Linear Elastic Body Research Articles

To modeling the auxetic materials: some fundamental aspects

The auxetic materials are considered from the point of view of correspondence to the classical theory of elasticity. It is shown that some classical postulates relative to the elastic constants should be refined. Three cases of description of auxetic materials — by the model of linear elastic isotropic body, by the model of linear elastic transversally isotropic body, by the nonlinear elastic isotropic body (Murnaghan potential) — are analyzed shortly. The initial assumption on positivity of internal energy of deformation is saved and then the uniform stress states (unilateral tension, omnilateral compression, pure shear) are used to analyze the elastic constants. This allows to describe the new mechanical effects: expansion of the standard sample-rod-prism under unilateral tension and expansion of the standard sample-cube under hydrostatic compression as well as an existence of the arbitrary negative values of Poisson ratios, what is accompanied by the negative values of the Lame $\lambda$, Young $E$ and compression $k$ moduli, for the linear isotropic case and some elastic constants in the linear transversely isotropic case. The case of nonlinear description shows that the auxetic materials should be defined by the primary physical effect — observation in the standard for mechanics of materials experiment of longitudinal tension of a prism that the transverse deformation of prism is positive (a material as if swells) in contrast to the classical materials, where it is negative.

1108 領域積分型の形状微分公式(形状最適化)

The present paper describes an evaluation method of the shape derivatives, which is defined as the Frechet derivative with respect to the domain variation, of cost functions in a shape optimization problem of a domain in which a boundary value problem of partial differential equation is defined. Instead of the known method using the equation of boundary integral type, an equation of domain integral type is proposed to evaluate the shape derivative. This method is applied to the mean compliance minimization problem of linear elastic body. Numerical example shows the validity of the method.

Cavity/Inclusion Detection in Plane Linear Elastic Bodies Using Linear Sampling Method

In this paper, solution of inverse problems in elastostatic fields is investigated. For this purpose, we propose a qualitative inverse approach based on linear sampling method (LSM) for cavity/inclusion detection in a two-dimensional (2D) isotropic linear elastic body using measurement of data on the boundary. The LSM is an effective approach to image the geometrical features of unknown targets. Although the LSM has been used in the context of inverse scattering problems such as acoustics, and electromagnetism, there is no specific attempt to apply this method for identification of cavities/inclusions in inverse elastostatic problems. This study emphasizes the implementation of the LSM coupled with the finite element method (FEM). A set of numerical simulations on 2D elastostatic problems is presented to highlight many effective features of the proposed LSM fast qualitative identification method.

Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization

Abstract For an isotropic linear elastic body, only displacement or traction boundary conditions are given on a part of its boundary, whilst all of displacement and traction vectors are unknown on the rest of the boundary. The inverse problem is different from the Cauchy problems. All the unknown boundary conditions on the whole boundary must be determined with some interior points' information. The preconditioned conjugate gradient method (PCGM) in combination with the boundary element method (BEM) is developed for reconstructing the boundary conditions, and the PCGM is compared with the conjugate gradient method (CGM). Morozov's discrepancy principle is employed to select the iteration step. The analytical integral algorithm is proposed to treat the nearly singular integrals when the interior points are very close to the boundary. The numerical solutions of the boundary conditions are not sensitive to the locations of the interior points if these points are distributed along the entire boundary of the considered domain. The numerical results confirm that the PCGM and CGM produce convergent and stable numerical solutions with respect to increasing the number of interior points and decreasing the amount of noise added into the input data.

Numerical Simulation of the Effect of Structure-Structure Interaction in the Frequency Response

Various vibration phenomena of buildings occurred in recent earthquakes. Some of them were very complicated and might be caused by unknown structure-structure interactions through foundations and/or ground soil. The aim of this research is to investigate such complicated structure-structure interaction by numerical simulation. After formulating the problem mathematically, we develop a simulation algorithm for linear elastic bodies that have distributional material constants. As for the discretization procedure, we adopt the finite element method (FEM) combined with the perfectly matched layer (PML) technique. Using the algorithm, we succeeded in computing several interesting phenomena of frequency response of the system caused by structure-structure interaction.

Identification of small well-separated defects in an isotropic elastic body using boundary measurements

An inverse problem of identification of a finite number of small, well-separated defects in an isotropic linear elastic body is considered. It is supposed that the defects are cavities or inclusions (rigid or linear elastic). If the defects are cavities then their boundaries are supposed unloaded. If the defects are inclusions it is supposed complete bonding between the matrix and inclusions. It is assumed also that as a result of static test the loads and displacements are measured on the external boundary of the body. A method for determination of centers of the defects projections on an arbitrary plane is developed. If the defects are ellipsoids their geometrical parameters (directions and magnitudes of the ellipsoids axes) are determined also. Numerical examples illustrating efficiency of the developed method are considered.

