Complex Microstructure Research Articles

Superior Resistance and Ductility through Novel Quench-and Partitioning-Path in Complex Refined Microstructure

A well-designed complex microstructure containing both soft and hard micro-constituents can enhance the mechanical properties of steel. In this study, commercial AISI 9254 steel was annealed at 900 °C, rapidly cooled to 550 °C for 500 s to promote approximately 50% fine pearlitic transformation, quenched to 125 °C for partial martensitic transformation, and finally heated to 375 °C for 1800 s to complete the partitioning stage in a novel quench and partitioning (Q&P) process. Tensile testing revealed a yield strength (YS) of ≈ 1500 MPa, an ultimate tensile strength (UTS) of ≈ 1570 MPa, and a total elongation of ≈ 13.85%. This high yield strength indicates the ability of the material to support the development of lightweight, yet high-strength components for demanding applications. Additionally, the balanced total elongation helps mitigate the risk of brittle failure, enhancing fracture toughness and reducing the likelihood of premature failures in critical structural applications. These results indicate an increase of approximately 8.3% in strength and 34.5% in ductility compared to the as-received 9254 steel. X-ray analysis revealed that the complex microstructure had fewer crystallographic defect densities than the as-received sample. Secondary electron images showed ultrafine martensite laths and cementite lamellae within the body-centered cubic (BCC) matrix, with some proeutectoid ferrite found at prior austenite grains. Electron backscattered diffraction (EBSD) analysis estimated low internal distortion in martensite laths, with average crystal defect densities around 2.25 × 1014 m−2. The BCC matrix contained ferrite and martensite, with carbide particles and a small amount of retained austenite detected by transmission electron microscopy (TEM). These findings confirm the enhanced mechanical properties of commercial 9254 steel through the novel Q&P processing.

Multispectral laser-scanning pulse-sampling fluorescence lifetime system for large-scale tissue imaging.

We report a multispectral laser-scanning pulse-sampling fluorescence lifetime imaging (LSPS-FLIm) system designed for rapid, high-resolution imaging of large tissue specimens. This system provides a substantial imaging field of view (FoV) of 6 × 15 cm2 with a high spatial resolution of ∼17.5 µm. The LSPS-FLIm system has been tested on a range of fluorescent dyes, endogenous tissue fluorophores, and tissue specimens with varied sizes and properties. These tests demonstrate the system's versatility in resolving morphological and molecular features, enabling centimeter-scale FoV imaging, and distinguishing complex microstructures in tissue specimens. With the capability to maintain high imaging quality and acquisition speed while minimizing tissue damage, LSPS-FLIm represents a promising advancement in the field of fluorescence lifetime imaging for biological and clinical applications.

Research on multi-source microstructure image recognition of foam ceramics using convolutional network combine with frequency domain

Foam ceramics are widely used in industrial applications due to their unique properties, including high porosity, lightweight, and high-temperature resistance. However, their complex microstructure presents significant challenges for image analysis. Traditional machine learning methods often fall short in capturing both global feature dependencies and detailed representations. To address this, a novel artificial intelligence recognition model, FD-Conv, is proposed, which combines the global information processing capabilities of Transformers with the local feature extraction strengths of convolutional neural networks. Additionally, a frequency domain block detail enhancement mechanism is introduced to improve recognition accuracy. Experimental results demonstrate that the FD-Conv model enhances recognition accuracy by at least 7.6% compared to state-of-the-art methods. Furthermore, the model effectively identifies foam ceramics with varying compositions and formulations and quantifies their microstructural phase characteristics. This research aims to advance the application of foam ceramic microstructure image analysis by improving recognition accuracy, particularly in multi-source microscopic image feature learning and pattern recognition.

