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}
Pub Date : 2025-11-15DOI: 10.1016/j.jmps.2025.106426
Ce Chen , Liujun Wu , Chenyang Xin , Wenbin Liu , Xin Yi , Huiling Duan
The mechanical response of nonlinear heterogeneous materials is strongly influenced by the deformation-dependent spatial variation of properties in the matrix and inclusions. Conventional micromechanical approaches, typically based on linearization techniques and uniform moduli within each material phase, often fail to capture the effective response of such nonlinear systems, where the local tangent modulus acts as a deformation-dependent measure of stiffness rather than an intrinsic material property. Here, we present a neural-network-enhanced micromechanical framework built upon an evolving nonlinear reference medium with spatially non-uniform tangent moduli, for composites comprising an isotropic matrix and isotropic spherical inclusions with nonlinear interfacial effects. Building on a single-inclusion configuration, where an inclusion is embedded in a reference medium, we introduce two physics-guided neural networks that capture the spatial variation with local deformation states. One network models the inclusions with prescribed properties, while the other represents the reference medium, whose material properties evolve with macroscopic deformations. By enforcing the interfacial displacement–traction condition, we identify the varying properties of the reference medium and determine the effective tangent modulus of the composite. Applied to nonlinear particle-reinforced elastomers at high volume fractions, the framework significantly outperforms classical micromechanical approaches. Moreover, the trained model demonstrates remarkable generalization across diverse nonlinear behaviors of inclusions and matrix, interfacial conditions, loading modes, and volume fractions—without retraining. The framework also extends naturally to plasticity problems, yielding accurate predictions for porous plastic solids. This work establishes a new pathway for integrating neural networks into the derivation of micromechanical relations for complex nonlinear composites.
{"title":"A neural-network-enhanced micromechanical framework with evolving reference medium for nonlinear heterogeneous materials","authors":"Ce Chen , Liujun Wu , Chenyang Xin , Wenbin Liu , Xin Yi , Huiling Duan","doi":"10.1016/j.jmps.2025.106426","DOIUrl":"10.1016/j.jmps.2025.106426","url":null,"abstract":"<div><div>The mechanical response of nonlinear heterogeneous materials is strongly influenced by the deformation-dependent spatial variation of properties in the matrix and inclusions. Conventional micromechanical approaches, typically based on linearization techniques and uniform moduli within each material phase, often fail to capture the effective response of such nonlinear systems, where the local tangent modulus acts as a deformation-dependent measure of stiffness rather than an intrinsic material property. Here, we present a neural-network-enhanced micromechanical framework built upon an evolving nonlinear reference medium with spatially non-uniform tangent moduli, for composites comprising an isotropic matrix and isotropic spherical inclusions with nonlinear interfacial effects. Building on a single-inclusion configuration, where an inclusion is embedded in a reference medium, we introduce two physics-guided neural networks that capture the spatial variation with local deformation states. One network models the inclusions with prescribed properties, while the other represents the reference medium, whose material properties evolve with macroscopic deformations. By enforcing the interfacial displacement–traction condition, we identify the varying properties of the reference medium and determine the effective tangent modulus of the composite. Applied to nonlinear particle-reinforced elastomers at high volume fractions, the framework significantly outperforms classical micromechanical approaches. Moreover, the trained model demonstrates remarkable generalization across diverse nonlinear behaviors of inclusions and matrix, interfacial conditions, loading modes, and volume fractions—without retraining. The framework also extends naturally to plasticity problems, yielding accurate predictions for porous plastic solids. This work establishes a new pathway for integrating neural networks into the derivation of micromechanical relations for complex nonlinear composites.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106426"},"PeriodicalIF":6.0,"publicationDate":"2025-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145546298","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 : 2025-11-14DOI: 10.1016/j.jmps.2025.106424
F. Vicentini, J. Heinzmann, P. Carrara, L. De Lorenzis
Variational phase-field models of brittle fracture are powerful tools for studying Griffith-type crack propagation in complex scenarios. However, as approximations of Griffith’s theory — which does not incorporate a strength criterion — these models lack flexibility in prescribing material-specific strength surfaces. Consequently, they struggle to accurately capture crack nucleation under multiaxial stress conditions. In this paper, inspired by Alessi et al. (2014), we propose a variational phase-field model that approximates cohesive fracture. The model accommodates an arbitrary (convex) strength surface, independent of the regularization length scale, and allows for flexible tuning of the cohesive response. Our formulation results in sharp cohesive cracks and naturally enforces a sharp non-interpenetration condition, thereby eliminating the need for additional energy decomposition strategies. It inherently satisfies stress softening and produces ”crack-like” residual stresses by construction. To ensure strain hardening, the ratio of the regularization length to the material’s cohesive length must be sufficiently small; however, if crack nucleation is desired, this ratio must also be large enough to make the homogeneous damaged state unstable. We investigate the model in one and three dimensions, establishing first- and second-order stability results. The theoretical findings are validated through numerical simulations using the finite element method, employing standard discretization and solution techniques.
