Granular materials are ubiquitous in nature and are used extensively in daily life and in industry. The modeling of these materials remains challenging; therefore, finding models with acceptable predictive accuracy that at the same time also reflect the complexity of the granular dynamics is a central research theme in the field. Soft particle packings present additional modeling challenges, as it has become clear that soft particles also have particle-level relaxation timescales that affect the packing behavior. We construct a simple one-dimensional, one-timescale model that replicates much of the essence of compressed hydrogel packing mechanics. We verify the model performance against both 3D and 2D packings of hydrogel particles, under both controlled strain and stress deformation conditions. We find that the modification of a Standard Linear Solid model with a strain dependent prefactor for the relaxation captures the time-history and rate dependence, as well as the necessary absence of cohesion effectively. We also indicate some directions of future improvement of the modeling.

This paper presents a variationally consistent model for a one-dimensional continuously distributed interface. For simplicity and to emphasize the consequences of the present approach, the mechanical system is assumed to consist of an elastic axially deformable truss connected to a fixed environment. In other words, we model an elastic truss anchored to a rigid support by means of a bed of axially distributed damage-elasto-plastic springs. The bed of springs is equipped with (i) two plastic accumulation fields, one for each relative direction of the one-dimensional space in which the model is embedded, and with (ii) a damage field. The thresholds of damage and plastic fields are coupled in the dissipation potential. Additional coupling between plastic and damage is also considered in the dissipation potential. This constitutive coupling makes each loading cycle slightly and progressively more damaging so that fatigue effects are achieved. The evolution problem is stated as a hemivariational inequality which ensures irreversibility of the internal fields. The numerical examples include: (i) a pull-out test; (ii) monotonic growth of damage and plastic descriptors at each point of the one-dimensional domain; (iii) progressive narrowing and degradation of hysteretic loops; (iv) Wöhler curves and identification of Basquin-Coffin-Manson-type relation parameters; and (v) propagation of de-bonding and identification of Paris-law-type parameters. The proposed formulation can therefore be regarded as minimal, variationally consistent, and physically interpretable.

The work reported in “Granular micromechanics-based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations" misidentified key terms in the grain-pair objective relative displacement when accounting for the second gradient of placement. In this paper, we correct that oversight by deriving a revised expression for the grain-pair objective relative displacement within the granular micromechanics framework. The amended terms, which resemble Christoffel symbols expressed in terms of strain gradients, modify the contributions of both the normal and tangential components to the strain energy and, consequently, alter the identified strain gradient elastic parameters. Importantly, the identification of the standard (first gradient) elastic tensor remains unchanged. This brief paper presents the corrected derivation, the resulting stiffness tensors for anisotropic strain gradient elasticity, and updated analytical expressions for the material parameters in both 2D and 3D isotropic settings.

Orthodontic treatments apply controlled displacement or mechanical forces (quasi-static but persistent) to teeth via dental braces. These forces are transmitted through the periodontal ligament (PDL) to the surrounding alveolar bone, initiating a bone remodeling process that allows the teeth to gradually shift position. The PDL transmits load to the alveolar bone and initial stress peaks relax with time as bone resorbs in regions of compression and deposits in regions of tension. Bone tissue, although mostly elastic on short time scales, exhibit therefore adaptive behaviors over longer durations due to cellular remodeling. In this paper we use viscoelastic Maxwell models to capture such an adaptive behavior. Two models are considered. The first is a two-degrees-of-freedom (2DOF) model. The second investigates the continuum two-dimensional case. We derive them, first time in this context, with the use of a Hamilton-Rayleigh principle. For both cases, we show that the load free configuration changes according to the experienced load and the models quantify these concept. The models are general and can be applied to any load history. Besides, the second 2D model can also be applied to any geometric configuration. The examples are nevertheless sufficiently simple to be solved analytically. The target is to help to predict optimal loading schedules to avoid excessive stress or damage.

