Pub Date : 2025-11-30DOI: 10.1016/j.jmps.2025.106449
Paolo Maria Mariano , Domenico Mucci
For simple elastic bodies in small strain regime, the convexity of the energy allows one to discuss equilibrium problems under stress constraints (the specification of an admissible convex region in the stress space) in terms of the complementary energy. In the presence of large strains, the necessary lack of energy convexity does not allow one to retrace the same path. A significant concept of complementary energy in large strain regime rests on the Legendre transform of the energy with respect to the deformation gradient, its cofactor and determinant. The related minimum problem necessarily requires that constraints be assigned to the derivatives of the energy density with respect to the variables already listed (these derivatives are required to be in a convex subset of an appropriate linear space). A problem is to characterize the related stresses in terms of a constrained energy. We tackle this problem for radially symmetric simple bodies under radial deformations and show how the resulting stress is bounded. We also prove the existence of radially symmetric minimizers for the constrained elastic energy under Dirichlet boundary conditions.
{"title":"Stress boundedness and existence of radial minimizers in constrained nonlinear elasticity","authors":"Paolo Maria Mariano , Domenico Mucci","doi":"10.1016/j.jmps.2025.106449","DOIUrl":"10.1016/j.jmps.2025.106449","url":null,"abstract":"<div><div>For simple elastic bodies in small strain regime, the convexity of the energy allows one to discuss equilibrium problems under stress constraints (the specification of an admissible convex region in the stress space) in terms of the complementary energy. In the presence of large strains, the necessary lack of energy convexity does not allow one to retrace the same path. A significant concept of complementary energy in large strain regime rests on the Legendre transform of the energy with respect to the deformation gradient, its cofactor and determinant. The related minimum problem necessarily requires that constraints be assigned to the derivatives of the energy density with respect to the variables already listed (these derivatives are required to be in a convex subset of an appropriate linear space). A problem is to characterize the related stresses in terms of a constrained energy. We tackle this problem for radially symmetric simple bodies under radial deformations and show how the resulting stress is bounded. We also prove the existence of radially symmetric minimizers for the constrained elastic energy under Dirichlet boundary conditions.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106449"},"PeriodicalIF":6.0,"publicationDate":"2025-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145651076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-29DOI: 10.1016/j.jmps.2025.106459
Qiancheng Ren , Yilan Xu , Jinglan Liu , Xiaochu Chen , Qi Yang , Jiayuan Fang , Pei Zhao
The evolution of a solid interface from coupling to friction and its mechanisms still face challenges. Here, we use large-twist-angle bilayer graphene combined with isotope-labeling-assisted Raman spectroscopy to measure the mechanical behaviors of its two layers from coupling to friction. Results show that as the strain of the bottom graphene layer increases, the interfacial interaction gradually weakens from the edge region and finally achieves the superlubricity state. A modified multi-adhesive shear-lag model is established based on the experiments, and its numerical analysis supports the experimental data. Molecular simulations demonstrate that after a critical strain, the interfacial force of large-twist-angle bilayer graphene decreases rapidly to enter the multiple adhesive state and finally stabilizes for friction, attributed to the generation and movements of interfacial dislocations, which reduce the interfacial interaction and promote the layer sliding.
