Fibrous networks form the structural basis of many biological and synthetic materials such as collagen and self-assembling peptide hydrogels (SAPHs). While the macroscopic mechanical response of these materials is known to emerge from their mesoscale structure, the specific impact of fibre bundling (fibres associating into bundles) remains under explored. Using computational and analytical modelling, we analyze the effects of bundling on the bulk mechanical behaviour of fibrous networks. We develop a framework that generates 2D and 3D stochastic Mikado-type networks with a controllable bundling strength and demonstrate that bundling influences network mechanics by reshaping the network geometry and through internal mechanical coupling within bundles. We define two distinct modes of bundling: merged bundles, which behave as singular, mechanically coupled, thick fibres, and proximal bundles, where the fibres remain distinct. We find that bundling generally leads to a reduction in small strain stiffness and modifies large deformation strain stiffening by postponing the onset of stiffening and changing its rate. Notably, networks with identical topologies can exhibit different stiffness levels and stiffening behaviour depending on the degree of internal coupling within their bundles. These results identify bundling strength and bundle coupling as two independent design parameters, offering new strategies for tuning the mechanical properties of engineered hydrogels and fibrous materials.
To elucidate the interactions among dislocation slipping, twinning deformation and microcrack initiation and propagation in magnesium (Mg) alloys, a crystal plasticity-twinning-fracture coupled phase field model is established. By incorporating the energy evolution associated with these physical processes into a unified free energy functional, the model naturally captures their intrinsic coupling through energy redistribution and competition. The model is applied to single-crystal Mg alloys with various crystalline orientations, as well as texture-free and basal-textured polycrystalline systems. The results indicate that crystalline orientation and texture regulate the spatial distribution of dislocations and twins, thereby affecting microcrack evolution. In the single crystals, the anisotropic microcrack propagation is governed by the magnitude and spatial inhomogeneity of plastic deformation where symmetric plastic fields enable stable propagation along the pre-microcrack direction, whereas asymmetric fields induce a microcrack deviation toward low-plasticity regions. Twinning enhances fracture resistance by increasing the energy dissipation near pre-microcrack tips, while the stress concentrations arising from the strain incompatibility and dislocation pile-up at twin boundaries (TBs), as well as the twin-twin and twin-dislocation interactions, promote microcrack propagation. In the polycrystalline systems, microcracks preferentially initiate at grain boundaries (GBs) due to the twin-GB, dislocation-GB, and twin-dislocation-GB interactions, while propagate and coalesce along GBs driving by the local plastic accumulation. The microcracks within grains propagate mainly along slip bands or TBs. Furthermore, the complex stress states near the microcrack tips and GBs intensify the interactions between plastic deformation and fracture by regulating local deformation modes. These findings provide insights into the microstructure-dependent deformation and fracture mechanism of Mg alloys.
Optimizing fracture resistance in micro-architected materials remains challenging due to the complex interaction of microstructure-induced toughening mechanisms and the inherently history-dependent nature of crack propagation. In this work, we develop a gradient-based topology optimization framework to enhance the fracture resistance of micro-architected materials by maximizing effective fracture energy, defined as the maximum J-integral attained during crack growth. The framework couples a heterogeneous domain, where crack propagation and microstructure-induced toughening mechanisms are explicitly resolved, with an effective homogeneous domain to evaluate a path-independent J-integral serving as the optimization objective. A reverse adjoint formulation is derived to compute history-dependent sensitivities associated with crack irreversibility. Mechanism-guided strategies are incorporated through tailored initial configurations and optimization constraints. The results demonstrate that substantial enhancement of effective fracture energy can be achieved through the synergistic activation of multiple toughening mechanisms, including crack blunting, branching, deflection, and distributed microcracking. An intermediate relative-density regime (ρ¯∈[0.5,0.7]) is identified, within which multiple mechanisms coexist and fracture energy significantly exceeds that of both the base material and canonical lattices. Experimental validation using additively manufactured specimens shows close agreement between predicted and measured crack paths, peak loads, and fracture energies. Notably, the optimized architectures exhibit strong geometric similarity to deep-sea glass sponge skeletons, highlighting the capability of the proposed framework to rediscover evolution-inspired toughening strategies through a purely mechanics-driven optimization process.
