Heterogeneous Solids Research Articles

Evaluation of the fracture toughness of butt-welded joints using the boundary effect model

Yielding loads (Py) from small three-point-bending (3-p-b) specimens with notch tips in the heat-affected zone, fusion zone, weld metal, and base metal of Q235 (common carbon steel in China) were used to estimate the corresponding fracture toughnesses. To model the elastic and plastic fracture around the Q235 welded joint, the boundary effect model initially developed for quasi-brittle fracture of heterogeneous solids with a large crack-tip fracture process zone was adopted in this study, bypassing the stringent ASTM standard requirements on the specimen size, initial crack length, and un-notched ligament. It was found that the weld metal had the highest fracture toughness value around 130 MPa√m, followed by the heat-affected zone of 109 MPa√m, the base metal of 83 MPa√m, and the fusion zone of 77 MPa√m. Microstructures and notch-tip plastic zones in each section of the welded joint were measured and used to explain the fracture toughness measurements.

Possibilities of Full Waveform Inversion as an imaging method in heterogeneous solids

Ultrasonic imaging of single objects in heterogeneous solids such as concrete is mainly performed using linearized methods that are limited to one wave type and neglect geometric reflections and mode conversions. This limits the reconstruction accuracy and can induce artifacts. By considering more information of the complete resulting wavefield, full waveform inversion (FWI) promises a more accurate imaging. The method compares the measured signals with the results of the forward calculation of the scattered field of the source signals. The distribution of all three isotropic material parameters is iteratively adjusted to minimize the difference between the calculated and measured signals. By using a viable forward model, no limiting assumptions about wave propagation are necessary. Since the problem is ill-posed, regularization techniques are used for stabilization. The paper explains the operation and possibilities of the model-based approach considering both synthetic and experimental measurement data. First results on idealized scatterer geometries in homogeneous and heterogeneous solids are presented. Beginning with synthetic measurement results for the identification of anomalies in the solid body, the method is then applied to laboratory experimental data in the ultrasonic frequency regime. In both cases the same concrete structure is adressed where a known inclusion made of extruded polystyrene foam (XPS) is representing the anomaly to be detected.

Tensile strength and fracture toughness of steel fiber reinforced concrete measured from small notched beams

Steel fiber reinforced concrete (SFRC) is a kind of multiphase composite material and its tensile strength (ft) and fracture toughness (KIC) cannot be easily measured using small samples due to its highly heterogeneous microstructure. In this study, a novel evaluation is proposed for calculating ft and KIC of brittle heterogeneous solids. The results obtained by this method have been confirmed by three-point bending (TPB) test results of 75 notched SFRC beams in which the length of the corrugated steel fibers is 36 mm. Considering the heterogeneity and discontinuous fracture failure characteristic of SFRC samples, a virtual crack propagation calculation model is established, and the accurate prediction of ft and KIC can be obtained by incorporating the model into the Hu-Duan boundary effect model (BEM). The reasonable range of tensile strength of SFRC used in this paper is about 3.77–5.66 MPa. Compared with the median value of this range, the calculation error of BEM is only 7.69 %. By discussing the influence of α-ratio (defined as the ratio between the height W and the initial notch depth a0) on the predicted results, a value that is less affected by the front and rear boundary is obtained, and the value range is 0.16–0.24. Meanwhile, the exact ft and KIC can be obtained by using only 1–2 groups of samples with reasonable α-ratio. Based on the failure curve of SFRC established by using ft and KIC, the theoretical minimum size of samples with different α-ratio is determined under the condition of linear elastic fracture mechanics (LEFM). This method provides a valuable reference for the estimation of concrete mechanical parameters.

The impact of the adsorbent energy heterogeneities by multidimensional-multicomponent PC-SAFT-DFT

Over the past few decades, the classical density functional theory (cDFT) models have gained importance in modeling inhomogeneous fluids. Notably, adsorption databases produced by non-local DFT (NLDFT), quenched solid DFT (QSDFT) and 2D-NLDFT approaches are used in different characterization methodologies to recover the pore size distribution (PSD) for a various nanoporous materials. This family of models has a much lower computational cost when compared to molecular simulation. However, it lacks formulations that use DFT for multidimensional-multicomponent systems with heterogeneous solids. In addition to the complexity of representing the interactions between confined molecules and their confined mixtures, it is challenging to represent surface heterogeneities. Here we study materials with equivalent external potential but presenting energy heterogeneities varying throughout the material using a multidimensional classical DFT approach consistent with the SAFT equation of state. For this purpose, we applied three external potentials to model three types of slit-like cavities. Among them, a classical slit-like pore was generated, in which the energy sites did not vary along with the material. In the other two cavities, the energy sites could vary along the cavity, simulating a cavity in which the external potential varies bidimensionally. Finally, we evaluate the effect of the energy heterogeneity of these cavities on the adsorption of Ar and Kr and their mixing. The results show that local heterogeneities can alter the shape of isotherm curves and adsorption selectivities. Solids with heterogeneity showed higher selectivity but lower capacity.

