Taking a discrete beam lattice as a starting point, continuum models are derived using continualisation and asymptotic series equivalence. The models contain higher-order spatial gradients of the displacements and, therefore, belong to the class of so-called generalised continua. Furthermore, the continuum models are anisotropic, not only regarding the lower-order terms (i.e. the classical elasticity terms) but also the higher-order terms (i.e. the gradient-enrichment terms). We show that the resulting continuum models can be interpreted as particular cases of the Theory of Critical Distances, which itself is a special case nonlocal elasticity. Two minor simplifications are suggested in order to facilitate straightforward finite element implementation. Taking the compact tension test as a numerical example, the resulting models are shown avoid singularities in the stress fields around sharp crack tips. Finally, a comparison is carried out with the results of the associated discrete beam lattice.

Starting from the finite element equations of a periodic beam lattice, a continuum theory is derived via Taylor series expansions. The continualised equations of motion are of the micropolar type and thus contain stiffness and inertia contributions in terms of translational as well as rotational degrees of freedom. Scrutiny of the underlying energy functionals and of the model’s response to excitation via harmonic waves confirms that the continualised model is unstable. In order to stabilise the model, Padé approximation is applied to the rotational equation of motion. As a result of this process, micro-inertia terms appear in the model that are expressed as the spatial gradients of the standard rotational inertia terms. It is demonstrated by means of an analysis of dispersive waves that the Padé approximation not only stabilises the model but also improves the accuracy with which the continuum model approximates the original discrete model.

This work presents a new methodology to model elastic lattice systems through a two-step approach that permits to reliably capture its dynamic behavior with a lower computational cost than modeling the lattice explicitly. The first step consists of a non-standard continualization accounting for scale effects. Several methods are explored to derive new continuum models, whose dispersive and vibrational behaviors are compared to that of the lattice, considered as a reference. Non-classical models with micro-inertia reveal high accuracy, not presenting physical inconsistencies. The second step follows a FEM spatial discretization of the developed continua, accounting for micro inertia terms in the mass matrix. Finally, the FEM formulation allows the use of element sizes larger (up to four times) than the physical length scale of the lattice system, thus significantly reducing the computational cost while maintaining accuracy and enabling a versatile application to materials, geometries and boundary conditions. The methodology is tested here for a 2D system with displacements in the plane, but can be extended to other lattice typologies as well.

Periodic lattice models of beams are homogenised to obtain equivalent continuum models. Various modelling aspects for the discrete model are discussed, such as the selection of a beam theory and the assumption of mass distribution. Straightforward continualisation by means of Taylor series leads, upon truncation, to a hierarchical series of models of increasing accuracy: the lowest order of these belong to the category of couple stress theories, whereas in the higher-order models the equations of motion are extended with strain gradients and acceleration gradients of the displacement and rotational degrees of freedom. Some of the resulting models are affected by instabilities. These instabilities can be eliminated by alternative approaches to continualisation based on Padé approximations, as will be demonstrated. Throughout, a systematic comparison between lumped mass and distributed mass is made.

This paper presents a methodology where a macroscopic linear material response incorporates microscopic variations, such as transient interactions and micro-inertia effects. This is achieved by implementing the temporal coupling between macro and microstructures, along with the spatial coupling, within a dynamic computational homogenisation framework. In the context of dynamic multiscale modelling, the temporal coupling method offers significant advantages by effectively reducing deviations emerging from micro-inertia effects and transient phenomena. The effectiveness of the developed procedure is validated by a comparison of the macroscopic results with the solutions of direct numerical simulation for a one-dimensional periodic laminate bar with different contrast levels. The homogenised results obtained using the developed procedure indicate that a better prediction of the macroscopic requires a larger Representative Volume Element (RVE) which improves the estimation of multiscale strain energy and a larger time window which improves the estimation of multiscale kinetic energy. The simultaneous increase in the RVE size and the time averaging window yields the best results in predicting the macroscopic response.

In this contribution, a variationally consistent novel Elishakoff beam theory considering long-range interactions has been developed, which succeeds in emulating the dynamic behavior of CNTs with both (5,5) and (10,10) armchairs, something that has not been attained previously. This theory is based on the modification of the Timoshenko one, and is taken here for the first time as a basis for two finite element implementations, their equations of motion been written either as a coupled set of equations in terms of displacement and rotation (reducible formulation) or as a single equation in terms of displacement only (irreducible formulation). Both stability and accuracy analyses are performed, concluding that the irreducible form allows longer time steps, while providing appropriate accuracy throughout the whole wavenumber range.