{"title":"Transition from coupling to friction at the interface of large-twist-angle bilayer graphene","authors":"Qiancheng Ren , Yilan Xu , Jinglan Liu , Xiaochu Chen , Qi Yang , Jiayuan Fang , Pei Zhao","doi":"10.1016/j.jmps.2025.106459","DOIUrl":"10.1016/j.jmps.2025.106459","url":null,"abstract":"<div><div>The evolution of a solid interface from coupling to friction and its mechanisms still face challenges. Here, we use large-twist-angle bilayer graphene combined with isotope-labeling-assisted Raman spectroscopy to measure the mechanical behaviors of its two layers from coupling to friction. Results show that as the strain of the bottom graphene layer increases, the interfacial interaction gradually weakens from the edge region and finally achieves the superlubricity state. A modified multi-adhesive shear-lag model is established based on the experiments, and its numerical analysis supports the experimental data. Molecular simulations demonstrate that after a critical strain, the interfacial force of large-twist-angle bilayer graphene decreases rapidly to enter the multiple adhesive state and finally stabilizes for friction, attributed to the generation and movements of interfacial dislocations, which reduce the interfacial interaction and promote the layer sliding.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106459"},"PeriodicalIF":6.0,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145614043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-29DOI: 10.1016/j.jmps.2025.106456
Wei Tang , Lingfeng Wang , Shen Sun , Liucheng Zhou , Marco Paggi , Min Yi
Laser additive manufacturing (LAM) is increasingly employed as an in-situ repair technique for restoring the structural integrity and fatigue performance of metallic components. The fatigue and fracture behavior of LAM repaired components are significantly affected by defects introduced during the repair process, which poses challenges for predicting fatigue properties after LAM repair. Herein, we demonstrate the fatigue strength enhancement and fatigue crack growth (FCG) mechanisms in LAM repaired titanium-alloy blades by integrating vibration-based bending fatigue experiments with phase-field modeling (PFM). It is found that LAM repair of the notched TC17 forged blade could improve the fatigue strength by 94%. Fatigue cracks are revealed to initiate at internal defects within the LAM repair and propagate along transgranular paths influenced by defect clusters, deviating from the surface-initiated cracks in the forged counterparts. X-ray computed tomography reveals that the defect is dominated by small pores, with over 80% exhibiting an equivalent diameter below 60 µm. Furthermore, a macroscopic PFM incorporating fatigue life model that considers repair-induced pore defects is applied to predict the fatigue performance after LAM repair. Phase-field simulation results are shown to agree well with the experimental ones in terms of fatigue strength (error < 6%), critical crack length (error < 8%), and fracture surface morphology. Impact of defect features, material and model parameters on fatigue properties are investigated using our PFM, and the repair-induced pore size is shown to govern fatigue crack initiation and growth behavior of LAM repaired blade. Our work highlights the governing role of LAM repair-induced pore defects in high-cycle fatigue performance and enables a predictive PFM framework applicable to the fatigue evaluation of LAM repaired metallic components.
{"title":"Additive-manufacturing repair towards restoring fatigue performance of metallic component: Experiment and phase-field model prediction","authors":"Wei Tang , Lingfeng Wang , Shen Sun , Liucheng Zhou , Marco Paggi , Min Yi","doi":"10.1016/j.jmps.2025.106456","DOIUrl":"10.1016/j.jmps.2025.106456","url":null,"abstract":"<div><div>Laser additive manufacturing (LAM) is increasingly employed as an in-situ repair technique for restoring the structural integrity and fatigue performance of metallic components. The fatigue and fracture behavior of LAM repaired components are significantly affected by defects introduced during the repair process, which poses challenges for predicting fatigue properties after LAM repair. Herein, we demonstrate the fatigue strength enhancement and fatigue crack growth (FCG) mechanisms in LAM repaired titanium-alloy blades by integrating vibration-based bending fatigue experiments with phase-field modeling (PFM). It is found that LAM repair of the notched TC17 forged blade could improve the fatigue strength by 94%. Fatigue cracks are revealed to initiate at internal defects within the LAM repair and propagate along transgranular paths influenced by defect clusters, deviating from the surface-initiated cracks in the forged counterparts. X-ray computed tomography reveals that the defect is dominated by small pores, with over 80% exhibiting an equivalent diameter below 60 µm. Furthermore, a macroscopic PFM incorporating fatigue life model that considers repair-induced pore defects is applied to predict the fatigue performance after LAM repair. Phase-field simulation results are shown to agree well with the experimental ones in terms of fatigue strength (error < 6%), critical crack length (error < 8%), and fracture surface morphology. Impact of defect features, material and model parameters on fatigue properties are investigated using our PFM, and the repair-induced pore size is shown to govern fatigue crack initiation and growth behavior of LAM repaired blade. Our work highlights the governing role of LAM repair-induced pore defects in high-cycle fatigue performance and enables a predictive PFM framework applicable to the fatigue evaluation of LAM repaired metallic components.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106456"},"PeriodicalIF":6.0,"publicationDate":"2025-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145619692","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-25DOI: 10.1016/j.jmps.2025.106444
Hugh Dorward , Mahmoud Mostafavi , David M. Knowles , Matthew J. Peel
Surrogate models are a useful tool in enabling efficient modelling and propagation of uncertainty through material process-structure-property linkages. One promising application is in the modelling the dependence of macroscopic material properties on the microstructure of polycrystalline materials. However, this requires parameterisation of complex microstructural features such as crystallographic texture.