The domain-independent interaction-integral (DII-integral) has been widely used to evaluate stress intensity factors (SIFs) in inhomogeneous and interfacial material systems. However, as it is derived within local continuum mechanics, its domain independence when computed from nonlocal fields has not been established. This study examines the applicability of the DII-integral within the ordinary state-based peridynamics (OSB-PD) framework under quasi-static conditions. We prove that, if the interface is perfectly bonded, the DII-integral remains domain-independent even when the integration domain intersects material interfaces. The proposed PD-based evaluation is then examined in problems featuring evolving interfaces and environmental degradation, including thermal barrier coatings with rough and evolving thermally grown oxide (TGO) layers, corrosion-degraded isotropic solids, and metal matrix composites subjected to galvanic corrosion. Numerical results demonstrate domain-insensitive SIFs across representative scenarios and quantify the influence of degradation on crack tip intensities. The developed framework provides a useful basis for fracture characterization in nonlocal settings and future extensions to coupled multi-physical degradation–fracture problems.
Structural evolution is an intrinsic feature of plastic deformation in glassy polymers, yet its microscopic origin and continuum representation remain elusive. Using molecular dynamics (MD) simulations, this work investigates deformation-induced structural evolution of microscopic plastic processes. Plastic deformation is found to initiate through spatially isolated non-affine rearrangements and to evolve into correlated, spatially extended structures after the onset of yielding. Non-affine displacement, bond rotation, and local volumetric fluctuation are strongly co-localized, indicating that they represent different manifestations of the same plastic events. Dilative and contractive volumetric strains coexist at the monomer scale, giving rise to distinct atomic-scale volume gradients. The evolution of spatial correlations reveals a progressive spreading of volumetric disturbance during yielding and post-yield flow, governed by competing generation and relaxation processes with distinct characteristic times. Based on these observations, a scalar volumetric disturbance variable is introduced to describe deformation-induced structural evolution. Its stochastic evolution is formulated in terms of plasticity-driven generation, thermally activated relaxation, and gradient-driven spatial redistribution. Upon ensemble averaging, this description yields a continuum-scale internal variable with a diffusion-type contribution. Incorporating this physically derived evolution equation, a thermodynamically consistent constitutive framework is established that captures strain softening and its evolution under deformation. At the representative volume element scale, the formulation recovers a mean-field description consistent with the observed macroscopic response. At the spatially resolved finite-element scale, the gradient term, with a coefficient estimated from the MD second-moment measure, regularizes the spatial heterogeneity of the structural variable while leaving its averaged evolution essentially unchanged. These results establish deformation-induced structural evolution in glassy polymers as an intrinsic physical process that can be coherently represented at the continuum scale through microscopic configurational dynamics.
The relationship between microstructural fabric and the symmetry class of the homogenized elasticity tensor is investigated within a rigorous measure-theoretic framework where fabric tensors are defined as tensorial moments of a positive measure on the unit sphere, and constitutive mappings are assumed to be isotropic. Within such a context, it is shown that isotropic mappings depending on the sole second-order fabric moment cannot generate effective elasticity with symmetry lower than orthotropy; such a limitation, previously acknowledged as a practical bound of second-order models, is here established as a structural theorem. In order to overcome such a limitation, it is proved that inclusion of the fourth-order moment is mandatory. In particular, under a non-degeneracy condition generically typical of isotropic mappings, triclinic effective elasticity is equivalent to absence of rotational invariances for the fouth-order fabric tensor. Accordingly, a complete spectral classification of the eight classical symmetry classes of linear elasticity is provided in terms of the joint spectral structure of the second- and fourth-order fabric tensors. It is further proved that triclinic symmetry is the generic condition in the space of completely symmetric fourth-order tensors, so that higher symmetry classes are exceptional rather than typical. The developed theoretical framework is validated numerically on eight three-dimensional lattice truss models designed to approximate prescribed symmetry classes, thus confirming that the fourth-order fabric tensor governs the symmetry of the effective elastic response both qualitatively and quantitatively.
We investigate how active microstructures select the operatorial structure of thermal evolution. Rather than postulating alternative heat-transport laws, we derive families of thermal evolution equations from the balance structure associated with active microstructural interactions under different classes of constraints. The constitutive architecture of the material determines not only transport coefficients but also the very class of the thermal evolution operator. The analysis reveals a hierarchy of constitutively selected thermal regimes, each characterized by specific coercive quantities and associated natural functional topologies. In this perspective, admissible evolution spaces are not imposed as external analytical choices but emerge as constitutive consequences of the underlying microstructural interactions. The proposed viewpoint provides a unified framework in which classical Fourier conduction, generalized non-Fourier evolution, and microstructure-induced pseudoparabolic effects appear as different manifestations of a common constitutive mechanism. To preserve the physical character of the theory, no artificial regularization devices are introduced, and the analytical conclusions are restricted to the coercive structures generated by the physical balances themselves.
