Strain Gradient Material Research Articles

Strain gradient thermopiezoelectric materials

Results from the linear theory of thermopiezoelectricity are discussed in this article. A brief introduction to dielectric materials and the phenomenon of direct and inverse piezoelectricity is given. Then, a general theory of thermopiezoelectricity for strain gradient materials is given, where the second gradient of the displacement field and electric potential enter into the set of independent constitutive variables. The second law of thermodynamics is formulated following an entropy production inequality proposed by Green and Laws. The thermodynamics restrictions are obtained, then the linear constitutive equations and the mixed initial-boundary value problem are established. On the basis of this theory, a uniqueness result and a variational principle are obtained.

Nonlinear finite element free and forced vibrations of cellular plates having lattice-type metamaterial cores: a strain gradient plate model approach

The main objective of this paper is to develop a theoretically and numerically reliable and efficient methodology based on combining a finite element method and a strain gradient shear deformation plate model accounting for the nonlinear free and forced vibrations of cellular plates having lattice-type metamaterial cores. The proposed model provides a modelling framework computationally more efficient than framework of the conventional three-dimensional elasticity for the simulation of the underlying nonlinear dynamics of cellular plates with microarchitectures. The corresponding governing equations follow Mindlin’s strain gradient elasticity theory including the micro-inertia effect applied to the first-order shear deformation plate theory along with the nonlinear von Kármán kinematics. Standard and higher-order computational homogenization methods determine the classical and strain gradient material constants, respectively. A higher-order C1-continuous 6-node finite element is adopted for the discretization in the spatial domain, and an arc-length continuation technique along with time-periodic discretization is implemented to solve the resulting nonlinear time-dependent problem. Through a set of comparative studies with 3D full-field models as reliable validation references for triangularly prismatic example cores, the accuracy and efficacy of the proposed dimension reduction methodology are demonstrated for a diverse range of problem parameters for analyzing the large-amplitude dynamic structural response. The results show that the proposed strain gradient plate model can accurately capture the microarchitecture effects of cellular plates with low computational costs, which signifies a relevant contribution particularly to the computational analysis of nonlinear dynamics.

Gradient of mechanical properties in polymer nanocomposites: From atomistic scale to the strain gradient effective continuum

It is well known that glassy polymer nanocomposites (PNC) exhibit heterogeneous mechanical properties, with a distinct polymer/nanoparticle (NP) interfacial region. The interphase rigidity in particular can significantly influence the overall mechanical response of glassy PNCs. Continuum theories typically describe the mechanical behavior of the interphase such to depend only on the deformation and, thus, they discard the strong internal deformation gradients within the interphase, that are necessary for enriched continuum models of the PNC’s. Here we formulate a higher-gradient elasticity constitutive law for the polymer/NP interphase region through a systematic micro–meso–macro coupling, using data from atomistic molecular dynamics (MD) simulations. From a physical standpoint, we show that in order to accurately consider the polymer/NP interphase is necessary to describe it as a strain gradient material, whereas the bulk material may still behave as a Cauchy material, experiencing much lower strain gradient effects, compared to the polymer/NP interphase. Using atomistic MD simulations, the polymer/NP interphase thickness in PNCs is examined by probing the density profile distribution at equilibrium. By upscaling the molecular response within the interphase towards a continuum model, a strain gradient effective model for the interphase emerges, whereby the hyperstress tensor and the conjugate strain gradient, together with the stress and strain, are computed for each atom using the MD simulations. In a further upscaling step, a continuum-based homogenization scheme based on the unfolding method is elaborated for the identification of the strain-gradient elastic moduli of polymer composite materials, according to which the interphase obeys a second-gradient linear elastic constitutive law, with an inner reinforcement and an outer surrounding matrix obeying a first-gradient elasticity model.

Isogeometric topology optimization of strain gradient materials

In this paper optimal topologies of isotropic linear elastic strain gradient materials are investigated by means of isogeometric topology optimization. The employment of strain gradient theory allows not only to capture the microstructural effects of materials but also to regularize stress/strain concentration phenomena and to address the so-called wedge forces. Isogeometric analysis is used in order to meet the requirements of higher-order continuity of strain gradient elasticity theory. For the purpose of determining the constitutive parameters of strain gradient materials, a novel experiment is conceived by using the Digital Image Correlation (DIC) technique. The so-called SIMP (Solid Isotropic Material with Penalization) method is applied to interpolate the material stiffness tensors (including the fourth-order and the sixth-order stiffness tensor) by power law with penalty. Benchmark numerical experiments are conducted to illustrate the computational effectiveness and numerical robustness of the proposed method.

