Elastic Continua Research Articles

On angular and surface interactions in two-dimensional elastic lattices

This paper deals with two-dimensional elastic lattices which asymptotically converge towards isotropic linear elastic continua. The square lattice composed of central and angular interactions, pioneered by Gazis et al. (1960) for 3D cubic lattices, allows the modelling of isotropic linear elastic continua (in the continuous asymptotic limit) with equivalent Poisson's ratios smaller than 1/3 in a plane stress assumption. This paper presents an alternative or complementary elastic model based on surface elastic interactions, which can be viewed as a complementary lattice model for values of Poisson's ratios greater than 1/3 in plane stress. The physical background of this non-central potential is discussed, basing on the existence of a dual surface pressure. The model can be also introduced from variational arguments. This surface model of Fuchs type is shown to be equivalent to a perimetric elastic model in the linear range. Additionally, this paper explores the possibility of coupling both surface and angular lattice elasticity for square lattices, with the potential to cover a large variety of elastic continua in the asymptotic limit. Gradient elasticity and lattice-based nonlocal elastic continua are constructed, starting from this simple square lattice composed of angular, surface and central interactions. The static behaviour of these square lattices (with central, angular and surface interaction) is investigated in pure compression and pure shear loading, with exact analytical solutions for the discrete displacement field.

Nonlinear continuum mechanics with defects resembles electrodynamics—A comeback of the aether?

AbstractThe dynamics of an incompressible, isotropic elastic continuum are discussed. Starting from the Lorentz‐invariant motion of defects in elastic continua, MacCullagh's aether theory (1839) of an incompressible elastic solid is reconsidered. Since MacCullagh's theory, based on linear elasticity, cannot describe charges, particular attention is given to a topological defect that causes large deformations and, therefore, requires a nonlinear description. While such a twist disclination can take the role of a charge, the deformation field of a large number of these defects produces a microstructure of deformation related to a Cosserat continuum (1909). On this microgeometric level, a complete set of quantities can be defined that satisfies equations equivalent to Maxwell's. From a broader perspective, the elastic continuum approach appears to be intriguing, yet it has considerable difficulties in describing Dirac's large numbers, another unresolved mystery in fundamental physics. Despite this shortcoming, I think, however, that this paper may stimulate further research.

On the role of the microstructure in the deformation of porous solids

This study explores the role that the microstructure plays in determining the macroscopic static response of porous elastic continua and exposes the occurrence of position-dependent nonlocal effects that are strictly correlated to the configuration of the microstructure. Then, a nonlocal continuum theory based on variable-order fractional calculus is developed in order to accurately capture the complex spatially distributed nonlocal response. The remarkable potential of the fractional approach is illustrated by simulating the nonlinear thermoelastic response of porous beams. The performance, evaluated both in terms of accuracy and computational efficiency, is directly contrasted with high-fidelity finite element models that fully resolve the pores’ geometry. Results indicate that the reduced-order representation of the porous microstructure, captured by the synthetic variable-order parameter, offers a robust and accurate representation of the multiscale material architecture that largely outperforms classical approaches based on the concept of average porosity.

Vibration of general symmetric systems using group theory

This work presents group-theory-based vibration analysis to determine the highly structured, symmetry-related modal properties of symmetric systems and classify them into specific types. No eigenvalue problem solutions are required. In fact, the properties are obtained without a mathematical model. They derive only from symmetry and the degrees of freedom chosen to describe the deformation. Dynamic or static response is similarly decomposed into the sum of multiple response components associated with the mode types. One knows a priori, without any numerical computation, which mode types can or can not be excited by a given input excitation. Group theory divides the full system equations of motion and eigenvalue problem into multiple, smaller, decoupled problems associated with each mode type, which is computationally efficient. The method applies for general three-dimensional systems having any type of symmetry where components of the system are modeled using any mixture of lumped-parameter (including finite element) and elastic continuum models. Only basic elements of group theory are required, and they are introduced. Among three examples of the method, one derives the modal properties for general cyclically symmetric systems having any number of central components and substructures that can be modeled as any of rigid bodies, finite element meshes, and elastic continua.

Inverse problem techniques for multiple crack detection in 2D elastic continua based on extended finite element concepts

Two efficient methods are presented to detect multiple cracks in 2D elastic bodies, based on the insights from Extended Finite Element. Adetection mesh is assigned to the cracked body and the responses are measured at the nodes. A finite element model with the same mesh is used to represent the uncracked state of the physical body. In the first method which is called Crack Detection based on Residual Error (CDRE), the residual error norm is calculated based on the uncracked body stiffness matrix and the cracked body responses. The contour of the error norm would show the crack pattern; the method is computationally efficient. In the second method that is coined as Crack Detection based on Stiffness Residual (CDSR), the crack locations are found based on the difference between the stiffness matrix of the cracked body and the uncracked body. The stiffness matrix of the cracked body is found by solving a dynamic inverse problem based on a modified Tikhonov regularization. The efficiency and accuracy of the identification method are enhanced by predicting the crack pattern by the CDRE method. Several examples are presented to show the accuracy and robustness of the methods in the presence of high noise levels.