A key challenge in material theory is the formulation of models that satisfy all common mechanical constitutive conditions while retaining sufficient flexibility. In this context, several important modeling aspects remain unresolved for polyconvex anisotropic hyperelasticity. We address some of these challenges and apply our results for physics-augmented neural network (PANN) constitutive modeling. The main contributions of this paper are as follows: (1) We propose a new constitutive modeling framework for polyconvex anisotropic hyperelasticity based on triclinic invariants and group symmetrization, and apply it to a PANN model. For finite symmetry groups, this PANN model fulfills all common mechanical constitutive conditions a priori. (2) We propose a new group symmetrization-based method for the construction of polyconvex invariants for finite symmetry groups. Based on this, we derive a new integrity basis for a tetragonal symmetry group and a new functional basis for a cubic symmetry group. To the best of our knowledge, these are the first polyconvex integrity or functional bases for symmetry groups characterized by structural tensors of order higher than two. (3) We benchmark different polyconvex PANN constitutive models with highly nonlinear homogenization data of cubic metamaterials. (4) We provide an introduction to the construction of polyconvex invariants, integrity and functional bases and review established polyconvex bases alongside the newly derived ones.
This paper deals with the multi-material topology design of three-dimensional elastic microstructures. The macroscopic response of the material is predicted using a well-established asymptotic homogenization theory for elliptic equations with rapidly oscillating coefficients. The elasticity system in periodic media is considered, and the underlying microstructure is perturbed by the nucleation of small spherical inclusions with material properties different from those of the background. The topological asymptotic analysis of the homogenized properties is performed with respect to the radius of the inclusions. The resulting topological derivative is written in the form of a symmetric fourth-order tensor field defined over the microstructure, measuring the sensitivity of the effective elasticity tensor to topological perturbations. This result is used to devise a multi-material topology optimization algorithm for elastic microstructures, combining the associated topological derivative with a level-set representation method. The proposed framework enables the design of microstructures with tailored effective properties, including bulk modulus maximization and Poisson’s ratio minimization.
Soft granular materials are porous packings of highly deformable particles found in systems ranging from blood clots and tissue scaffolds to starchy foods and 3D-printing inks. They start as collections of discrete, loosely packed motifs, but then transition to a dense (confluent) solid under load. Understanding when that porous-to-dense solid transition occurs is critical because it controls whether a biomaterial can be injected through a needle, how easily cells and drugs infiltrate a healing wound, and how nutrients are transported through engineered tissue. Classical approaches typically rely on counting inter-particle contacts per particle (i.e., the coordination number), but this becomes rather ambiguous as contact points evolve into contact surfaces when soft grains undergo large, nonlinear changes in size and shape. Here, we propose a fundamental shift in the prevailing conceptual framework. Instead of tracking the granular contact, we track the void population. Through computational simulations, we discover that overall porosity evolves through a universal sequence: as the material compresses, interstitial voids undergo a systematic cascade of fragmentation (proliferation) followed by collapse (elimination). The resulting peak in void population is identified as a topological marker that defines a confluence threshold in soft granular compaction. At this threshold, void fragmentation gives way to void evanescence, the macroscopic force–displacement response changes scaling, and stress distribution appears distributed through the deformed particle network. The critical threshold of this topological marker is independent of the particle size distribution or elasticity, offering a robust geometry-based descriptor for the densification of soft matter.
The fracture behavior of geomaterials is largely governed by their internal microcrack structures, and establishing a link between microcrack evolution and macroscopic fracture remains a key challenge. In this work, a multiscale anisotropic phase field model is proposed to simulate the complex fracture processes of geomaterials with randomly distributed microcracks. Within the framework of asymptotic homogenization, the system energy under brittle fracture during microcrack opening and ductile fracture induced by frictional sliding during microcrack closure is characterized. By statistically describing the distribution of microcracks, the microstructural features are incorporated into the homogenized stiffness, plastic yield criterion, and phase field structure tensor, enabling the model to naturally capture the anisotropy of stiffness, plasticity, and fracture toughness arising from the preferential orientation of microcracks. Under the principle of energy conservation and the stability condition, the variationally consistent governing equations of the proposed model are derived. Numerical examples demonstrate that the model can accurately reproduce the brittle or ductile fracture behavior of geomaterials under tensile, compressive, and mixed-mode loading conditions, and can reasonably predict fracture anisotropy induced by the preferential orientation of microcracks.