This study compares two methods for parameterising texture for use in reduced-order models: a principal component analysis reduction of generalised spherical harmonics (GSH-PCA), and a simpler, scalar parameterisation using the Taylor factor. The effectiveness of each method is demonstrated by applying each to a Gaussian process (GP) regression surrogate of material deformation, trained on data from crystal plasticity simulation.
The GSH-PCA parameterisation reduces the number of variables required to capture texture to between 5–10 for cubic-orthorhombic symmetry and has the advantage of allowing reconstruction of the original texture from the GSH-PCA coefficients. In comparison, the Taylor factor offers a simpler surrogate model with a single input parameter, however this model has less overall predictive accuracy with more uncertainty in the input variable space. Despite this, the use of GP regression as the surrogate model with functional outputs allows the uncertainties from both texture parameterisations to be propagated through to the prediction of macroscopic mechanical behaviour.
{"title":"Reduced-order representations of crystallographic texture for application to surrogate modelling of austenitic stainless steel","authors":"Hugh Dorward , Mahmoud Mostafavi , David M. Knowles , Matthew J. Peel","doi":"10.1016/j.jmps.2025.106444","DOIUrl":"10.1016/j.jmps.2025.106444","url":null,"abstract":"<div><div>Surrogate models are a useful tool in enabling efficient modelling and propagation of uncertainty through material process-structure-property linkages. One promising application is in the modelling the dependence of macroscopic material properties on the microstructure of polycrystalline materials. However, this requires parameterisation of complex microstructural features such as crystallographic texture.</div><div>This study compares two methods for parameterising texture for use in reduced-order models: a principal component analysis reduction of generalised spherical harmonics (GSH-PCA), and a simpler, scalar parameterisation using the Taylor factor. The effectiveness of each method is demonstrated by applying each to a Gaussian process (GP) regression surrogate of material deformation, trained on data from crystal plasticity simulation.</div><div>The GSH-PCA parameterisation reduces the number of variables required to capture texture to between 5–10 for cubic-orthorhombic symmetry and has the advantage of allowing reconstruction of the original texture from the GSH-PCA coefficients. In comparison, the Taylor factor offers a simpler surrogate model with a single input parameter, however this model has less overall predictive accuracy with more uncertainty in the input variable space. Despite this, the use of GP regression as the surrogate model with functional outputs allows the uncertainties from both texture parameterisations to be propagated through to the prediction of macroscopic mechanical behaviour.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106444"},"PeriodicalIF":6.0,"publicationDate":"2025-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145598603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-24DOI: 10.1016/j.jmps.2025.106430
Guozhan Xia , Renwei Mao , Weiqiu Chen , Yipin Su
This study explores the tunable wrinkling of a magneto-active hydrogel (MAH) block immersed in a tank filled with a magnetic solvent. An electromagnet, with two poles positioned at the top and bottom of the tank, generates a uniform magnetic biasing field via energized coils. Within the framework of nonlinear magneto-elastic theory and non-equilibrium thermodynamics of hydrogels, the mechanical behaviors of small-amplitude wrinkling are modeled, incorporating the influence of external Maxwell stress. By employing the surface impedance matrix method combined with the Stroh formulation, the bifurcation relations governing wrinkling onset are decoupled into symmetric and antisymmetric modes. For the first time, explicit expressions are established in a compact form for compressible soft materials with general free energy functions. As representative examples, generalized neo-Hookean and Gent ideal MAH blocks are examined. Numerical results show that blocks with a smaller permeability than that of the surroundings (i.e., the normalized permeability ), exhibit wrinkling behaviors largely similar to those without an external magnetic field, albeit with minor differences in certain details. Notably, a distinct “mutation” in wrinkling is observed for extremely thick blocks, which is attributed to a geometric constraint associated with the normalized thickness of the block relative to the tank. However, for blocks with higher permeability (), the tensile Maxwell stress significantly destabilizes the system, resulting in distinct wrinkling patterns that differs from those in low-permeability blocks. Intriguingly, wrinkling can emerge during the simultaneous thickening and area expansion of the block, but this phenomenon may be suppressed by geometric constraints. These findings offer valuable insights into the behavior of magneto-active hydrogels, potentially advancing their theoretical understanding and practical applications.
