Macroscopic Elasticity Research Articles

Stimulus-Induced Dynamic Behavior Regulation of Flexible Crystals through the Tuning of Module Rigidity.

Introducing dynamic behavior into periodic frameworks has borne fruit in the form of flexible porous crystals. The detailed molecular design of frameworks in order to control their collective dynamics is of particular interest, for example, to achieve stimulus-induced behavior. Herein, by varying the degree of rigidity of ditopic pillar linkers, two isostructural flexible metal-organic frameworks (MOFs) with common rigid supermolecular building bilayers were constructed. The subtle substitution of single (in bibenzyl-4,4'-dicarboxylic acid; H2BBDC) with double (in 4,4'-stilbenedicarboxylic acid; H2SDC) C-C bonds in pillared linkers led to markedly different flexible behavior of these two MOFs. Upon the removal of guest molecules, both frameworks clearly show reversible single-crystal-to-single-crystal transformations involving the cis-trans conformation change and a resulting swing of the corresponding pillar linkers, which gives rise to Flex-Cd-MOF-1a and Flex-Cd-MOF-2a, respectively. Strikingly, a more favorable gas-induced dynamic behavior in Flex-Cd-MOF-2a was verified in detail by stepwise C3H6/C3H8 sorption isotherms and the corresponding in situ powder X-ray diffraction experiments. These insights are strongly supported by molecular modeling studies on the sorption mechanism that explores the sorption landscape. Furthermore, a consistency between the macroscopic elasticity and microscopic flexibility of Flex-Cd-MOF-2 was observed. This work fuels a growing interest in developing MOFs with desired chemomechanical functions and presents detailed insights into the origins of flexible MOFs.

A computational approach to identify the material parameters of the relaxed micromorphic model

We determine the material parameters in the relaxed micromorphic generalized continuum model for a given periodic microstructure in this work. This is achieved through a least squares fitting of the total energy of the relaxed micromorphic homogeneous continuum to the total energy of the fully-resolved heterogeneous microstructure, governed by classical linear elasticity. We avoid establishing exact micro–macro transition relations, as in classical homogenization theory, because defining a representative volume element is not feasible in the absence of scale separation, as such an element does not exist. The relaxed micromorphic model is a generalized continuum that utilizes the Curl of a micro-distortion field instead of its full gradient as in the classical micromorphic theory, leading to several advantages and differences. The most crucial advantage is that it operates between two well-defined scales. These scales are determined by linear elasticity with microscopic and macroscopic elasticity tensors, which respectively bound the stiffness of the relaxed micromorphic continuum from above and below. While the macroscopic elasticity tensor is established a priori through standard periodic first-order homogenization, the microscopic elasticity tensor remains to be determined. Additionally, the characteristic length parameter, associated with curvature measurement, controls the transition between the micro- and macro-scales. Both the microscopic elasticity tensor and the characteristic length parameter are here determined using a computational approach based on the least squares fitting of energies. This process involves the consideration of an adequate number of quadratic deformation modes and different specimen sizes. We conduct a comparative analysis between the least square fitting results of the relaxed micromorphic model, the fitting of a skew-symmetric micro-distortion field (Cosserat-micropolar model), and the fitting of the classical micromorphic model with two different formulations for the curvature; one simplified formulation involving only one single characteristic length and a simplified isotropic curvature with three parameters. The relaxed micromorphic model demonstrates good agreement with the fully-resolved heterogeneous solution after optimizing only four parameters. The “simplified” full micromorphic model, which includes isotropic curvature and involves the optimization of seven parameters, does not achieve superior results, while the Cosserat model exhibits the poorest fitting.

Numerical modeling of edge inhomogeneity in a macro cantilever beam

This study explores the inhomogeneity problem near the edge of a cantilever beam. Common in gears and turbine blades, inhomogeneities under local edge contact, coupled with macroscale bending effects, result in extreme and complex stress concentration. This model employs the Conjugate Gradient Method to combines the macroscopic beam theory and microscopic elasticity. The Eshelby inclusion method for simulating inhomogeneities. The Hetényi image method is used for address edge contacts in the beam. Enhanced by the Discrete Convolution fast Fourier Transform algorithm, the model’s accuracy and efficiency are validated with the Finite element method. Parametric analysis highlights the influence of distance between indenter and edge, stress components distribution and potential failure in cross-scale challenges.

