Generalized Continuum Theories Research Articles

Generalized continuum theory for nematic elastomers: Non-affine motion and characteristic behavior

We develop a physically-motivated mechanical theory for predicting the behavior of nematic elastomers — a subset of liquid crystal elastomers (LCEs). We begin with a statistical description of network geometry that naturally incorporates independent descriptors for the mesogens, which create the nematic phase, and the polymer chains, which are assumed to not deform affinely with global deformations. From here, we develop thermodynamically consistent constitutive laws based on classical continuum mechanics principles and ultimately provide simple governing equations with transparent physical interpretation. We found that our framework converges identically to two previously developed mechanical theories, including the well-known neo-classical theory, when considering the extreme ends of our parametric space. We then explore the new predictive capabilities of our model inside these two extremes and illustrate its unique predictions at finite strains, which are distinct in form from other theories. We validate our model using published experimental data from four monodomain nematic liquid crystal elastomers.

Unravelling Size-Dependent and Coupled Properties in Mechanical Metamaterials: A Couple-Stress Theory Perspective.

The lack of material characteristic length scale prevents classical continuum theory (CCT) from recognizing size effect. Additionally, the even-order material property tensors associated with CCT only characterize the materials' centrosymmetric behavior and overlook their intrinsic chirality and polarity. Moreover, CCT is not reducible to 2D and 1D space without adding couples and higher-order deformation gradients. Despite several generalized continuum theories proposed over the past century to overcome the limitations of CCT, the broad application of these theories in the field of mechanical metamaterials has encountered significant challenges. These obstacles primarily arise from a limited understanding of the material coefficients associated with these theories, impeding their widespread adoption. Implementing a bottom-up approach based on augmented asymptotic homogenization, a consistent and self-sufficient effective couple-stress theory for materials with microstructures in 3D, 2D, and 1D spaces is presented. Utilizing the developed models, material properties associated with axial-twist, shear-bending, bending-twist, and double curvature bending couplings in mechanical metamaterials are characterized. The accuracy of these homogenized models is investigated by comparing them with the detailed finite element models and experiments performed on 3D-printed samples. The proposed models provide a benchmark for the rational design, classification, and manufacturing of mechanical metamaterials with programmable coupleddeformation modes.

A quadratic finite element for the relaxed micromorphic model

AbstractIn this work we discuss the relaxed micromorphic model and implementation details for a full three‐dimensional formulation entailing a quadratic Lagrangian‐Nédélec finite element and appropriate boundary conditions in the discrete setting.The relaxed micromorphic model is a generalized continuum theory with the capacity to capture more complex kinematical behaviour than in the classical Cauchy continua. Such behaviour is commonly found in materials with a pronounced micro‐structure such as porous media and metamaterials. The theory introduces the microdistortion field, encompassing nine additional degrees of freedom for each material point in the continuum, effectively turning each material point into a deformable micro‐body. The uncommon discrete formulation stems from the employment of the Curl operator in the energy functional of the relaxed micromorphic model, thus requiring H(curl)‐conforming finite elements for well‐posedness to be inherited in the discrete setting. The model further introduces the so called consistent coupling condition, which requires some technical considerations in order to be upheld correctly.This work demonstrates the finite element formulation, culminating with a numerical example.

Micromorphic theory as a model for blood in the microcirculation: correction and analysis

This paper analyzes the applicability of Eringen’s Generalized Continuum Theories as a model for human blood in the microcirculation. The applied theory considers a fluid with a fully deformable substructure, namely a micromorphic fluid. This analysis is motivated by the fact that blood itself can be considered a suspension of deformable particles, i.e., red blood cells (RBCs), suspended in a Newtonian fluid, i.e., blood plasma. As a consequence, non-Newtonian phenomena such as shear-thinning are observed in blood. To test the micromorphic fluid as a model for blood, the solution for the velocity and the motion of substructure is determined for a cylindrical pipe flow and compared to experimental results of blood flow through narrow glass capillaries representing idealized blood vessels. A similar analysis was also conducted by Kang and Eringen in 1976, but it contains some misprints and minor errors regarding the mathematical expressions and subsequent discussion which are corrected in this paper. For certain material parameters, the micromorphic fluid models capture high-shear blood flow in narrow glass capillaries very well. This concerns both the velocity profiles and the shear-thinning behavior. Furthermore, a parameter study reveals that the flexibility of substructure governs the micromorphic shear-thinning. In this regard, parallels can be drawn to the shear-thinning of human blood, which is also induced by the deformability of RBCs. This makes the micromorphic fluid a complex but accurate model for human blood, at least for the considered experiments.