Review on nonlocal continuum mechanics: Physics, material applicability, and mathematics

The classical continuum mechanics assumes that a material is a composition of an infinite number of particles each of which is a point that can only move and interact with its nearest neighbors. This classical mechanics has limited applications where it fails to describe the discrete structure of the material or to reveal many of the microscopic phenomena, e.g., micro-deformation and micro-dislocation. This observation motivated the need for a general point of view that instills the fact that the material particle is a volume element that would deform and rotate, and the material is generally a multiscale material. In addition, the particle's equilibrium should not be considered in isolation from its nonlocal interactions with other particles of the material. Material models with these features are the nonlocal microcontinuum theories.Whereas review articles and books on microcontinuum theories and nonlocal mechanics would be found in the literature, no review that extensively deals with the fundamentals of nonlocal mechanics from the physics, material, and mathematical points of view has been presented so far. There is a current scientific debate on the benefits of applying nonlocal theories to various fields of mechanics. This is due to a lack of understanding of the physics behind these theories. In addition, questions on the applicability of nonlocal mechanics for various materials are not answered yet. Furthermore, mathematicians revealed paradoxes and complications of finding solutions of nonlocal field problems. In this review, we shed light on these folders. We give extensive interpretations on the physics of nonlocal mechanics of particles and elastic continua, and the applicability of nonlocal mechanics to multiscale materials and single-scale materials is interpreted. In addition, the existing complications of solving nonlocal field problems, and the various methods and approaches to overcome these complications are collected and discussed from the physical and material points of view. Furthermore, we define the open forums that would be considered in future studies on nonlocal mechanics.

Cellular structure design based on free material optimization under connectivity control

Cellular structures exhibit exceptional multifunction performance at relatively low densities and have wide industrial applications, but their effective design still remains challenging. The study aims to find a manufacturable cellular structure of valid geometry under certain volume usage budget, given a fixed design domain under external loadings. It is achieved via first introducing the concept of free material optimization (FMO), which provides an ultimately best structure among all possible elastic continua. After this, two novel control strategies are developed in combination with inverse homogenization to ultimately produce a cellular structure of valid geometric connectivity. It mainly includes approaches of material space reduction via hierarchical clustering, and a novel physics-based connectivity control to tailor the geometric connectivity between neighboring microstructures. Performance of the approach is tested on various 2D examples, in comparison with structures generated via classical (multiscale) topology optimization approaches.

ПСЕВДОТЕНЗОРНАЯ ФОРМУЛИРОВКА МЕХАНИКИ ГЕМИТРОПНЫХ МИКРОПОЛЯРНЫХ СРЕД

The possibility of applications of relative tensors concepts to the mechanics of micropolar continuum and, in particular, for the hemitropic micropolar continua is considered. The fundamental tensors and orienting relative scalars in three-dimensional space are introduced. Permutation symbols and absolute Levi-Civita tensors are investigated in further details. Algebraic and differential properties of the relative tensors are discussed. The weights of the fundamental kinematic tensors are determined. The wryness tensor and the asymmetric strain tensor are constructed in terms of the vectors of micro-rotation and displacements. Notions of force and couple traction vectors, associated force and associated couple stress vector, force and couple stresses tensors are discussed in the frameworks of relative tensors algebra. The weights of the basic micropolar elasticity tensors are determined and discussed. The constitutive form of the micropolar elastic potential is introduced as an absolute scalar in order to obtain micropolar constitutive equations. In the linear case, the elastic potential is a quadratic form whose coefficients are pseudoscalars. The weights of the constitutive pseudoscalars are calculated. The dimensionless constitutive micropolar constants and constitutive constants with physical dimensions are discriminated. Statics and dynamics of micropolar elastic continua are developed in terms of relative tensors. Dynamic equations involving displacements and microrotations in the case of semi-isotropic (hemitropic) symmetry are derived and represented by the pseudotensor technique. The paper can be considered as a script of fundamental formulas and concepts related to the algebra and differentiation of relative tensors of arbitrary rank.