On the identification of material parameters in the relaxed micromorphic continuum

AbstractFor modeling materials with size‐effects, the classical Cauchy linear elasticity theory and first‐order homogenization procedures are insufficient. Enriched continua are able to capture such phenomena in principle. In this work, we model the size‐effects of metamaterial beams via the relaxed micromorphic continuum. The material parameters of the relaxed micromorphic model are evaluated. We present a simple procedure for the determination of the material parameters based on the fact that the model operates between two well‐defined scales that are the classical continuum scales of macroscopic linear elasticity, whose stiffness is given by a known macroscopic elasticity tensor, and a microscopic scale with a microscopic elasticity tensor yet to be identified. The size‐effects of metamaterial cantilevers under traction load are captured successfully.

Sensitivity-matrix-based identification of anisotropic elastic constants and microstructure features in plasma sprayed coatings utilizing ultrasonic back-reflection

With the unique lamellar microstructure, macroscopic elasticity of plasma sprayed coatings shows either positive or inverse anisotropy due to complex microstructure of inter-layer pores and intra-layer cracks. It is extremely difficult to nondestructively evaluate anisotropic elastic constants of these coatings with thin thickness. Moreover, it is an absolute challenge to reveal the elastic anisotropy transitions between positive and inverse. It is also difficult to clarify the nonlinear relationship between macroscopic elasticity and microstructure, which is necessary for microstructural identification. In present work, a sensitivity-matrix-based method was developed to determine five independent elastic constants of plasma sprayed coating by ultrasonic back-reflection method, with the relative errors of inverted elastic constants less than 1.33 %. Then, an optimized physical-descriptor-based microstructure characterization and reconstruction method was proposed to model “pore and crack” coatings with different microstructural features, and relative errors of microstructural features between the reconstructed and preset values were less than 4.35 %. Based on above, the dependences between six microstructural features and elastic constants were established. The inverse elastic anisotropy was explained systematically as the increase of vertical crack content, the “roundness” of horizontal pore and its “weakening” preferred orientation. Finally, the sensitivity-matrix-based method was used to simultaneously identify the porosity, crack content, pore average aspect ratio, pore orientation factor, and crack orientation factor of plasma sprayed Al2O3 coating based on elastic constants inverted. The relative errors in these identifications were less than 14.85 %. This work provides a new sensitivity-matrix-based solving strategy for multi-parameter integrated quantitative identification of complex microstructural features in materials.

Homogenized elasticity and domain of linear elasticity of 2D architectured materials

This article presents a generic method to determine analytically the equivalent homogeneous linear elastic behavior of any 2D architectured material and to determine its macroscopic linear elasticity domain. The method is applied throughout the paper in the equilateral triangular case, as an example. Additional examples of square, isosceles right-angled triangular and hexagonal architectured materials are presented as a conclusion. The proposed method combines a periodic homogenization for the equivalent homogeneous linear elastic behavior and three criteria to establish the macroscopic linear elasticity domain. The first criterion corresponds to the limit strength of the bars. The second one, which corresponds to the apparition of a periodic buckling, involving a finite number of cells, is studied on a single primitive unit cell using Bloch wave analysis. The buckling modes that appear under equibiaxial compression for the equilateral triangle architectured material are similar to experimental results reported in the literature. The third criterion is the aperiodic buckling, or macroscopic instability, which corresponds to the apparition of shear bands or a macroscopic buckling. The resultant macroscopic linear elasticity domain resulting from these three criteria is presented, for the first time, in the case of the equilateral triangle, isosceles right-angled triangle architectured material and also for two other geometries, square and hexagonal architectured material. It is shown that the theoretical properties of these architectured materials are, in an Ashby chart, in the domain of light but relatively strong materials, a domain in which man-made materials are currently poorly represented.