脆性断裂的变分相场模型是研究复杂情况下griffith型裂纹扩展的有力工具。然而，作为格里菲斯理论的近似值（不包含强度标准），这些模型在规定材料特定强度表面方面缺乏灵活性。因此，他们很难准确地捕捉多轴应力条件下的裂纹形核。在本文中，受Alessi et al.（2014）的启发，我们提出了一个近似于内聚断裂的变分相场模型。该模型可容纳任意（凸）强度表面，独立于正则化长度尺度，并允许灵活调整内聚响应。我们的配方产生尖锐的粘性裂缝，并自然地强制执行尖锐的非相互渗透条件，从而消除了对额外能量分解策略的需要。它本质上满足应力软化，并通过施工产生“裂纹状”残余应力。为保证应变硬化，正则化长度与材料内聚长度之比必须足够小；然而，如果想要裂纹成核，这个比率也必须足够大，以使均匀损伤状态不稳定。我们在一维和三维上研究了模型，建立了一阶和二阶稳定性结果。采用有限元方法，采用标准离散化和求解技术，通过数值模拟验证了理论结果。
{"title":"Variational phase-field modeling of cohesive fracture with flexibly tunable strength surface","authors":"F. Vicentini, J. Heinzmann, P. Carrara, L. De Lorenzis","doi":"10.1016/j.jmps.2025.106424","DOIUrl":"10.1016/j.jmps.2025.106424","url":null,"abstract":"<div><div>Variational phase-field models of brittle fracture are powerful tools for studying Griffith-type crack propagation in complex scenarios. However, as approximations of Griffith’s theory — which does not incorporate a strength criterion — these models lack flexibility in prescribing material-specific strength surfaces. Consequently, they struggle to accurately capture crack nucleation under multiaxial stress conditions. In this paper, inspired by Alessi et al. (2014), we propose a variational phase-field model that approximates cohesive fracture. The model accommodates an arbitrary (convex) strength surface, independent of the regularization length scale, and allows for flexible tuning of the cohesive response. Our formulation results in sharp cohesive cracks and naturally enforces a sharp non-interpenetration condition, thereby eliminating the need for additional energy decomposition strategies. It inherently satisfies stress softening and produces ”crack-like” residual stresses by construction. To ensure strain hardening, the ratio of the regularization length to the material’s cohesive length must be sufficiently small; however, if crack nucleation is desired, this ratio must also be large enough to make the homogeneous damaged state unstable. We investigate the model in one and three dimensions, establishing first- and second-order stability results. The theoretical findings are validated through numerical simulations using the finite element method, employing standard discretization and solution techniques.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106424"},"PeriodicalIF":6.0,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145545471","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 : 2025-11-14DOI: 10.1016/j.jmps.2025.106425
Mohammadali Behboodi, Yida Zhang
Adsorption-induced swelling occurs in a wide spectrum of natural and engineered porous materials. A key underlying mechanism is the monotonic reduction of solid-fluid surface energy upon fluid adsorption, which lowers the contractive adsorption stress and causes the porous skeleton to swell (Bangham and Fakhoury, 1928). Some mesoporous materials, however, deviate from the monotonic swelling pattern predicted by this mechanism, exhibiting an abrupt shrinkage at intermediate adsorbate partial pressures before swelling resumes and continues to full saturation. This behavior is commonly attributed to capillary condensation of the adsorbate from the vapor to the liquid phase within the pores. Understanding the stresses and the shrinkage induced by capillary condensation is critical in various industrial applications including micro-/nanofabrication, geotechnical engineering in collapsible soils, and sorption-driven actuation technologies. This work aims to develop a unified poromechanics theory that captures the full sequence of adsorption-induced deformation, including initial swelling, contraction during capillary condensation, and resumed expansion near full saturation. The formulation begins with a thermodynamic analysis of an unsaturated deformable porous solid, acknowledging the energetics of the solid-fluid (sl), solid-vapor (sv), and liquid-vapor (lv) interfaces. The resulting free energy balance permits the simultaneous derivation of the liquid retention characteristics curve and the coupled mechanical effects driven by adsorption and partial saturation. Within this framework, two strategies for constructing constitutive relations are examined: one explicitly resolves the dynamic evolution of sl-sv-lv interfacial areas to emphasize the underlying physics, while the other partially lumps the surface energies into a macroscopic capillary potential to facilitate model calibration using standard laboratory tests. The models are evaluated using datasets from two markedly different solid-fluid systems: N2 gas adsorption on a hierarchical porous silica at 77 K and water adsorption on a carbon xerogel at 298 K. Both approaches effectively capture the complex, non-monotonic strain isotherms exhibited by the adsorbent. The adsorption-desorption hysteresis is also addressed in a thermodynamically consistent framework. The proposed theory demonstrates both robustness and unifying power in explaining the complex strain isotherms of porous materials along adsorption and desorption paths, covering the entire spectrum from vacuum-dry to fully liquid-saturated states.