Variational-based energy–momentum schemes of higher-order for elastic fiber-reinforced continua

This paper contributes to the improvement of recently published higher-order accurate energy–momentum schemes for an anisotropic hyperelastic material formulation based on the concept of structural tensors. These Galerkin-based energy–momentum schemes are developed by means of time finite elements with the focus on improving numerical stability and robustness in the presence of stiffness and large rotations. They show advantages over conventional time stepping schemes in combination with anisotropic materials formulated with polyconvex strain energy density functions. In this paper, higher-order energy–momentum schemes are developed by using a differential variational principle in order to remedy the drawbacks of these Galerkin-based schemes, namely a variationally inconsistent time approximation of the strains in the higher-order case, no coordinate free superimposed stress tensor and an unseparated time approximation of the fibers and the matrix material. These new structure-preserving time integrators preserve all conservation laws of a free flying hyperelastic anisotropic continuum, namely the total linear and the total angular momentum conservation as well as the total energy conservation. Numerical examples demonstrate dynamic behavior of anisotropic continua, discrete conservation properties of flying continua and the effect of higher-order accuracy subject to transient loads.

Effect of Variation in Parameters of a Compliant Micro–Gripper Design

At micro-level operations, to make precise microscopic movements, there is need of aalternative mechanism other than conventional mechanism like compliant mechanism. It is rapidly increasing field where a device that uses the compliance of its structure to achieve mechanical tasks such as force and motion transmission.Due to lack of available space, the new mechanisms should be compact without compromise in their accurate & precise performance. Unlike conventional mechanisms, compliant mechanism is monolithic, joint-free & elastic continua.This paper chiefly focuses on the results of variation of parameters within the design of a compliant micro-gripper that uses semi-circular hinge to transmit motion. According to the requirement of an application, the various dimensions of a micro-gripper can be varied. PRBM (Pseudo Rigid Body Model) approach is used to design the micro-gripper. This paper contains analysis work using FEA software (ANSYS). Keywords -Compliant mechanism;PRBM;monolithic;elastic continua.

Transition radiation excited by a surface load that moves over the interface of two elastic layers

Transition radiation is emitted when a perturbation source such as an electric charge or mechanical load, which has no inherent frequency, moves along a straight line at constant velocity in or near an inhomogeneous medium. Transition radiation of elastic waves is emitted, for example, by the wheels of a train due to track inhomogeneities such as non-uniform subsoil. This type of radiation was analyzed in the framework of several 1-D and 2-D elastic systems, but very few studies focus on elastic continua. Here, we consider a continuum consisting of two elastic layers (i.e., waveguides) that are coupled at a vertical interface. Both layers are in plane strain, have a free surface and are fixed to a rigid bottom, while the load is assumed to move along the free surface and over the layer interface. The response in each layer consists of the stationary eigenfield that moves with the load and a free field that propagates independently. The latter is expanded into a set of propagating and evanescent guided modes, while all fields are coupled at the interface. Orthogonality relations are employed to find the modal coefficients. Results show that the transition radiation energy (i.e., the energy flux through a surface far away from the layer interface) becomes powerful for load velocities approaching the critical velocity or for high contrast in material parameters. Furthermore, the free-field contribution to the energy flux through a circular surface close to the layer interface exhibits peculiar directivities. Depending on the contrast in material parameters and the load velocity, it can be extreme along the free surface, along the layer interface or into the medium. The free-field contribution can dominate that of the eigenfield for high load velocities, as is the case for the transition radiation energy. To study the features of the generated wave field in pure form, we take the load velocity sub-critical throughout the paper, excluding other radiation effects. However, the adopted solution is not restricted to that and is expected to work also for layered media, even when connected to other elastic structures like beams.

Energy-momentum conserving higher-order time integration of nonlinear dynamics of finite elastic fiber-reinforced continua

Since many years the relevance of fibre-reinforced polymers is steadily increasing in fields of engineering, especially in aircraft and automotive industry. Due to the high strength in fibre direction, but the possibility of lightweight construction, these composites replace more and more traditional materials as metals. Fibre-reinforced polymers are often manufactured from glass or carbon fibres as attachment parts or from steel or nylon cord as force transmission parts. Attachment parts are mostly subjected to small strains, but force transmission parts usually suffer large deformations in at least one direction. Here, a geometrically nonlinear formulation is necessary. Typical examples are helicopter rotor blades, where the fibres have the function to stabilize the structure in order to counteract large centrifugal forces. For long-run analyses of rotor blade deformations, we have to apply numerically stable time integrators for anisotropic materials. This paper presents higher-order accurate and numerically stable time stepping schemes for nonlinear elastic fibre-reinforced continua with anisotropic stress behaviour.

Minimum Principles for Interacting Cosserat Elastic Continua

This paper is concerned with the linear theory of composites modelled as mixtures of two Cosserat elastic continua. First, we present a minimum principle in the case of equilibrium. Then, we consider the dynamic theory and establish a minimum principle of Reiss type for a mixed problem.