Ultrahigh Poisson's ratio glasses

The manner in which metallic glasses fail under external loading is known to correlate well with those glasses' Poisson's ratio $\ensuremath{\nu}$: Low-$\ensuremath{\nu}$ (compressible) glasses typically feature brittle failure patterns with scarce plastic deformation, while high-$\ensuremath{\nu}$ (incompressible) glasses typically fail in a ductile manner, accompanied by a high degree of plastic deformation and extensive liquidlike flow. Since the technological utility of metallic glasses depends on their ductility, materials scientists have been concerned with fabricating high-$\ensuremath{\nu}$ glassy alloys. To shed light on the underlying micromechanical origin of high-$\ensuremath{\nu}$ metallic glasses, we employ computer simulations of a simple glass-forming model with a single tunable parameter that controls the interparticle potential's stiffness. We show that the presented model gives rise to ultrahigh-$\ensuremath{\nu}$ glasses, reaching $\ensuremath{\nu}=0.45$ and thus exceeding the most incompressible laboratory metallic glass. We discuss the possible role of the so-called unjamming transition in controlling the elasticity of ultrahigh-$\ensuremath{\nu}$ glasses. To this aim, we show that our higher-$\ensuremath{\nu}$ computer glasses host relatively softer quasilocalized glassy excitations, and establish relations between their associated characteristic frequency, macroscopic elasticity, and mechanical disorder.

Improving catalytic efficiency via tailoring macroscopic elasticity of nanoporous materials

Improving catalytic efficiency via tailoring macroscopic elasticity of nanoporous materials

Ultrasonic characterization of elastic constants of plasma sprayed Al2O3 coatings based on simulated annealing algorithm

Macroscopic elasticity of plasma sprayed thermal barrier coatings (TBCs) is often anisotropic. Effective characterization of elastic constants is crucial for prediction and evaluation of TBC mechanical behaviors. However, elastic constants of TBCs sprayed on matrix is usually difficult to be determined. In this paper, an inversion method of TBC elastic constants is developed through ultrasonic back-reflection technique combined with Christoffel equation. An algorithm for calculating the coating ultrasonic velocities (longitudinal and transverse wave velocities) in different directions is proposed. A control device to adjust and measure the ultrasonic incident angle from water/coating/matrix/water multi-interface structure is designed. Five elastic constants of coating are inverted using the determined velocities combined with simulated annealing algorithm. Results show that the inverted elastic constants of plasma sprayed Al2O3 coating C11, C13, C33, C44, and C66 are 104.99 GPa, 22.09 GPa, 185.78 GPa, 28.43 GPa, and 26.64 GPa, respectively. The deviations between experiment and inversion velocities were less than 2.37%, and Young's modulus in three orthogonal directions show anisotropy. Then, statistical features of pore and micro-crack, including content, orientation, etc., were analyzed using laser scanning confocal microscope and scanning electron microscope, which primarily indicates that micro-cracks with preferred orientation (mainly parallel to x3 axis) are responsible for that elastic constant in the direction normal to the matrix is higher than that in the transverse direction.

All Structures Great and Small: Nanoscale Modulations in Nematic Liquid Crystals.

The nature of the nanoscale structural organization in modulated nematic phases formed by molecules having a nonlinear molecular architecture is a central issue in contemporary liquid crystal research. Nevertheless, the elucidation of the molecular organization is incomplete and poorly understood. One attempt to explain nanoscale phenomena merely “shrinks down” established macroscopic continuum elasticity modeling. That explanation initially (and mistakenly) identified the low temperature nematic phase (NX), first observed in symmetric mesogenic dimers of the CB-n-CB series with an odd number of methylene spacers (n), as a twist–bend nematic (NTB). We show that the NX is unrelated to any of the elastic deformations (bend, splay, twist) stipulated by the continuum elasticity theory of nematics. Results from molecular theory and computer simulations are used to illuminate the local symmetry and physical origins of the nanoscale modulations in the NX phase, a spontaneously chiral and locally polar nematic. We emphasize and contrast the differences between the NX and theoretically conceivable nematics exhibiting spontaneous modulations of the elastic modes by presenting a coherent formulation of one-dimensionally modulated nematics based on the Frank–Oseen elasticity theory. The conditions for the appearance of nematic phases presenting true elastic modulations of the twist–bend, splay–bend, etc., combinations are discussed and shown to clearly exclude identifications with the nanoscale-modulated nematics observed experimentally, e.g., the NX phase. The latter modulation derives from packing constraints associated with nonlinear molecules—a chiral, locally-polar structural organization indicative of a new type of nematic phase.