Recent works have shown that extended 2D Cosserat-type or micropolar continuum models predict macro-scale mechanical chirality due to coupling of stretching deformations and micro-rotation. This type of macro-scale mechanical chirality has been demonstrated to emerge in 2D materials with granular microstructures due to specific grain-interaction mechanisms. To elucidate this otherwise rarely reported macro-scale behavior, which cannot be modeled using classical Cauchy continuum model, we have utilized granular micromechanics approach (GMA) to link the micro-scale (grain) and macro-scale (collection of grains). The present paper investigates the efficacy of these theoretical micro-macro linkages through full-field micro-scale finite element (FE) simulations based upon classical continuum model. A 2D microstructure of granular motif designed using GMA is chosen to illustrate the concepts. The granular microstructure consists of grains arranged in square pattern with interconnected nearest neighbors to achieve a pre-designed grain-pair behavior. The deformation behavior of this microstructure is simulated under uniaxial extension, compression and simple shear using full-field micro-scale finite element model based upon classical Cauchy continuum. The deformation behavior of an equivalent continuum domain is also simulated using the GMA based extended 2D anisotropic micropolar model (or extended 2D Cosserat model). The comparison of results from the GMA based model with the classical micro finite element simulations strongly indicates the need for higher-order continuum models and their further refinement. It is also shown that the chiral effect and its consequent Poisson’s effect are strong and can be modulated by modifying the grain-pair interconnections.

Granular materials undergoing large deformations exhibit nonlinearity arising from both geometrical and material effects. Here we extend the earlier Granular Micromechanics Approach (GMA)-based continuum models by enriching the grain-pair elastic energy functional and dissipation potential with new tangential terms and a coupling between normal and tangential parameters to better describe these nonlinear effects. These new terms are expected to better represent various deformation mechanisms that grains experience including grain crushing, rearrangement, rotation/rolling, sliding, loss of lateral support, neighborhood densification and strain localization. The elastic energy of a generic grain-pair direction of the GMA is reformulated consisting of a complete quadratic and quartic (duffing) dependencies on normal and tangential relative displacements to capture various nuances of material nonlinearity. Further, the grain-pair dissipation potential formulation is modified by introducing (i) plastic dissipation in the tangential direction and (ii) coupling between the plastic dissipation in tension and compression. It is shown that the coupling of this nonlinear elastic and the damage-plastic behavior yields an apparent molecular-type, such as Lennard-Jones type, potential for the granular system. The GMA framework is exploited to link the grain-scale behavior to the macroscopic response. In this framework, the evolution of grain-scale dissipation parameters is obtained using a hemi-variational principle. To motivate the adoption of GMA framework, we present a brief review of grain-scale deformation mechanisms that lead to the emergent dissipative macroscale behavior and the evolution of induced anisotropy. The enriched model is used to simulate one-dimensional compression of elasto-frictional granular media. Classical behavior of lateral earth pressure coefficient, the ratio of lateral to axial stress, is predicted by the model. The obtained results are expounded with the aid of the evolution of two sets of parameters during the compaction process, one at the grain-scale and the other at the emergent macroscale. At the grain scale, we investigate the directional evolution of elastic energy, damage and plasticity. At the macroscale, we follow the evolution of macroscale induced anisotropy by considering the eigenvalues and harmonic decomposition of equivalent stiffness.

We propose an evolutionary algorithm that seeks to determine the optimal anisotropy of a deforming body in response to a given applied mechanical load, under the constraint of assigned mass. The algorithm is based for the first time upon a granular micromechanics approach to determine the effective material behavior, making use of an orientation-dependent distribution of normal and tangential elastic grain-grain interactions, whose associated stiffnesses are assumed to depend on an orientation-dependent angular mass density. This novel idea is intrinsically simple and takes advantage of both those penalization techniques, that are generally used in topological optimization, and on those basic concepts of continuum granular micromechanics that are particularly prone to be used in this field. The algorithm is initialized with an isotropic distribution of mass such that the total mass exceeds the desired one. Grain-grain interaction stiffnesses along orientations that are only lightly stressed by the applied load are penalized and the angular mass density is accordingly reduced along these orientations. This yields a non-uniform orientation-dependent mass density and, in turn, an anisotropic constitutive law. The proposed algorithm is numerically evaluated for two load cases implying homogeneous deformations and the response to loading produced by the optimal effective fourth-rank elasticity tensor is compared with that obtained by isotropic angular mass density reduction. The proposed algorithm can be employed for engineering microstructures and as a building block for topology optimization algorithms.