Precipitation-controlled grain boundary engineering in a cast & wrought Ni-based superalloy

Grain boundary engineering (GBE) describes the various approaches to control the fraction and composition of special boundaries in designing engineering materials. For example, complex thermo-mechanical processing routes are harnessed to increase the fraction of Σ3 twin boundaries, bringing about improvements to strength, toughness, and corrosion properties in austenitic stainless steels. However, high-temperature gas turbine engine components designed for the most demanding applications must be manufactured from Ni-based superalloys. To provide the required high-temperature strength, their microstructures are highly complex and consist of an austenitic γ-matrix, γ' precipitates, and grain boundary (GB) carbides, and/or borides. GBE approaches for Ni-based superalloys have been reported to reduce in-service GB cracking via targeted engineering of the GB microstructure. However, the same complex microstructures severely limit the processability of the most advanced Ni-based superalloys required to develop the next generation gas turbine engines.To tackle this challenge, we propose precipitation-controlled GBE for Ni-based superalloys with limited formability, via formation of GB serrations and precipitation upon slow cooling. We showcase our approach by achieving controlled GB precipitation of GB-γ' and M6C carbides in the cast & wrought Ni-based superalloy René 41 via a simple heat treatment. Ductility improvements are predicted via crystal plasticity modeling due to increased slip transmission compatibility. Instead of generating additional Σ3 twin boundaries only, these interfaces are directly incorporated into the GB network. It is shown that precipitation-controlled GBE is enabled by GB-γ' nucleating heterogeneously on M6C whereby competitive coarsening of GB-γ' provides the driving force. The fundamental mechanisms of our GBE approach are discussed and summarized in a qualitative microstructural model.

Prediction of Mechanical Properties and Fracture Behavior of TC17 Linear Friction Welded Joint Based on Finite Element Simulation.

TC17 titanium alloy is widely used in the aviation industry for dual-performance blades, and linear friction welding (LFW) is a key technology for its manufacturing and repair. However, accurate evaluation of the mechanical properties of TC17-LFW joints and research on their joint fracture behavior are still not clear. Therefore, this paper used the finite element numerical simulation method (FEM) to investigate the mechanical behavior of the TC17-LFW joint with a complex micro-structure during the tensile processing, and predicted its mechanical properties and fracture behavior. The results indicate that the simulated elastic modulus of the joint is 108.5 GPa, the yield strength is 1023.2 MPa, the tensile strength is 1067.5 MPa, and the elongation is 1.98%. The deviations from measured results between simulated results are less than 2%. The stress and strain field studies during the processing show that the material located at the upper and lower edges of the joint in the WZ experiences stress and strain concentration, followed by the extending of the stress and strain concentration zone toward the center of the WZ. And finally, the strain concentration zone covered the entire WZ. The fracture behavior studies show that the material necking occurs in the TMAZ of TC17(α + β) and WZ, while cracks first appear in the WZ. Subsequently, joint cracks propagate along the TC17(α + β) side of the WZ until fracture occurs. There are obvious tearing edges formed by the partial tearing of the WZ structure in the simulated fracture surface, and there are fracture surfaces with different height differences at the center of the joint crack, indicating that the joint has mixed fracture characteristics.

Computational Characterization of the Structure, Energy, Strengths, and Fracture Resistances of Symmetric Tilt Grain Boundaries in Ice.

Using an interatomic potential that can capture the tetrahedral configuration of water molecules (H2O) in ice without the need to explicitly track the motion of the O and H atoms, coarse-grained (CG) atomistic simulations are performed here to characterize the structures, energy, cohesive strengths, and fracture resistance of the grain boundaries (GBs) in polycrystalline ice resulting from water freezing. Taking the symmetric tilt grain boundaries (STGBs) with a tilting axis of ⟨0001⟩ as an example, several main findings from our simulations are (i) the GB energy, EGB, exhibits a strong dependence on the GB misorientation angle, θ. The classical Read-Shockley model only predicts the EGB - θ relation reasonably well when θ < 20° or θ > 45° but fails when 20° < θ < 45°; (ii) two "valleys" appear in the EGB-θ landscape. One occurs at θ = 22° for Σ14(2̅31̅0) GB, and the other is at θ = 32° for Σ26(3̅41̅0) GB. These two GBs might be the most common in polycrystalline ice; (iii) all the STGB structures under consideration here are found to be a collection of edge dislocations with a Burgers vector of b = 1/3⟨112̅0⟩. The core structure of this edge dislocation is composed of a pentagon and a heptagon atomic ring. The separation and orientation of the structure units (SUs) at the GB exhibit a strong dependence on θ; (iv) the length of an atomic bond within the SUs, rather than EGB and θ which are often used in the literature, is identified as one controlling parameter that dictates the intrinsic GB cohesive strength; (v) characterization of the fracture resistance of the GB containing an initial crack is beyond the reach of nanoscale atomistic simulations but is feasible in concurrent atomistic-continuum (CAC) simulations that can simultaneously retain the atomic GB structure together with the long-range stress field within one model. The above findings provide researchers with a stepping stone to understand the complex microstructure of polycrystalline ice and its response to external forces from the bottom up. Such knowledge may be consolidated into constitutive rules and then transferred into the higher length scale models, such as cohesive zone finite element models (CZFEMs), for predicting how polycrystalline ice fractures at laboratory and even geophysical length scales.