Surface energy and wetting behavior on high-conditioned aggregates

Surface energy and its dispersive and polar components are considered good indicators of adhesion, cohesion and other mass-transfer phenomena where its behavior and dependency play a huge role in the determination of the total surface free energy. The main purpose of this paper is to describe the impact of surface topology on contact angle measurements and to study the behavior of the dispersive and polar components on the total surface free energy for industrial-relevant materials. Understanding this relationship will yield a great fundamental and practical basis for applications in heterogeneous solids such as limestone and gravel, two of the most used aggregates in construction and filling material. The dispersive component of the surface energy is the most wide-range contributor and its influence at high-energy regions becomes the main source of surface free energy, however the acidic component acts as a primary contributor at low-energy regions with steep a short-range participation, and the basic component can be considered as a mid-range contributor with a synergistic interaction with the acidic component. It should be noted that the aggregate surface sample is considered as a highly energetic surface implying that the surface conditioning has a great impact in terms of measurement’s reliability due to voids and topological imperfections.

An efficient and quantitative phase-field model for elastically heterogeneous two-phase solids based on a partial rank-one homogenization scheme

This paper presents an efficient and quantitative phase-field model for elastically heterogeneous alloys that ensures the two mechanical compatibilities—static and kinematic, in conjunction with chemical equilibrium within the interfacial region. Our model contrasts with existing phase-field models that either violate static compatibility or interfacial chemical equilibrium or are computationally costly. For computational efficiency, the partial rank-one homogenization (PRH) scheme is employed to enforce both static and kinematic compatibilities at the interface. Moreover, interfacial chemical equilibrium is ensured by replacing the composition field with diffusion potential field as the independent variable of the model. Its performance is demonstrated by simulating four single-particle and one multi-particle cases for two binary two-phase alloys: Ni–Al γ′/γ and UO2/void. Its accuracy is then investigated against analytical solutions. For the single-particle γ′/γ alloy, we find that the accuracy of the phase-field results remain unaffected for both planar and non-planar geometries, when the PRH scheme is employed. Fortuitously, in the UO2/void simulations, despite a strong elastic heterogeneity – the ratio of Young’s modulus of the void phase to that of the UO2 phase is 10−4 – we find that the PRH scheme shows significantly better convergence compared to the Voigt–Taylor scheme (VTS) for both planar and non-planar geometries. Nevertheless, for the same interface width range as in the γ′/γ case, the interface migration in these simulations shows dependence on interface width. Contrary to the γ′/γ simulations, we also find that the simulated elastic fields show deviations from the analytical solution in the non-planar UO2/void case using the PRH scheme.

A mixed FFT-Galerkin approach for incompressible or slightly compressible hyperelastic solids under finite deformation

In this study, we propose a mixed Fast Fourier Transform (FFT)-based homogenization approach to analyze finite deformations of heterogeneous solids with different incompressible or slightly compressible hyperelastic phases. The proposed mixed formulations are constructed from the minimization of the specialized total elastic strain energy using trigonometric polynomials and an adapted Green’s operator. Unlike the original form of the standard strain-based FFT-Galerkin approach, our proposed formulation using both the deformation gradient and hydrostatic pressure as unknowns successfully solves the convergence issue corresponding to the incompressibility limit of elastic materials, without loss of accuracy and efficiency. A general explicit matrix form of the system of linear equations to be solved during the Newton–Raphson iteration procedure for the mixed FFT-Galerkin approach is also provided, making the algorithm easy to implement. Examples of simple shear and uniaxial traction of heterogeneous solids consisting of slightly compressible phases, incompressible phases, or a hybrid of incompressible/totally compressible phases validated the developed FFT-Galerkin approach, demonstrating its potential for possible applications in accurately calculating the mechanical responses of materials such as composites, polymer foams, hydrogels, and other meta-materials consisting of incompressible phases.

