The work presented in a recent paper by the authors [35] for a thermodynamically consistent and kinematic assumption free plate and shell formulation for small deformation and small strain based on the conservation and balance laws of classical continuum mechanics (CCM) is extended here for non-classical continuum mechanics (NCCM). This formulation incorporates additional physics due to internal rotations that arise due to the deformation gradient tensor. This physics is neglected in CCM, hence is absent in the plate and shell formulation of reference [35]. Consideration of this new physics requires modifications of the current balance laws as well as consideration of a new balance law “balance of moment of moments” (BMM) [2, 3]. Cauchy stress tensor becomes non-symmetric. Cauchy moment tensor is conjugate to the symmetric part of the rotation gradient tensor which exists now due to new physics. Balance of angular momenta yields additional three differential equations as part of the mathematical model. The new balance law (BMM) establishes symmetry of the Cauchy moment tensor. The new physics considered here exists in all deforming solid continua as it is due to the deformation gradient tensor, but is ignored in CCM. The consequence of this new physics is additional stiffness, hence additional strain energy storage and change in the time history of displacements and stress field compared to formulations based on CCM. The basic mathematical model for the plate and shell deformation consists of conservation and balance laws in $$\mathbb {R}^3$$ based on NCCM incorporating internal rotations. The associated finite element formulations for obtaining the solution of the mathematical model consists of : (i) geometry of the plate or shell described by the flat or curved middle surface (as done conventionally) and nodal vectors locating the top and bottom faces of the plate/shell (ii) the displacement field approximation that is p-version hierarchical in the plane as well as in the transverse direction (iii) integral form is constructed using Galerkin Method with Weak Form (GM/WF) and the corresponding element equations. The formulation presented here remains valid and accurate for thin as well as thick plate/shell and naturally reduces to the formulation of reference [35] based on classical continuum mechanics. Model problem studies and comparisons with the studies based on CCM formulation [35] will be presented in a follow up paper.
The focus of the present work is to present an analytical approach for buckling and free vibrations analysis of thick functionally graded nanoplates embedded in a Winkler-Pasternak medium. The equations of motion are derived according to both the third-order shear deformation theory, proposed by Reddy, and the nonlocal elasticity Eringen's model. For the first time, the equations are solved analytically for plates with two simply supported opposite edges, the solutions also turning helpful as shape functions in the analysis of structures with more complex geometries and boundary conditions. Sensitivity analyses are finally performed to highlight the role of nonlocal parameters, aspect and side-to-thickness ratios, boundary conditions, and functionally graded material properties in the overall response of plates and cylindrical shells. It is felt that the proposed strategy could be usefully adopted as benchmark solutions in numerical routines as well as for predicting some unexpected behaviors, for instance, in terms of buckling load, in thick nanoplates on elastic foundations.
In this paper, the subject of the principle of causality and the physical realizability in the wave motion characteristics within a linear nonlocal elastic medium is examined. The principle of primitive causality is examined via Kramers–Kronig (K–K) relations and the principle of relativistic causality or the Einstein causality is examined via wave motion responses. Gradient as well as integral type nonlocality has been considered. Methodology here involves a Fourier frequency domain based spectral analysis and the wave motion characteristics include: wave modes, group speeds and frequency response function. In general, due to existence of atleast one of the non-physical features in the characteristics, violation of the causality is observed. The non-physical features include: existence of infinitesimally small or zero speeds; existence of very large or infinite speeds; existence of negative speeds; and absence of attenuation of waves. Violation to the primitive causality takes place as a disagreement to the K–K relations, either due to existence of negative and/or zero group speeds or due to absence of wave attenuation in the possible wave modes. Violation to Einstein causality is observed due to existence of infinitely large group speeds. Agreement to the primitive causality is achieved due to the presence of both wave dispersion and wave attenuation in the wave modes. Although existence of infinitely large group speeds violates Einstein causality, however, violation of the primitive causality is not observed. Upon considering only the physically realizable wavemodes, it is observed that, a local Neumann type boundary condition may be sufficient to conduct a wave motion study in a class of nonlocal boundary value problems. As an application of the primitive causality to the Fourier domain analysis, the wavenumbers from the K–K relations are utilized to demonstrate a mitigation effect of certain non-physical features in the wave motion responses.