A micromechanics-based continuum damage model was used to predict the ductility of polyamide-6 (PA-6) in the as-received (unaged) and photo-oxidized (UV-aged) states. The relationship between ductility and triaxiality was experimentally characterized using round notched bars loaded to failure. The unaged bars failed by cavitation, and the ductility was strongly sensitive to the triaxiality. The UV-aged bars exhibited significantly reduced ductility, as oxidation embrittles the material through (1) chain scission at the macromolecular scale and (2) chemo-mechanical cracking at the microscale. In the model, the state of damage is defined by three independent microstructural variables with clear physical meaning. The initial state of damage was characterized using synchrotron X-ray tomography. Damage consisted of moderately oblate, penny-shaped cavities for unaged PA-6 and extremely oblate, chemo-mechanical cracks for UV-aged PA-6. Based only on differences in the initial state of damage between unaged and UV-aged PA-6, the model successfully predicts catastrophic embrittlement from photo-oxidation. For unaged PA-6, quantitative agreement between the model and experiments is realized without fitting parameters. For UV-aged PA-6, quantitative agreement is achieved by introducing a parameter to scale the rate of void opening, which phenomenologically accounts for the effects of oxidation on the dense matrix, such as chain scission and chemi-crystallization. The success of the damage model for both unaged and UV-aged PA-6 suggests (1) that chemo-mechanical damage is critical for understanding oxidation-induced embrittlement and (2) that failure of oxidized polymers follows a ductile fracture process.
Thermal stress evolution in laser directed energy deposition (DED) involves complex interplay between macroscopic thermo-mechanical response and grain-scale stress heterogeneity, posing significant difficulties for quantitative multi-scale evaluation. In this study, a three-scale numerical framework is developed for evaluating thermal stress evolution in DED. The framework couples macro-scale thermo-mechanical analysis, meso-scale thermo-fluid simulation, and micro-scale crystal plasticity, and enables explicit separation of Type I stress, Type II stress, and intragranular stress variation corresponding to macroscopic, intergranular, and intragranular scales. Data operators are introduced to transfer heat source input, clad geometry, temperature, material distribution, displacement boundary conditions, and grain structure between different scales, ensuring physically consistent macro–micro fields and boundary condition transfer. To the best of the authors’ knowledge, it is the first three-scale computational framework that enables quantitative evaluation of Type I stress, Type II stress, and intragranular stress variation in additively manufactured polycrystals under physically consistent macro–micro field and boundary-condition transfer. The framework is validated through comparisons of temperature history, melt pool morphology, macro-scale stress distribution, and macro–micro stress consistency, demonstrating agreement with available experimental data. Further numerical results demonstrate that the micro-scale stress can reach approximately twice the macro-scale tensile level and exhibits pronounced tensile–compressive heterogeneity even when the macro-scale field remains entirely tensile. Type II stress reveals intergranular tensile–compressive incompatibility reaching 164.3–204.8 MPa across representative grain boundaries, while intragranular stress variation captures intragranular stress contrasts approaching 298 MPa associated with local stress distribution. These findings demonstrate that explicit separation of scale-dependent stress components is essential for identifying critical stress regions that cannot be resolved from macro-scale or micro-scale stress magnitude alone. In addition, orientation and size-dependent analyses indicate that grain orientation and morphology influence stress heterogeneity, and strengthening the ⟨001⟩ texture reduces both intergranular and intragranular stress contrast. The proposed framework therefore provides a powerful tool for quantitative evaluation of macro–micro stress interplay in DED and offers new insight into microstructure-informed stress regulation in DED.
Bond exchange reactions (BER) is a type of chemical process that rearranges the molecular topology of polymer networks through breaking and reforming polymer chains. BER imparts reprocessing capability into conventionally unreprocessable thermosets, and make them reconfigurable, repairable, and recyclable. However, since reprocessing (especially reconfiguring) thermosets through BER is a highly complex chemo-mechanically coupled process, constitutive models that can directly connect BER-enabled topological rearrangement with chain-level kinetics and macroscopic response remain limited. Here, we develop a chemo-mechanically coupled statistical chain model for reconfigurable thermosets governed by BER. The chain density function is taken as the central internal variable, and its evolution is described by the coupled effects of affine deformation, BER-governed interconversion among associated, dangling, and free chains, and free chain diffusion in chain space. Within a thermodynamically consistent framework, chain free energy, reaction kinetics, and kinematic constraints are linked to the macroscopic stress. We further incorporate thermal activation through an activation energy and a reduced-time description, which provides a unified treatment of thermo-temporal coupling. The model captures temperature- and time-dependent responses, including stress relaxation and loading-unloading hysteresis, and reproduces the BER-governed evolution of chain orientation from an initially isotropic state to a deformation-induced anisotropic state and its subsequent relaxation, consistent with the small-angle X-ray scattering (SAXS) observations. The framework provides a physically interpretable basis for parameter identification, prediction of thermo-temporal thermomechanical behavior, and design of reconfigurable thermosets.
