Gradient Continuum Research Articles

A variational formulation for three-dimensional linear thermoelasticity with ‘thermal inertia’

AbstractA variational model has been developed to investigate the coupled thermo-mechanical response of a three-dimensional continuum. The linear Partial Differential Equations (PDEs) of this problem are already well-known in the literature. However, in this paper, we avoid the use of the second principle of thermodynamics, basing the formulation only on a proper definition (i) of kinematic descriptors (the displacement and the entropic displacement), (ii) of the action functional (with kinetic, internal and external energy functions) and (iii) of the Rayleigh dissipation function. Thus, a Hamilton–Rayleigh variational principle is formulated, and the cited PDEs have been derived with a set of proper Boundary Conditions (BCs). Besides, the Lagrangian variational perspective has been expanded to analyze linear irreversible processes by generalizing Biot’s formulation, namely, including thermal inertia in the kinetic energy definition. Specifically, this implies Cattaneo’s law for heat conduction, and the well-known Lord–Shulman model for thermo-elastic anisotropic bodies is then deduced. The developed variational framework is ideal for the perspective of analyzing the thermo-mechanical problems with micromorphic and/or higher-order gradient continuum models, where the deduction of a coherent system of PDEs and BCs is, on the one hand, not straightforward and, on the other hand, natural within the presented variational deduction.

Exploring novel mechanical metamaterials: Unravelling deformation mode coupling and size effects through second-order computational homogenisation

Architected materials and mechanical metamaterials are known for their unique macroscopic properties and complex behaviour that often defy conventional continuum mechanics. Therefore, in this contribution, a recent multi-scale second-order computational homogenisation method (Dos Santos et al., 2023) is employed to explore these materials under finite strains. The approach combines a second gradient continuum theory at the macro-scale and a representative volume element (RVE) with classical first-order continuum mechanics at the micro-scale. The Method of Multi-scale Virtual Power ensures a consistent scale transition. The predictive capability and applicability of the second-order computational strategy are evaluated through coupled multi-scale numerical simulations. These simulations involve two- and three-dimensional problems, with a strong focus on the development of novel metamaterials, while also accounting for diverse loading conditions, such as tension/compression-induced undulation, bending, and compression-induced torsion. Comparisons with first-order homogenisation and Direct Numerical Simulations validate the approach. Analysis of homogenised consistent tangents reveals valuable insights into macroscopic properties. Overall, the results highlight the capability of the second-order strategy to capture significant phenomena, including second-order deformation modes, coupling deformation mechanisms, and size effects.

Nonlinear in-plane thermomechanical stability of shallow sandwich micro-arches including strain gradient tensors

The microstructural-dependent nonlinear in-plane stability characteristics of functionally graded (FG) sandwich shallow micro-arches subjected to a uniformly distributed lateral load in conjunction with a temperature rise are investigated in the current study. In this regard, the third-order shear flexible arch formulations are established within the framework of the modified strain gradient continuum mechanics. The considered sandwich micro-arches are made of nanocomposites reinforced with graphene nanofillers based upon various FG lamination patterns. The Chebyshev–Gauss–Lobatto type of gridding scheme together with the pseudo arc-length continuation technique are employed to employ a numerical solution strategy for tracing the size-dependent lateral load–deflection and lateral load–axial load nonlinear stability plots. It is observed that by adding a rise in the temperature to the applied uniform lateral load, the first bifurcation lateral load enhances, while the second one reduces. Also, it is dedicated that there is a unique value of the applied uniform lateral load corresponding to which the axial resultant load as well as the induced lateral deflection in the micro-arch are the same for all values of the temperature rise. Moreover, it is revealed that by increasing the value of the rise in the temperature, the size dependency in the value of the upper limit lateral load becomes less prominent, while it enhances in the value of the lower limit lateral load.