Dynamic analysis on generalized linear elastic body subjected to large scale rigid rotations

The dynamic analysis of a generalized linear elastic body undergoing large rigid rotations is investigated. The generalized linear elastic body is described in kinematics through translational and rotational deformations, and a modified constitutive relation for the rotational deformation is proposed between the couple stress and the curvature tensor. Thus, the balance equations of momentum and moment are used for the motion equations of the body. The floating frame of reference formulation is applied to the elastic body that conducts rotations about a fixed axis. The motion-deformation coupled model is developed in which three types of inertia forces along with their increments are elucidated. The finite element governing equations for the dynamic analysis of the elastic body under large rotations are subsequently formulated with the aid of the constrained variational principle. A penalty parameter is introduced, and the rotational angles at element nodes are treated as independent variables to meet the requirement of C1 continuity. The elastic body is discretized through the isoparametric element with 8 nodes and 48 degrees-of-freedom. As an example with an application of the motiondeformation coupled model, the dynamic analysis on a rotating cantilever with two spatial layouts relative to the rotational axis is numerically implemented. Dynamic frequencies of the rotating cantilever are presented at prescribed constant spin velocities. The maximal rigid rotational velocity is extended for ensuring the applicability of the linear model. A complete set of dynamical response of the rotating cantilever in the case of spin-up maneuver is examined, it is shown that, under the ultimate rigid rotational velocities less than the maximal rigid rotational velocity, the stress strength may exceed the material strength tolerance even though the displacement and rotational angle responses are both convergent. The influence of the cantilever layouts on their responses and the multiple displacement trajectories observed in the floating frame is simultaneously investigated. The motion-deformation coupled model is surely expected to be applicable for a broad range of practical applications.

Direct numerical simulation of fracture behaviour for random short wood fibre-reinforced composites in comparison with digital image correlation experiments

The present work is to predict the fracture behaviour of biocomposites from the tensile properties of its components. In this work, a direct numerical simulation was realized for fracture behaviour of random short spruce fibre-reinforced composites. For calculations, wood fibres were considered as linear elastic bodies and polypropylene matrix as an elastic–plastic material. Then, the numerical results were compared with the experimental results obtained by digital image correlation (DIC). This comparison indicates that random fibre finite element (FE) model can be able to fairly reflect the influence of random fibre microstructure on the fracture behaviour of the composites. The calculation and comparison show that in a region in the front of the crack tip, the average values of J-integral of both random fibre and homogeneous FE model are in good agreement with that of DIC experiment under the same extension load.

Nonlinear buckling of a spherical shell embedded in an elastic medium with imperfect interface

This work analyzes nonlinear buckling of a single spherical shell imperfectly bonded to an infinite elastic matrix under a compressive remote load. The inclusion is modeled using a nonlinear shell formulation and the matrix is treated as a linear elastic body. Imperfect bonding conditions are realized through a linear spring interface model. A variational method is used to derive the governing differential equations, which are cast into a tractable set of nonlinear algebraic equations using the Galerkin method. An incremental iterative technique based on the modified Newton–Raphson method is employed to find the critical load of the system. The accuracy and convergence properties of the proposed method are validated through finite element analysis. The study is relevant to the analysis of compressive failure of syntactic foams used in marine and aerospace applications. Results are specialized to glass particle-vinyl ester matrix syntactic foams to test the hypothesis as to whether microballoons’ buckling is a dominant failure mechanism in such composites under compression. Parametric studies are conducted to understand the effect of interfacial properties and inclusion wall thickness on the overall mechanical behavior of the composite. Comparisons between analytical findings and experimental results on compressive response of syntactic foams and isolated microballoons indicate that inclusion buckling is unlikely a determinant of compressive failure in vinyl ester-glass systems. In particular, the matrix is found to exert a beneficial stabilizing effect on the inclusions, which fail under brittle fracture before the onset of buckling.