Boosting the electro-mechanical coupling of piezoelectric polyvinyl alcohol–polyvinylidene fluoride blends by dispersing nano-graphene platelets

Water-soluble polyvinyl alcohol (PVA) mixed with piezo-active polyvinylidene fluoride (PVdF) micro-grains constitute hybrid blends for transfusing mechanical energy to electrical energy. In principle, the value of the piezoelectric coefficient is a portion of the value that neat PVdF exhibits. In the present work, we investigate the possibility of augmenting the total electromechanical coupling by dispersing nano-graphene platelets (NGPs). Mechanical stress applied on structurally and compositionally heterogeneous solids results in increased values of the internal local stress field: NGPs are likely to amplify the local stress exercised on the surface of individual piezoelectric polymer grains. PVA–PVdF (3:1 w/w), cast from a water solution, loaded with various fractions of NGPs boost the value of the overall piezoelectric coefficient by 150% per weight fraction of NGPs and becomes superior to values reported for neat electro-active PVdF.

An open source peridynamics code for dynamic fracture in homogeneous and heterogeneous materials

We present a novel open source peridynamics software for modeling two- and three-dimensional dynamic fracture in homogeneous and heterogeneous solids. The software and its implementation is described in detail and made available on GitHub and under MIT license. The capability of the software is demonstrated through several numerical examples including a wave propagation problem. Furthermore, 3D fracture simulations in a polymer-matrix composite at the meso-scale is shown to demonstrate the capability of the code to model complex microstructure including the interaction between interface fracture and fracture in the bulk.

Threshold damage mechanisms in brittle solids and their impact on advanced technologies

Threshold damage mechanisms in brittle covalent–ionic solids are outlined. Fracture and deformation modes are analyzed in terms of classical contact mechanics. Distinctions are made between brittle, ductile and quasiplastic mechanisms in both axial and translational contact. Special attention is devoted to the relatively unexplored subthreshold region where macrofracture is largely suppressed, a region of increasing relevance in the relentless move toward ever smaller devices and precision shaping technologies in the manufacturing sector. Cross-section micrographic images illustrate the fundamental nature of shear events within the hardness deformation zone responsible for crack initiation and propagation. Basic analytical relations for the strengths of surfaces with contact-induced damage in the postthreshold and subthreshold regions are presented, with emphasis on concept rather than fine detail. Strength data for a prototypical brittle material after sharp-indenter damage are presented to highlight the vital role of microstructure in determining transitions between brittle and quasiplastic responses. Pristine defect-free solids are shown to be highly vulnerable to contact damage, even in the subthreshold region. Heterogeneous solids with granular microstructures have lower initial strengths, but are more flaw tolerant. Brittle solids are also highly susceptible to degradation by surface removal processes in wear and machining settings, to a large extent depending again on microstructure. Implications of these findings concerning advanced technological applications of covalent–ionic solids are discussed.

Impact damage of the surface layer of α-quartz amorphized by the Ar+ ions implantation

The goal of the performed experiments was to assess the effect of ion bombardment on the mechanical properties of α-quartz. The 40 keV Ar+-ions were implanted with fluences of 1014 and 1016 ion/cm2 were used. The accumulation of broken interatomic bonds in the sample surface layer was traced with the reflectance IR spectroscopy. The sample surface was subjected to the localized impact damaging by a pointed striker prior to and after the Ar+ implantation. The fractoluminescence (FL) excited in damaged samples was recorded with a 20 ns resolution during 1 ms. The statistical analysis of the energy distributions in impact-induced FL time series detected in unirradiated samples showed a random accumulation of newly formed point defects. The FL energy distributions in the ion-implanted samples followed a power law typical for the fracture of heterogeneous solids such as rocks, ceramics, etc.

Supersonic wave propagation in heterogeneous solids and the evolution of energy distribution

Dynamic homogenization of heterogeneous materials has received a lot of attention recently, due to the fact that it can avoid expensive direct simulations with high space-time resolution. In this paper, the applicability of effective homogenous media on some peculiar dynamic phenomena is discussed. Longitudinal wave velocity has been believed to be the upper limit of mechanical signal speeds in a homogenous elastic solid according to the classical elasticity theory. However, for heterogeneous materials, we demonstrate numerically and theoretically that if the wave impedances of the constituent phases are not equal, there must be the supersonic phenomenon such that some wave may propagate faster than the longitudinal wave in the effective homogenous medium. Especially, if two materials with the same longitudinal wave velocity are used to compose a heterogeneous medium, the effective longitudinal wave speed will almost definitely be lower than its local counterpart. We also investigate the evolution of energy distribution over time when the wave propagates through heterogeneous media with various gradients, which indicates, interestingly, that the smoother the change in wave impedances of the heterogeneous medium, the more energy in “before-wave” region and the less energy in “after-wave” region. The corresponding dispersion curves and group speed curves also show dramatic change for heterogeneous media with different smoothness, which can partially explain but not quantitively predict these phenomena. We attempt to develop an effective medium model to reproduce the energy evolution but fail. All these findings pose a big challenge to the proper establishment of dynamic homogenization models.