Second-order multi-scale modelling of natural and architected materials in the presence of voids: Formulation and numerical implementation

This contribution proposes a second-order computational homogenisation formulation for natural and architected materials in the presence of voids. The macro-scale is described by a second gradient continuum theory in the finite strain regime, and the micro-scale is modelled by the concept of representative volume element (RVE) within the classical first-order continuum mechanics. The Method of Multi-scale Virtual Power is employed to link the two scales, ensuring a variationally consistent scale transition. Unlike the multi-scale model proposed by Rodrigues Lopes and Andrade Pires (2022), the developed formulation allows the consideration of voids in the RVE domain and boundary, which is critical for modelling cellular materials, lattice structures, and metamaterials, among others. This is achieved by defining the kinematical quantities only in the solid domain of the RVE and postulating a new homogenisation relation for the second-order gradient. The kinematic constraints are imposed on the RVE by the Lagrange multiplier method and particularised for minimal (lower bound), periodic and direct (upper bound) conditions. It is demonstrated that the homogenised macroscopic stress tensors can be expressed in terms of the Lagrange multipliers. The finite element method is adopted for the numerical solution of the micro and macro equilibrium. The Newton–Raphson scheme is employed to solve the non-linear systems of equations at both scales and the consistent macroscopic tangents required for the FE2 framework are derived. Several numerical examples of porous solids, lattice structures and metamaterials illustrate the consistency and applicability of the formulation for two and three-dimensional problems.

Strain gradient viscoelasticity theory of polymer networks

A physically-based strain gradient viscoelasticity theory is proposed specially for polymer networks, accounting for the strain gradient effect and the history-dependent behavior, where the microstructure-dependence and history-dependence of stress and hyperstress are interpreted physically and quantitatively. The chain representation of the Helmholtz free energy density is transferred to the strain gradient continuum field representation through the geometric (spatial) connection between the chain stretch ratio and the continuum deformation measure. The necessity of using strain gradients to characterize asymmetric deformation of the microstructure of polymer networks serves as the origin of microstructure-dependent hyperstress, which is work-conjugated to the strain gradient field. The free energy per chain is assumed history-dependent to account for chain-environment interactions (e.g., friction between segments), which ultimately results in the history-dependent behavior of polymeric solids. A general constitutive relation is provided, which together with the assumed network structure, gives a concrete one. It is shown that independent of the assumed network structure, the stress, and the hyperstress share the same dimensionless relaxation function. A strain gradient viscoelastic constitutive relation based on the eight-chain network is applied to the analysis of polymer nanocomposites, where the complex moduli are rationally defined and calculated numerically. It is found that the strain gradient effect can be viewed as an amplifier that transforms the strain-induced stress into the effective stress.

Mesoscale DVC analyses and parameter calibration for pantographic block in 3-point flexure

In the present work, the mechanical response of a fiber network metamaterial was studied when subjected to 3-point flexure. To account for the initial deformation of the sample due to the fabrication process, backtracking (i.e., the nominal mesh was accurately repositioned in the reference configuration) was implemented in the Digital Volume Correlation (DVC) procedure. Kinematic fields were then measured via DVC at two different scales. Finite element based DVC was validated thanks to a series of initialization steps. The displacement fields were successfully measured and a peculiar deformation mechanism was observed. These measurements were employed to calibrate the constitutive parameters of a second gradient continuum model introduced for modeling this type of metamaterial. Numerical simulations are shown to be in good agreement with experimental measurements at the macroscopic scale.

Effect of damage on Rayleigh wave propagation in second gradient lattice materials

The frequencies of the Rayleigh waves propagating in periodic lattice materials prone to internal damage is analysed, using a second gradient displacement effective model as an enriched substitution continuum to capture the dispersive wave features. In this work, we model the damage by randomly eliminating beams from the full networks constructed from periodic unit cells. The apparent mechanical properties of progressively damaged structures are evaluated using dedicated homogenization techniques in the context of second gradient continuum mechanics. The wave frequency is computed, for different relative densities, versus the amount of damage for the square and hexagonal networks. Results reveal that the hexagonal lattice has higher values of frequency compared to the square lattice. Charts of the wave frequencies versus wavenumber for different amounts of global lattice damage can serve as a non-destructive technique to infer the amount of damage that exists in beam-lattice materials.

A shear lag theory integrated with second strain gradient continuum model for the composite reinforced with extensible nano-fibers

A shear lag theory integrated with second strain gradient continuum model for the composite reinforced with extensible nano-fibers

Nonlocal strain gradient-based meshless collocation model for nonlinear dynamics of time-dependent actuated beam-type energy harvesters at nanoscale

There is a diverse and well-constructed application of energy harvesting systems due to their innovate technology to provide the necessary power for low-energy electronics. In this regard, the prime objective of the current study is to analyze the size-dependent nonlinear dynamic performance of piezoelectric beam-type energy harvesters at nanoscale subjected to a time-dependent mechanical uniform load. A laminated structure containing an agglomerated nanocomposite core integrated with top and bottom piezoelectric surface layers is considered for the nanoscale bridge-type energy harvesters. To take the size dependency into account, the nonlocal strain gradient continuum elasticity is formulated based upon a quasi-3D beam theory incorporating the both features of size effects. Thereafter, the size-dependent nonlinear problem is then solved numerically relevant to simply supported and clamped end conditions via employing the meshless collocation technique. Accordingly, the numerical solving procedure is established without any background meshes as well as eliminating the integration and singularity by using proper multiquadric radial basis functions.