{"title":"Wrinkling mechanics of immersed magneto-active hydrogels","authors":"Guozhan Xia , Renwei Mao , Weiqiu Chen , Yipin Su","doi":"10.1016/j.jmps.2025.106430","DOIUrl":"10.1016/j.jmps.2025.106430","url":null,"abstract":"<div><div>This study explores the tunable wrinkling of a magneto-active hydrogel (MAH) block immersed in a tank filled with a magnetic solvent. An electromagnet, with two poles positioned at the top and bottom of the tank, generates a uniform magnetic biasing field via energized coils. Within the framework of nonlinear magneto-elastic theory and non-equilibrium thermodynamics of hydrogels, the mechanical behaviors of small-amplitude wrinkling are modeled, incorporating the influence of external Maxwell stress. By employing the surface impedance matrix method combined with the Stroh formulation, the bifurcation relations governing wrinkling onset are decoupled into symmetric and antisymmetric modes. For the first time, explicit expressions are established in a compact form for compressible soft materials with general free energy functions. As representative examples, generalized neo-Hookean and Gent ideal MAH blocks are examined. Numerical results show that blocks with a smaller permeability than that of the surroundings (i.e., the normalized permeability <span><math><mrow><mover><mi>μ</mi><mo>¯</mo></mover><mo><</mo><mn>1</mn></mrow></math></span>), exhibit wrinkling behaviors largely similar to those without an external magnetic field, albeit with minor differences in certain details. Notably, a distinct “mutation” in wrinkling is observed for extremely thick blocks, which is attributed to a geometric constraint associated with the normalized thickness of the block relative to the tank. However, for blocks with higher permeability (<span><math><mrow><mover><mi>μ</mi><mo>¯</mo></mover><mo>></mo><mn>1</mn></mrow></math></span>), the tensile Maxwell stress significantly destabilizes the system, resulting in distinct wrinkling patterns that differs from those in low-permeability blocks. Intriguingly, wrinkling can emerge during the simultaneous thickening and area expansion of the block, but this phenomenon may be suppressed by geometric constraints. These findings offer valuable insights into the behavior of magneto-active hydrogels, potentially advancing their theoretical understanding and practical applications.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106430"},"PeriodicalIF":6.0,"publicationDate":"2025-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145593063","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-23DOI: 10.1016/j.jmps.2025.106445
Dinghuai Yang , Zhichao Liu , Jian Cheng , Mingjun Chen , Linjie Zhao , Shengfei Wang , Feng Geng , Yazhou Sun , Qiao Xu
Spallation and thermal damage limit the application of fused silica under extremely intense lasers. Herein, the unclear underlying mechanisms, including extreme-irradiation-induced plasticity-related behaviors were studied based on first-constructed cross-scale models, molecular dynamics simulation, and multimodal characterization. Material spallation originated from the anomalous “quasiplasticity” and phased propagation of micro-cracks under quadruplex elastoplastic waves. Although the fastest primary wave could not cause macroscopic deformation, it could lead to micro-plasticity phenomena (ring-structure transformation and point-defect proliferation) due to material phase transformation and destabilizing effects. Subsequently, conjugate secondary and head elastoplastic waves governed initialization processes of micro-cracks, where primary-wave-induced E’-Center and NBOHC defects played roles of “damage precursors”. Concomitantly, transitional deformation zones containing massive strip-like-distributed cavities (similar to “immature” micro-cracks) were generated around micro-cracks. There was a cascading evolution process of point defects, cavities, and micro-cracks under phased energy input from waves, causing an anomalous “quasiplasticity” process within brittle fused silica. It differs from transient fracture processes of brittle materials. Finally, the Rayleigh waves trapped on surfaces attracted micro-cracks towards them, causing disastrous surface damage. The thermal damage originated from the volcanic vents formed within 3∼4 ns, which was induced under the comprehensive action of the impact of elastoplastic waves, cascading solid-liquid-gas phase transition, GPa-level pressure difference between ablated zones and air, and fluidic flow disturbances. The whole time-evolution sequence axis diagram of the material failure process was drawn based on these. Summarily, this work could offer novel insights into the anomalous “quasiplasticity”, spallation, and thermal damage phenomena of fused silica under intense lasers.