In the present work, flexural response of functionally graded plates subjected to transverse loads have been investigated using the meshless natural neighbor Galerkin method (NNGM). The plate formulation has been developed based on the Reddy’s (Mechanics of laminated composite plates and shells: theory and analysis, 2nd edition, CRC Press, Boca Raton, 2014) third-order shear deformation theory (TSDT) using the von Kármán nonlinear strains. The governing equations of the TSDT have been derived accounting for the length scale/size effects considering the Eringen’s nonlocal stress-gradient model (Eringen in Microcontinuum filed theories—I: foundations and solids, Springer-Verlag, 1998). The C 1 continuous shape functions have been computed using the sibson’s interpolant and generalizing a Bezier patch over the domain. The nonlocal nonlinear model of the resulting governing equations has been developed, and Newton’s iterative procedure is used for the solution of nonlinear algebraic equations. The mechanical properties of functionally graded plate are assumed to vary continuously through the thickness and obey a power-law distribution of the volume fraction of the constituents. The variation of volume fractions through the thickness have been computed using two different homogenization techniques, namely, the rule of mixtures and the Mori–Tanaka scheme. A detailed parametric study to show the effect of side-to-thickness ratio, power-law index, and nonlocal parameter on the load-deflection characteristics of plates have been presented. The central deflections obtained using (NNGM) have been compared with the results from literature based on finite element method. The results have been compared with the two homogenization schemes and also with results computed with the first-order shear deformation theory (FSDT) to show the accuracy of nonlocal nonlinear formulation based on TSDT.
The paper investigates the post-buckling behavior of micro-cantilevers within the framework of consistent couple stress theory. The theoretical foundation for carrying out the analysis is formulated in terms of moment and curvature which is apt for solving thin beam problems. A differential equation governing the deflection of micro-cantilever subjected to different boundary conditions is obtained from the developed moment-curvature relationship. Axially loaded micro-cantilevers are solved by using an efficient semi-analytical method which involves removal of singularity. On the other hand solutions to eccentrically loaded micro-cantilevers are determined with the help of purely numerical approach based on iterative shooting method. Profiles of post-buckled micro-cantilevers subjected to different loading conditions are furnished. The obtained results for the micro-cantilever display strong size-dependency due to presence of the material length scale parameter in the developed model. Also the non-classical theory results tend to those from classical theory when the length scale parameter is negligible as compared to the characteristic length of the structure.
Kirchhoff type shells are continuum models used to study the mechanics of thin elastic bodies; these are largely based on the theory of surfaces. Here, we report a reformulation of Kirchhoff shells using the theory of moving frames. This reformulation permits us to treat the deformation and the geometry of the shell as two separate entities. The structure equations which represent the familiar torsion and curvature free conditions (of the ambient space) are used to combine deformation and geometry in a compatible way. From such a perspective, Kirchhoff type theories have non-classical features which are similar to the equations of defect mechanics (theory of dislocations and disclinations). Using the proposed framework, we solve a boundary value problem and thus demonstrate, to an extent, the importance of moving frames.
The stability of a granular column composed of a finite number of grains is investigated through an exact and some approximated continuum models. Shear and rotational interactions are taken into account at the rigid grain interfaces. This system can be considered as a discrete Cosserat chain with two independent degrees-of-freedom, namely the deflection and the rotation of each grain. The buckling of this discrete granular system on elastic foundation with translational and rotational stiffness (to account for some possible transversal grain interactions) is calculated whatever the number of grains. The formulation of the discrete boundary value problem is based on the exact resolution of a fourth-order linear difference equation. This solution is compared to the one of a continuous Cosserat chain asymptotically obtained for an infinite number of grains. In this last case, the asymptotic solution converges towards the one of a Bresse–Timoshenko beam under Winkler–Pasternak foundation. A more refined Cosserat continuum is built by continualization of the difference equations valid for the discrete Cosserat medium. It is shown that this more refine continuous model can be classified as a gradient elasticity Cosserat continuum, which is able to reproduce the scale effects observed for the buckling of the discrete granular system. These scale effects are related to the grain size, as compared to the structural length of the granular system. The key role played by the shear interaction in the instabilities of granular structural system is revealed, especially when the bending interaction can be neglected.
Modeling the formation and evolution of microstructure in phase transforming materials presents challenges to traditional continuum mechanics approaches. This is mainly because they do not account for effects arising from the discreteness of the underlying lattice. Such effects can be described by non-classical approaches based on discrete particle models. We study the propagation of an austenite-martensite phase boundary using a Frenkel–Kontorova model. The model is based on a one dimensional chain of atoms on the phase boundary under the influence of a temperature dependent substrate potential. Using this model we derive the kinetic relation as a function of temperature.