Second gradient continuum model for anisotropic elastic and piezoelectric structures calibrated based on phonon dispersion relations

In this paper, we present a variant of the second gradient continuum model that allows description of the spatial dispersion effects of high-frequency waves in anisotropic crystals. The considered model takes into account the strain and inertia gradient effects and the classical electro-mechanical coupling. The simplified constitutive equations of the model contain seven and four length scale parameters for piezoelectric hexagonal crystals and elastic cubic crystals, respectively. It is shown that all except one length scale parameter can be identified based on the fitting of the continuum model to the lattice dynamics calculations for the dispersion relations of the bulk acoustic phonons. Examples of the identification are presented for hexagonal piezoelectric ZnO and AlN and for cubic diamond and Si crystals. Using a calibrated model, we then obtain the dispersion relations for the high-frequency Bleustein–Gulyaev wave in piezoelectric crystals and for the shear horizontal surface wave in elastic crystals. The last one is predicted by the gradient models only for the high frequencies, while at low frequencies this wave degenerates to the bulk shear wave. Examples of calculations are also provided for the Rayleigh wave propagating through the homogeneous piezoelectric half-space and layered ZnO(AlN)/diamond/Si structures. It is shown that this kind of surface wave can be used to identify the last unknown length scale parameter of the model.

A comparison between the finite element method and a kinematic model derived from robot swarms for first and second gradient continua

In this paper, we consider a deformable continuous medium and its discrete representation realized by a lattice of points. The former is solved using the classical variational formulation with the finite element method. The latter, a 2D discrete “kinematic” model, instead is conceived to determine the displacements of the lattice points depending on interaction rules among them and thus provides the final configuration of the system. The kinematic model assigns the displacements of some points, so-called leaders, by solving Newton’s law; the other points, namely followers, are left to rearrange themselves according to the lattice structure and the flocking rules. These rules are derived from the effort to describe the behaviour of a robot swarm as a single whole organism. The advantage of the kinematic model lies in reducing computational cost and the easiness of managing complicated structures and fracture phenomena. In addition, generalizing the discrete model to non-local interactions, such as for second gradient materials, is easier than solving partial differential equations. This paper aims to compare and discuss the deformed configurations obtained by these two approaches. The comparison between FEM and the kinematic model shows a reasonable agreement even in the case of large deformations for the standard case of the first gradient continuum.

Nonlinear dynamical performance of microsize piezoelectric bridge-type energy harvesters based upon strain gradient-based meshless collocation approach

In the current exploration, the size-dependent nonlinear dynamical performance of piezoelectric bridge-type energy harvesters at microscale is analyzed. The proposed microbeam-type energy harvesters are made of a lightweight agglomerated nanocomposite polymeric passive bulk reinforced with carbon nanotubes (CNTs) and integrated with piezoelectric facesheets. In order to perform a highly accurate analysis, the modified strain gradient continuum elasticity is adopted within the framework of the quasi-3D beam theory incorporating various microstructural-dependent strain gradient tensors. Thereafter, the size-dependent nonlinear coupled electromechanical differential equations are solved numerically via employing the meshless collocation technique using a combination of the radial basis and polynomial basis functions to avoid any possible singularity. It is demonstrated that by increasing the amount of CNTs inside clusters, the effect of strain gradient tensors on the reduction in the value of induced deflection in microsized energy harvester decreases from 25.57% to 22.65% for simply supported boundary conditions, and from 33.96% to 31.56% for clamped boundary conditions. Also, the average of achieved voltage by the microsized energy harvesters increases from 245.1 mv to 262.8 mv for simply supported boundary conditions, and from 79.5 mv to 96.7 mv for clamped boundary conditions.

