Heterogeneous Solids Research Articles

Percolation versus depinning transition: The inherent role of damage hardening during quasibrittle failure.

The intermittent damage evolution preceding the failure of heterogeneous brittle solids is well described by scaling laws. In deciphering its origins, failure is routinely interpreted as a critical transition. However at odds with expectations of universality, a large scatter in the value of the scaling exponents is reported during acoustic emission experiments. Here we numerically examine the precursory damage activity to reconcile the experimental observations with critical phenomena framework. Along with the strength of disorder, we consider an additional parameter that describes the progressive damageability of material elements at mesoscopic scale. This hardening behavior encapsulates the microfracturing processes taking place at lower length scales. We find that damage hardening can not only delay the final failure but also affect the preceding damage accumulation. When hardening is low, the precursory activity is strongly influenced by the strength of the disorder and is reminiscent of damage percolation. On the contrary, for large hardening, long-range elastic interactions prevail over disorder, ensuring a rather homogeneous evolution of the damage field in the material. The power-law statistics of the damage bursts is robust to the strength of the disorder and is reminiscent of the collective avalanche dynamics of elastic interfaces near the depinning transition. The existence of these two distinct universality classes also manifests as different values of the scaling exponent characterizing the divergence of the precursor size on approaching failure. Our finding sheds new light on the connection between the level of quasibrittleness of materials and the statistical features of the failure precursors. Finally, it also provides a more complete description of the acoustic precursors and thus paves the way for quantitative techniques of damage monitoring of structures-in-service.

Multiscale contact homogenisation: A novel perspective through the method of multiscale virtual power

The interaction between deformable bodies and rigid foundations undergoing finite strains is explored in this work with the Method of Multiscale Virtual Power, unlocking novel insights into contact homogenisation theories. The focus lies in establishing the foundational kinematical links across scales and achieving seamless homogenisation of the traction vector through rigorous duality arguments. Two distinct families of contact homogenisation models are formulated: surface–surface and surface–volume models. These families emerge from the virtual power balance established between the target macroscopic surface and the corresponding microscopic contact surface or volume. They not only represent fundamental kinematical entities at each scale but also enable the replacement of the heterogeneous contact traction distribution with a smooth, homogenised version. Microscale equilibrium problems and homogenisation relations, along with the relevant macro- and microscale quantities, are derived by means of straightforward variational arguments. Furthermore, potential finite size effects are addressed in the homogenised response, enhancing the reliability and applicability of the strategy across diverse scenarios. Promising submodels have been identified within both families, broadening the understanding of multiscale contact phenomena and including existing models as particular cases. The proposed continuum-based variational families of models naturally lead to generic finite element-based frameworks for contact homogenisation of heterogeneous solids, as described in a forthcoming paper.

ASSESSMENT OF SPECIFIC SURFACE AREA AND POROSITY OF SUBSOILS

Soils and subsoils have a complex structure consisting of several elements with different morphology and chemical composition, combining micro-, meso-, and macroporous substructures. Classical approaches, such as Brunauer-Emmett-Teller (BET) or Barrett-Joyner-Halenda (BJH) methods, cannot provide an accurate assessment of morphological characteristics of such solids, nor can do modern sophisticated numerical methods which depend on idealized models of pores that hardly reflect a real morphology of heterogeneous solids in question. The present study describes a way to overcome this difficulty by applying a combination of the αS-plot technique and a modified Remy-Poncelet method. The proposed approach allowed estimating the volumes of the untramicropores, supermicropores and the sum of meso- and macropores as well as the specific surface areas of the solid without ultramicropores and of the solid without all micropores. The former specific surface area is a more reasonable estimate of the area available for layer coverage than the BET specific surface area, and all together these characteristics provide a detail description of solid’s morphology. Subsoil samples excavated from different depths were used as exemplary solids. They mostly consisted of quartz, feldspar, and clay minerals and had different sorts of pores associated with the clay fraction and soil aggregates as well as a relatively large fraction of non-porous sand particles. Relations of the morphological properties with subsoil profile and mineralogical composition are discussed. For citation: Asnin L.D., Samoylov M.S., Pershina M.V., Tselishchev Yu.G., Sliusar N.N. Assessment of specific surface area and porosity of subsoils. ChemChemTech [Izv. Vyssh. Uchebn. Zaved. Khim. Khim. Tekhnol.]. 2024. V. 67. N 6. P. 55-64. DOI: 10.6060/ivkkt.20246706.6996.