Inferring material parameters of soft materials, especially soft tissues from experiments, has always been a challenge because of the inaccuracy of specimen geometry, difficulty in proper gripping and difficulty in obtaining reliable displacement data. If one is able to obtain full field data with Digital Image Correlation or other similar techniques, we show that in spite of poor quality experimental data, it is still possible to get the material parameters using Virtual Fields Method with carefully chosen virtual fields. We demonstrate the approach by two cases. The first is Ecoflex (silicone rubber) under biaxial deformations at controlled loading rates, and the second is rat skin samples with imperfect shape under biaxial deformations at controlled loading rates. In both cases, rather than trying to obtain homogeneous deformation conditions, we use a technique based on the Virtual Fields Method to extract the material properties from the deformation of the specimen collected by Digital Image Correlation (DIC) and the force load measured by the load sensor. In order to apply loads in two principal directions simultaneously, a custom biaxial set-up was built and mounted into a uniaxial Instron tensile set-up. We show here that in spite of significant inhomogeneity in the deformation, errors, and missing data in the measured displacement field, we are still able to recover the material properties of the soft solid by suitable choices of virtual fields.
In this paper, two novel versions of weak form quadrature elements are developed for bending, free vibration and stability analysis of non-classical first strain gradient Kirchhoff plate theory. In the first version, Lagrange interpolations are assumed in orthogonal directions to approximate the field variables and in the second, mixed interpolations are used, with Lagrange in one direction and Hermite in another. The elements are formulated with the aid of variational principles and the Gauss–Lobatto–Legendre quadrature points are considered as element nodes and also used for numerical integration of element matrices. The multi-degrees degrees of freedom associated with the non-classical plate theories are accounted into the formulation in a simplified way, which facilitates the application of classical and non-classical boundary conditions. The procedure to compute the weighting coefficients by incorporating the classical and non-classical degrees of freedom are explained for both element versions. Detailed mathematical formulation of the elements and their numerical implementation is presented. The efficiency of the proposed elements is demonstrated by solving numerical examples and comparing the results with the exact solutions available in the literature. Further, new results are presented for strain gradient plates with different support conditions, which can serve as reference for other researchers in this field.
The strength of laser-welded web-core sandwich plates is often limited by buckling. In design of complex thin-walled structures the combination of possible structural and material combinations is basically infinite. The feasibility of these combinations can be assessed by using analytical, numerical and experimental methods. At the early design stages such as concept design stage, the role of analytical methods is significant due to their capability for parametric description and extremely low computational efforts once the solutions have been established for prevailing differential equations. Over the recent years significant advances have been made on analytical strength prediction of web-core sandwich panels. Therefore, aim of the present paper is to show impact of this development to the design space of web-core sandwich panels in buckling. The paper reviews first, briefly the differential equations of a 2-D micropolar plate theory for web-core sandwich panels and the Navier buckling solution for biaxial compression recently derived by Karttunen et al. (Int J Solids Struct 170(1):82–94, 2019) by exploiting energy methods. By comparing the micropolar and widely-used classical first-order shear deformation plate theory (FSDT) solutions, it is shown that the different equivalent single layer (ESL) formulations and plate aspect ratios have a significant impact on the practical outcomes of the feasible design space and this way motivating further developments for micropolar formulations from practical structural engineering viewpoint.
When a Unidirectional fiber reinforced polymer (UDFRP) composite is subjected to transverse loading, there is spatial variation of stresses in the constituents. The failure in matrix initiates at the location of maximum stress. Stress distribution and failure initiation in constituents of UDFRP composites is usually studied through finite element (FE) analysis of representative volume element (RVE) which is computationally expensive and time consuming. The present study proposes an analytical model through which stress variation and failure initiation in the constituents of UDFRP composite can be obtained in simple and reliable way and it can be readily used in designing. For this model, RVE is idealized in the form of springs arranged in parallel and series. These springs represent the stiffness of constituents (fiber and matrix). The results of analytical model are compared with FE simulations and good agreement is observed. Influence of fiber volume fraction on failure initiation of UDFRP composites is also studied through FE analysis of RVE and analytical model.
A formulation is presented for the 2D dynamic analysis and earthquake response simulation of base isolation systems. The approach is force-based and consists of casting the computation in each time increment as a convex optimization problem. Interaction between the two horizontal components of response is considered in an elegant and simple way through yield functions appearing as constraints of the optimization problem. Numerical examples are carried out to illustrate the approach. These comprise bidirectional shearing of a high damping rubber bearing and earthquake simulations of a real-world base isolation system.
A high-order theory is developed for the analysis of beams with general mono-symmetric cross-sections. The theory represents the nonlinear distribution of the longitudinal normal stress across the section depth by a polynomial series expansion up to any order as specified by the analyst. The corresponding shear and transverse normal stresses are obtained by satisfying the 2D infinitesimal stress-flow equilibrium conditions. The resulting statically admissible stress fields are then used in conjunction with the principle of stationary complementary strain energy to formulate the governing compatibility equations and boundary conditions. Closed form solutions are then developed for general loading and boundary conditions. Comparisons with the theory of elasticity and 3D finite element analysis predictions showcase the ability of the present theory to naturally capture shear deformation effects, transverse normal stress effects, nonlinear longitudinal normal stress distributions in deep beams, as well as the effect of support height. Unlike conventional beam solutions that are based on postulated kinematic assumptions, which tend to converge to the displacement response from below, the present theory avoids introducing any kinematic assumptions and is shown to converge to the solution from above. The theory is applicable to beams with doubly symmetric or mono-symmetric cross-sections, with isotropic or orthotropic materials, and subjected to general loading and boundary conditions. The theory is shown to offer advantages compared to other theories when modelling deep beams and/or beams with supports that are offset from the centroidal axis.