An isogeometric boundary element formulation for stress concentration problems in couple stress elasticity

An isogeometric boundary element method (IGABEM) is developed for the analysis of two-dimensional linear and isotropic elastic bodies governed by the couple stress theory. This theory is the simplest generalized continuum theory that can effectively model size effects in solids. The couple stress fundamental solutions are explicitly derived and used to construct the boundary integral equations. A new boundary integral equation arises to obtain the moments and rotations introduced by the couple stress formulation. A new analytical solution is also derived in the present work for an elliptical opening in an infinite sheet under uniaxial far-field stress. Several stress concentration problems are examined to illustrate and validate the application of the IGABEM in couple stress elasticity. It is shown that the IGABEM scheme exhibits advantageous convergence properties in comparison with the conventional BEM for boundary value problems within the framework of couple stress elasticity.

Selecting Generalized Continuum Theories for Nonlinear Periodic Solids Based on the Instabilities of the Underlying Microstructure

In the context of architected materials, it has been observed that both long-wavelength instabilities leading possibly to localization and short-wavelength instabilities leading to the apparition of a deformation pattern could occur. This work proposes for the first time a comparison of the ability of two families of higher order equivalent media, namely strain-gradient and micromorphic media, to capture both patterning and long-wavelength macroscopic instabilities in those materials. The studied architected material consists in a simple one-dimensional arrangement of non-linear springs, thus allowing for analytical or nearly analytical treatment of the problem, avoiding any uncertainties or imprecisions coming from a numerical method. A numerical solution of the problem is then used to compare the post-buckling prediction of both models. The study concludes that, micromorphic media are the appropriate choice of equivalent continuum models to emply when dealing with the possibility of patterning inside a structured medium, but if long-wavelength global instability is of interest, a strain-gradient type equivalent medium is well suited.

Wave Propagation in Timoshenko–Ehrenfest Nanobeam: A Mixture Unified Gradient Theory

Abstract A size-dependent elasticity theory, founded on variationally consistent formulations, is developed to analyze the wave propagation in nanosized beams. The mixture unified gradient theory of elasticity, integrating the stress gradient theory, the strain gradient model, and the traditional elasticity theory, is invoked to realize the size effects at the ultra-small scale. Compatible with the kinematics of the Timoshenko–Ehrenfest beam, a stationary variational framework is established. The boundary-value problem of dynamic equilibrium along with the constitutive model is appropriately integrated into a single function. Various generalized elasticity theories of gradient type are restored as particular cases of the developed mixture unified gradient theory. The flexural wave propagation is formulated within the context of the introduced size-dependent elasticity theory and the propagation characteristics of flexural waves are analytically addressed. The phase velocity of propagating waves in carbon nanotubes (CNTs) is inversely reconstructed and compared with the numerical simulation results. A viable approach to inversely determine the characteristic length-scale parameters associated with the generalized continuum theory is proposed. A comprehensive numerical study is performed to demonstrate the wave dispersion features in a Timoshenko–Ehrenfest nanobeam. Based on the presented wave propagation response and ensuing numerical illustrations, the original benchmark for numerical analysis is detected.

Some closed form series solutions to peridynamic plate equations

Peridynamics is a generalized continuum theory which is developed to account for long range internal force/moment interactions. Peridynamic equations of motion are of integro-differential type and only several closed-form analytical solutions are available for elementary structures. The aim of this paper is to derive analytical solutions to peridynamic Kirchhoff type (shear rigid) plates equations for both static and dynamic cases. Applying trigonometric series with respect to in-plane coordinates, solutions for the deflection function of a rectangular simply supported plate are derived. The coefficients in the series are presented in a closed analytical form such that the equations of motion are exactly satisfied. For the dynamic case the solution is derived applying the variable separation with respect to the time and the in-plane coordinates. Several numerical cases are presented to illustrate the derived solutions. Furthermore, results of peridynamic plate equations are compared against results obtained from the classical plate theory. A very good agreement between these two theories is obtained for the case of the small horizon sizes which shows the capability of the presented approach.