{"title":"Anomalous quasiplasticity, spallation, and thermal damage in fused silica under laser-induced quadruple stress waves and multi-field coupling effects","authors":"Dinghuai Yang , Zhichao Liu , Jian Cheng , Mingjun Chen , Linjie Zhao , Shengfei Wang , Feng Geng , Yazhou Sun , Qiao Xu","doi":"10.1016/j.jmps.2025.106445","DOIUrl":"10.1016/j.jmps.2025.106445","url":null,"abstract":"<div><div>Spallation and thermal damage limit the application of fused silica under extremely intense lasers. Herein, the unclear underlying mechanisms, including extreme-irradiation-induced plasticity-related behaviors were studied based on first-constructed cross-scale models, molecular dynamics simulation, and multimodal characterization. Material spallation originated from the anomalous “quasiplasticity” and phased propagation of micro-cracks under quadruplex elastoplastic waves. Although the fastest primary wave could not cause macroscopic deformation, it could lead to micro-plasticity phenomena (ring-structure transformation and point-defect proliferation) due to material phase transformation and destabilizing effects. Subsequently, conjugate secondary and head elastoplastic waves governed initialization processes of micro-cracks, where primary-wave-induced E’-Center and NBOHC defects played roles of “damage precursors”. Concomitantly, transitional deformation zones containing massive strip-like-distributed cavities (similar to “immature” micro-cracks) were generated around micro-cracks. There was a cascading evolution process of point defects, cavities, and micro-cracks under phased energy input from waves, causing an anomalous “quasiplasticity” process within brittle fused silica. It differs from transient fracture processes of brittle materials. Finally, the Rayleigh waves trapped on surfaces attracted micro-cracks towards them, causing disastrous surface damage. The thermal damage originated from the volcanic vents formed within 3∼4 ns, which was induced under the comprehensive action of the impact of elastoplastic waves, cascading solid-liquid-gas phase transition, GPa-level pressure difference between ablated zones and air, and fluidic flow disturbances. The whole time-evolution sequence axis diagram of the material failure process was drawn based on these. Summarily, this work could offer novel insights into the anomalous “quasiplasticity”, spallation, and thermal damage phenomena of fused silica under intense lasers.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106445"},"PeriodicalIF":6.0,"publicationDate":"2025-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145575496","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-22DOI: 10.1016/j.jmps.2025.106431
Hui Li, Shanyong Wang
Phase-field modelling of compressive–shear fracture in anisotropic materials with tension-compression asymmetry remains a major challenge, despite its significance in geomechanics and structural engineering. To this end, a novel hybrid phase-field model with an orthogonality-based strain decomposition is proposed for modelling of mixed-mode brittle fracture in orthotropic/anisotropic materials. In this model, the strain is first mapped into an auxiliary space via the square root of the stiffness (ℂ1/2) and thus is orthogonally decomposed into volumetric tensile, volumetric compressive, and deviatoric parts. The deviatoric strain is further partitioned by spectral decomposition into positive and negative items. This volumetric–deviatoric–spectral strain split results in a fivefold partition of the strain energy. A new driving force is then proposed by combining the Mohr–Coulomb criterion and three fracture energies with the five energy components within the AT1 phase-field finite-element formulation. The present model is validated through 2D tension and shear tests on single-notched plates and 2D/3D compression tests on single-hole plates. It is found that the simulated results agree well with published numerical and experimental data, and the accuracy and capability of the model for modelling mixed-mode fracture in anisotropic materials under shear/compression are well validated.