{"title":"Interfacial evolution explains the complex swelling-shrinkage responses of porous materials from vacuum-dry to full liquid saturation","authors":"Mohammadali Behboodi, Yida Zhang","doi":"10.1016/j.jmps.2025.106425","DOIUrl":"10.1016/j.jmps.2025.106425","url":null,"abstract":"<div><div>Adsorption-induced swelling occurs in a wide spectrum of natural and engineered porous materials. A key underlying mechanism is the monotonic reduction of solid-fluid surface energy upon fluid adsorption, which lowers the contractive adsorption stress and causes the porous skeleton to swell (Bangham and Fakhoury, 1928). Some mesoporous materials, however, deviate from the monotonic swelling pattern predicted by this mechanism, exhibiting an abrupt shrinkage at intermediate adsorbate partial pressures before swelling resumes and continues to full saturation. This behavior is commonly attributed to capillary condensation of the adsorbate from the vapor to the liquid phase within the pores. Understanding the stresses and the shrinkage induced by capillary condensation is critical in various industrial applications including micro-/nanofabrication, geotechnical engineering in collapsible soils, and sorption-driven actuation technologies. This work aims to develop a unified poromechanics theory that captures the full sequence of adsorption-induced deformation, including initial swelling, contraction during capillary condensation, and resumed expansion near full saturation. The formulation begins with a thermodynamic analysis of an unsaturated deformable porous solid, acknowledging the energetics of the solid-fluid (<em>sl</em>), solid-vapor (<em>sv</em>), and liquid-vapor (<em>lv</em>) interfaces. The resulting free energy balance permits the simultaneous derivation of the liquid retention characteristics curve and the coupled mechanical effects driven by adsorption and partial saturation. Within this framework, two strategies for constructing constitutive relations are examined: one explicitly resolves the dynamic evolution of <em>sl-sv-lv</em> interfacial areas to emphasize the underlying physics, while the other partially lumps the surface energies into a macroscopic capillary potential to facilitate model calibration using standard laboratory tests. The models are evaluated using datasets from two markedly different solid-fluid systems: N<sub>2</sub> gas adsorption on a hierarchical porous silica at 77 K and water adsorption on a carbon xerogel at 298 K. Both approaches effectively capture the complex, non-monotonic strain isotherms exhibited by the adsorbent. The adsorption-desorption hysteresis is also addressed in a thermodynamically consistent framework. The proposed theory demonstrates both robustness and unifying power in explaining the complex strain isotherms of porous materials along adsorption and desorption paths, covering the entire spectrum from vacuum-dry to fully liquid-saturated states.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106425"},"PeriodicalIF":6.0,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145545472","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 : 2025-11-13DOI: 10.1016/j.jmps.2025.106419
Yanni Chen , Zhongxuan Yang , Itai Einav
Stresses and pressures are used to represent the hydromechanical state of deformable porous media. Past formulations often adopt the effective stress principle, usually in an empirical and energetically inconsistent way. Using the rigorous hydrodynamic procedure, this study pursues an alternative energy-consistent formulation for the full characterisation of both saturated and unsaturated porous materials. An elastic stress is consistently linked to its energy-conjugated elastic strain and, in the absence of viscous stress, has a structure that was previously interpreted as an effective stress. Here, it is emphasised that this similarity does not imply that the elastic stress is ‘effective’ in the classical sense, namely that it can replace total stress in dry soils to represent the mechanical behaviour of saturated or unsaturated soils. The dependence of the elastic stress on