Variational formulation of dynamical homogenization towards nonlocal effective media

The dynamical homogenization of heterogeneous materials is performed in the low and high frequency regimes using a variational formulation of dynamical equilibrium combined with the Laplace or Fourier-Floquet-Bloch decomposition of the mechanical fields (displacement, velocity, momentum, stress). The assumed periodicity of the initial heterogeneous structure entails that the homogenization process can be carried out at the scale of a reference unit cell over which periodicity conditions apply. A dynamical Hill type macrohomogeneity condition is formulated that plays a central role in this work. The weak form of the unit cell dynamical equilibrium is formulated as the minimization of the mean field action integral expressed versus the Floquet components of the displacement and velocity. The effective dynamical moduli are evaluated in reciprocal space as unit cell averages of the microscopic properties and localization operators relating the local velocity and strain to their macroscopic counterpart. More specific dynamical models involving locality in space (thus valid for long wavelengths) but nonlocality in time (thus describing both low and high frequency regimes) are formulated, considering Cauchy type and strain gradient effective models. The range of validity of the developed dynamical generalized continuum theories is assessed by comparing the phase velocity-wavenumber plots for inclusion based composites with reference plots obtained from Bloch's theorem. The benchmark of dynamical models evidences that the nonlocal dynamical model proves able to predict with a good accuracy the dynamical behavior of 2D periodic inclusion based composites.

Microstructural effects on the response of a multi-layered elastic substrate

The size dependent response of a multi-layered elastic substrate under surface loading is investigated in the present work. The effect of material microstructure is modeled using the generalized continuum theory of couple stress elasticity. In the formulation, the displacement field is chosen as a primary unknown and the generalized Navier’s equation is established for a generic material layer. The Fourier integral transform method is directly applied to derive the general solution of the elastic field for the generic layer, and such solution is subsequently utilized to form the layer stiffness equation in the transform space. Boundary conditions together with the field continuity across the material interfaces are enforced, via the standard assembly procedure, to obtain a system of linear algebraic equations governing all unknowns of the multi-layered system. An efficient quadrature is adopted to carry out all involved integrals arising from the integral transform inversion. Detailed results are reported that confirm the validity of our results through the comparison with well-known solutions and demonstrate the capability of the proposed model to simulate various scenarios of layered media including those made of functionally graded materials. The important role of the material length scale effect on the predicted response is also elucidated.

Rayleigh wave equations with couple stress: Modeling and dispersion characteristic

In elastostatics, the scale effect is a phenomenon in which the elastic parameters of a medium vary with the specimen size when the specimen is sufficiently small. Linear elasticity cannot explain the scale effect because it assumes that the medium is a continuum and does not consider microscopic rotational interactions within the medium. In elastodynamics, wave-propagation equations are usually based on linear elasticity. Thus, nonlinear elasticity must be introduced to study the scale effect on wave propagation. We have developed one of the generalized continuum theories, a so-called couple-stress theory, into solid earth geophysics to build a more practical model of the underground medium. The first-order velocity-stress wave equation is derived to simulate the propagation of Rayleigh waves. Body and Rayleigh waves are compared using elastic theory and couple-stress theory in a homogeneous half-space and a layered space. The results indicate that couple stress causes the dispersion of surface waves and S-waves even in a homogeneous half-space. The effect is enhanced by increasing the source frequency and characteristic length, despite its insufficiently clear physical meaning. Rayleigh waves are more sensitive to the couple-stress effect than are body waves. Based on the phase-shifting method, it is determined that Rayleigh waves exhibit different dispersion characteristics in the couple-stress theory than in the conventional elastic theory. For the fundamental mode, dispersion curves tend to move to a lower frequency with an increase in characteristic length [Formula: see text]. For the higher modes, the dispersion curve energy is stronger with a greater characteristic length [Formula: see text].

Analysis of planes within reduced micromorphic model

A plane within reduced micromorphic model subjected to external static load is studied using the finite element method. The reduced micromorphic model is a generalized continuum theory which can be used to capture the interaction of the microstructure. In this approach, the microstructure is homogenized and replaced by a reduced micromorphic material model. Then, avoiding the complexity of the microstructure, the reduced micromorphic model is analyzed to reveal the interaction of the microstructure and the external loading. In this study, the three-dimensional formulation of the reduced micromorphic model is dimensionally reduced to address a plane under in-plane external load. The governing system of partial differential equations with corresponding consistent boundary conditions are discretized and solved using the finite element method. The classical and nonclassical deformation measures are then demonstrated and discussed for the first time for a material employing the reduced micromorphic model.