Modeling resonant ultrasound of synthetic polycrystalline microstructures produced by Dream3D

Resonant ultrasound spectroscopy (RUS) is established for determining elastic constants of homogeneous solids. Often applied to polycrystalline materials like metals, RUS presumes homogeneity based on the microstructure's scale. This presumption is challenged with large grain sizes and higher-order modes, complicating homogeneity testing in real samples due to the need for 3D metallurgical analysis. An alternative is RUS modeling on digitized synthetic microstructures. This presentation investigates the RUS variational model with Rayleigh-Ritz solutions on microstructures created using Dream3D. It starts with a proof for solutions that minimize the Lagrangian in heterogeneous solids. Subsequently, Rayleigh-Ritz volume integrals are numerically evaluated for synthetic polycrystalline microstructures, as spatial heterogeneity precludes analytical solutions. No homogeneity assumption is made. The 3D integrals, requiring advanced GPU processing for a detailed mesh, lead to forward model results. These will be compared against common homogenization schemes, highlighting new insights when using RUS for material characterization of polycrystalline materials.

Data-driven homogenisation of the response of heterogeneous ductile solids with isotropic damage

We propose a computational procedure to derive a data-driven surrogate constitutive model capturing the elastic–plastic response and progressive damage of a heterogeneous solid. This is demonstrated by analysing the response to deformation of a volume element of a non-linear, n-phase random composite, used in this study as a model material. Finite Element simulations are conducted, imposing pseudo-random, multiaxial, non-proportional histories of macroscopic strain to such volume element. The corresponding predicted histories of macroscopic stresses and other variables are recorded, to form part of a training dataset for the surrogate model. Essential additional training data is obtained by recording the changes in the homogenised stiffness matrix of the volume element during the deformation, by performing a series of linear perturbation analyses. Supervised machine learning is applied to the data, proposing suitable sets of inputs and outputs and implementing a phenomenological constitutive model based on simple neural networks. This results in a data-driven model of high accuracy.

The Porothermoelastic Stress Evolution of Inhomogeneous Solid due to Near-Surface Reservoir

Anisotropic and heterogeneous solids, comprising a minimum of two or more elements with different properties, appear pervasively in rock materials including the pore structure and mineral composition of quartz and granite and usually have extensive applications in construction aggregates, dimension stone, geotechnical engineering, and petrology/geology. The elastic stress fields of inhomogeneous materials or composites inevitably change due to the presence of heterogeneous microstructure under applied external conditions, and hence the total mechanical and physical properties are affected by the temperature and pore pressure fluctuations beneath the surface. In this paper, the thermoporoelastic (TPE) approach on the basis of eigenstrain concept is introduced to predict the stress fields in fluid-saturated porous geologic materials like hydrocarbon reservoirs or aquifers by accounting for the coupling between thermal, poroelastic, and mechanical effects. It is an extension of the micromechanical theory that also incorporates thermal effects like pore pressure changes and temperature alternations. In addition, the TPE approach provides an important multiphysical modeling tool for understanding subsurface fluid-rock interactions and stress states in applications like unconventional oil/gas and geothermal energy.

Nonempirical Range-Separated Hybrid Functional with Spatially Dependent Screened Exchange.

Electronic structure calculations based on density functional theory (DFT) have successfully predicted numerous ground-state properties of a variety of molecules and materials. However, exchange and correlation functionals currently used in the literature, including semilocal and hybrid functionals, are often inaccurate to describe the electronic properties of heterogeneous solids, especially systems composed of building blocks with large dielectric mismatch. Here, we present a dielectric-dependent range-separated hybrid functional, screened-exchange range-separated hybrid (SE-RSH), for the investigation of heterogeneous materials. We define a spatially dependent fraction of exact exchange inspired by the static Coulomb-hole and screened-exchange (COHSEX) approximation used in many-body perturbation theory, and we show that the proposed functional accurately predicts the electronic structure of several nonmetallic interfaces, three- and two-dimensional, pristine, and defective solids and nanoparticles.