Correlative image learning of chemo-mechanics in phase-transforming solids.

Constitutive laws underlie most physical processes in nature. However, learning such equations in heterogeneous solids (for example, due to phase separation) is challenging. One such relationship is between composition and eigenstrain, which governs the chemo-mechanical expansion in solids. Here we developed a generalizable, physically constrained image-learning framework to algorithmically learn the chemo-mechanical constitutive law at the nanoscale from correlative four-dimensional scanning transmission electron microscopy and X-ray spectro-ptychography images. We demonstrated this approach on LiXFePO4, a technologically relevant battery positive electrode material. We uncovered the functional form of the composition-eigenstrain relation in this two-phase binary solid across the entire composition range (0 ≤ X ≤ 1), including inside the thermodynamically unstable miscibility gap. The learned relation directly validates Vegard's law of linear response at the nanoscale. Our physics-constrained data-driven approach directly visualizes the residual strain field (by removing the compositional and coherency strain), which is otherwise impossible to quantify. Heterogeneities in the residual strain arise from misfit dislocations and were independently verified by X-ray diffraction line profile analysis. Our work provides the means to simultaneously quantify chemical expansion, coherency strain and dislocations in battery electrodes, which has implications on rate capabilities and lifetime. Broadly, this work also highlights the potential of integrating correlative microscopy and image learning for extracting material properties and physics.

An improved Multiscale FEM for the free vibrations of heterogeneous solids

We present an improvement of the Multiscale Finite Element Method (MsFEM). MsFEM incorporates fine-scale features of a domain through special oscillatory coarse-scale trial functions and its essential step is an online computation of these special functions defined using local auxiliary problems. Our enhancement makes use of the dynamic condensation that transmits the construction of the oscillatory basis functions exclusively to the coarse mesh skeleton (element interfaces) where we defined special 1D eigenproblems. Consequently, both the approximation error and computational cost are reduced. Moreover, arbitrary coarse mesh element geometries, like polytopal ones, may be used without any additional modifications. We illustrate the accuracy and convergence of the improved method with higher order approximation through selected plane strain examples.

Cu nanoparticles embedded on reticular chitosan-derived N-doped carbon: Application to the catalytic hydrogenation of alkenes, alkynes and N-heteroarenes

Applying biomass-waste as catalysts and catalytic supports is gaining a tremendous interest owing to its expected outcomes in terms of cost effectiveness and sustainability. In this context, we herein disclose a straightforward encapsulation of nanosized copper on hierarchically porous, biomass-derived nitrogen-containing carbon framework. Our approach uses chitosan - derived from the marine shell-fish wastes - as a cheap, sustainable carbon and nitrogen source, melamine as nitrogen provider and ethylenediaminetetraacetic acid as a cross-linker to induce the reticular network, much suitable for restricting the growth of the metal seeds. The resulting copper grown on nitrogen-doped carbon, bearing relatively large surface area (106 m2·g−1) and a large group of well-dispersed Cu nanoparticles (average of 2 nm) even with high Cu loading (41 wt%), exhibits catalytic activity for the hydrogenation of unsaturated double and triple carbon-carbon bonds and heteroaryles. This sustainable design of catalyst, using affordable copper and cheap biowaste, could discard palladium and other expensive elements loaded on tedious synthetic supports from the library of heterogeneous solids intended for fine chemical synthesis.

Correlated accumulation of microcracks in impact damaged surface of α-quartz amorphized by Ar-=SUP=-+-=/SUP=- ions implantation

The microcrack accumulation in impact damaged α-quartz plates prior to and after the Ar+-ion implantation was studied with the acoustical emission (AE) method. The 40 keV implantation doses of 1014 and 1016 ion/cm2 were applied. The statistical analysis of the energy distribution in impact-induced AE time series detected in unirradiated samples showed a random (poissonian type) accumulation of defects which is specific for mechanical destruction of homogeneous brittle materials. The energy distribution in the AE time series excited when damaging the preliminary implanted samples followed a power law typical for the fracture of heterogeneous solids such as rocks, ceramics, etc. The optical photography evidenced a transition from brittle (prior to implantation) to ductile damage formation caused by the disturbed interconnectivity of the crystalline structure in irradiated specimens. Keywords: ion implantation, alpha quartz, impact damaging, acoustical emission.