Mechanical Properties of Cobalt Chromium Alloy Manufactured by Direct Energy Deposition Technology

Abstract Cobalt chromium (CoCr), a well-known biocompatible material, is additively manufactured using direct energy deposition (DED) technology in this study. This study investigates some important mechanical characteristics of the additively manufactured CoCr using two different numerical simulation methods in addition to mechanical tests and experiments. Mechanical experiments such as hardness, wear, and flexural bending test were conducted on DED-processed samples. All experiments were also conducted on conventionally processed CoCr specimens for comparison purposes. DED-processed CoCr samples exhibited a complex microstructure with a variety of features such as cellular, columnar, and equiaxed grains within their melt pools. While the DED-processed sample had a lower hardness compared to the conventionally processed one, it exhibited a higher wear resistance. The tensile strength obtained from resonance frequency testing was higher for the DED-processed CoCr sample compared to the conventionally fabricated one. The out-of-plane mechanical strength of CoCr samples was measured by conducting flexural bending test, and the conventional sample showed a higher flexural modulus than the DED sample. The bend tests were also numerically simulated using two different finite element analysis (FEA) procedures. The FEA results for the conventionally processed samples are in good agreement with the ones obtained from the experimental flexural bending test. The results of the FEA studies on the DED-processed samples were within 10-20 % of the experimental ones, showing the potential of numerical methods in estimating this property without the need of mechanical testing.

Autonomous particle-shape-based classification and identification of calcareous soils through machine learning

Recent geotechnical and geological applications have highlighted numerous challenges associated with conventional particle-size based soil classification, particularly for special calcareous soils widely used in marine and offshore constructions. Calcareous soils are always characterized by complex microstructures in terms of loose fabric, irregular particle shapes and abundant intra-particle pores. Therefore, this study aims to establish an autonomous particle-shape categorization criteria for calcareous soils, and further facilitate their intelligent particle identification and classification using image-based machine learning (ML) techniques. First, an extensive collection of images of individual calcareous sand particles were acquired through microscopic photography to establish the datasets for ML training. Four key shape parameters including circularity, roundness, aspect ratio and convexity were measured based on image processing and analysis. Subsequently, an unsupervised ML algorithm, K-means clustering, was applied to these shape parameters to autonomously determine optimal particle-shape categories for calcareous soils. Finally, based on the obtained particle shape classes, two supervised CNN algorithms, AlexNet and YOLO-v3, were trained and applied to automatically identify and classify individual particles or particle assemblies of calcareous soils from various raw image types. The predictive results demonstrate the high accuracy and efficiency of ML models in identifying and classifying the particle shapes of calcareous soils.

Fabrication of 3D Functional Nanocomposites Through Post-Doping of Two-Photon Microprinted Nanoporous Architectures.

Two-photon lithography (TPL) enables the fabrication of complex 3D structures with sub-micrometer precision. Incorporation of new functionalities into TPL-printed structures is key to advance their applications. A prevalent approach to achieve this is by directly adding functional nanomaterials into the photoresist (called "pre-doping"), which has several inherent challenges including material compatibility, light scattering, and nanoparticle agglomeration. Here, a conceptually different "post-doping" strategy is proposed, where the functionality of the TPL-printed architectures is achieved by impregnating functional materials into their nanoporous 3D mimics. Using the principle of polymerization-induced phase separation, TPL printing of complex microarchitectures with well-defined nanoporous structures having pores of ≈420nm is realized, which allows spontaneous impregnation of functional liquids via capillary effect. Importantly, unlike the "pre-doping" approach that requires printing optimization for each photoresist, this strategy is highly versatile in terms of functionalities possible. As a proof-of-concept, the impregnation of several functional liquids into TPL-printed porous microstructures is demonstrated: a fluorinated-lubricant, an ionic liquid, and three types of fluorescent liquids, conferring the microstructures with slippery, conductive, and localized fluorescence properties, respectively. Such versatility to fabricate complex microstructures with tailorable and localized functionalities is expected to open new possibilities in wide fields including bionics, electronics, and cell biology.