Phase-field modeling of stochastic fracture in heterogeneous quasi-brittle solids

Owing to the random nature of heterogeneity, damage and fracture behavior of quasi-brittle materials exhibits a considerable degree of uncertainty. Computational modeling of stochastic fracture in quasi-brittle materials has become an indispensable tool for analysis and design of engineering structures. To this end, we present in this paper a computational framework to capture probabilistic fracture in heterogeneous quasi-brittle solids by combining the random field theory and the phase-field cohesive zone model (PF-CZM). The spatial variation of the material strength and fracture energy is represented by a cross-correlated bivariate random field generated by the Karhunen–Loève expansion. The recently proposed PF-CZM is employed to simulate the stochastic crack nucleation and propagation in quasi-brittle solids. The objectivity of the Monte-Carlo simulation is achieved by imposing a specific condition on the phase-field length scale parameter and the correlation length of the random field. In particular, upon this condition the width of the fracture process zone (FPZ) is considerably smaller than the correlation length of the random field such that the material inside the FPZ does not exhibit significant spatial variations of mechanical properties. As the fracture energy is intrinsically incorporated in the PF-CZM, it is unnecessary in this case to explicitly consider the FPZ width. The resulting probabilistic PF-CZM together is applied to the Monte-Carlo simulations of fracture in concrete structures of different geometries. It is shown that the stochastic simulation results are insensitive to both the phase-field length scale parameter and the finite element mesh discretization as in the previous deterministic analyses. Enhanced with the specific condition on the involved characteristic lengths, the PF-CZM provides a viable tool for stochastic simulations of damage and fracture in quasi-brittle structures.

An explicit dynamic FFT method for homogenizing heterogeneous solids under large deformations

We propose an explicit displacement-based FFT method for computational homogenization and multi-scale modeling of heterogeneous solids. Unlike existing FFT methods that solve the static equilibrium equation using implicit iterative techniques, our proposed explicit method solves dynamic equations with the Central Difference method. The voxel-like discretizations of the periodic domain and the direct calculation of gradient and divergence operations using Discrete Fourier Transform and its inverse result in a straightforward algorithm that avoids convergence issues and does not require material consistent tangents. The stability condition of this explicit method is also addressed. Four numerical examples (2D particle-reinforced composites of hyperelasticity and viscoelasticity, a 3D fiber-reinforced composite, and a 3D polycrystalline metal) illustrate that the proposed method calculates as accurately and efficiently as implicit FFT methods for quasi-static problems. Furthermore, the proposed explicit displacement-based FFT method outperforms implicit FFT methods when dealing with microstructures comprising phases with very high contrast. Moreover, since the method does not require material consistent tangents, it extends the applicability of FFT methods to the homogenization of materials with complex or black-boxed constitutive models, where consistent tangents are challenging to compute.

Distillation of non-locality in porous solids

Non-local interactions are intrinsic to multiscale heterogeneous solids. The strength of the interactions exhibits a position-dependent character whose spatial distribution strongly correlates with the underlying microstructure. In this work, a variable-order fractional calculus-based framework to distill the position-dependent non-local effects is developed. By considering the example of porous plates, theoretical and numerical analyses will illustrate the ability of variable-order mechanics to enable parsimonious and causal models via a synthetic variable-order map that naturally measures the heterogeneous non-locality induced by the microstructure; in other terms, the non-classical role of the microstructure in determining the macroscopic response. The characteristic measure for non-locality, different from the local entanglement resulting from the porosity map, allows distillation of the macroscopic non-locality, analogous to quantum non-locality distillation. The macroscopic distillation also enables the additivity of the order map, that is the ability of assembled order maps to capture the response of combined porous plates.

Integrated digital image correlation for micro-mechanical parameter identification in multiscale experiments

Micromechanical constitutive parameters are important for many engineering materials, typically in microelectronic applications and material design. Their accurate identification poses a three-fold experimental challenge: (i) deformation of the microstructure is observable only at small scales, requiring SEM or other microscopy techniques; (ii) external loadings are applied at a (larger) engineering or device scale; and (iii) material parameters typically depend on the applied manufacturing process, necessitating measurements on material produced with the same process. In this paper, micromechanical parameter identification in heterogeneous solids is addressed through multiscale experiments combined with Integrated Digital Image Correlation (IDIC) in conjunction with various possible computational homogenization schemes. To this end, some basic concepts underlying multiscale approaches available in the literature are first reviewed, discussing their respective advantages and disadvantages from the computational as well as experimental point of view. A link is made with recently introduced uncoupled methods, which allow for identification of material parameter ratios at the microscale, still lacking a proper normalization. Two multiscale methods are analysed, allowing to bridge the gap between microstructural kinematics and macroscopically measured forces, providing the required normalization. It is shown that an integrated experimental--computational scheme provides relaxed requirements on scale separation. The accuracy and performance of the discussed techniques are analysed by means of virtual experimentation under plane strain and large strain assumptions for unidirectional fibre-reinforced composites. The robustness against image noise is also assessed.

