The elastic properties, elastic models and elastic perspectives of metallic glasses

Bulk metallic glass (BMG) provides plentiful precise knowledge of fundamental parameters of elastic moduli, which offer a benchmark reference point for understanding and applications of the glassy materials. This paper comprehensively reviews the current state of the art of the study of elastic properties, the establishments of correlations between elastic moduli and properties/features, and the elastic models and elastic perspectives of metallic glasses. The goal is to show the key roles of elastic moduli in study, formation, and understanding of metallic glasses, and to present a comprehensive elastic perspectives on the major fundamental issues from processing to structure to properties in the rapidly moving field.A plentiful of data and results involving in acoustic velocities, elastic constants and their response to aging, relaxation, applied press, pressure and temperature of the metallic glasses have been compiled. The thermodynamic and kinetic parameters, stability, mechanical and physical properties of various available metallic glasses especially BMGs have also been collected. A survey based on the plentiful experimental data reveals that the linear elastic constants have striking systematic correlations with the microstructural features, glass transition temperature, melting temperature, relaxation behavior, boson peak, strength, hardness, plastic yielding of the glass, and even rheological properties of the glass forming liquids. The elastic constants of BMGs also show a correlation with a weighted average of the elastic constants of the constituent elements. We show that the elastic moduli correlations can assist in selecting alloying components with suitable elastic moduli for controlling the elastic properties and glass-forming ability of the metallic glasses, and thus the results would enable the design, control and tuning of the formation and properties of metallic glasses.We demonstrate that the glass transition, the primary and secondary relaxations, plastic deformation and yield can be attributed to the free volume increase induced flow, and the flow can be modeled as the activated hopping between the inherent states in the potential energy landscape. We then propose an extended elastic model to understand flow in metallic glass and glass-forming supercooled liquid, and the model presents a simple and quantitative mathematic expression for flow activation energy of various glasses. The elastic perspectives, which consider all metallic glasses exhibit universal behavior based on a small number of readily measurable parameters of elastic moduli, are presented for understanding the nature and diverse properties of the metallic glasses.     

A simple viscoelastic model for soft tissues the frequency range 6-20 MHz

In this paper, we present measurements of the shear properties of porcine skeletal muscle, liver, and kidney and a novel model describing them. Following a previously used method, shear mechanical impedances are measured, and complex shear moduli are obtained in the frequency range 6-20 MHz. As indicated in previous results, negative storage moduli are obtained in some measurements, which yield negative shear moduli in traditional linear viscoelastic models such as the Maxwell model, the Voigt model, and the Kelvin model. To resolve this problem, we propose a simple extension of the Voigt model. A mass is introduced into the model to account for the extra phase shift that apparently produces the negative moduli, and the shear stress thereby is related to the inertia of the material. The observed negative storage moduli are predicted by the new model when the relaxation time of the material is large and the working frequency is high. The model is fitted to experimental data to obtain values for material constants.     

A mechanistic model for transverse damage initiation,evolution, and stiffness reduction in laminated composites

A constitutive model to predict stiffness reduction due to transverse matrix cracking is derived for laminae with arbitrary orientation, subject to in-plane stress, embedded in laminates with symmetric but otherwise arbitrary laminate stacking sequence. The moduli of the damaged laminate are a function of the crack densities in the damaging laminae, which are analyzed one by one. The evolution of crack density in each lamina is derived in terms of the calculated strain energy release rate and predicted as function of the applied load using a fracture mechanics approach. Unlike plasticity-inspired formulations, the proposed model does not postulate damage evolution functions and thus there is no need for additional experimental data to adjust material parameters. All that it is needed are the elastic moduli and critical energy release rates for the laminae. The reduction of lamina stiffness is an integral part of the model, allowing for stress redistribution among laminae. Comparisons with experimental data and some results from the literature are presented.     

A simple viscoelastic model for soft tissues in the frequency range 6-20 MHz.