the deformability of the solid is incorporated constitutively using a general elastic strain energy of pressure- and density-dependent media, excluding energy costs from solid density changes due to volumetric elastic straining. By adopting the resulting internal energy that is convex for physically realistic porous materials, the proposed formulation yields a rigorous quantification of the elastic stress, and the pressures of the air, water, and solid required for characterising saturated and unsaturated soils, including the Biot stress correction coefficient for deformable porous media at variable saturation. The formulation also reveals the intrinsic dependence of the stress coefficients on material elasticity and the characteristics of water retention responses.
{"title":"Hydrodynamics of stresses and pressures in saturated and unsaturated deformable porous media","authors":"Yanni Chen , Zhongxuan Yang , Itai Einav","doi":"10.1016/j.jmps.2025.106419","DOIUrl":"10.1016/j.jmps.2025.106419","url":null,"abstract":"<div><div>Stresses and pressures are used to represent the hydromechanical state of deformable porous media. Past formulations often adopt the effective stress principle, usually in an empirical and energetically inconsistent way. Using the rigorous hydrodynamic procedure, this study pursues an alternative energy-consistent formulation for the full characterisation of both saturated and unsaturated porous materials. An elastic stress is consistently linked to its energy-conjugated elastic strain and, in the absence of viscous stress, has a structure that was previously interpreted as an effective stress. Here, it is emphasised that this similarity does not imply that the elastic stress is ‘effective’ in the classical sense, namely that it can replace total stress in dry soils to represent the mechanical behaviour of saturated or unsaturated soils. The dependence of the elastic stress on the deformability of the solid is incorporated constitutively using a general elastic strain energy of pressure- and density-dependent media, excluding energy costs from solid density changes due to volumetric elastic straining. By adopting the resulting internal energy that is convex for physically realistic porous materials, the proposed formulation yields a rigorous quantification of the elastic stress, and the pressures of the air, water, and solid required for characterising saturated and unsaturated soils, including the Biot stress correction coefficient for deformable porous media at variable saturation. The formulation also reveals the intrinsic dependence of the stress coefficients on material elasticity and the characteristics of water retention responses.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106419"},"PeriodicalIF":6.0,"publicationDate":"2025-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519062","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 : 2025-11-12DOI: 10.1016/j.jmps.2025.106417
Amit Ashkenazi, Adi Shultz, Lee Jordan, Dana Solav
<div><div>Accurate quantification of soft tissue material parameters is essential for tissue mechanics simulations, medical device design, surgical planning, and non-invasive diagnostics. Finite element analysis (FEA) is commonly employed, but generating accurate simulations often requires patient- and location-specific tissue material parameters. Although soft tissue constitutive models are well-developed, practical implementation is limited by the invasive nature of experiments required for fitting model parameters. Non-invasive methods, such as indentation and suction, offer in vivo applicability but typically lack analytical solutions that would allow direct fitting of material parameters. Consequently, parameter identification becomes an inverse problem solved via FEA, which is often ill-posed, yielding multiple sets of seemingly optimal parameters, especially with limited experimental data. This non-uniqueness undermines the reliable prediction of tissue response under varying loads. This study investigates the identifiability of transversely isotropic hyperelastic material parameters through macro-scale indentation, combining simultaneous measurements of force and full-field surface deformation. We use a simplified two-parameter constitutive model to represent a soft composite phantom and compare the homogenized parameters identified through indentation with those obtained from separate analyses of the matrix and fiber materials. Our findings indicate that a measurement error of 5% leads to certainty bounds of <span><math><mrow><mo>±</mo><mn>5</mn><mo>.