Strain-gradient theory for shear deformation free-form microshells: Governing equations of motion and general boundary conditions

A first-order shear deformation (FOSD) free-form microshell model described in general curvilinear coordinates was developed within the complete framework of Mindlin’s form II linear isotropic strain-gradient theory (SGT), considering both strain-gradient and micro-inertia effects. The high-order governing equations of motion and consistent boundary conditions were simultaneously obtained through a variational formulation based on Hamilton’s principle. The established microshell model contains five strain-gradient material constants and one length parameter resulting from micro-inertia effects in addition to two classical Lamé constants, thereby capturing microstructure-induced size-dependent phenomena in both static and dynamic analyses. The constructed model within the general SGT can be flexibly reduced to those based on the modified strain-gradient, modified couple stress, and simplified strain-gradient theories. Moreover, by enforcing geometric restrictions, the general formulation for free-form microshells can be specialized for doubly curved microshells. Finally, the forced vibration results of a simply supported closed cylindrical microshell confirmed the reliability and accuracy of the proposed model and strain-gradient and high-order inertia effects on the size-dependent dynamic characteristics of microshells. Altogether, the presented mathematical formulations can provide a foundation for the in-depth understanding of microscale shell structures.

On a hemi-variational formulation for a 2D elasto-plastic-damage strain gradient solid with granular microstructure

<abstract><p>We report a continuum theory for 2D strain gradient materials accounting for a class of dissipation phenomena. The continuum description is constructed by means of a (reversible) placement function and by (irreversible) damage and plastic functions. Besides, expressions of elastic and dissipation energies have been assumed as well as the postulation of a hemi-variational principle. No flow rules have been assumed and plastic deformation is also compatible, that means it can be derived by a placement function. Strain gradient Partial Differential Equations (PDEs), boundary conditions (BCs) and Karush-Kuhn-Tucker (KKT) type conditions are derived by a hemi variational principle. PDEs and BCs govern the evolution of the placement descriptor and KKT conditions that of damage and plastic variables. Numerical experiments for the investigated homogeneous cases do not need the use of Finite Element simulations and have been performed to show the applicability of the model. In particular, the induced anisotropy of the response has been investigated and the coupling between damage and plasticity evolution has been shown.</p></abstract>

Verification of strain gradient elasticity computation by analytical solutions

AbstractAs there are different computational methods for simulating problems in generalized mechanics, we present simple applications and their closed‐form solutions for verifying a numerical implementation. For such a benchmark, we utilize these analytical solutions and examine three‐dimensional numerical simulations by the finite element method (FEM) using IsoGeometric Analysis (IGA) with the aid of open source codes, called tIGAr, developed within the FEniCS platform. A study for the so‐called wedge forces and double tractions help to comprehend their roles in the displacement solution as well as examine the significance by comparing to the closed form solutions for given boundary conditions. It is found that numerical results are in a good agreement with the analytical solutions if wedge forces and double tractions are considered. It is also presented how the wedge forces become necessary in order to maintain equilibrium in strain gradient materials.

Analyzing size effects in a cracked orthotropic layer under antiplane shear loading

Scale-dependent stress intensity factors in an anti-plane cracked orthotropic material layer are evaluated using strain gradient theory. Both volumetric and surface strain gradient material characteristic lengths represented as l and $$l^{'}$$ , respectively, are employed to obtain semi-analytical solutions. The surface strain gradient effect is considered for both positive and negative $$l^{'}$$ values. The layer edges are assumed stress-free and oriented parallel to the crack plane. The presence of orthotropy can either increase or decrease the stress intensity factors depending on if it is greater or smaller than unity. The volumetric strain gradient effect reduces the stress intensity factor and it is more pronounced for smaller layer thickness. It was found that the negative surface gradient leads to a more complaint crack, while the positive surface gradient increases crack stiffness. Overall, the surface gradient effect is less significant in comparison with the volumetric gradient effect.

Computation of brittle fracture propagation in strain gradient materials by the FEniCS library

Strain gradient continuum damage modelling has been applied to quasistatic brittle fracture within an approach based on a maximum energy-release rate principle. The model was implemented numerically, making use of the FEniCS open-source library. The considered model introduces non-locality by taking into account the strain gradient in the deformation energy. This allows for stable computations of crack propagation in differently notched samples. The model can take wedges into account, so that fracture onset can occur at wedges. Owing to the absence of a damage gradient term in the dissipated energy, the normal part of the damage gradient is not constrained on boundaries. Thus, non-orthogonal and non-parallel intersections between cracks and boundaries can be observed.