In this paper, the finite element discretization of non-classical continuum models with micro-inertia is analysed. The focus is on micro-inertia extensions of the one-dimensional rod model, the beam bending theories of Euler-Bernoulli and Rayleigh, and the two-dimensional membrane model. The performance of a variety of mass matrices is assessed by comparing the natural frequencies and their modes with those of the associated discrete systems, and it is demonstrated that the use of higher-order mass matrices reduces errors and improves convergence rates. Furthermore, finite element sizes larger than the corre-sponding physical length scale are shown to be sufficient to capture the natural frequencies, thus facili-tating numerical models that are not only reliable but also computationally efficient.(c) 2022 The Author. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Phononic crystals (PCs) consist of a periodic arrangement of inclusions in a matrix material, and have garnered a great deal of interest owing to a phenomenon known as band gap frequencies in which particular frequency ranges are not able to propagate through the PCs. The aim of this work is to study the effects of magneto-elastic coupling and other parameters such as randomness in geometrical properties, volume fraction and size of inclusions on longitudinal wave propagation and, in particular, on the appearance of stop-band frequencies. The results indicate that the most important parameters deciding whether a frequency is in a stop-band or a pass-band are the randomness in geometrical properties and piezomagnetic coupling. It was observed that piezomagnetic coupling can lead to a widening of the first stop-band range for a periodic microstructure. Moreover, while randomness in particle size leads to a stop-band range and reduced wave transmission in the second pass region, randomness in particle position leads to removal of the pass band ranges compared to periodic structures. Additionally, the influence of piezomagnetic coupling becomes insignificant in fully random structures.

A coupled thermoelastic model with fractional order derivative which incorporates the microstructural effects and thermoelastic coupling effects simultaneously at small scale is provided and is used to study wave dispersion and bandgap features of Bloch waves in one-dimensional phononic crystals. Dipolar gradient elasticity is used to account for the effects of microstructure while the non-Fourier heat conduction with fractional order derivatives is used to model thermal conduction at small scale. The interaction of thermo-elastic coupled waves with a periodic structure leads to Bloch waves, and the transfer matrix method is used to obtain the dispersion equation of the Bloch waves based on the Bloch theorem. A parameter study is performed in the numerical example to investigate the influence of the strain gradient parameter, the micro-inertial parameter, the relaxation time and the fractional order on the dispersion and bandgap of Bloch waves.

We used a novel single-notch triangle (SNT) sample to conduct dynamic fracture tests and investigate the effect of a circular cavity on the dynamic fracture mechanisms that occur in fractured rock masses subjected to a single crack under dynamic loads. Dynamical loads were applied with a drop hammer weight device. A modified finite difference method code was developed to simulate the dynamic cracking behavior of SNT samples under the action of impact loading. Next, we calculated the maximum principal stress and circumferential stress around the circular cavity and investigated dynamic crack initiation toughness (DCIT) by applying both experimental and numerical approaches. We also investigated the mechanisms that occur in the circular cavity with regards to the initiation of dynamic cracking and the cracking behavior by applying numerical models. Analysis demonstrated that the circular cavity promotes cracking initiation behavior and that the most significant impact was detected at a horizontal offset distance of 15 mm. Furthermore, the crack initiation time and DCIT were used effectively to evaluate the dynamic stability of cracks. However, the DCIT is accurate parameter for investigating how a mass of fractured rock changes according to different dynamic loads. We found that the roughness of the fractured surface around the crack tip was more significant than that of the fractured surface behind the circular cavity; this demonstrated that the circular cavity enhanced the energy consumed during crack propagation.

The accuracy of the Theory of Critical Distances, Gradient Elasticity and the Averaged Strain Energy Density criterion in estimating fatigue lifetime of notched additively manufactured Ti6Al4V is assessed against numerous experimental results generated under load ratios equal to -1 and 0.1. The 3D-printed Ti-alloy under investigation was tested by keeping the notches in the as-manufactured condition. The common feature of the considered design approaches is that they all make use of a material length scale. The validation exercise based on the generated experimental results demonstrates that the length scale concept can be extended successfully also to the fatigue assessment of notched 3D-printed metallic components.

This paper suggests a methodology to obtain homogenised material properties from a transient dynamic numerical model. The standard Hill–Mandel Principle, based on spatial averages, is extended with time averaging. Thus, in addition to a sufficiently large Representative Volume Element (RVE) to carry out the averaging in space, a sufficiently large time window is required to carry out the time averaging. The space–time averaging procedure is validated for a periodic laminate bar subjected to a variety of boundary conditions, impedance contrasts and loading conditions. The homogenised results converge to the analytical solutions and confirm that having a higher impedance contrast between laminate components requires not only larger RVE sizes but also longer time averaging windows. The most efficient macroscopic approximation is obtained by a balanced increase in RVE size and time averaging window.

The mechanical behaviour of metamaterials typically depends on their microstructural configuration and composition, in addition to their relative density. The design of these materials requires extensive experiments or complex finite element models which tend to be numerically demanding. In order to understand, control and optimise the macroscopic mechanical behaviour, in this paper numerical homogenisation is applied to a simple square unit cell with a single inclusion using a combination of elastic and viscoelastic responses on the micro level. Through a systematic analysis of unit cell behaviour with increasingly complex microstructural configurations, it is shown how certain macroscale constitutive laws can be obtained in a controlled and controllable manner.