</mo><mn>2</mn><mtext>%</mtext></mrow></math></span> and <span><math><mrow><mo>±</mo><mn>28</mn><mtext>%</mtext></mrow></math></span> for the isotropic and anisotropic parameters, respectively, when utilizing combined force–deformation data. In contrast, when only force data is considered, they are <span><math><mrow><mo>±</mo><mn>22</mn><mo>.</mo><mn>5</mn><mtext>%</mtext></mrow></math></span> and <span><math><mrow><mo>±</mo><mn>210</mn><mtext>%</mtext></mrow></math></span>, respectively. These results demonstrate that surface deformation measurements are crucial for uniquely identifying anisotropic hyperelastic parameters through indentation. Further research is needed to evaluate identifiability in more complex models and in vivo indentation scenarios.</div><div><strong>Statement of significance</strong></div><div>Understanding how anisotropic soft tissues respond to loads is important for designing better medical devices, improving surgical planning, and developing new diagnostic tools. However, it is challenging to model and quantify the mechanical properties of these tissues without destructive procedures. This study demonstrates that combining indentation tests with 3D imaging to track surface deformations enables the identification of transversely isotropic hyperelastic material parameters with substantially smaller uncertainty compared to standard indentation. These findings can help
{"title":"Indentation-based anisotropic material parameter identifiability: Validation on a synthetic soft tissue phantom","authors":"Amit Ashkenazi, Adi Shultz, Lee Jordan, Dana Solav","doi":"10.1016/j.jmps.2025.106417","DOIUrl":"10.1016/j.jmps.2025.106417","url":null,"abstract":"<div><div>Accurate quantification of soft tissue material parameters is essential for tissue mechanics simulations, medical device design, surgical planning, and non-invasive diagnostics. Finite element analysis (FEA) is commonly employed, but generating accurate simulations often requires patient- and location-specific tissue material parameters. Although soft tissue constitutive models are well-developed, practical implementation is limited by the invasive nature of experiments required for fitting model parameters. Non-invasive methods, such as indentation and suction, offer in vivo applicability but typically lack analytical solutions that would allow direct fitting of material parameters. Consequently, parameter identification becomes an inverse problem solved via FEA, which is often ill-posed, yielding multiple sets of seemingly optimal parameters, especially with limited experimental data. This non-uniqueness undermines the reliable prediction of tissue response under varying loads. This study investigates the identifiability of transversely isotropic hyperelastic material parameters through macro-scale indentation, combining simultaneous measurements of force and full-field surface deformation. We use a simplified two-parameter constitutive model to represent a soft composite phantom and compare the homogenized parameters identified through indentation with those obtained from separate analyses of the matrix and fiber materials. Our findings indicate that a measurement error of 5% leads to certainty bounds of <span><math><mrow><mo>±</mo><mn>5</mn><mo>.</mo><mn>2</mn><mtext>%</mtext></mrow></math></span> and <span><math><mrow><mo>±</mo><mn>28</mn><mtext>%</mtext></mrow></math></span> for the isotropic and anisotropic parameters, respectively, when utilizing combined force–deformation data. In contrast, when only force data is considered, they are <span><math><mrow><mo>±</mo><mn>22</mn><mo>.</mo><mn>5</mn><mtext>%</mtext></mrow></math></span> and <span><math><mrow><mo>±</mo><mn>210</mn><mtext>%</mtext></mrow></math></span>, respectively. These results demonstrate that surface deformation measurements are crucial for uniquely identifying anisotropic hyperelastic parameters through indentation. Further research is needed to evaluate identifiability in more complex models and in vivo indentation scenarios.</div><div><strong>Statement of significance</strong></div><div>Understanding how anisotropic soft tissues respond to loads is important for designing better medical devices, improving surgical planning, and developing new diagnostic tools. However, it is challenging to model and quantify the mechanical properties of these tissues without destructive procedures. This study demonstrates that combining indentation tests with 3D imaging to track surface deformations enables the identification of transversely isotropic hyperelastic material parameters with substantially smaller uncertainty compared to standard indentation. These findings can help","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"208 ","pages":"Article 106417"},"PeriodicalIF":6.0,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145509551","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 : 2025-11-12DOI: 10.1016/j.jmps.2025.106423