Modeling progressive failure and crack evolution in a randomly distributed fiber system via a coupled phase-field cohesive model

Despite the high microstructural heterogeneity of fiber-reinforced composites, few modeling framework provides a comprehensive and detailed understanding of the failure mechanisms of these materials. The aim of this work is to present a coupled phase-field cohesive-modeling framework that can precisely capture the progressive failure and damage behaviors of multiphasic microstructures and multifiber systems. Here, the phase-field method captures crack evolution in the matrix, and a coupled cohesive-zone model is introduced to characterize interfacial debonding. The novel model framework comprises the following novel aspects. (1) A newly developed scalar indicator that directly extracts inelastic strain from the total strain field and couples the cohesive traction-separation law with the phase-field model to determine the regularized interfacial displacement jump. (2) The periodic boundary conditions in the coupled phase-field cohesive framework are incorporated to characterize crack evolution in random fiber systems. (3) A complete set of failure modes, namely crack initiation, propagation, kinking, and coalescence are characterized in highly heterogeneous solids. Parametric studies of the novel framework yield numerical results that are highly consistent with experimental findings and reveal the effects of fiber distributions, fiber volume fractions, and boundary conditions on the nonlinear mechanical behaviors of fiber-reinforced composites. The results demonstrate the excellent potential of the novel numerical framework to evaluate the mechanical performances of composite materials in engineering applications.

Ultrasonic imaging in highly heterogeneous backgrounds

This work formally investigates the differential evolution indicators as a tool for ultrasonic tracking of elastic transformation and fracturing in randomly heterogeneous solids. Within the framework of periodic sensing, it is assumed that the background at time t ∘ contains (i) a multiply connected set of viscoelastic, anisotropic and piecewise homogeneous inclusions, and (ii) a union of possibly disjoint fractures and pores. The support, material properties and interfacial condition of scatterers in (i) and (ii) are unknown, while elastic constants of the matrix are provided. The domain undergoes progressive variations of arbitrary chemo-mechanical origins such that its geometric configuration and elastic properties at future times are distinct. At every sensing step t ∘ , t 1 , … , multi-modal incidents are generated by a set of boundary excitations, and the resulting scattered fields are captured over the observation surface. The test data are then used to construct a sequence of wavefront densities by solving the spectral scattering equation. The incident fields affiliated with distinct pairs of obtained wavefronts are analysed over the stationary and evolving scatterers for a suit of geometric and elastic evolution scenarios entailing both interfacial and volumetric transformations. The main theorem establishes the invariance of pertinent incident fields at the loci of static fractures and inclusions between a given pair of time steps, while certifying variation of the same fields over the modified regions. These results furnish a basisfor theoretical justification of differential evolution indicators for imaging in complex composites which, in turn, enable the exclusive tomography of evolution in a background endowed with many unknown features.

Effects of Localized Interface Phonons on Heat Conductivity in Ingredient Heterogeneous Solids

Phonons are the primary heat carriers in non-metallic solids. In compositionally heterogeneous materials, the thermal properties are believed to be mainly governed by the disrupted phonon transport due to mass disorder and strain fluctuations, while the effects of compositional fluctuation induced local phonon states are usually ignored. Here, by scanning transmission electron microscopy electron energy loss spectroscopy and sophisticated calculations, we identify the vibrational properties of ingredient-dependent interface phonon modes in Al x Ga1 – x N and quantify their various contributions to the local interface thermal conductance. We demonstrate that atomic-scale compositional fluctuation has significant influence on the vibrational thermodynamic properties, highly affecting the mode ratio and vibrational amplitude of interface phonon modes and subsequently redistributing their modal contribution to the interface thermal conductance. Our work provides fundamental insights into understanding of local phonon-boundary interactions in nanoscale inhomogeneities, which reveal new opportunities for optimization of thermal properties via engineering ingredient distribution.

Interfacial fracture analysis for heterogeneous materials based on phase field model

Simulating crack initiation/propagation in heterogeneous materials has long been a challenge due to their complex topology and the complicated interactions between each phase. In order to address this task, an interfacial debonding phase field model is established for interfacial fracture behavior of highly heterogeneous solids in this work. Inspired by the phase field framework, a discrete interface is transformed into a smeared interface using interface phase field. The developed model has the following novelties: (1) interface properties are smeared such that singularity of material property can be eliminated; (2) due to the interaction between characteristic length scale parameters of the interface and the crack, critical energy release rate of interface is regularized; and (3) it can predict complex cracking phenomena such as multiple crack initiation, merging and branching. Particularly, the proposed model is validated experimentally and numerically. With the proposed model, influence of interface properties such as Young’s modulus and critical energy release rate on fracture behavior is studied in detail. The proposed model is potential in predicting interfacial debonding and the crack kinking path in complicated quasi-brittle heterogeneous materials.