Postcritical Behavior of Nonlocal Strain Gradient Arches: Formulation and Differential Quadrature Solution

Arches are important components in nanostructures and systems. Because of the remarkable strength of nanomaterials, nanoarches are slender, thus enhancing their vulnerability to geometric instability, such as buckling. Although molecular simulations are often employed to analyze nanostructures, their use in routine analysis/design is formidable due to prohibitively exhaustive computation. Equivalent continuum models were developed as alternatives. The buckling and postcritical phenomena of the classical arch also form an important benchmark problem in nonlinear mechanics. This study investigates the buckling and the postcritical behavior of nanoarch subjected to inward pressure using nonlocal (NL), strain gradient (SG) continuum theory. The governing equations are derived in the form of a sixth-order nonlinear integrodifferential equation, unlike the fourth-order for a classical arch. The equation is then solved numerically using a differential quadrature (DQ) with appropriate boundary conditions. An incremental-iterative arc-length continuation is employed for the solution of the resulting algebraic system of equations. The equilibrium paths are traced for the possible instability modes, such as symmetric/antisymmetric bifurcations, snap-through and limit point instability, ascertained with the internal-external force diagrams. A particular instability mode is triggered at a certain threshold/range of the slenderness ratio of the arch, which is significantly influenced by the NL and SG interactions. These interactions not only cause quantitative changes in the instability behavior but also lead to qualitative changes, such as cessation, shift, and conversion of modes, more prominently for SG arches. Similar to classical arches, the prebuckling nonlinearity is shown to be significant.

Padé-Based Strain Gradient Modeling of Bandgaps in Two-Dimensional Acoustic Lattice Metamaterials

A strain gradient (SG) continuum theory of two-dimensional (2D) lattice metamaterials based on Padé approximation has been proposed, called PSGM hereafter, to predict their acoustic dispersion characteristics. Square and triangular elastic lattices have been investigated for a demonstrating purpose. By applying Padé approximation to the Taylor expansion of displacement field, the fourth- and eighth-order SG continuum models are established. The dispersion relations obtained by the proposed model are examined by comparing with the results by discrete analyses as well as existing SG theories. It is confirmed that the proposed theory is always more accurate than the existing SG counterpart with the same SG order. Furthermore, the present formulation is free of any unrealistic instability issue in dispersion, which has challenged the existing SG theories. The results also show that the accuracy of PSGM will be enhanced with increasing SG orders. Within the present parameter settings, it can be found that the eighth-order PSGM can successfully capture the dispersive properties of both lattices throughout the first irreducible Brillouin zone.

Complete 1D continuum model for a pantographic beam by asymptotic homogenization from discrete elements with shear deformation measure

Starting from a discrete model of a pantographic beam based on beam elements with 6 DOF in 2D space, 2 translational and 1 rotational for each end node, and proposing an energy quadratic in elongation, shear, bending and torsion deformation measures, we find by asymptotic homogenization a model of a complete 1D second gradient continuum. Compared to previous models, the discrete model of the present work includes shear deformation by considering the beam element’s end nodes relative orientation. The resulting continuum model differs in a stiffness coefficient stemming from the shear deformation, but it coincides in the limit of shear stiff elements.

Gradient continuum model of nonlocal metamaterials with long-range interactions

Compared with classical metamaterials, nonlocal metamaterials have distributed long-range interactions. In this paper, a gradient continuum model is developed to properly predict the dispersive behaviour of a one-dimensional nonlocal metamaterial with long-range interactions. First, a discrete monoatomic model is reconstructed into a supercell model. Then, a Taylor expansion based on supercell model is applied to the continuous displacement field, resulting in a gradient continuum model. The dispersive relation of the gradient continuum model is obtained and compared with discrete supercell model to evaluate its suitability. The proposed gradient continuum model with the eighth-order truncation is found to be enough to capture the dispersion behaviours all over the first Brillouin zone. The results indicate that the proposed gradient continuum model can predict the dispersion behaviour of the one-dimensional nonlocal system very well. Furthermore, the gradient continuous model of two mass-in-mass system with long-range interactions are verified.