Data-Driven Framework for the Prediction of PEGDA Hydrogel Mechanics.

Poly(ethylene glycol) diacrylate (PEGDA) hydrogels are biocompatible and photo-cross-linkable, with accessible values of elastic modulus ranging from kPa to MPa, leading to their wide use in biomedical and soft material applications. However, PEGDA gels possess complex microstructures, limiting the use of standard polymer theories to describe them. As a result, we lack a foundational understanding of how to relate their composition, processing, and mechanical properties. To address this need, we use a data-driven approach to develop an empirical predictive framework based on high-quality data obtained from uniaxial compression tests and validated using prior data found in the literature. The developed framework accurately predicts the hydrogel shear modulus and the strain-stiffening coefficient using only synthesis parameters, such as the molecular weight and initial concentration of PEGDA, as inputs. These results provide simple and reliable experimental guidelines for precisely controlling both the low-strain and high-strain mechanical responses of PEGDA hydrogels, thereby facilitating their design for various applications.

Multimodal defect analysis and application of virtual machining for solid-state manufactured aluminium structure

AbstractAdditive friction stir deposition (AFSD) is an emerging solid-state non-fusion additive manufacturing (AM) technology, which produces parts with wrought-like material properties, high deposition rates, and low residual stresses. However, impact of process interruption on defect formation and mechanical properties has not yet been well addressed in the literature. In this study, Al6061 aluminium structure with two final heights and deposition interruption is successfully manufactured via AFSD and characterised. Defect analysis conducted via optical microscopy, electron microscopy, and X-ray computed tomography reveals &gt; 99% relative density with minimal defects in centre of the parts. However, tunnel defects at interface between substrate and deposit as well as kissing bonds are present. Edge of deposit contains tunnel defects due to preference for greater material deposition on advancing side of rotating tool. Virtual machining highlights the ability to remove defects via post-processing, avoiding mechanical performance impact of stress concentrating pores. Electron backscatter diffraction revealed regions with localised shear bands that contain 1–5 µm equivalent circular diameter grains. Kissing bonds are exhibited in areas separated by large grain size difference. Meanwhile, Vickers hardness testing reveals hardness variation with deposit height. This work advances the understanding of complex microstructure development, material flow, and mechanical behaviour of AFSD Al6061 alloy.

Additive Manufacturing of Binary and Ternary Oxide Systems Using Two-Photon Polymerization and Low-Temperature Sintering.

Multicomponent oxide systems have many applications in different fields such as optics and medicine. In this work, we developed new hybrid photoresists based on a combination of an organic acrylate resin and an inorganic sol, suitable for 3D printing via two-photon polymerization (2PP). The inorganic sol contained precursors of a binary SiO2-CaO or a ternary SiO2-CaO-P2O5 system. Complex microstructures were 3D printed using these hybrid photoresists and 2PP. The obtained materials were characterized using thermogravimetric analysis (TGA), Fourier transform infrared spectroscopy (FTIR), and scanning electron microscopy (SEM) techniques. Our results revealed that the produced microstructures were able to endure sintering at 700 °C without collapsing, leading to scaffolds with 235 and 355 nm resolution and pore size, respectively. According to the TGA analysis, there was no significant mass loss beyond 600 °C. After sintering at 500 °C, the FTIR spectra showed the disappearance of the characteristic bands associated with the organic phase, and the presence of bands characteristic of the binary and ternary oxide systems and carbonate groups. The SEM images showed different morphologies of agglomerated nanoparticles with mean sizes of about 20 and 60 nm for ternary and binary systems, respectively. Our findings open the way towards precise control of bioglass scaffold fabrication with tremendous design flexibility.

A Finite Operator Learning Technique for Mapping the Elastic Properties of Microstructures to Their Mechanical Deformations

ABSTRACTTo obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics‐informed neural networks and concept of neural operators. We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics‐based training yields higher accuracy compared with purely data‐driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high‐resolution solutions, avoiding the need to use basic interpolation techniques. The first one is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid points. Next, Fourier‐based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. The proposed approach draws from finite element and deep energy methods but generalizes and enhances them by learning parametric solutions without relying on external data. Compared with other operator learning frameworks, it leverages finite element domain decomposition in several ways: (1) it uses shape functions to construct derivatives instead of automatic differentiation; (2) it automatically includes node and element connectivity, making the solver flexible for approximating sharp jumps in the solution fields; and (3) it can handle arbitrary complex shapes and directly enforce boundary conditions. We provided some initial comparisons with other well‐known operator learning algorithms, further emphasize the advantages of the newly proposed method.