In this paper, we present measurements of the shear properties of porcine skeletal muscle, liver, and kidney and a novel model describing them. Following a previously used method, shear mechanical impedances are measured, and complex shear moduli are obtained in the frequency range 6-20 MHz. As indicated in previous results, negative storage moduli are obtained in some measurements, which yield negative shear moduli in traditional linear viscoelastic models such as the Maxwell model, the Voigt model, and the Kelvin model. To resolve this problem, we propose a simple extension of the Voigt model. A mass is introduced into the model to account for the extra phase shift that apparently produces the negative moduli, and the shear stress thereby is related to the inertia of the material. The observed negative storage moduli are predicted by the new model when the relaxation time of the material is large and the working frequency is high. The model is fitted to experimental data to obtain values for material constants.     

Effect of reinforcing layer on shape fixity and time-dependent deployment in shape-memory polymer textile composites

The purpose of the present study is to model shape fixity and time-dependent deployment in shape-memory polymer composites (SMPCs) and to evaluate the effect of textiles’ tensile and bending moduli on these properties. We constructed an SMPC model by combining SMP layers and a reinforcing layer. We also considered the thermo-viscoelasticity of SMP and the difference in values between the tensile and bending moduli of the reinforcing layer. Employing this model, we simulated deployment under pure bending conditions. Comparison with experimental results confirmed that our proposed model is able to simulate shape fixity and time-dependent deployment in SMPCs. We also confirmed that the bending modulus is an important factor for shape fixity and time-dependent deployment, whereas the tensile modulus has nothing to do with these properties.     

Elastic field and effective moduli of periodic composites with arbitrary inhomogeneity distribution

This paper develops a semi-analytic model for periodically structured composites, of which each period contains an arbitrary distribution of particles/fibers or inhomogeneities in a three-dimensional space. The inhomogeneities can be of arbitrary shape and have multiple phases. The model is developed using the Equivalent Inclusion Method in conjunction with a fast Fourier Transform algorithm and the Conjugate Gradient Method. The interactions among inhomogeneities within one computational period are fully taken into account. An accurate knowledge of the stress field of the composite is obtained by setting the computational period to contain one or more structural periods of the composite. The effective moduli of the composite are calculated from average stresses and elastic strains. The model is used to analyze the stress field and effective moduli of anisotropic composites that have cubic symmetry. It shows that the bulk and shear moduli predicted by the present model are well located within the Hashin-Shtrikman bounds. The study also shows that the stress field of the composite can be significantly affected by the distribution of inhomogeneities even though the effective moduli are not affected much.     

Elastic Constants of Particle and Fiber Reinforced Metal Matrix Composites

Abstract

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropics between directions of the fiber rich plane and that normal to the plane could be predicted by the model.     

Young’s moduli of functionalized single-wall carbon nanotubes under tensile loading

This paper quantitatively investigates the effect of chemical functionalization on the axial Young’s moduli of single-walled carbon nanotubes (SWCNTs) based on molecular mechanics (MM) simulation, in which the COMPASS force field is used to model the interatomic interactions in a nonfunctionalized nanotube or a functionalized nanotube grafted with vinyl groups. We obtain the axial Young’s moduli of both functionalized and nonfunctionalized SWCNTs. The influences of the number and distribution density of the sp³-hybridized carbon atoms and the radius and chirality of the SWCNTs on Young’s moduli are studied. The results indicate that Young’s moduli depend strongly on the chirality of the SWCNTs and the distribution density of the sp³-hybridized carbon atoms. A 37.50% content of sp³-hybridized carbon atoms may degrade Young’s modulus by up to 33.36%. In addition, MM simulations show that the functionalization of SWCNTs results in a decrease of Young’s moduli of the corresponding SWCNT/polyethylene composites.     

Elastic constants of particle and fiber reinforced metal matrix composites

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropies between directions of the fiber rich plane and that normal to the plane could be predicted by the model.This article is dedicated to Professor Dr. Paul Höller on the occasion of his 65th birthday.     