Nonlinear Dynamical Instability Characteristics of FG Piezoelectric Microshells Incorporating Nonlocality and Strain Gradient Size Dependencies

In the present exploration, the nonlocal stress and strain gradient microscale effects are adopted on the nonlinear dynamical instability feature of functionally graded (FG) piezoelectric microshells under a combination of axial compression, electric actuation, and temperature. To perform this objective, a unified unconventional shell model based on the nonlocal strain gradient continuum elasticity is established to capture the size effects as well as the influence of the geometrical nonlinearity together with the shear deformation along with the transverse direction on the dynamic stability curves. With the aid of an efficient numerical strategy incorporating the generalized differential quadrature strategy and pseudo arc-length continuation technique, the extracted unconventional nonlinear differential equations in conjunction with the associated edge supports are discretized and solved to trace the dynamic stability paths of FG piezoelectric microshells. It is revealed that the nonlocal stress and strain gradient effects result in, respectively, higher and lower values of the nonlinear frequency ratio in comparison with the conventional one due to the stiffening and softening characters associated with the nonlocality and strain gradient size dependency, respectively. In addition, it is observed that within the prebuckling territory, the softening character of nonlocality is somehow more than the stiffening character of strain gradient microsize dependency, while by switching to the postbuckling domain, this pattern becomes vice versa.

Constructing continuum models of acoustic metamaterials via the symbiotic organisms search (SOS) optimization

Based on the symbiotic organisms search (SOS) optimization algorithm, a robust strain gradient (SG) continuum model has been proposed to accurately capture the broadband dispersion relations of one-dimensional acoustic metamaterials. Under the continuous assumption, an unavoidable key step is the Taylor expansion of displacements, which directly influences the accuracy of the corresponding continuum theory. When the wavelength becomes comparable to the periodic characteristic size, the coefficients of Taylor expansions need necessary adjustments due to the discreteness of the microstructure. Thus, the continuum theories still face critical challenges in predicting the broadband dispersion feature. This remains widely open so far. In this study, we attempt to adopt the SOS optimization to determine the optimal Taylor expansion coefficients to guarantee the dispersion diagrams causing the minimal error throughout the first Brillouin zone. The robustness of the SOS-based SG continuum model is demonstrated with three benchmark examples, i.e., the monoatomic, diatomic, and mass-in-mass lattices. Such an attempt of constructing continuum models with the help of optimization tools may shed some new light on continuum mechanics of structure media.

Three-dimensional nonlinear stability analysis of axial-thermal-electrical loaded FG piezoelectric microshells via MKM strain gradient formulations

The present study focuses on the moving Kriging meshfree (MKM) technique combined with the three-dimensional strain gradient continuum mechanics to analyze three-dimensional nonlinear stability response of thermo-electro-mechanical loaded functionally graded (FG) piezoelectric cylindrical microshells. The derived three-dimensional MKM model has the capability to present the transition of buckling mode containing different microsize-dependent gradient tensors. The microshells are made of a mixture containing PZT-4 and PZT-5H piezoelectric phases, the material properties of which vary continuously along the shell thickness and captured based on the power law composition scheme. With the aid of MKM strain gradient-based shell model, it is possible to satisfy the function property associated with Kronecker delta by putting the required boundary conditions directly at the associated nodes without any predefined mesh. The microsize-dependent nonlinear stability plots are obtained in the presence of modal transition relevant to various environment temperatures, external electric voltages, microstructural size dependency tensors, and power-law indexes. It is found that the stiffening character related to all three microstructural strain gradient tensors is more significant for the second nonlinear stability mechanical/electrical load in comparison with the first one. However, for the three-dimensional mechanical/electrical bifurcation load, the stiffening character of these gradient tensors becomes even lower than the first postbuckling load. Moreover, by applying an actuation via a positive voltage, the first three-dimensional critical buckling load and the associated end shortening reduce, while an actuation via a negative voltage results in to increase them. It is observed that by shifting to the postbuckling territory, the role of the electric actuation gets negligible.

Analysis of surface effects based on first and second strain gradient mechanics

The present contribution provides an analysis of surface effects in the mechanical response of architected materials modeled in the framework of strain gradient mechanics. The classical and strain gradient properties are evaluated by relying on a dedicated discrete homogenization method to upscale the microstructural information towards an effective strain gradient continuum at the macroscopic level. The formulation of the strain gradient model formulated via Hill extended macrohomogeneity condition allows a proper surface expression of the effective strain gradient kinematic and static variables. The scaling law of the strain gradient moduli with the edge contribution is obtained from their closed-form expressions versus the lattice microstructural parameters, and recoursing to the notion of shape derivative. The sensitivity of the strain gradient moduli to the relative amount of bulk versus surface effects is evaluated, showing that absolute size effects are well captured by strain gradient moduli. The energetic formulation of a second strain gradient continuum allows to revisit the notion of anisotropic surface energy, thereby providing a generalization of Mindlin's model (Mindlin, 1965) of surface energy.