The aging phenomena are often associated with the diffusion of certain particles which then activate internal chemical reactions. In this paper, the system of such particles is modelled with an damaging fluid and a hemi-variational method is proposed in order to describe damage and deformation of a dam-shaped two-dimensional body, where a key point is the introduction of the coupling between the damage and the damaging fluid concentration. Another important key point of the present model is the presence of energetic thresholds for damage activation, which are assumed to be different in tension (lower) and in compression (higher). In this work, the body is subjected not only to the self-weight of the two-dimensional dam body and to the left-hand side water pressure, and therefore to the distributed external loads, dual of the displacement field, but also to the dual of the concentration of the mentioned damaging fluid, called the external distributed damaging fluid influx pressure. The parametric analyses are carried out in terms of diffusivity and damage-concentration coupling. The influx pressure drives the incoming flow of the damaging fluid, which is coupled with the damage variable, which is induced to evolve until the failure event, which characterises the lifetime of the structure. In order to evaluate the reasonableness of the model two cases are analysed: in the first one the body is modelled having a rectangular shape. In the second one a realistic trapezoidal dam shape is considered. Both have the same area and height in order to compare the results. This comparison yields the following intuitive considerations. First of all, at the beginning of the time history, i.e., at the time at which the structure is assumed to be built, the tensile state, evaluated by the positive part of the trace of the deformation tensor, of the rectangular structure, is much higher then that of the trapezoidal one. The evolution of damage is faster in the rectangular model and, as expected, the present trapezoidal shapes for the design of the dams possess a longer lifetime and are valid not only for the structural response at the time of construction but also, within the presented model, for a better aging performance. It is worth noting that this is a standard result for dam engineers, and justifies the shapes that are used in the present dam design.

The investigation of stress transfer phenomena at the interface of composite materials has been a critical topic in mechanical engineering. This research examines the mechanisms governing the stress distribution at the interfaces and the exchange between different phases. Aiming at exploiting the complementary contributions of diverse materials within each phase, the analysis of bond behaviour could optimize the overall performance and functionality of composite materials. Numerous documents in literature provide efficient computational simulations that replicate the non-linear behaviour of interface contacts. However, many models are dependent on factors like grid-mesh configuration and step interaction. The purpose of this research is to develop an interface model, based on the elasto-plastic response of a single spring. The damage formulation includes contributions such as irreversible damage, plastic kinematic descriptors, and a hemivariational approach. Karush-Kuhn-Tucker conditions are derived to govern the evolution of descriptors, damage, and plastic variables. This model’s suitability and reliability are verified by comparing numerical results with experimental data from pull-out tests conducted on different systems of a single steel cord embedded in an inorganic mortar matrix. Finally, the evolution of damage and plasticity is also demonstrated.

Standard finite element analyses (FEM) based on tension-compression symmetric constitutive laws are not always well suited to capture the structural response of masonry arches, whose behavior is largely governed by compressive action and very limited tensile resistance. In this context, Heyman’s assumptions and limit analysis principles often provide an effective reference framework. However, classical limit analysis formulations offer limited flexibility in material characterization and cannot be directly extended to account for variations in stiffness, finite tensile capacity, or heterogeneity in masonry properties. This work proposes a novel application of a granular micromechanics-based continuum formulation to the analysis of masonry arches and related structures within a finite element framework. The originality of the approach lies in extending the granular micromechanics formalism, originally developed for bimodulus and granular-like materials, to the thrust-line assessment of masonry arches and to the reproduction of no-tension-type structural responses. The formulation is particularly suited to masonry-like non-uniform media, where orientation-dependent interactions at the local scale govern the effective mechanical response at the structural scale. In particular, the formulation is used to investigate the conditions leading to the thinnest statically admissible arch configuration under prescribed loading and boundary conditions. The constitutive response is derived from an orientation-dependent (micro-)interaction energy and incorporates tension/compression asymmetry through distinct (micro-)stiffnesses in extension and compression, governed by a single parameter n, defined as the ratio of compressive to tensile stiffness. By calibrating n, the model can reproduce the unilateral response associated with Heyman’s no-tension hypothesis while retaining the flexibility of continuum mechanics to accommodate alternative material behaviors and stress-based analyses beyond classical limit formulations. From a numerical standpoint, the approach provides a finite-element-based treatment of the underlying boundary-value problem, while preserving a stress-based interpretation of the structural response. A thrust-line reconstruction procedure is developed by post-processing the computed stress field to obtain sectional resultants, eccentricity, and the corresponding line of pressure along the arch. The approach is assessed through a set of benchmark problems including: (i) the minimum-thickness condition for a semicircular arch under self-weight, used for the calibration of parameter n; (ii) families of segmental arches to evaluate transferability across geometries; (iii) combined gravity and distributed horizontal loading, for which the predicted critical multiplier closely matches reference values; and (iv) concentrated vertical loading at varying positions, capturing the associated thrust-line migration and loss of admissibility when such a line exits the thickness. For sufficiently large values of n, the proposed framework yields thrust-line predictions comparable to those of no-tension masonry theory. At the same time, the model parameters can be adjusted to account for finite tensile capacity and stiffness variability, making it possible to distinguish between different masonry constituents and mortar qualities.