基于五相球模型确定含涂层空心微球复合材料的有效模量

利用推广的五相球模型得到了含涂层空心微球填充复合材料的有效体积模量、剪切模量和杨氏模量预测的理论公式。分析了复合材料有效模量同空心微球壁的厚度、填充体积分数、涂层厚度等参数的关系。为了说明本文结果的有效性, 将五相球模型退化为不含涂层空心球填充复合材料的情况, 并与文献中的实验数据进行对比。算例计算表明: 涂层较薄时, 填料体积分数越大, 空心微球壁相对越厚, 弹性模量就越大。当填料体积分数最大时, 在空心微球壁相对最薄处, 弹性模量最低。      

Evaluating the development of elastic anisotropy with plastic flow

It has been observed that many initially isotropic materials show the development of anisotropic elastic response after plastic flow. It is desirable to be able to model this change in the elastic properties as a function of the extent of plastic flow. This is particularly important when considering the traveling of waves in some glassy polymers that exhibit large differences in the wave moduli along the different directions resulting from unequal plastic flow in these directions. A thermodynamically based model of plasticity is developed and used to evaluate the elastic moduli associated with infinitesimal elastic deformations around the unloaded configuration. It is shown that for this model there are at least four independent material functions describing the elastic moduli of an initially isotropic material. These moduli are functions of the isotropic invariants of the right plastic Cauchy stretch tensor.     

Evaluation of thermo-viscoelastic property of CFRP laminate based on a homogenization theory

The thermo-viscoelastic constitutive equation of unidirectional carbon fiber reinforced plastic (CFRP) is evaluated using a numerical approach based on the finite element method (FEM) and homogenization theory. The constitutive equation of the CFRP is considered in the Laplace-transformed domain, and it is discussed on the basis of the correspondence principle, which is satisfied by each of the Laplace-transformed elastic moduli. Homogenization theory is employed to estimate the ‘homogenized elastic moduli’ of the composite composed of matrix resin and carbon fibers. Using the approximation of a generalized Maxwell model, the relaxation moduli of CFRP are obtained by numerical computation using the FEM. From the relaxation modulus of epoxy resin and elastic moduli of carbon fiber, thermo-viscoelastic properties of CFRP laminates at several temperatures can be estimated using the FEM with homogenization theory. The effectiveness of the present study is verified by comparing the experimental results and numerical calculations for the relaxation moduli of the CFRP laminates.     

Simulation of the hydration kinetics and elastic moduli of cement mortars by microstructural modelling

The ability of the VCCTL microstructural model to predict the hydration kinetics and elastic moduli of cement materials was tested by coupling a series of computer simulations and laboratory experiments, using different cements. The novel aspects of this study included the fact that the simulated hydration kinetics were benchmarked using real-time measurements of the early-age phase composition during hydration by in situ X-ray diffraction. Elastic moduli are measured both by strain gauges (static approach) and by P-wave propagation (dynamic approach). Compressive strengths were measured by loading mortar prisms until rupture. Virtual samples were generated by VCCTL, using particle size distribution and phase composition as input. The hydration kinetics and elastic moduli were simulated and the numerical results were compared with the experimental observations. The compressive strength of the virtual mortars were obtained from the elastic moduli, using a power-law relation. Experimentally measured and simulated time-dependence of the major cement clinker phases and hydration product phases typically agreed to within 5%. Also, refinement of the input values of the intrinsic elastic moduli of the various phases enabled predictions of effective moduli, at different ages and different water-to-cement mass ratios, that are within the 10% uncertainty in the measured values. These results suggest that the VCCTL model can be successfully used as a predictive tool, which can reproduce the early age hydration kinetics, elastic moduli and mechanical strength of cement-based materials, using different mix designs.     

Nonlinear large deflection buckling analysis of compression rod with different moduli

Many novel materials exhibit a property of different elastic moduli in tension and compression. One such material is graphene, a wonder material, which has the highest strength yet measured. Investigations on buckling problems for structures with different moduli are scarce. To address this new problem, first, the nondimensional expression of the relation between offset of neutral axis and deflection curve is derived based on the phased integration method, and then using the energy method, load–deflection relation of the rod is determined; second, based on the improved constitutive model for different moduli, large deformation finite element formulations are developed, and combined with the arc-length method, finite element iterative program for rods with different moduli is established to obtain buckling critical loads; third, material mechanical properties testing of graphite, which is the raw material of graphene, is performed to measure the tensile and compressive elastic moduli; moreover, buckling tests are also conducted to investigate the buckling behavior of this kind of graphite rod. By comparing the calculation results of the energy method and finite element method with those of laboratory tests, the analytical model and finite element numerical model are demonstrated to be accurate and reliable. The results show that it may lead to unsafe results if the classic theory was still adopted to determine the buckling loads of those rods composed of a material having different moduli. The proposed models could provide a novel approach for further investigation of nonlinear mechanical behavior for other structures with different moduli.     