{"title":"Orthogonality-based energy split for anisotropic compressive-shear brittle fracture: A hybrid phase-field model","authors":"Hui Li, Shanyong Wang","doi":"10.1016/j.jmps.2025.106431","DOIUrl":"10.1016/j.jmps.2025.106431","url":null,"abstract":"<div><div>Phase-field modelling of compressive–shear fracture in anisotropic materials with tension-compression asymmetry remains a major challenge, despite its significance in geomechanics and structural engineering. To this end, a novel hybrid phase-field model with an orthogonality-based strain decomposition is proposed for modelling of mixed-mode brittle fracture in orthotropic/anisotropic materials. In this model, the strain is first mapped into an auxiliary space via the square root of the stiffness (ℂ<sup>1/2</sup>) and thus is orthogonally decomposed into volumetric tensile, volumetric compressive, and deviatoric parts. The deviatoric strain is further partitioned by spectral decomposition into positive and negative items. This <em>volumetric–deviatoric–spectral</em> strain split results in a fivefold partition of the strain energy. A new driving force is then proposed by combining the Mohr–Coulomb criterion and three fracture energies with the five energy components within the AT<sub>1</sub> phase-field finite-element formulation. The present model is validated through 2D tension and shear tests on single-notched plates and 2D/3D compression tests on single-hole plates. It is found that the simulated results agree well with published numerical and experimental data, and the accuracy and capability of the model for modelling mixed-mode fracture in anisotropic materials under shear/compression are well validated.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106431"},"PeriodicalIF":6.0,"publicationDate":"2025-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145575504","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-20DOI: 10.1016/j.jmps.2025.106429
Kshiteej J. Deshmukh , Graeme W. Milton
Electromagnetic metamaterials with a uniaxial effective permittivity tensor, characterized by its transverse () and axial (ϵ∥) components, play a central role in the design of advanced photonic and electromagnetic materials including hyperbolic metamaterials, and biological imaging platforms. Tight bounds on the complex effective permittivity of such metamaterials are critical for predicting and optimizing their macroscopic electromagnetic response. While rigorous tight bounds exist for isotropic two-phase composites, corresponding results for uniaxial composites remain relatively unexplored. In this work, we systematically investigate the attainable range of and ϵ∥ in the quasistatic regime for two-phase metamaterials with isotropic homogeneous phases. By analyzing known microgeometries and constructing hierarchical laminates (HLs), we demonstrate that the classical bounds on are not optimal. We conjecture improved bounds based on numerically fitted circular arcs derived from convex hulls of values obtained from HLs, and we identify optimal rank-4 HL structures that achieve all points on the conjectured bounds. Additionally, we quantify the correlation between and ϵ∥ for fixed volume fractions, and propose a design algorithm to construct HL microstructures achieving prescribed values of . Leveraging the Cherkaev-Gibiansky transformation and the translation method, we extend recent techniques developed for isotropic composites by Kern-Miller-Milton to derive translation bounds on the uniaxial complex effective permittivity tensor. Using the trace bounds we also numerically obtain the correlated bounds on when and ϵ∥ differ by a fixed proportionality constant. Finally, bounds on the sensitivity of the effective permittivity tensor of low-loss composites are obtained and their optimality is shown in two-dimensions. Our results advance the theoretical understanding of uniaxial metamaterials and provide practical tools for the design of tailored anisotropic metamaterials.