Computational second-order homogenization of materials with effective anisotropic strain-gradient behavior

A computational homogenization method to determine the effective parameters of Mindlin’s Strain Gradient Elasticity (SGE) model from a local heterogeneous Cauchy linear material is developed. The devised method, which is an extension of the classical one based on the use of Quadratic Boundary Conditions, intents to correct the well-known non-physical problem of persistent gradient effects when the Representative Volume Element (RVE) is homogeneous. Those spurious effects are eliminated by introducing a microstructure-dependent body force field in the homogenization scheme together with alternative definitions of the localization tensors. With these modifications, and by a simple application of the superposition principle, the higher-order stiffness tensors of SGE are computed from elementary numerical calculations on RVE. Within this new framework, the convergence of SGE effective properties is investigated with respect to the size of the RVE. Finally, a C1-FEM procedure for simulating the behavior of the effective material at the macro scale is developed. We show that the proposed model is consistent with the solutions arising from asymptotic analysis and that the computed effective tensors verify the expected invariance properties for several classes of anisotropy. We also point out an issue that the present model shares with asymptotic-based solutions in the case of soft inclusions. Applications to anisotropic effective strain-gradient materials are provided, as well as comparisons between fully meshed structures and equivalent homogeneous models.

Continuum damage model for strain gradient materials with applications to 1D examples

In complex materials there exist different, often independent, mechanisms for the evolution of anelastic phenomena at the micro- and at the macroscale. Consequently, gradient plasticity has been introduced for a large variety of non-local materials. In a similar way, continuum damage has been extended to micromorphic and other non-local models, introducing internal damage variables and their gradient. In the paper is proposed a new continuum damage model for strain gradient materials characterized by two independent damage internal variables, each evolving with its own law. In this way, no additional field equation for the damage variable is needed. The model is intended to be used for metamaterials. Some important characteristics of the model are analysed with numerical analyses of a 1D example. In particular, the ability of the model to prevent strain localization is discussed.

A micromorphic approach to stress gradient elasticity theory with an assessment of the boundary conditions and size effects

AbstractBasing on thermodynamics principles, a stress gradient elasticity theory is presented whereby the material is modeled as a centro‐symmetric micromorphic anisotropic material. A principle of minimum complementary energy and a stationarity principle of Hellinger–Reissner type are provided, by which the boundary conditions are uniquely determined in a form consistent with continuum mechanics. For 3‐D solids, these conditions include, beside the three ordinary boundary conditions for the assigned boundary data (displacements and/or tractions), six extra boundary conditions by which a microstructure/continuum compatibility condition is enforced, generally through the vanishing of the normal derivative of the stress, , on the whole boundary surface.A typical boundary‐value problem for isotropic materials with a single length scale parameter under static loads is discussed, showing that it is governed by the same Navier PDEs of classical elasticity, along with a set of Helmholtz PDEs as constitutive equations. Size effects are shown to arise from two distinct sources, that is, the spatial fluctuating of the body force and the double curl of the Hookean stress field. It is proved that a formulation of stress gradient elasticity theory based on the pure notion of nonsimple material (like the ones existing in the literature) leads to anomalies in the ordinary and extra boundary conditions. A comparison of the present theory with that by Eringen is also discussed.Two examples of stress gradient structural models are analytically worked out, namely, a thick‐walled cylinder and a Kirchhoff–Love circular plate. For comparison, the solutions for strain gradient material are also presented. Graphic illustrations show that in both examples stiffening size effects like with the ancillary strain gradient models are found.

Comparison of anti-plane surface waves in strain-gradient materials and materials with surface stresses*

Here we discuss the similarities and differences in anti-plane surface wave propagation in an elastic half-space within the framework of the theories of Gurtin–Murdoch surface elasticity and Toupin–Mindlin strain-gradient elasticity. The qualitative behaviour of the dispersion curves and the decay of the obtained solutions are quite similar. On the other hand, we show that the solutions relating to the surface elasticity model are more localised near the free surface. For the strain-gradient elasticity model there is a range of wavenumbers where the amplitude of displacements decays very slowly.

Strain gradient fracture in an anti-plane cracked material layer

Fracture mechanics analysis in terms of evaluating stress intensity factors of an anti-plane isotropic cracked layer is carried out using strain gradient elasticity theory. The crack plane is assumed parallel to the layer edges. Both volumetric and surface strain gradient material characteristic lengths are considered in formulations and numerical solutions. Two boundary value problems corresponding to “stress-free” and “clamped” boundaries are considered in which each solution is reduced to the dual integral equations. The Fredholm integral equation, proceeding from the dual integral equations, is numerically solved to evaluate crack tip stress intensity factor. Stress intensity factors for stress-free boundary conditions are higher with smaller height (or with a longer crack) and vice versa for clamped boundaries. Volumetric strain gradient effect reduces stress intensity factor and demonstrates strong size effect on a smaller scale. Crack stiffness becomes more pronounced with positive surface strain gradient, while negative surface gradient leads to a more complaint crack. In general, the contribution of volumetric strain gradient is shown to be more dominant than that of surface strain gradient.