Interaction of Ultrasonic SH-Polarization Waves with Two-Dimensional Strain Fields in a Plane Layer

Within the framework of the previously developed mathematical model of propagation of weak elastic perturbations in nonuniformly strained elastic continua, we formulate a stationary problem of interaction of ultrasonic SH-polarization waves with an elastic layer in the state of plane deformation. The problem is solved by the iterative method. We study the influence of the parameters of plane deformation in the mean-integral approximation on the processes of reflection and refraction of waves in the “elastic half space –layer–elastic half space” system.

Determinants of plasma membrane wounding by deforming stress

Once excess liquid gains access to air spaces of an injured lung, the act of breathing creates and destroys foam and thereby contributes to the wounding of epithelial cells by interfacial stress. Since cells are not elastic continua, but rather complex network structures composed of solid as well as liquid elements, we hypothesize that plasma membrane (PM) wounding is preceded by a phase separation, which results in blebbing. We postulate that interventions such as a hypertonic treatment increase adhesive PM-cytoskeletal (CSK) interactions, thereby preventing blebbing as well as PM wounds. We formed PM tethers in alveolar epithelial cells and fibroblasts and measured their retractive force as readout of PM-CSK adhesive interactions using optical tweezers. A 50-mOsm increase in media osmolarity consistently increased the tether retractive force in epithelial cells but lowered it in fibroblasts. The osmo-response was abolished by pretreatment with latrunculin, cytochalasin D, and calcium chelation. Epithelial cells and fibroblasts were exposed to interfacial stress in a microchannel, and the fraction of wounded cells were measured. Interventions that increased PM-CSK adhesive interactions prevented blebbing and were cytoprotective regardless of cell type. Finally, we exposed ex vivo perfused rat lungs to injurious mechanical ventilation and showed that hypertonic conditioning reduced the number of wounded subpleural alveolus resident cells to baseline levels. Our observations support the hypothesis that PM-CSK adhesive interactions are important determinants of the cellular response to deforming stress and pave the way for preclinical efficacy trials of hypertonic treatment in experimental models of acute lung injury.

243 大変形を考慮した接触する弾性体の形状同定(OS9.逆問題解析手法の開発と最新応用(2),オーガナイズドセッション)

243 大変形を考慮した接触する弾性体の形状同定(OS9.逆問題解析手法の開発と最新応用(2),オーガナイズドセッション)

Balance laws and energy release rates for cracks in dipolar gradient elasticity

It is the purpose of this work to derive the balance laws (in the Günther–Knowles–Sternberg sense) pertaining to dipolar gradient elasticity. The theory of dipolar gradient (or grade 2) elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain–energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the gradient of both strain and rotation (additional terms). The balance laws are derived here through a more straightforward procedure than the one usually employed in classical elasticity (i.e. Noether’s theorem). Indeed, the pertinent balance laws are obtained through the action of the standard operators of vector calculus ( grad, curl and div) on appropriate forms of the Hamiltonian of the system under consideration. These laws are directly related to the energy release rates in the processes of crack translation, rotation and self-similar expansion. Under certain conditions, they are identified with conservation laws and path-independent integrals are obtained.

On the restrictions imposed by non-affine material symmetry groups for elastic rods: application to helical substructures

Wire ropes, DNA strands and helical springs are among those bodies which can be modeled as an elastic rod with a helical substructure. The resulting form of the strain-energy function is a matter of material symmetry. This symmetry is explored using a novel treatment which combines non-affine transformations and a relabeling of the material coordinates. The restrictions this treatment imposes on the strain-energy function include a periodic dependency on torsional strain. In addition, comparisons are made with results from a recent treatment of helical symmetry by Healey. Finally, conclusions applicable to material symmetry restrictions for other polar elastic continua are presented.

Energy theorems and the J-integral in dipolar gradient elasticity

Within the framework of Mindlin’s dipolar gradient elasticity, general energy theorems are proved in this work. These are the theorem of minimum potential energy, the theorem of minimum complementary potential energy, a variational principle analogous to that of the Hellinger–Reissner principle in classical theory, two theorems analogous to those of Castigliano and Engesser in classical theory, a uniqueness theorem of the Kirchhoff–Neumann type, and a reciprocal theorem. These results can be of importance to computational methods for analyzing practical problems. In addition, the J-integral of fracture mechanics is derived within the same framework. The new form of the J-integral is identified with the energy release rate at the tip of a growing crack and its path-independence is proved. The theory of dipolar gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory considered here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. The latter case is also treated in the present study.

SPACE-TIME DEFECTS AS A SOURCE OF CURVATURE AND TORSION

Space time is described as a continuum four-dimensional medium similar to ordinary elastic continua. Exploiting the analogy internal stress states are considered. The internal "stress" is originated by the presence of defects. The defects are described according to the typical Volterra process. The case of a point defect in an otherwise isotropic four-dimensional medium is discussed showing that the resulting metric tensor corresponds to an expanding (or contracting) universe filled up with a non-zero energy-momentum density.