I. Doghri , M. Haddad , G. Tsilimidos , S. Haouala
A coupled time and space homogenization formulation is proposed for heterogeneous micro-structures with viscoelastic–viscoplastic (VE–VP) constituents and subjected to large numbers of cycles. A time homogenization theory is presented in a general setting, based on two time scales and asymptotic time expansion of the fields. It leads to a macro-time VE–VP problem being fed with stress fluctuations computed from a micro-time VE problem. New theoretical results are discussed. Coupling with space homogenization is detailed for the incremental-secant mean-field homogenization (MFH) formulation. The latter takes into account per phase residual strains and stresses upon virtual VE unloading and leads to an incremental stiffness operator which is naturally isotropic for an isotropic VE–VP constituent. Coupling with time homogenization brings new terms which are not present in the original MFH method. Computational algorithms are proposed based on implicit time integration schemes, and numerical simulations illustrate the remarkable performance of the proposed formulation and algorithms.
{"title":"Coupled time and space homogenization of viscoelastic–viscoplastic composite materials under large numbers of loading cycles","authors":"I. Doghri , M. Haddad , G. Tsilimidos , S. Haouala","doi":"10.1016/j.jmps.2025.106423","DOIUrl":"10.1016/j.jmps.2025.106423","url":null,"abstract":"<div><div>A coupled time and space homogenization formulation is proposed for heterogeneous micro-structures with viscoelastic–viscoplastic (VE–VP) constituents and subjected to large numbers of cycles. A time homogenization theory is presented in a general setting, based on two time scales and asymptotic time expansion of the fields. It leads to a macro-time VE–VP problem being fed with stress fluctuations computed from a micro-time VE problem. New theoretical results are discussed. Coupling with space homogenization is detailed for the incremental-secant mean-field homogenization (MFH) formulation. The latter takes into account per phase residual strains and stresses upon virtual VE unloading and leads to an incremental stiffness operator which is naturally isotropic for an isotropic VE–VP constituent. Coupling with time homogenization brings new terms which are not present in the original MFH method. Computational algorithms are proposed based on implicit time integration schemes, and numerical simulations illustrate the remarkable performance of the proposed formulation and algorithms.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106423"},"PeriodicalIF":6.0,"publicationDate":"2025-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145515558","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 : 2025-11-11DOI: 10.1016/j.jmps.2025.106422
Jan Eliáš , Gianluca Cusatis
This article answers the question of whether homogenization of discrete fine-scale mechanical models, such as particle or lattice models, gives rise to an equivalent continuum that is of Cauchy-type or Cosserat-type. The study employs the machinery of asymptotic expansion homogenization to analyze discrete mechanical models with rotational degrees of freedom commonly used to simulate the mechanical behavior of heterogeneous solids. The proposed derivation has general validity in both stationary (steady-state) and transient conditions (assuming wavelength much larger that particle size) and for arbitrary nonlinear, inelastic fine-scale constitutive equations. The results show that the unit cell problem is always stationary, and the only inertia term appears in the linear momentum balance equation at the coarse scale. Depending on the magnitude of the local bending stiffness, mathematical homogenization rigorously identifies two limiting conditions that correspond to the Cauchy continuum and the Cosserat continuum. A heuristic combination of these two limiting conditions provides very accurate results also in the transition from one limiting case to the other. Finally, the study demonstrates that cases for which the Cosserat character of the homogenized response is significant are associated with non-physically high fine-scale bending stiffness and, as such, are of no interest in practice.