Phonon Hall viscosity from phonon-spinon interactions

Motivated by experimental observations, Samajdar et al. [Nature Physics 15, 1290 (2019)] have proposed that the insulating Neel state in the parent compounds of the cuprates is proximate to a quantum phase transition to a state in which Neel order coexists with semion topological order. We study the manner in which proximity to this transition can make the phonons chiral, by inducing a significant phonon Hall viscosity. We describe the spinon-phonon coupling in a lattice spinon model coupled to a strain field, and also using a general continuum theory constrained only by symmetry. We find a nonanalytic Hall viscosity across the transition, with a divergent second derivative at zero temperature.

Classification of first strain-gradient elasticity tensors by symmetry planes

First strain-gradient elasticity is a generalized continuum theory capable of modelling size effects in materials. This extended capability comes from the inclusion in the mechanical energy density of terms related to the strain-gradient. In its linear formulation, the constitutive law is defined by three elasticity tensors whose orders range from four to six. In the present contribution, the symmetry properties of the sixth-order elasticity tensors involved in this model are investigated. If their classification with respect to the orthogonal symmetry group is known, their classification with respect to symmetry planes is still missing. This last classification is important since it is deeply connected with some identification procedures. The classification of sixth-order elasticity tensors in terms of invariance properties with respect to symmetry planes is given in the present contribution. Precisely, it is demonstrated that there exist 11 reflection symmetry classes. This classification is distinct from the one obtained with respect to the orthogonal group, according to which there exist 17 different symmetry classes. These results for the sixth-order elasticity tensor are very different from those obtained for the classical fourth-order elasticity tensor, since in the latter case the two classifications coincide. A few numerical examples are provided to illustrate how some different orthogonal classes merge into one reflection class.

Wave transmission and reflection analysis through complex media based on the second strain gradient theory

The scattering of guided waves through a complex structure is of great importance in predicting vibration energy transmission and reflection across structures. Multi-scale models based on generalized continuum theory have been developed recently to investigate the vibration behavior of complex media in the frame of continuum mechanics. In this study, models based on the second strain gradient theory are employed together with a scattering matrix method to study the wave transmission and reflection through non-classical coupling regions between two waveguides. Numerical results are presented for longitudinal and bending waves through 1-D coupling interfaces. Upon which we discuss the local behavior of the internal heterogeneity in the complex coupling region and its impact on wave scattering properties at coupling interfaces. The results obtained also show that the proposed approach is of great potential in the investigation of vibration transmission control by enriching a waveguide’s internal structure.

Asymptotically consistent size-dependent plate models based on the couple-stress theory with micro-inertia

Several beam and plate models have been recently developed in the literature to accommodate for size-dependence. These are usually obtained starting from a generalized continuum theory (such as the couple-stress, strain-gradient or non-local theory or their modifications) and then deducing the governing equations through Hamilton’s principle and ingenuous kinematical assumptions. This approach, originated by Kirchhoff, usually fails to reproduce the dispersion features of the equivalent 3D theory. Besides, it produces a variety of models, in dependence of the different assumptions, such as Kirchhoff’s or Mindlin’s. In contrast, in this paper we adopt asymptotic reduction: moving from the couple-stress linear theory of elasticity with micro-inertia, we deduce new models for elongation and flexural deformation of microstructured plates. The resulting models are consistent, in the sense that they reproduce the dispersion features of the corresponding 3D body. Also, models are unique, for they may only differ by the order of the approximation. We find that microstructure especially affects inertia terms, which can be hardly captured by a-priori kinematical assumptions. For static flexural deformations, our results match those already obtained assuming plane cross-sections within the modified couple-stress theory. In fact, we show that couple-stress, reduced couple-stress and strain gradient theories all lead to equivalent results. Higher order models are also given, that describe the near first-cut-off behaviour and account for thickness deformations in the spirit of Timoshenko.