BEM solution of delamination problems using an interface damage and plasticity model

The problem of quasistatic and rate-independent evolution of elastic-plastic-brittle delamination at small strains is considered. Delamination processes for linear elastic bodies glued by an adhesive to each other or to a rigid outer surface are studied. The energy amounts dissipated in fracture Mode I (opening) and Mode II (shear) at an interface may be different. A concept of internal parameters is used here on the delaminating interfaces, involving a couple of scalar damage variable and a plastic tangential slip with kinematic-type hardening. The so-called energetic solution concept is employed. An inelastic process at an interface is devised in such a way that the dissipated energy depends only on the rates of internal parameters and therefore the model is associative. A fully implicit time discretization is combined with a spatial discretization of elastic bodies by the BEM to solve the delamination problem. The BEM is used in the solution of the respective boundary value problems, for each subdomain separately, to compute the corresponding total potential energy. Sample problems are analysed by a collocation BEM code to illustrate the capabilities of the numerical procedure developed.

New invariants in the mechanics of deformable solids

A system of invariants of symmetric three-dimensional tensors of second order is proposed. The system contains three classical invariants of a tensor and three new invariants which depend on the components of two or three tensors. The system proposed allows one to express the strain energy density of a linear elastic anisotropic body and virtual work done by internal stresses in terms of the invariants for any constitutive law of the material. Application of the invariants to the derivation of the tetrahedron finite element of anisotropic solids is discussed.

Identification of Defects in an Elastic Body by Means of the Boundary Measurements

A problem of identification of a single defect (a crack, a cavity or an inclusion) in an anisotropic, linear elastic body using boundary measurements is considered. An analytical solution of the problem of ellipsoidal defect identification in an infinite elastic solid is presented in the conditions when arbitrary constant stresses are applied at the infinity and the loads and displacements are known on a closed surface, containing the defect inside. The problem is solved using reciprocity gap functional method. The obtained analytical solution is used for solving the problem of ellipsoidal defect identification in a bounded elastic body by means of the results of one static test. It is assumed that the loads and displacements are measured on the external boundary of the body. Numerical examples illustrating efficiency of the developed method are considered. Stability of the results relative to the noise in the data and variation of the defect shape is studied.

Towards integral inequalities of surface deformation in two-dimensional linear elasticity

This work is inspired to achieve an integral inequality for a linear elastic body in which the surface integral of squares of surface strains or surface displacement gradients is bounded from above by surface integral of squares of boundary tractions with a finite bound constant independent of boundary tractions. Here, all candidate displacements of the proposed variational quotients are required to meet the equations of equilibrium without body forces, and the boundary tractions are defined based on the displacements. Detailed results are given for two-dimensional anti-plane shear and plane-strain of circular domains, which confirm that the integral of surface strains or surface displacement gradients can be bounded from above by the integral of boundary tractions.

Evaluation of a nonlinear Hertzian‐based model reveals prostate cancer cells respond differently to force than normal prostate cells

Understanding how the mechanical properties of cells alter with disease may help with the development of novel diagnostics and treatment regimes. The emergence of tools such as the atomic force microscope (AFM) has enabled us to physically measure the mechanical properties of cells. However, suitable models for the analysis of real experimental data are either absent, or fail to provide a simple analysis tool in which experimental data can be analyzed quickly and reliably. The Hertz model has been widely used to study AFM data on living cells, however it makes assumptions that are untrue for cells, namely that cells behave as linear elastic bodies. This article presents and evaluates an alternative nonlinear Hertz model, which allows the Young's modulus to vary according to a second order polynomial function of indentation depth. Evaluation of the model revealed that prostate cancer cells (PC3) responded more uniformly to force compared to the normal PNT2 cells. Also, more energy (J) was needed to deform the normal prostate cells compared to the prostate cancer cells. Finally, the model described here suggests that overall the normal prostate cells behave in a more linear fashion to applied force compared to the prostate cancer cells.

Finite‐volume stress analysis in multi‐material linear elastic body

SUMMARYCorrect calculation of stresses at the interface of bonded or otherwise joined materials plays a significant role in many applications. It is therefore important that traction at the material interface is calculated as accurately as possible. This paper describes procedures that can be employed to achieve this goal by using centre‐based finite‐volume method. Total traction at the interface is calculated by decomposing it into normal and tangential components, both being calculated at each side of the interface, and applying the continuity assumption. The way in which the traction approximation is achieved depends on calculation of tangential gradient of displacement at the interface. To this end, three different methods are proposed and validated against problems with known solutions. It was shown that all methods can be successfully used to simulate problems with multi‐material domains, with the procedure based on finite area method being most accurate. Copyright © 2012 John Wiley & Sons, Ltd.