Mori–Tanaka estimates of the effective elastic properties of stress-gradient composites

A stress-gradient material model was recently proposed by Forest and Sab (Mech. Res. Comm.40, 16–25, 2012) as an alternative to the well-known strain-gradient model introduced in the mid 60s. We propose a theoretical framework for the homogenization of stress-gradient materials. We derive suitable boundary conditions ensuring that Hill–Mandel’s lemma holds. As a first result, we show that stress-gradient materials exhibit a softening size-effect (to be defined more precisely in this paper), while strain-gradient materials exhibit a stiffening size-effect. This demonstrates that the stress-gradient and strain-gradient models are not equivalent as intuition would have it, but rather complementary. Using the solution to Eshelby’s spherical inhomogeneity problem that we derive in this paper, we propose Mori–Tanaka estimates of the effective properties of stress-gradient composites with spherical inclusions, thus opening the way to more advanced multi-scale analyses of stress-gradient materials.

Continuum model of microstructure induced softening for strain gradient materials

Gradient plasticity has been introduced to predict localization bands induced by softening or damage, and has been extended to micromorphic models used for materials with microstructure and scale effects. Motivated by the consideration that in complex materials there may exist different mechanisms for the evolution of anelastic deformation at the small and at the large scale, and that they may interact, in the paper a model of second gradient material with two different evolution laws for the macroscopic and for the microscopic plastic strain is proposed, the latter related to the presence of second gradient deformation. The model is intended to be used for metamaterials, and its relevant properties can be obtained from an homogenization analysis at the microlevel.

Symmetry conditions in strain gradient elasticity

We study the variational significance of the “order-of-differentiation” symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.

Surface effects, boundary conditions and evolution laws within second strain gradient plasticity

The principle of the virtual power (PVP) is used in conjunction with the concepts of “energy residual” and “insulation condition” to address second strain gradient plasticity. The energy residual with its typical divergence format is an extra stress power playing the role of basic state variable to describe the gradient effects, whereas the insulation condition constitutes a global energy characterization of the body as part of the body/environment system. The microstructure of a second strain gradient material (but not of a first strain gradient one) is shown to exhibit surface effects with the formation of a thin boundary layer. This boundary layer is in local (and global) equilibrium according to the principles of the material surface mechanics and supports the boundary microtractions, except a part (Cauchy-like traction) transmitted to the bulk microstructure; it works as a structured two-dimensional manifold replacing the conventional purely geometrical concept of boundary surface. By the insulation condition the higher order boundary conditions are determined (for first and second strain gradient plasticity), including those for the moving elastic/plastic interface. Besides the usual continuity boundary conditions, some extra (at most three) “rest” boundary conditions are required to fix the current location of the interface. The restrictions on the constitutive equations and the evolution laws are also addressed, whereby a mixed modeling scheme is used to model the dissipative stresses. An application to an elastic-softening bar in extension shows the ability of the proposed model to capture and describe the formation of a multi-waved deformation pattern within the localization band.

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.

A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior

A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the “surface effects” produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body suitably modelled as a simple material cell system, in association with the principle of the virtual power (PVP) for quasi-static actions, an equivalent structural system is derived, featured by a (macro-scale) PVP having the typical format as for a second strain gradient material model. Due to the surface effects, the latter model does work as a combination of two subsystems, i.e. the bulk material behaving as a Cauchy continuum, and the boundary surface operating as a membrane-like boundary layer, each subsystem being in (local and global) equilibrium by its own. Further, the applied (ordinary) boundary traction splits into two (response-dependent) parts, i.e. the “Cauchy traction” transmitted to the bulk material and the “Gurtin–Murdoch traction” acting, together with all other boundary tractions, upon the boundary layer. The role of the boundary layer as a two-dimensional manifold enclosing a Cauchy continuum is elucidated, also with the aid of a discrete model. A strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary-value problem for quasi-static loads and its (unique) solution. A simple application is presented. Two appendices are included, one reports the proof of the global equilibrium of the boundary layer, the other is concerned with double and triple stresses. The paper is complemented by a companion Part II one on dynamics. Previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137] are improved and extended.