Size effects in two-dimensional layered materials modeled by couple stress elasticity

In the present study, the effect of material microstructure on the mechanical response of a two-dimensional elastic layer perfectly bonded to a substrate is examined under surface loadings. In the current model, the substrate is treated as an elastic half plane as opposed to a rigid base, and this enables its applications in practical cases when the modulus of the layer (e.g., the coating material) and substrate (e.g., the coated surface) are comparable. The material microstructure is modeled using the generalized continuum theory of couple stress elasticity. The boundary value problems are formulated in terms of the displacement field and solved in an analytical manner via the Fourier transform and stiffness matrix method. The results demonstrate the capability of the present continuum theory to efficiently model the size-dependency of the response of the material when the external and internal length scales are comparable. Furthermore, the results indicated that the material mismatch and substrate stiffness play a crucial role in the predicted elastic field. Specifically, the study also addresses significant discrepancy of the response for the case of a layer resting on a rigid substrate.

Higher-Order Peridynamic Material Correspondence Models for Elasticity

Higher-order peridynamic material correspondence model can be developed based on the formulation of higher-order deformation gradient and constitutive correspondence with generalized continuum theories. In this paper, we present formulations of higher-order peridynamic material correspondence models adopting the material constitutive relations from the strain gradient theories. Similar to the formulation of the first-order deformation gradient, the weighted least squares technique is employed to construct the second-order and the third-order deformation gradients. Force density states are then derived as the Frechet derivatives of the free energy density with respect to the deformation states. Connections to the second-order and the third-order strain gradient elasticity theories are established by realizing the relationships between the energy conjugate stresses of the higher-order deformation gradients in peridynamics and the stress measures in strain gradient theories. In addition to the horizon, length-scale parameters from strain gradient theories are explicitly incorporated into the higher-order peridynamic material correspondence models, which enables application of peridynamics theory to materials at micron and sub-micron scales where length-scale effects are significant.

Non-standard contact conditions in generalized continua: microblock contact model for a Cosserat body

Generalized continuum theories involve non-standard boundary conditions that are associated with the additional kinematic variables introduced in those theories, e.g., higher gradients of the displacement field or additional kinematic degrees of freedom. Accordingly, formulation of a contact problem for such a continuum necessarily requires that adequate contact conditions are formulated for the additional kinematic variables and/or for the respective generalized tractions. In this paper, we address several related open problems, namely, how to enhance the classic contact conditions to include the effects of the additional kinematic variables, how to link the enhanced contact model to the underlying microstructure of the solid, and how to do it in a consistent manner. As a first step towards a new class of contact models for generalized continua, a microblock contact model is derived for a Cosserat solid based on simple micromechanical considerations. To illustrate the non-trivial effects introduced by the non-standard boundary conditions, the problem of compression of an infinite strip with nonaligned microblocks is considered, and the analytical solution is derived for the corresponding boundary layers. A Hertz-like contact problem is also solved numerically with the focus on non-standard features of the solution and on the related size effects.

Three-dimensional nonlocal anisotropic elasticity: a generalized continuum theory of Ångström-mechanics

In this work, based on Eringen’s theory of nonlocal anisotropic elasticity, the three-dimensional nonlocal anisotropic elasticity of generalized Helmholtz type is developed. The derivation of a new three-dimensional nonlocal anisotropic kernel, which is the Green function of the three-dimensional anisotropic Helmholtz equation, enables to capture anisotropic length scale effects by means of a length scale tensor, which is a symmetric tensor of rank two. The derived nonlocal kernel function possesses up to six internal characteristic lengths on the Angstrom-scale. The presented theory of nonlocal elasticity possesses the appropriate property to be a generalized continuum theory of Angstrom-mechanics, since the range of its validity and applicability is up to the Angstrom-scale. The connection between the theory of nonlocal anisotropic elasticity and lattice theory is established. The tensor function of nonlocal elastic moduli as well as the nonlocal kernel function is given in terms of the Hessian matrix in the lattice approach. In the framework of the considered theory, the modeling of dislocations in anisotropic materials taking into consideration anisotropic dislocation core effects is presented. Important dislocation key formulas, namely the anisotropic Peach–Koehler stress formula, the Peach–Koehler force and the anisotropic Blin’s formula, are derived. A major tool used in deriving the expression of anisotropic Blin’s formula is Kirchner’s so-called $${\varvec{F}}$$-tensor, which is here generalized toward nonlocal anisotropic elasticity. The main feature and advantage of the derived fields, compared with the corresponding ones in classical anisotropic elasticity, is that they are free of singularities. Numerical applications to straight dislocations in bcc Fe are given, revealing the ability and advantage of the considered theory to describe adequately nonsingular anisotropic stress and self-energy fields capturing the effects of anisotropy on the Angstrom-scale.