Thermodynamic Constitutive Model for Sands with Consideration of Distributed Fabric

A micromechanics-inspired fabric-dependent thermodynamic model is developed to simulate the critical-state and phase-transformation behavior of sands. A generalized fabric tensor that considers the statistical distribution of granular fabric is proposed. The fabric tensor is the sum of three families of a transversely isotropic tensor, which is assumed to be characterized by the probability density function of the normal distribution. The new tensor variable is incorporated into the reversible and irreversible thermodynamic models, and the fabric-dependent hyperelastic and plastic constitutive relations are then derived. The hyperelastic instability leads to a yield criterion that depends on fabric distribution and its evolution. Plastic strain development is linked to fabric distribution by a concept of accumulative yielding probability transitions, which in turn provides smooth transition from macroscopic elasticity to plasticity and finally to the critical state upon shearing. Using this theoretical framework, the physical mechanism underlying the phase-transformation and critical-state behavior of sands are interpreted from the evolution of fabric distribution. The performance of the model is verified by simulating various shear tests of Toyoura sand and a discrete element method (DEM)-modeled sand. The effects of b-value and principal stress direction on the stress-strain behavior are related to the evolution of fabric distribution. The noncoaxiality among stress, strain, and fabric is also captured by the model.

Does mesoscopic elasticity control viscous slowing down in glassforming liquids?

The dramatic slowing down of relaxation dynamics of liquids approaching the glass transition remains a highly debated problem, where the crux of the puzzle resides in the elusive increase in the activation barrier ΔE(T) with decreasing temperature T. A class of theoretical frameworks-known as elastic models-attribute this temperature dependence to the variations of the liquid's macroscopic elasticity, quantified by the high-frequency shear modulus G∞(T). While elastic models find some support in a number of experimental studies, these models do not take into account the spatial structures, length scales, and heterogeneity associated with structural relaxation in supercooled liquids. Here, we propose and test the possibility that viscous slowing down is controlled by a mesoscopic elastic stiffness κ(T), defined as the characteristic stiffness of response fields to local dipole forces in the liquid's underlying inherent structures. First, we show that κ(T)-which is intimately related to the energy and length scales characterizing quasilocalized, nonphononic excitations in glasses-increases more strongly with decreasing T than the macroscopic inherent structure shear modulus G(T) [the glass counterpart of liquids' G∞(T)] in several computer liquids. Second, we show that the simple relation ΔE(T) ∝ κ(T) holds remarkably well for some computer liquids, suggesting a direct connection between the liquid's underlying mesoscopic elasticity and enthalpic energy barriers. On the other hand, we show that for other computer liquids, the above relation fails. Finally, we provide strong evidence that what distinguishes computer liquids in which the ΔE(T) ∝ κ(T) relation holds from those in which it does not is that the latter feature highly fragmented/granular potential energy landscapes, where many sub-basins separated by low activation barriers exist. Under such conditions, it appears that the sub-basins do not properly represent the landscape properties relevant for structural relaxation.

