using particle swarm optimization (PSO) technique. The face sheets of sandwich panel are considered to be thin and the sandwich panel is subjected to a uniformly distributed normal load. The Navier type solution and the closed-form expressions for the macroscopic in-plane elastic constants of the honeycombs are used to predict the deflection of the panel. Also, the maximum stress in the composite sandwich panel is determined using the macroscopic stress–strain relation of the honeycomb core and the plate. Niching Memetic PSO (NMPSO) and Locally Informed Particle Swarm (LIPS) variants of PSO are examined to minimize the weight of the panel. Numerical results showed that the optimal geometry of the honeycomb cell has this property that its radius and thickness converge to their lower bounds while its length converges to its upper bound. This means that to have an optimal panel, each cell should have the allowable minimum cross section (highest number of cells) and maximum allowable length. Moreover the effects of panel aspect ratio, cell width and applied load were examined. Also, the numerical results confirm the efficiency and effectiveness of the NMPSO in finding optimal solution on the constrained and unconstrained objective functions.

Effect of fibre arrangement on the multiaxial fatigue of fibrous composites: a micromechanical computational model

Structural components made of fibre-reinforced materials are frequently used in engineering applications. Fibre-reinforced composites are multiphase materials, and complex mechanical phenomena take place at limit conditions but also during normal service situations, especially under fatigue loading, causing a progressive deterioration and damage. Under repeated loading, the degradation mainly occurs in the matrix material and at the fibre-matrix interface, and such a degradation has to be quantified for design structural assessment purposes. To this end, damage mechanics and fracture mechanics theories can be suitably applied to examine such a problem. Damage concepts can be applied to the matrix mechanical characteristics and, by adopting a 3-D mixed mode fracture description of the fibre-matrix detachment, fatigue fracture mechanics concepts can be used to determine the progressive fibre debonding responsible for the loss of load bearing capacity of the reinforcing phase. In the present paper, a micromechanical model is used to evaluate the unixial or multiaxial fatigue behaviour of structures with equi-oriented or randomly distributed fibres. The spatial fibre arrangement is taken into account through a statistical description of their orientation angles for which a Gaussian-like distribution is assumed, whereas the mechanical effect of the fibres on the composite is accounted for by a homogenization approach aimed at obtaining the macroscopic elastic constants of the material. The composite material behaves as an isotropic one for randomly distributed fibres, while it is transversally isotropic for unidirectional fibres. The fibre arrangement in the structural component influences the fatigue life with respect to the biaxiality ratio for multiaxial constant amplitude fatigue loading. One representative parametric example is discussed.

Polarization approximations for the macroscopic elastic constants of transversely isotropic multicomponent unidirectional fiber composites

The multicomponent transversely isotropic fiber composites, with unidirectional cylindrical fibers made from different component materials and embedded in a continuous binding matrix, are considered. Hashin–Shtrikman-type polarization trial fields developed earlier by us to bound effective moduli of composites are used to construct polarization approximations for four from the five effective elastic constants of the multicomponent composites, except the longitudinal Young’s modulus, for which the classical volume-weighted-average approximation appears sufficiently accurate and general. The approximations contain certain free parameters. Those parameters can be determined from dilute solutions for circular inclusions, which yield results coincided with Maxwell and Mori–Tanaka approximations in the case, or from the reference effective moduli at some finite volume proportions of the components numerically or experimentally. The simple polarization approximations obey Hashin–Shtrikman bounds over all volume proportions of the components and agree well with numerical results on benchmark models.

Closed-form expressions for the macroscopic in-plane elastic and creep coefficients of brick masonry

Approximate expressions for the macroscopic in-plane elastic and creep coefficients of brick masonry with a regular pattern are derived in closed form, using a homogenization approach for periodic media. A microscopic displacement field fulfilling suitable periodicity boundary conditions, and depending on a limited number of degrees of freedom, is formulated over any masonry Representative Volume Element (RVE). According to this field, closed-form expressions for the macroscopic elastic constants are obtained at various degrees of approximation, either using a Method of Cells-type approach, or minimizing the potential energy of the RVE subjected to any given macroscopic stress. Eventually, the results are extended to the description of the global creep behavior of brickwork under service loads, assuming the creep laws of units and mortar to be expressed by Prony series. Using the FE solution as a benchmark, the proposed approach is found to accurately match both the macroscopic constitutive law in linear elasticity and the time evolution of the macroscopic strains of brickwork under sustained macroscopic stress.

Meso-scale site-bond model for elasticity: theory and calibration

A meso-scale site-bond model is proposed to simulate the macroscopic elastic properties of isotropic materials. The microstructure of solids is represented by an assembly of truncated octahedral cells with sites at the cell centres and bonds linking the nearest neighbouring sites. Based on the equivalence of strain energy stored in a unit cell to strain energy stored in a continuum of identical volume, the normal and shear stiffness coefficients of bonds are derived from the given macroscopic elastic constants: Young’s modulus and Poisson’s ratio. To validate the obtained spring constants, benchmark tests including uniaxial tension and plane strain are performed. The simulated macroscopic elastic constants are in excellent agreement with the theoretical values. As a result, the proposed site-bond model can be used to simulate the macroscopic elastic behaviour of solids with Poisson’s ratios in the range from −1 up to 1/2.

Site-bond Modelling of Structure-failure Relations in Quasi-brittle Media

The non-linear behaviour of quasi-brittle media emerges from distributed micro-cracking. This is analysed conveniently by discrete lattice models. A 3D site-bond model is specialised here for materials with three-phase microstructures: stiff inclusions in a compliant matrix containing pores. The deformation behaviour is based on analytically derived relations between bond properties, length scale and macroscopic elastic constants. The microstructure-model mapping is based on size distributions and volume densities of inclusions and pores, typically obtained through analyses of 3D images. Inclusions data is used to calculate the required length scale. Pores data is used to define the failure behaviour of individual bonds. Applications of the methodology to cement-based materials and nuclear graphite are presented separately in this volume.