Effective elastic moduli of porous solids

The principles of continuum mechanics can be extended to porous solids only if the effective moduli are known. Although the effective bulk modulus has already been determined by approximating the geometry of a porous solid to be a hollow sphere, bounds could only be established for the other moduli. This problem of indeterminacy of the moduli is solved in this study using a particular model from the variation of the effective Poisson's ratio. In addition to this, the results are extended for the hollow sphere to real geometry by introducing a porositydependent factor. These results are compared with experimental data and the agreement is found to be good. As the effective Poisson's ratio cannot be determined accurately using experiments, the derived equation is verified using finite element analysis.     

Models for the elastic deformation of honeycombs

A theoretical model has been developed for predicting the elastic constants of honeycombs based on the deformation of the honeycomb cells by flexure, stretching and hinging. This is an extension of earlier work based on flexure alone. The model has been used to derive expressions for the tensile moduli, shear moduli and Poisson's ratios. Examples are given of structures with a negative Poisson's ratio. It is shown how the properties can be tailored by varying the relative magnitudes of the force constants for the different deformation mechanisms. Off-axis elastic constants are also calculated and it is shown how the moduli and Poisson's ratios vary with applied loading direction. Depending on the geometry of the honeycomb the properties may be isotropie (for regular hexagons) or extremely anisotropic. Again, the degree of anisotropy is also affected by the relative magnitude of the force constants for the three deformation mechanisms.     

Linear elasticity of planar delaunay networks. Part II: Voigt and Reuss bounds,and modification for centroids

Summary The linear elastic Delaunay network model developed in a previous paper is used to obtain further results on mechanical properties of graph-representable materials. First, we investigate the error involved in the uniform strain approximation — a computationally inexpensive approach widely employed in the determination of effective moduli of granular and fibrous media. Although this approximation gives an upper bound on the macroscopic moduli, it results in very good estimates of their second order statistics. In order to derive a lower bound another window definition has to be introduced. Also, an energy-based derivation of both bounds is given. The final result relates to a modification of a Delaunay network so that its vertices correspond to the centroids of cells of the corresponding Voronoi tessellation; an increase of effective moduli and a decrease of their scatter are observed.     

A mechanical analysis of metallic tritide aging by helium bubble growth

A simple mechanical model is proposed for the aging of a metallic tritide. The material is assumed to be elastic–power law viscoplastic. Part of the helium atoms generated by tritium decay form spherical bubbles that weaken the elastic moduli of the overall material. By contrast, others can be stored in solid solution in the matrix and are likely to increase the moduli. Two variants of the model are compared, assuming either instantaneous or finite rate diffusion of helium. They predict globally similar evolutions of the gas pressure inside the bubbles, the geometrical parameters (bubble radius, overall swelling), as well as the matrix and overall elastic moduli. The results are in good agreement with atomistic calculations of the pressure evolution. Furthermore, recent experimental measurements of the Young modulus changes during aging are better reproduced when He diffusion rate is finite, thus supporting the second variant of the model.     

A generalized self-consistent method for solids containing randomly oriented spheroidal inclusions

Summary In this paper, a generalized self-consistent method is proposed to predict the effective moduli of a material containing single-phase and randomly oriented spheroidal inclusions, with same aspect ratios. This is achieved by using an energy equivalence framework, associated with a generalization of the classical three phase model to spheroidal inclusions. The localization problem (spheroidal duplex inclusion problem) is formulated with the Papkovitch-Neuber approach; this requires expansion formulae for the spheroidal potentials, which are derived in the Appendix. Finally, the determination of the effective moduli is equivalent to solving a system of nonlinear equations. Effective moduli are presented for various types of inclusions, and comparisons are made with the estimations obtained from the self-consistent and Mori-Tanaka methods. Moreover, the effects of inclusion geometry and spatial distribution of inclusions on the effective moduli are investigated and compared to each other.