On the inverse quenching technique applied to gelatin solutions

Gelatin gels are known to be nonequilibrium systems, because of the continuous growth and rearrangements of physical junctions, even in the solidlike state. Establishing a relationship between the relative degree of cross-linking and macroscopic elasticity would be crucial in understanding, modeling, and predicting the transformation processes of gelatin solutions. Performing rheological experiments on a distinct gel structure, with a definite number of cross-links, is, however, a challenging task. In isothermal conditions, indeed, the density of physical cross-links changes indefinitely, and network evolution cannot be arrested. Inspired by the inverse quenching technique applied in the past to semicrystalline polymers, we here apply an unusual thermal history to an aqueous solution of gelatin in the semiconcentrated regime (6.67%w pig-skin gelatin), in order to freeze the system in a metastable condition for a time sufficiently long to perform a rheological characterization. The solution, initially kept in the sol state at 60°C, is rapidly cooled below gelation temperature, and isothermal gelation is started at 10°C. After soaking at this low temperature for a given time, the sample is rapidly heated (inverse quenching) up to a value in the range 24–29 °C, where kinetics is monitored. If the waiting time at low temperature and the inverse quenching temperature are suitably chosen, sample elasticity will remain stationary for a relatively large time window, and rheological experiments can then be reliably performed.

A multiscale four-layer finite element model to predict the effects of collagen fibers on skin behavior under tension.

Human skin is a complex multilayered multiscale material that exhibits nonlinear and anisotropic mechanical behavior. It has been reported that its macroscopic behavior in terms of progression of wrinkles induced by aging is strongly dependent on its microscopic composition in terms of collagen fibers in the dermis layer. In the present work, a multiscale four-layer 2D finite element model of the skin was developed and implemented in Matlab code. The focus here was to investigate the effects of dermal collagen on the macroscopic mechanical behavior of the skin. The skin was modeled by a continuum model composed of four layers: the Stratum Corneum, the epidermis, the dermis, and the hypodermis. The geometry of the different layers of the skin was represented in a 2D model with their respective thicknesses and material properties taken from literature data. The macroscopic behavior of the dermis was modeled with a nonlinear multiscale approach based on a multiscale elastic model of collagen structure going from cross-linked molecules to the collagen fiber, combined with a Mori-Tanaka homogenization scheme. The model includes the nonlinear elasticity of the collagen fiber density, the fiber radius, the undulation, and the fiber orientation. An axial tension was applied incrementally to the lateral surfaces of the skin model. A parametric study was performed in order to investigate the effect of the collagen constituents on the macroscopic skin mechanical behavior in terms of the predicted macroscopic stress-strain curve of the skin. The results of the FE computations under uniaxial tension showed that the different layers undergo different strains, leading to a difference in the transversal deformation at the top surface. In addition, the parametric study revealed a strong correlation between macroscopic skin elasticity and its collagen structure.

Reversible Hydrogels with Switchable Diffusive Permeability

AbstractHydrogels are polymer networks swollen in water that are characterized by soft mechanics and high permeability. This makes them good candidates for separation and membrane technologies. The diffusion is controlled by the mesh size of the network, and this can be made tunable through the introduction of thermoresponsive polymers. However, this is still a developing field. To contribute to this development, a dual dynamic network is formed composed of four‐arm polyethylene glycol precursors in which each arm is functionalized with both a terpyridine moiety capable of forming reversible metal–ligand complexes along with branches of poly(N‐isopropylacrylamide) (pNIPAAm), which can be switched between expanded and collapsed states and therewith offers a path for switching the microscopic gel‐network mesh size, along with the macroscopic gel elasticity. The dually sensitive hydrogel has the capability of doubling its elastic modulus when going above the lower critical solution temperature (LCST) of pNIPAAm. In addition, the diffusive permeability of the hydrogel is switched upon change of temperature, whereby diffusants are trapped above the LCST of pNIPAAm. Moreover, it is possible to disassemble and reform the gel upon change of pH. With that, the hydrogel has potential application as a switchable and reversible membrane.

Correlation between Macroscopic Elasticity and Chain Dynamics of Natural Rubber during Vulcanization as Determined by a Unique Rheo-NMR Combination

Correlation between Macroscopic Elasticity and Chain Dynamics of Natural Rubber during Vulcanization as Determined by a Unique Rheo-NMR Combination

Macroscopic Elasticity of Porous Object: Systematic Analysis Based on Finite Element Analysis and Its Application

Macroscopic Elasticity of Porous Object: Systematic Analysis Based on Finite Element Analysis and Its Application

A generalized tube model of rubber elasticity.