Modelling fracture process zone width and length for quasi-brittle fracture of rock, concrete and ceramics

The crack-tip fracture process zone (FPZ) length with distributed cohesive stresses is commonly modelled for quasi-brittle fracture of heterogeneous solids. This study highlights the crack-tip blunting effect from FPZ width since FPZW > FPZL at the peak fracture load. Both FPZ width and length are linked to the microstructure (grain size for rock and ceramics, aggregate size for concrete, atomic diameter for single crystal silicon), providing a fresh perspective on quasi-brittle fracture phenomena. Importantly, FPZW at the peak fracture load bridges the gap between the fracture toughness KIC and tensile strength ft, i.e. KIC ↔ FPZW ↔ ft. The influence of a blunt notch on quasi-brittle fracture can also be explained by a widened FPZW. A closed-form model containing both FPZW and FPZL (approximately FPZW/FPZL ≈ 2 at the peak fracture load) is used to analyse experimental data of rock, concrete and ceramic with macro-/micro-sized FPZ, and single crystal silicon with FPZ-like critically stressed atomic bonds in front of atomic scale defects.

An adaptive multiscale finite element method for strain localization analysis with the Cosserat continuum theory

Strain localization analysis based on finite element method (FEM) usually requires an intensive computation to capture an accurate shear band and the limit stress, especially in the heterogeneous problems, and thus costs much computational resource and time. This paper proposes an adaptive multiscale FEM (AMsFEM) to improve the computational efficiency of strain localization analysis in heterogeneous solids. In the multiscale analysis, h- and p-adaptive strategies are proposed to update the fine and coarse meshes, respectively. The problem of mesh dependence is handled by the Cosserat continuum theory. In the fine-scale adaptive procedure, triangular elements are taken to discretize the fine-scale domain, and the newest vertex bisection is utilized for refinement based on the gradient of displacement. In the coarse-scale adaptive procedure, a multi-node coarse element technique is considered. By introducing a probability density function for each side of a coarse element, the optimal positions of the newly added coarse nodes can be determined. With the proposed adaptive multiscale procedure, the computational DOFs are reduced smartly and massively. Three representative heterogeneous examples demonstrate that the proposed method can accurately capture shear bands, with an improved computational efficiency and robust convergence.

Multiscale analysis of thermal problems in heterogeneous materials with Direct FE2 method

AbstractMultiscale thermal analysis is motivated by efficient modeling of complex microstructural response (nonlinear conduction, transient conduction, coupled thermomechanics, etc.) without resorting to fully resolved simulations, direct numerical simulation (DNS). It is typically conducted with an FE method that couples two levels of finite element simulations in a nested manner. This article presents a Direct FE method for modeling heat transfer problems in heterogeneous solids. The proposed method is developed based on the same theoretical framework as the FE method, namely downscaling and upscaling rules. However, the scale transitions are achieved through temperature/nodal thermal force in Direct FE rather than temperature gradient/heat flux in common FE implementations. The two‐level simulations can be solved in a monolithic scheme where the macroscopic and microscopic DOFs are coupled through multi‐point constraints. This feature allows an easy implementation in standard FE packages without resorting to repeated transfer of data between two scales. For transient thermal problems, the implementation provides an option to either incorporate or exclude thermal inertial effects and heat generation at the microscale. The performance of the Direct FE is demonstrated by several numerical examples including coupled thermal‐mechanical analysis and transient heat conduction, and the results are compared with DNSs.

A Multifeatured Data-Driven Homogenization for Heterogeneous Elastic Solids

A computational homogenization of heterogeneous solids is presented based on the data-driven approach for both linear and nonlinear elastic responses. Within the Double-Scale Finite Element Method (FE2) framework, a data-driven model is proposed to substitute the micro-level Finite Element (FE) simulations to reduce computational costs in multiscale simulations. The heterogeneity of porous solids at the micro-level is considered in various material properties and geometrical attributes. For material properties, elastic constants, which are Lame’s coefficients, are subjected to be heterogeneous in the linear elastic responses. For geometrical features, different numbers, sizes, and locations of voids are considered to reflect the heterogeneity of porous solids. A database for homogenized microstructural responses is constructed from a series of micro-level FE simulations, and machine learning is used to train and test our proposed model. In particular, four geometrical descriptors are designed, based on N-probability and lineal-path functions, to clearly reflect the geometrical heterogeneity of various microstructures. This study indicates that a simple deep neural networks model can capture diverse microstructural heterogeneous responses well when given proper input sources, including the geometrical descriptors, are considered to establish a computational data-driven homogenization scheme.