Identification of Structural Constituents in Advanced Multiphase High-Strength Steels Using Electron Back-Scattered Diffraction

This study addresses the characterization of the particular microstructural constituents of multiphase transformation-induced plasticity (TRIP)-aided steels belonging to the first and third generations of Advanced High Strength Steels (AHSS) to explore the possibilities of the EBSD method. Complex microstructures composed of ferrite, bainite, retained austenite and martensite were qualitatively and quantitatively assessed. Microstructural constituents with the same crystal structure were distinguished using characteristic EBSD parameters like confidence index (CI), image quality (IQ), kernel average misorientation (KAM) and specific crystallographic orientation relationships. A detailed linear analysis of the IQ parameter and misorientation angles was also performed. These tools are very helpful in linking different symmetric or asymmetric features of metallic alloys with a type of their structure and morphology details. Two types of samples were investigated: thermomechanically processed and subjected to 10% tensile strain to study the microstructural changes caused by plastic deformation.

Modeling of Characteristics of Complex Microstructure and Heterogeneity at the Core Scale

Complex pore structures and strong matrix heterogeneity distinguish carbonate rocks, but there is a lack of comprehensive methods to describe these characteristics. In this study, a integrated approach is proposed to improve the accuracy and adaptability of velocity prediction methods, using a modified squirt flow model based on microcrack structures to characterize complicated pore structures, and a mixed random medium model to represent significant heterogeneity. In addition, the microcrack structure is obtained by inversion, but different from the D-Z method, each group of microcracks corresponds to a different equivalent medium model, so as to improve the accuracy of the inversion results. And the modified squirt flow model takes into account the attenuation caused by local flow between microcracks. The random medium model simulates the inhomogeneous body in the core by adjusting the autocorrelation length a and b, the rounding coefficient n, and the angle θ. A comparative study of the measured data of five limestone and dolomite samples reveals that the P-wave prediction error of the new model is less than 5%, whereas the Biot model is less than 10%, implying that the prediction accuracy of the new model is better.

Trends in Photopolymerization 3D Printing for Advanced Drug Delivery Applications.

Since its invention in the 1980s, photopolymerization-based 3D printing has attracted significant attention for its capability to fabricate complex microstructures with high precision, by leveraging light patterning to initiate polymerization and cross-linking in liquid resin materials. Such precision makes it particularly suitable for biomedical applications, in particular, advanced and customized drug delivery systems. This review summarizes the latest advancements in photopolymerization 3D printing technology and the development of biocompatible and/or biodegradable materials that have been used or shown potential in the field of drug delivery. The drug loading methods and release characteristics of the 3D printing drug delivery systems are summarized. Importantly, recent trends in the drug delivery applications based on photopolymerization 3D printing, including oral formulations, microneedles, implantable devices, microrobots and recently emerging systems, are analyzed. In the end, the challenges and opportunities in photopolymerization 3D printing for customized drug delivery are discussed.

Advanced Computational Modeling with FVDAM for Homogenization of Reinforced Concrete Beams

Reinforced concrete is widely used in civil engineering due to its strength, durability, and versatility. This material is heterogeneous, as it combines concrete with steel bars, giving it a tensile strength much higher than that of simple concrete. However, the computational modeling of this type of material is challenging due to the need to consider the distinct properties of the phases, generating considerable computational costs. According to Almeida (2018), the simulation process of reinforced concrete beams requires a meticulous analysis of the interactions between concrete and steel, as well as the non-linear behavior of the material under different load conditions. This complexity significantly increases the time and resources needed to perform accurate computational analyses. An alternative to modeling reinforced concrete is to replace it with a homogeneous material with properties/behaviors equivalent to those of the original heterogeneous material.The Finite-Volume Direct Averaging Micromechanics (FVDAM) method emerges as a promising and efficient alternative for obtaining the effective elastic properties of composite materials with complex microstructures. In this study, the efficacy of FVDAM in the homogenization of reinforced concrete beams was evaluated. The stiffness using this technique was compared with that obtained using the calculations presented by Pinheiro (2007). In addition, two unit cell models were compared, differentiating the shape of the fiber: square and circular (Escarpini and Almeida, 2023).In this work, it is expected to understand the influence of the fiber shape in the unit cell and compare the results of FVDAM with those of Pinheiro, in terms of stiffness, evaluating issues in the context of safety.