Uniaxial or one-dimensional compression is widely used in granular material processing to produce granular compacts and is a key experimental method for evaluating the compressibility of granular materials. In this study, the granular micromechanics approach (GMA) was applied to simulate the one-dimensional uniaxial compression behavior of granular packing. GMA offers a continuum modeling paradigm that connects the macroscale response to the grain-scale mechanisms. To describe the stiffening behavior that elasto-frictional granular materials exhibit in confined compression, the elastic energy functional and dissipation potential were modified from previous versions of GMA-based models. The elastic energy of a grain pair was formulated to include a higher-order dependence on the normal relative displacement of the grain pair. The dissipation potential was also modified by incorporating coupling between plastic parameters that characterize plastic deformation accumulation as the grain-pair experiences compression and/or tension during loading process. The modified model, derived in the framework of geometrically nonlinear deformations, was then applied to replicate experimentally measured 1D compression behavior of granular materials undergoing large compression. The results reveal nearly universal scaling with respect to the model parameters for predicting the behavior of granular materials composed of different particle types or initial density. A parametric study was conducted to determine the evolution of the microscale stored elastic energy and dissipation parameters in terms of grain-pair directions. In addition, this paper provides a brief literature review of experimental observations and modeling approaches related to powder compaction behavior. MPa (Vertical Stress) One-dimensional Compression Piston Die Granular Material Dense-Stiff G (Vertical Strain)

Many materials exhibit a bimodulus behavior which manifests as tension–compression asymmetry under uniaxial loading conditions. From the viewpoint of engineering analyses and design, attempts have been made to properly account for this asymmetry by defining bimodulus models under various multiaxial loading conditions. The models available in the literature propose a complex set of cases valid for different combination of tensile/compressive loading characterized in either strain or stress space based on classical continuum mechanics. The key drawback of these propositions is the attempt to reduce an inherently nonlinear problem to a simpler linear one in opposition to the clear evidence of loading dependency. Moreover, the proposed approaches are generally conceived in an orthotropic framework even though the material is initially isotropic. It is only upon the deformation that an apparent anisotropy is induced due to the tension–compression asymmetry, which cannot be considered in the initial, reference, configuration. Indeed, from a fundamental viewpoint, the emergence of the macroscale asymmetry and induced anisotropy has its roots in the microscale mechanics of an otherwise isotropic (or other) microstructure. To further expatiate the fundamental view, we exploit the granular micromechanics paradigm to present a simple continuum model within the isotropic framework in reference configuration. Using this approach, the macroscale tension–compression asymmetry and induced anisotropy is shown to emerge naturally without the inconsistencies observed in the methods proposed in the literature.

Purely compressed shells are often elegant and highly efficient structural forms, but this leanness may create risk if they are subjected to unexpected patterns and magnitudes of loading, such as may arise due to seismic events. In the same way that historic masonry structures were designed to sustain loads by activating purely compressive force paths, a modern metamaterial can be designed for specific purposes following the same logic. Conventional analysis methods for compression-only shells and vaults, often developed for masonry structures, have tended not to model combined vertical and horizontal loads directly. This has created a significant challenge for engineers assessing historic vaults or designing new shells. To address this gap, this paper presents an enhanced method based on membrane equilibrium analysis (MEA) and the static theorem of limit analysis. This approach is the first application of MEA to directly consider vertical and horizontal body forces acting on a compression-only shell through a parametric formulation of an Airy stress function. The method is applied to a case study of a sail vault subjected to vertical and horizontal loads. Moreover, it is demonstrated how this approach can be used to define iso-resistant shapes that offer more sustainable design options while preserving structural capacity.