The method of fundamental solutions for the detection of rigid inclusions and cavities in plane linear elastic bodies

We investigate the numerical reconstruction of smooth star-shaped voids (rigid inclusions and cavities) which are compactly contained in a two-dimensional isotropic linear elastic body from a single non-destructive measurement of both the displacement and traction vectors (Cauchy data) on the external boundary. The displacement vector satisfying the Lamé system in linear elasticity is approximated using the meshless method of fundamental solutions (MFS). The fictitious source points are located both outside the (known) outer boundary of the body and inside the (unknown) void. The inverse geometric problem is then reduced to finding the minimum of a nonlinear least-squares functional that measures the gap between the given and computed data, penalized with respect to both the MFS constants and the derivative of the radial polar coordinates describing the position of the star-shaped void. The interior source points are anchored and move with the void during the iterative reconstruction procedure. The stability of the numerical method is investigated by inverting measurements contaminated with noise.

Validation of a heterogeneous elastic-biphasic model for the numerical simulation of the PDL

An elastic-biphasic model for the simulation of the periodontal ligament (PDL) and the adjacent tooth is presented and investigated. The PDL is modelled as a biphasic material following the work of Ehlers and Markert (2001), whereas the tooth is modelled as a linear elastic body. A spatial discretisation scheme is proposed based on mixed finite elements for the spatial discretisation. Due to nonlinearity in the model, a predictor–corrector scheme is employed as a temporal discretisation scheme. In order to validate the PDL model, in vitro measurements are compared with numerical simulations. The numerical simulations are performed using geometries resulting from micro-CT scanner of the same porcine tooth, which was employed for the invitro measurements.

Assessment of bonding stresses between FRP sheets and masonry pillars during delamination tests

Under the simplifying assumption of perfect adhesion, motivated by the wide adoption of highly resistant glues, delamination of FRP-reinforced masonry pillars turns out to be governed exclusively by the non-linear behavior of the quasi-brittle, heterogeneous support. However, at the structural level, the macroscopic delamination response can be described effectively by concentrating all the sources of dissipation and non-linearity at the masonry-FRP interface, and assuming the support to behave as a linear-elastic body. In this paper, a detailed and critical comparison between two different fully three-dimensional finite element models is developed: (i) a model in which only masonry (i.e. brick and mortar independently) is damageable whilst the FRP reinforcement adheres perfectly to the support (namely, exclusively bulk damage is accounted for), and an alternative model (ii) in which a cohesive, zero thickness interface between the FRP and the support is considered (i.e. interface damage), whilst masonry behaves as a heterogeneous linear elastic material. The overall response during delamination and local stress distributions at the interface are critically investigated, varying the FRP reinforcement width.

Mixed Formulation for a Signorini Problem

Problem statement: This study deals with the study of a Signorini Problem (SP) of unilateral contact between two a linear or non linear elastic bodies. Approach: We present two variational formulations noted P1, P2, of the considered problem, where P1 depends on the displacement field and P2 depends on the stress field. In the linear case, under assumptions, then the considered problem was equivalent to a mixed variational formulation problem, where the unknowns are the displacement field and the normal constraint stress on the contact area. Results: Problems P1 and P2 were formally equivalent to the Signorini Problem (SP) and using Lions-Stampachia Theorem, we shown the existence and uniqueness result. Conclusion: The Signorini problem (PS) has a unique variational solution.

Development of method for evaluating cell hardness and correlation between bacterial spore hardness and durability

BackgroundDespite the availability of conventional devices for making single-cell manipulations, determining the hardness of a single cell remains difficult. Here, we consider the cell to be a linear elastic body and apply Young’s modulus (modulus of elasticity), which is defined as the ratio of the repulsive force (stress) in response to the applied strain. In this new method, a scanning probe microscope (SPM) is operated with a cantilever in the “contact-and-push” mode, and the cantilever is applied to the cell surface over a set distance (applied strain).ResultsWe determined the hardness of the following bacterial cells: Escherichia coli, Staphylococcus aureus, Pseudomonas aeruginosa, and five Bacillus spp. In log phase, these strains had a similar Young’s modulus, but Bacillus spp. spores were significantly harder than the corresponding vegetative cells. There was a positive, linear correlation between the hardness of bacterial spores and heat or ultraviolet (UV) resistance.ConclusionsUsing this technique, the hardness of a single vegetative bacterial cell or spore could be determined based on Young’s modulus. As an application of this technique, we demonstrated that the hardness of individual bacterial spores was directly proportional to heat and UV resistance, which are the conventional measures of physical durability. This technique allows the rapid and direct determination of spore durability and provides a valuable and innovative method for the evaluation of physical properties in the field of microbiology.