Deep learning assisting construction of heat transfer constitutive relationships for porous media

Predicting heat transfer in porous materials is challenging due to their complex microstructure, and traditional experimental and theoretical methods often fall short. To address this, we present a machine learning-based approach using a single Descriptor-to-Property Network (D2P-Net) to identify the relationships between the effective thermal conductivity (ETC) of porous media with the structural descriptors. The D2P-Net, an ensemble of decision trees, uses eleven structural descriptors, such as porosity, pore size, and connectivity, derived from four distinct types of three-dimensional porous structures. These structures are computationally generated, and their ETC is computed using the Lattice Boltzmann Method (LBM) to create a robust dataset for training. The model is trained using mean squared error (MSE) employed as the loss function, and achieves an R2 value of 0.994 and a root mean square error (RMSE) of 0.229, indicating high predictive accuracy. Additionally, SHapley Additive exPlanations (SHAP) values are employed to analyze the importance of each structural descriptor, offering valuable insights into their contributions to ETC predictions. This approach provides a reliable and interpretable method for understanding and optimizing the thermal properties of porous materials, with potential applications in areas requiring efficient thermal management, such as hypersonic vehicles.

Coupled Hydro-Gas-Mechanical 3D Modeling of LASGIT Experiment

Gas transport simulation in bentonite for radioactive waste disposal poses challenges for numerical models due to its complex microstructure. Understanding the processes involved is a prerequisite for assessing gas flow's impact on repository layouts. The DECOVALEX23 (D-2023) Task B (Large Scale Gas Injection Test: LASGIT) project aims to advance numerical techniques for predicting gas flow in repository systems through gas injection tests on compacted bentonite at the British Geological Survey (BGS). This study develops a comprehensive coupled hydro-gas-mechanical 3D numerical model to simulate the test, considering heterogeneous initial permeability and embedded fractures. Addressing bentonite swelling, three gap closure scenarios for the canister-bentonite blocks gap interface were considered. The model reproduces observed test behaviors, capturing preferential gas flow paths. Sensitivity analysis explores variations in volume factor sensitivity, calibration, hydraulic conductivity of interfaces, heterogeneity, permeability, and model parameters, contributing to a deeper understanding of the phenomenon's complexity. The proposed hydraulic modeling, enriched by considerations of gap closure states, predicts measured evolution of gas injection trends. suggesting reliability and potential applicability for similar conditions and facilitating a comprehensive analysis of its impact on gas testing processes. Additionally, the embedded fracture models underscore the critical role of fracture behavior and dilatancy in determining the system's hydro-mechanical response, with significant sensitivity to these factors influencing stress and pore pressure evolution. Hydro-mechanical models demonstrate that modeling approaches involving embedded fractures and dilatancy significantly influence gas pathways and entry gas pressure. System volume plays pivotal role in the analysis, while sensitivity analysis of contact transmissivity reveals potential influences on preferential gas pathway formation. Hydraulic and hydro-mechanical modeling methods show promise for further numerical investigations, indicating potential for yielding meaningful insights in future studies.

Multiscale damage analysis based on the variational effective model

AbstractIn the damage analysis of material with complex microstructures, a full‐scale direct numerical simulation (DNS) could produce huge computational costs. Alternatively, multiscale modeling is a more effective method by using less degrees of freedom to balance the calculation accuracy and cost. In this paper, a new multiscale damage modeling method is proposed based on the framework of the variational effective model originally presented by Jezdan et al. and the relaxation damage model presented by Junker et al. A variational statement for the free energy and the dissipation potential for a coarse scale model is constructed by relating the free energy and dissipation potential of the fine scale model. In this way, the coarse scale damage variable is defined instead of directly solving in the fine scale model; therefore, the computational efficiency can be increased. Besides, the relaxation damage model can effectively avoid ill‐posed boundary value problems in damage analysis without use of gradients or integration techniques. Two numerical examples are provided to demonstrate the effectiveness of the proposed method.