{"title":"Do discrete fine-scale mechanical models with rotational degrees of freedom homogenize into a Cosserat or a Cauchy continuum?","authors":"Jan Eliáš , Gianluca Cusatis","doi":"10.1016/j.jmps.2025.106422","DOIUrl":"10.1016/j.jmps.2025.106422","url":null,"abstract":"<div><div>This article answers the question of whether homogenization of discrete fine-scale mechanical models, such as particle or lattice models, gives rise to an equivalent continuum that is of Cauchy-type or Cosserat-type. The study employs the machinery of asymptotic expansion homogenization to analyze discrete mechanical models with rotational degrees of freedom commonly used to simulate the mechanical behavior of heterogeneous solids. The proposed derivation has general validity in both stationary (steady-state) and transient conditions (assuming wavelength much larger that particle size) and for arbitrary nonlinear, inelastic fine-scale constitutive equations. The results show that the unit cell problem is always stationary, and the only inertia term appears in the linear momentum balance equation at the coarse scale. Depending on the magnitude of the local bending stiffness, mathematical homogenization rigorously identifies two limiting conditions that correspond to the Cauchy continuum and the Cosserat continuum. A heuristic combination of these two limiting conditions provides very accurate results also in the transition from one limiting case to the other. Finally, the study demonstrates that cases for which the Cosserat character of the homogenized response is significant are associated with non-physically high fine-scale bending stiffness and, as such, are of no interest in practice.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106422"},"PeriodicalIF":6.0,"publicationDate":"2025-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145492373","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}
The aim of this work is to study the effect of cyclic loadings including shear on the ductile behavior of porous materials. We use the recent model of Roubaud et al. (2024), based on the sequential limit-analysis of an ellipsoidal cell containing an ellipsoidal cavity, in which the heterogeneous distribution of hardening is accounted for by considering a finite number of ellipsoidal layers. The model is implemented numerically in order to study the combined effects of hardening and void shape on cyclic ductile behavior. The predictions of the model are compared to finite element micromechanical unit-cell calculations with initially spherical voids, for various loading cases and hardening laws. Under cyclic loadings at low stress triaxiality levels, significant ratcheting effects in porosity, void shape and void orientation are observed. Overall, the predictions of the model are in agreement with the results of unit-cell calculations.
{"title":"Combined effects of hardening and void shape on the plasticity of porous solids under cyclic loadings including shear","authors":"François Roubaud , Cihan Tekoğlu , Almahdi Remmal , Léo Morin , Jean-Baptiste Leblond","doi":"10.1016/j.jmps.2025.106415","DOIUrl":"10.1016/j.jmps.2025.106415","url":null,"abstract":"<div><div>The aim of this work is to study the effect of cyclic loadings including shear on the ductile behavior of porous materials. We use the recent model of Roubaud et al. (2024), based on the sequential limit-analysis of an ellipsoidal cell containing an ellipsoidal cavity, in which the heterogeneous distribution of hardening is accounted for by considering a finite number of ellipsoidal layers. The model is implemented numerically in order to study the combined effects of hardening and void shape on cyclic ductile behavior. The predictions of the model are compared to finite element micromechanical unit-cell calculations with initially spherical voids, for various loading cases and hardening laws. Under cyclic loadings at low stress triaxiality levels, significant ratcheting effects in porosity, void shape and void orientation are observed. Overall, the predictions of the model are in agreement with the results of unit-cell calculations.</div></div>","PeriodicalId":17331,"journal":{"name":"Journal of The Mechanics and Physics of Solids","volume":"207 ","pages":"Article 106415"},"PeriodicalIF":6.0,"publicationDate":"2025-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145478104","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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