The examination of stress transfer phenomena at the interface of composite materials has long been a focal point in mechanical engineering. This investigation involves an in-depth study at these contact surfaces, aiming at exploring the mechanisms governing stress distribution and exchange across the distinct phases. In order to exploit the interaction between diverse materials within each phase, leveraging their complementary attributes, the bond behaviour analysis would help to optimize the overall performance and functionality of the composite material. Many documents are available in literature providing efficient computational simulations, reproducing the non-linear bond behaviour at the interface contact. Nevertheless, those models are sensitive to underlying assumptions and modelling choices, which can significantly influence the results. The aim of this research is to include all the modelling assumptions in the formulation of an elastic and a dissipation energy functionals within the context of a hemi-variational principle. For the sake of simplicity, the interface interaction is reduced to a single damage-elasto-plastic spring in series to an elastic one, which represents the deformation of one of the two phases. The irreversible formulation was developed considering the following steps: (i) definition of two irreversible kinematic descriptors, i.e. a damage index and a plastic displacement, (ii) assumption of an energy functional and (iii) postulation of a hemi-variational principle. Governing equations are therefore derived, including the Karush-Kuhn-Tucker conditions that predict the evolution of irreversible damage and plastic descriptors. The suitability and reliability of this model have been verified comparing numerical and experimental results for the pullout tests carried out for different systems of a single steel cord embedded in an inorganic mortar matrix. Likewise, the evolution of damage and plasticity is calculated. Finally, a parametric investigation is implemented to evaluate to what extent a single factor of the energy functional affects the overall performance.

This work introduces a novel approach to modeling one-dimensional (1D) reinforcements within a two-dimensional (2D) second-gradient elastic matrix, suitable for describing various engineering structures undergoing small displacements and strains. The matrix obeys the first strain gradient elasticity in the Mindlin formulation and incorporates reinforcements, which are represented as zero-thickness interfaces with the elastic properties of one-dimensional (1D) extensional Euler-Bernoulli beams. The core innovation lies in the variational deduction of the generalized boundary conditions at these interfaces, which effectively capture the behavior of the reinforcements without requiring their full geometric representation. The proposed methodology is validated through finite element simulations of a reinforced structural element subjected to uniform bending.

In this study, we aim to analyze the dispersion of ultrasonic waves due to second-gradient contributions and attenuation within the framework of continuum mechanics. To investigate dispersive behavior and attenuation effects, we consider the influence of both higher-order gradient terms (second gradients) and Rayleigh-type viscoelastic contributions. To this end, we employ the extended Rayleigh–Hamilton principle to derive the governing equations of the problem. Using a wave-form solution, we establish the relationship between the phase velocity and the material’s constitutive parameters, including those related to the stiffness of both standard (first-gradient) and second-gradient types, as well as viscosity. To validate the model, we use data available in the literature to identify all the material parameters. Based on this identification, we observe that our model provides a good approximation of the experimentally measured trends of both phase velocity and attenuation versus frequency. In conclusion, this result not only confirms that our model can accurately describe both wave dispersion and attenuation in a material, as observed experimentally, but also highlights the necessity of simultaneously considering both second-gradient and viscosity parameters for a proper mechanical characterization of materials.

It is well known from the literature that the phase velocity of waves is directly correlated with the stiffness of the material; however, experimental practice shows that this velocity changes significantly with varying frequencies, despite the fact that the elastic modulus of the material is, by definition, a material constant. We explore the dependence on the frequency of longitudinal ultrasonic plane waves velocity in construction materials, both from experimental and modeling points of view. For the sake of simplicity, the dispersive features are modeled by considering the case of a 1D medium, and two different kinds of mechanical models capable of describing wave dispersion phenomena are employed: a non-dissipative strain-gradient elastic model, and a dissipative viscoelastic one. In both cases, by using the extended Rayleigh-Hamilton principle, we derive the governing equations for 1D bulk waves propagation; in particular, in the case of the dissipative viscoelastic model either classical linear damping or Kelvin-Voigt damping is considered. The comparison of theoretical results with experimental findings obtained by ultrasonic tests on natural (sandstone) and artificial (concrete) construction materials shows that both theoretical models can satisfactorily describe the experimental behavior. These results encourage further experimental investigations for a clear and quantitative identification of the model that can be better used for engineering purposes.