A non‐uniform rational B‐splines enhanced finite element formulation based on the scaled boundary parameterization for the analysis of heterogeneous solids

AbstractThe contribution is concerned with a finite element formulation for the nonlinear analysis of heterogeneous solids in boundary representation. It results in an element with an arbitrary number of curved boundary edges. The curved edges can be parametrized by, for example, non‐uniform rational B‐splines (NURBS). The presented element formulation is based on the scaling concept, which is adopted from the so‐called scaled boundary finite element method (SBFEM). In contrast to SBFEM, the proposed method uses a numerical approximation for the displacement response in scaling direction. This enables the analysis of geometrically and physically nonlinear problems in solid mechanics. The interpolation at the boundary in circumferential direction is independent of interpolation in scaling direction. Thus, different basis functions can be used for each direction, for example, NURBS basis functions in circumferential and Lagrange basis functions in radial direction. It allows the construction of polygonal elements with an arbitrary number of curved sides, which are described by either whole NURBS curves or NURBS curves' segments. The advantage of the presented element formulation is the flexibility in mesh generation. For example, using Quadtree algorithms, a fast and reliable mesh generation can be achieved. Furthermore, in connection with trimming algorithms, the element formulation allows a precise representation of the geometry even with coarse meshes. Some benchmark tests are presented to evaluate the accuracy of the proposed numerical method against analytical solutions, and a comparison to standard element formulations is given as well.

Multi-Modes Multiscale Approach of Heat Transfer Problems in Heterogeneous Solids with Uncertain Thermal Conductivity

Multi-Modes Multiscale Approach of Heat Transfer Problems in Heterogeneous Solids with Uncertain Thermal Conductivity

ФОРМИРОВАНИЕ МАРШРУТА МАГИСТРАЛЬНОЙ ТРЕЩИНЫ РАЗРУШЕНИЯ В МНОГОУРОВНЕВОЙ СТРУКТУРЕ КОНГЛОМЕРАТНЫХ СТРОИТЕЛЬНЫХ КОМПОЗИТОВ

In the publication, conglomerate building composites (concrete) are identified as heterogeneous solids with a hierarchically organized spatial-geometric structure with a characteristic dimension from 10-10 to 10-1 m, with a minimum of 5-6 scale levels and three types of substructure design that differ in scale, genesis and mechanics of properties manifestation. The first type is characteristic of the macro-, meso- and microscale levels and is taken in the form of a two-component “construction” of a spatially continuous matrix and discrete solid and gaseous (macropores) inclusions deterministically and stochastically distributed in it; the second type refers to the submicro-, ultra-micro- and nanoscale levels and is believed to be in the form of a “microscale spatial structure” of new formations of a cementitious substance from consolidated individual crystalline differences; the third type, finally, corresponds to the atomic-molecular structure of new formations of the cementing substance. Characteristics of the distinguished types of substructures are given according to the scale of their components, the peculiarities of formation, the mechanics of manifestation of properties, design criteria and means of synthesis of each substructure. The patterns of formation of the fracture route in substructures of all types and substances of each scale level are analyzed. In this case, the development of the stress-strain state of the conglomerate composite according to the principle of energy dissipation, localization and increase (concentration) of stress is realized in the direction from the macro- to the atomic-molecular level of the structure of the composite, and the destruction itself and, accordingly, the formation of the crack route in time and in space of the composite passes in the direction from the atomic-molecular level to the macrolevel in a cascade through all intermediate structural levels. Within the framework of an integrated mechano-physico-chemical approach, the place of thermofluctuation theory (fracture physics) at the stages of breaking single atomic-molecular bonds and crack mechanics at the stages of development of micro- and macrodamage is shown. The possibilities of using theoretical principles of crack route formation to formulate and solve practical problems of designing and synthesizing optimal structures of conglomerate building composites are discussed.

Improving the performance of biomass-based electrocatalysts by means of hot pressing

To address the problem of decoupling arising from weak binding of extrinsic agents to biomass, we propose a hot-pressing method by which heterogeneous solids become firmly integrated in confined spaces at elevated temperatures and pressures.

A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method

Physics informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. Here, the training of the network is equivalent to the minimization of a loss function that includes the governing (partial differential) equations (PDE) as well as initial and boundary conditions. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form (given governing equation) which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem (which can be obtained from the weak form) is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids and check the performance of the proposed PINN model against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). It is concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and data from physical FE simulations for a fast and accurate design of composite materials in future developments.