Smart composite materials have been an active field of research in the last few decades. Magnetorheological elastomers (MREs) are examples of such smart composites and they show a coupling between magnetism and elasticity. MREs are heterogeneous materials and they consist of magnetic particles and a silicone-based elastomer. To describe and predict the behaviour of the macroscopic continuum accurately, microstructural information needs to be taken into account in the analysis of heterogeneous materials. Therefore, a combined approach called multi-scale analysis is used to consider various scales of observation simultaneously. The concept of Representative Volume Element (RVE) is typically employed by multi-scale approaches to describe the micro scale, and thus the size of RVE becomes a model parameter in such techniques. This has motivated the determination of the RVE size and the derivation of magnetoelastic constitutive relations in terms of the RVE sizes in our paper. The finite element method and a statistical analysis based on the coefficient of variation have been used to determine the RVE size of MREs. The results show that it is possible to determine a lower bound of the RVE size for an MRE. Furthermore, a parametric study has been conducted to examine the sensitivity of the RVE size on the different material properties of the constituents. It was found that the RVE size is primarily set by the contrast of the different material properties, i.e. the stiffness, permeability and magnetoelastic coupling coefficients.

Bergan-Wang theory for thick plates is extended from statics to dynamics. In line with static theory, kinematic assumptions are developed and explored that allow the equations of motion to be expressed in terms of the transverse displacement only. These assumptions include approximations of the shear strains in terms of spatial and temporal derivatives of the transverse displacement, as well as a simplification of the rotational inertia. The equations of motion are derived systematically through variational principles. The resulting partial differential equations are eighth-order in space and, depending on the kinematic assumptions, can be second-, fourth- or sixth-order in time. An analysis of dispersive flexural waves is used to compare and contrast the various theories.

A generalized theory of nonlocal elasticity is elaborated. The proposed integral type nonlocal formulation is based on attenuation functions being assumed as the convolution product of n first order (Eringen type) kernels. The theory stems from a generalized higher-order constitutive relation between the nonlocal stress and the local strain. Inspired by the Eringen two-phase local/nonlocal integral model, this theory can also be thought of as the constitutive relation for an (n + 1)-phase material, in which one phase has local elastic behavior, and the remaining n phases comply with nonlocal elasticity of higher order. The theory is supported by a suitable thermodynamic framework. In the spirit of Eringen's 1983 paper, the particular family of attenuation functions adopted are the Green functions associated with generalized Helmholtz type differential operators of order n -which suggests denoting this model as a generalized nonlocal elasticity theory of n-Helmholtz type. Besides the integral type nonlocal formulation, elegant and compact expressions for the differential and integro-differential counterpart are derived. For n = 1 this formulation straightforwardly leads to the Aifantis 2003 implicit gradient elasticity theory with simultaneous stress gradients and strain gradients, which was postulated to eliminate stress and strain singularities from crack tips and dislocation lines. For n = 2 an implicit gradient elasticity formulation with bi-Helmholtz type stress and strain gradients is obtained. The paper is complemented by a companion Part II on the particularization of the generalized theory of nonlocal elasticity for the one-dimensional case, along with some applications in statics and dynamics.

This paper is the second in a series of two that deal with a generalized theory of nonlocal elasticity of n -Helmholtz type. This terminology is motivated by the fact that the attenuation function (kernel) of the integral type nonlocal constitutive equation is the Green function associated with a generalized Helmholtz differential operator of order n . In the first paper, the governing equations have been derived and supported by suitable thermodynamic arguments. In this second paper, the proposed nonlocal model is specialized for the one-dimensional case to solve boundary-value problems. First, the relevant higher-order nonstandard boundary conditions in the differential (or, more precisely, integro-differential) version of the theory are derived. These boundary conditions are consistent with the particular family of attenuation functions adopted in the integral formulation. Then, some simple applications to statics and dynamics problems are presented. In particular, the theory is used to capture the static response and to perform free vibration analysis of a discrete lattice model with periodic microstructure (mass-and-spring chain) featured by nearest neighbor and next nearest neighbor particle interactions. In the latter case, boundary effects arise at the two lattice ends that are well captured by the proposed nonlocal continuum formulation. The nonlocal material parameters are identified a priori by matching the dispersion curve of the discrete lattice model, and a comparison in terms of attenuation function is also presented.

A continuum model of piezomagnetic material with strain, magnetic and piezomagnetic coupling gradient effects is proposed using a variational principle in this work. This model is employed to an anti-plane problem, and a general solution is constructed in polar coordinates. Special attention is paid to microstructural effects on the magnetic and mechanical response in an infinite piezomagnetic medium with a void. It is found that the microstructural length scales have a significant influence on the mechanic and magnetic fields. The three length scales (corresponding to strain, magnetic and piezomagnetic coupling gradients) are indispensable to describe the nonlocal effects of piezomagnetism. Additionally, controlling the direction and magnitude of the magnetic field at the edge of the void can be achieved by adjusting the microstructural length scales of the piezomagnetic medium.

In this paper two partially complementary formulations of the simple phenomenological Kachanov-Rabotnov continuum damage constitutive model are presented. The models are based on a consistent thermodynamic formulation using proper expressions for the Helmholtz free energy or its complementary form of the dissipation potential. Basic features of the models are discussed and the behaviour in tensile test and creep problems is demonstrated.