{"title":"Bounds on the uniaxial effective complex permittivity tensor of two-phase composites and optimal or near optimal microstructures","authors":"Kshiteej J. Deshmukh , Graeme W. Milton","doi":"10.1016/j.jmps.2025.106429","DOIUrl":"10.1016/j.jmps.2025.106429","url":null,"abstract":"<div><div>Electromagnetic metamaterials with a uniaxial effective permittivity tensor, characterized by its transverse (<span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span>) and axial (ϵ<sub>∥</sub>) components, play a central role in the design of advanced photonic and electromagnetic materials including hyperbolic metamaterials, and biological imaging platforms. Tight bounds on the complex effective permittivity of such metamaterials are critical for predicting and optimizing their macroscopic electromagnetic response. While rigorous tight bounds exist for isotropic two-phase composites, corresponding results for uniaxial composites remain relatively unexplored. In this work, we systematically investigate the attainable range of <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span> and ϵ<sub>∥</sub> in the quasistatic regime for two-phase metamaterials with isotropic homogeneous phases. By analyzing known microgeometries and constructing hierarchical laminates (HLs), we demonstrate that the classical bounds on <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span> are not optimal. We conjecture improved bounds based on numerically fitted circular arcs derived from convex hulls of <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span> values obtained from HLs, and we identify optimal rank-4 HL structures that achieve all points on the conjectured bounds. Additionally, we quantify the correlation between <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span> and ϵ<sub>∥</sub> for fixed volume fractions, and propose a design algorithm to construct HL microstructures achieving prescribed values of <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span>. Leveraging the Cherkaev-Gibiansky transformation and the translation method, we extend recent techniques developed for isotropic composites by Kern-Miller-Milton to derive translation bounds on the uniaxial complex effective permittivity tensor. Using the trace bounds we also numerically obtain the correlated bounds on <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span> when <span><math><msub><mi>ϵ</mi><mo>⊥</mo></msub></math></span> and ϵ<sub>∥</sub> differ by a fixed proportionality constant. Finally, bounds on the sensitivity of the effective permittivity tensor of low-loss composites are obtained and their optimality is shown in two-dimensions. Our results advance the theoretical understanding of uniaxial metamaterials and provide practical tools for the design of tailored anisotropic metamaterials.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106429"},"PeriodicalIF":6.0,"publicationDate":"2025-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145560044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-17DOI: 10.1016/j.jmps.2025.106428
Yang Zhao , Anh T. Nguyen , Hoang T. Nguyen , Zdeněk P. Bažant
The geological genesis of natural cracks in sedimentary rocks such as shale is a problem that needs to be understood to improve the technology of hydraulic fracturing as well as deep sequestration of harmful fluids. Why are the vertical natural cracks roughly parallel and equidistant, and why is the spacing roughly 10 cm rather than 1 cm or 100 cm? Fracture mechanics of critical cracks cannot answer this question. Neither can the material heterogeneity. The growth of critical parallel cracks is impossible because the relative crack face displacements would immediately localize into one crack, leading to an earthquake. The cracks must have formed, on the tectonic time scale, by a slow growth of subcritical shear cracks governed by the Charles-Evans law. The idea advanced here is that what controls the crack spacing is the balance between the reduction, due to shear dilatancy, of the concentration of ions such as Na and Cl in each fracture process zone (PFZ), which decelerates the cracks, and the restoration of ion concentration by diffusion of ions from the space between the cracks into the FPZ. This diffusion of water is driven mainly by the osmotic pressure gradient, which offsets the deceleration and depends strongly on the crack spacing. A simple analytical solution of the steady state is rendered possible by approximating the ion concentration profiles between adjacent cracks by parabolic arcs. Applying this theory to Woodford shale yields the approximate crack spacing of 10 cm, which is realistic. The stability of unlimited parallel mode II frictional crack growth is proven by examining the second variation of the free energy. Water concentration drop in the FPZ due to shear dilatancy and its restoration by water diffusion from the inter-crack space have similar effect, although probably much weaker.