The paper deals with the derivation of non classical interface conditions in linear poroelasticity in the framework of the quasi-static diphasic Biot’s model. More precisely, we analyze the mechanical behavior of two linear isotropic poroelastic solids, bonded together by a thin layer, constituted by a linear isotropic poroelastic material, by means of an asymptotic analysis. After defining a small parameter \(\varepsilon\), which will tend to zero, associated with the thickness and the constitutive coefficients of the intermediate layer, we characterize three different limit models and their associated limit problems, the so-called soft, hard and rigid poroelastic interface models, respectively. First and higher order interface models are derived. Moreover, we identify the non classical transmission conditions at the interface between the two three-dimensional bodies in terms of the jump of the stresses, specific discharge, pressure and displacements.
A size-dependent model for bending and free vibration of functionally graded piezoelectric (FGP) microbeam is developed by using modified couple stress theory and a unified higher order beam theory. This model can be specialized to various beam models, such as Euler–Bernoulli, Timoshenko as well as Reddy beam ones and vice versa. The governing equations of motion and associated boundary conditions are derived from Hamilton’s principle. Only one material length scale parameter is introduced to capture the size effect. The analytical solutions of simply supported FGP microbeam are presented by using Navier approach to bring out the effect of the material length scale parameter on the bending and free vibration of microbeam. Numerical simulations are presented to account for the effect of various parameters, such as material length scale parameters, volume fraction indexes, and slenderness ratios on the responses of static bending and free vibration of FGP microbeam.
The paper investigates the accuracy, the stability and the computational efficiency of a mixed explicit–implicit time integration approach proposed for predicting the nonlinear response of base-isolated structures subjected to earthquake excitation. Adopting the central difference method for evaluating the response of the nonlinear base isolation system and the Newmark’s constant average acceleration method for estimating the superstructure linear response, the proposed partitioned solution approach is used to analyze a 3D seismically isolated structure subjected to a bidirectional earthquake excitation. Both isolation systems adopting lead rubber and friction pendulum bearings are considered. Numerical results show that the computational time required by the proposed method, in spite of its conditional stability arising from the use of the central difference method in the explicit integration substep, is clearly reduced in comparison to the widely used implicit time integration method adopted in conjunction with the pseudo-force approach (i.e., pseudo-force method). As a matter of fact, the typical low stiffness of the isolation system leads to a critical time step larger than the one used to define the ground acceleration accurately and the proposed method preserves its computational efficiency even in the case of isolators with very high initial stiffness (i.e., friction pendulum bearings) for which the critical time step size could become smaller.
The present paper deals with a general asymptotic theory aimed at deriving some imperfect interface models starting from thin interphases. The novelty of this work consists in taking into account some non-standard constitutive behaviors for the interphase material. In particular, micro-cracks, surface roughness and geometrical nonlinearity are included into the general framework of the matched-asymptotic-expansion theory. The elastic equilibrium problem of a three-composite body comprising two elastic adherents and an adhesive interphase is investigated. Higher order interface models are derived within the cases of soft and hard interphase materials. Simple FEM-based numerical applications are also presented.
This article presents a theory of collisions of continua either solid or not. The basic idea which is developed is that the system made of distinct continua is deformable because their relative positions change. The collisions we consider occur while the continua are evolving and the duration of the collisions is small compare to the duration of the whole motion. Thus they are assumed instantaneous. We do not focus on the fast and sophisticated phenomena which occur during collisions. We focus on summing up these phenomena in a coherent theory which gives the elements to pursue the description of the motion. This instantaneity assumption leaves large possibilities to engineers and scientists to develop numerous and useful predictive theories. The basis of the theory is illustrated with the collision of a point with an immobile obstacle. Then the theory is applied to collisions of solids, either rigid or deformable, then to collisions of solids and fluids. The thermal effects of collisions may produce phase change: the example of rain falling on a deeply frozen soil is investigated. Collisions may be so violent that they fracture the bodies: fracturation may also be predicted by the theory. From the theoretical point of view, let us mention that this theory proves that the paradoxes, i.e., illogic results, which are said to result from the equations of mechanics, as the Painlevé–Jellet and Klein paradoxes, may be overcome in a clear and logic manner. Moreover these results are supported by experiments.