In the present paper, a new type of micro-mechanically motivated chain network model for rubber-like materials is proposed. The model captures topological constraints of polymer network chains, in particular, entanglements. The model demonstrates how the local molecular packing constraints modify under deformation and shows the impact of these changes on the macroscopic elasticity of the material. To this end, we combine concepts of a confining tube and a slip-link (reptation) model. In these models, entanglements of polymer chains play an important role. The nature of entanglements is discussed, and relationships governing entanglements are formulated in terms of molecular physics. In the context of nonlinear elasticity, we apply a non-affine concept which captures the liquid-like behavior of polymer networks at smaller scales in a more realistic way. Model predictions show good agreement with experimental results from uniaxial and biaxial tension tests.

ORIGIN OF HEAT EFFECTS AND SHEAR MODULUS CHANGES OF A Cu-BASED AMORPHOUS ALLOY

Shear modulus is one of important parameters to control the viscous flow, diffusion and structural relaxation of amorphous alloy. Macroscopic shear elasticity determines the change of heat flow. The correlation between the mechanical properties and the heat flow during the structural relaxation and glass transition is one of the important issues to understand the origin of mechanical properties of amorphous alloy. In the framework of the interstitialcy theory, the shear modulus, heat flow and viscosity of Cu$_{49}$Hf$_{42}$Al$_{9}$ amorphous alloy were used to probe the correlation between shear modulus and heat flow. In parallel, evolution of interstitialcy defects concentration was determined by shear modulus in initial and relaxed states. From the perspective of energy, the influence of interstitialcy defects concentration on the thermodynamic properties of amorphous alloy was investigated by activation energy spectrum (AES). Dynamic mechanical analysis (DMA) was used to investigate the dynamic mechanical process of the Cu$_{49}$Hf$_{42}$Al$_{9 }$ amorphous alloy. Structural relaxation induced by physical aging and the evolution of internal friction in-situ annealing was discussed. The results demonstrated that interstitialcy theory could accurately describe the relaxation kinetics, shear softening and other phenomena induced by structural relaxation of amorphous alloy. Temperature dependence of the shear modulus in initial and relaxed states can be well predicted by the data of differential scanning calorimetry (DSC). Activation energy spectrum directly reflects the interstitialcy defects concentration, which can be activated per unit activation energy. Structural relaxation leads to a reduction of the defect concentration, which indicates the structure transforms to a more stable state. During the glass transition process, defect concentration rapidly increases, which corresponds to the shear softening accompanied by heat absorption. Structural relaxation induces decrease of both internal friction and atomic mobility of amorphous alloy. However, the atomic mobility is high enough to eliminate the influence of structural relaxation on defect concentration in the supercooled liquid region.

Experimental determination of the elasticity of porous ceramics under the gas saturation and temperature treatment

After the formulation of the theory of poroelasticity, which describes the elasticity of fluid saturated porous media, many experimental studies have been reported on the determination of the elasticity of porous materials under the various physical and structural conditions. However, there are limited studies on how the elasticity behave under the variation of the physical conditions of the saturated fluid. This study presents an attempt to characterize the influence of the pore-saturated gas media and their physical properties on the elasticity of porous ceramic materials. Resonant ultrasound spectroscopic measurements were performed on test specimens of alumina with ∼40% porosity, zirconia with ∼48% porosity, and sintered fully dense zirconia to determine the hydrostatic pressure-dependent macroscopic elasticity. Here, we report the variation of elasticity of porous and full dense samples over approximately five orders of magnitude (800–0.02 psi) in absolute pressure. A material softening with increased pressure was observed from the porous specimens and the rate of softening is quantified with different gas saturation by helium, nitrogen and argon. The time evolution of mechanical equilibrium of the porous materials at low pressure (0.02 psi) and moderately high-temperature (150 °C) conditions will also be discussed.