{"title":"Osmotic control of the spacing of parallel shear cracks in shale growing subcritically in geologic past","authors":"Yang Zhao , Anh T. Nguyen , Hoang T. Nguyen , Zdeněk P. Bažant","doi":"10.1016/j.jmps.2025.106428","DOIUrl":"10.1016/j.jmps.2025.106428","url":null,"abstract":"<div><div>The geological genesis of natural cracks in sedimentary rocks such as shale is a problem that needs to be understood to improve the technology of hydraulic fracturing as well as deep sequestration of harmful fluids. Why are the vertical natural cracks roughly parallel and equidistant, and why is the spacing roughly 10 cm rather than 1 cm or 100 cm? Fracture mechanics of critical cracks cannot answer this question. Neither can the material heterogeneity. The growth of critical parallel cracks is impossible because the relative crack face displacements would immediately localize into one crack, leading to an earthquake. The cracks must have formed, on the tectonic time scale, by a slow growth of subcritical shear cracks governed by the Charles-Evans law. The idea advanced here is that what controls the crack spacing is the balance between the reduction, due to shear dilatancy, of the concentration of ions such as Na<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup></math></span> and Cl<span><math><msup><mrow></mrow><mrow><mo>−</mo></mrow></msup></math></span> in each fracture process zone (PFZ), which decelerates the cracks, and the restoration of ion concentration by diffusion of ions from the space between the cracks into the FPZ. This diffusion of water is driven mainly by the osmotic pressure gradient, which offsets the deceleration and depends strongly on the crack spacing. A simple analytical solution of the steady state is rendered possible by approximating the ion concentration profiles between adjacent cracks by parabolic arcs. Applying this theory to Woodford shale yields the approximate crack spacing of 10 cm, which is realistic. The stability of unlimited parallel mode II frictional crack growth is proven by examining the second variation of the free energy. Water concentration drop in the FPZ due to shear dilatancy and its restoration by water diffusion from the inter-crack space have similar effect, although probably much weaker.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106428"},"PeriodicalIF":6.0,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145546347","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-17DOI: 10.1016/j.jmps.2025.106421
Anh Tuan Le , Xavier Bruant , Ngoc Tram Phung , François Ozanam , Michel Rosso , Laurent Guin
We report operando measurements and concurrent modeling of the stress dependence of the chemical potential of lithium in a silicon electrode. An experimental study is carried out on hydrogenated amorphous silicon thin films in which the electrode stress state is modified operando during electrochemical lithiation and delithiation by applying an external mechanical load. During galvanostatic cycling, the electrode is periodically subjected to a tensile strain, inducing stress variations that are reflected in voltage changes. The measured stress-induced voltage changes are interpreted using a well-established chemomechanical model of lithium insertion in silicon. Comparison of voltage measurements with model predictions allows us to determine the concentration-dependent Young’s modulus (from 29 GPa to 26 GPa with increasing lithium content) and some of the viscoplastic parameters of lithiated silicon. The calibrated model shows good predictive capability when applied to lithiation cycles performed at a C-rate different from that of the calibration cycle. However, it shows limitations in explaining voltage changes under delithiation. These results show that thermodynamically-consistent chemomechanical models of lithiation not only adequately describe the effect of lithium insertion and deinsertion on stress, as already shown in the literature, but also capture the reverse effect of stress on lithium chemical potential in silicon. In this respect, this work opens up new perspectives for the quantitative validation and calibration of existing diffusion-deformation theories, notably by highlighting their possible limitations.
{"title":"Stress dependence of the chemical potential of lithium in a silicon electrode","authors":"Anh Tuan Le , Xavier Bruant , Ngoc Tram Phung , François Ozanam , Michel Rosso , Laurent Guin","doi":"10.1016/j.jmps.2025.106421","DOIUrl":"10.1016/j.jmps.2025.106421","url":null,"abstract":"<div><div>We report <em>operando</em> measurements and concurrent modeling of the stress dependence of the chemical potential of lithium in a silicon electrode. An experimental study is carried out on hydrogenated amorphous silicon thin films in which the electrode stress state is modified <em>operando</em> during electrochemical lithiation and delithiation by applying an external mechanical load. During galvanostatic cycling, the electrode is periodically subjected to a tensile strain, inducing stress variations that are reflected in voltage changes. The measured stress-induced voltage changes are interpreted using a well-established chemomechanical model of lithium insertion in silicon. Comparison of voltage measurements with model predictions allows us to determine the concentration-dependent Young’s modulus (from 29 GPa to 26 GPa with increasing lithium content) and some of the viscoplastic parameters of lithiated silicon. The calibrated model shows good predictive capability when applied to lithiation cycles performed at a C-rate different from that of the calibration cycle. However, it shows limitations in explaining voltage changes under delithiation. These results show that thermodynamically-consistent chemomechanical models of lithiation not only adequately describe the effect of lithium insertion and deinsertion on stress, as already shown in the literature, but also capture the reverse effect of stress on lithium chemical potential in silicon. In this respect, this work opens up new perspectives for the quantitative validation and calibration of existing diffusion-deformation theories, notably by highlighting their possible limitations.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106421"},"PeriodicalIF":6.0,"publicationDate":"2025-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145546297","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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