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Fiber-reinforced composites are materials that display a great potentiality in many applications for their multifaceted features. The employment of curved fibers contributes to enhancing this already wide range of possibilities further. In this paper, we propose two particular fiber arrangements with the intention of realizing ultra-stiff and ultra-soft composites. For this purpose, we explore the texture based on isostatic lines, which is bio-inspired to obtain a high strength composite and the pantographic pattern, which enables considerable displacement with very low resistance. To model these particular lattice plates, we consider an in-plane orthotropic lamina within the framework of the strain-gradient elasticity theory. Numerical simulations have been performed using the finite element method to investigate the mechanical behavior of the considered textures. A regular straight texture with fibers parallel to the edges of a rectangular sample has been taken into account as reference texture for comparison. Through some tests, we show that for the considered textures, namely the isostatic and the pantographic ones, the displacement force graphs differ by several orders of magnitude, justifying the characterization of their behavior.

International audience; We revisit the classical theory of indentation for very soft materials. Many experiments consist in extracting the stiffness of a membrane from the cubic answer of the force versus indentation depth. However, this law is restricted to a perfect membrane under a sharp point loading. In biophysical experiments, where this technique recovers some success at low scales thanks to AFM, the thin samples are highly deformable, hyper-elastic, pre-stretched and often attached to a substrate which cannot be neglected. In addition microscopic tiny pores may exist or may be created by the indenter. This diversity requires specific studies with the correct elasticity: here we choose the Neo-Hookean elastic model at large membrane deformations. We show that, the weak loading regime is extremely sensitive to any physical properties of the thin layer but also of the indenter geometry. In addition, when a hole exists, at finite forcing or finite indentation depth, we discover a topological bifurcation with abrupt dynamical jump variation of typical quantities such as the hole size. This bifurcation is similar to the famous catenoid instabilities which are also due to a topological bifurcation, when the two rings of support are pulled apart. This bifurcation is robust in the sense that it always exists whatever the physical properties of the sample.

We study instability of a Newtonian Couette flow past a gel-like film in the limit of vanishing Reynolds number. Three models are explored including one hyperelastic (neo-Hookean) solid, and two viscoelastic (Kelvin–Voigt and Zener) solids. Instead of using the conventional Lagrangian description in the solid phase for solving the displacement field, we construct equivalent “differential” models in an Eulerian reference frame, and solve for the velocity, pressure, and stress in both fluid and solid phases simultaneously. We find the interfacial instability is driven by the first-normal stress difference in the base-state solution in both hyperelastic and viscoelastic models. For the neo-Hookean solid, when subjected to a shear flow, the interface exhibits a short-wave (finite-wavelength) instability when the film is thin (thick). In the Kelvin–Voigt and Zener solids where viscous effects are incorporated, instability growth is enhanced at small wavenumber but suppressed at large wavenumber, leading to a dominant finite-wavelength instability. In addition, adding surface tension effectively stabilizes the interface to sustain fluid shear.

We study the equilibrium and the local stability for an incompressible elastic solid under arbitrary dead-load surface tractions on the boundary. We particularize the analysis to homogeneous deformations of a homogeneous, isotropic parallelepiped, finding necessary and sufficient algebraic local stability conditions consistent with the further requirement of the zero moment condition. We specialize the study for a uniform distribution of dead-load surface tractions s > 0 on two pairs of faces and –s on the remaining two faces, and we find that two classes of equilibrium solutions may occur: symmetric and asymmetric solutions, respectively. For the symmetric solutions we also determine local stability inequalities. For the special case of a Gent-Thomas material we show that both equilibrium symmetric and asymmetric solutions may occur if the material parameters satisfy certain inequalities. Then, we completely describe the response of the parallelepiped in a loading process starting from the unloaded state for five ranges of the values of the material parameters. In particular, for one of these ranges we show that symmetric solutions are the unique locally stable homogenous equilibrium deformations until a critical value scr of the load; at scr, symmetric solutions lose their uniqueness, and a bifurcation into asymmetric solutions may occur.

The propagation of finite-amplitude time-harmonic shear horizontal waves, in a pre-stressed compressible elastic layer of finite thickness embedded between two identical compressible elastic half-spaces, is investigated. This is accomplished by combining finite-amplitude linearly polarized inhomogeneous transverse plane wave solutions in the half-spaces and finite-amplitude linearly polarized unattenuated transverse plane wave solutions in the layer. The layer and half-spaces are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. The special case where the interfaces between the layer and the half-spaces are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the shear horizontal wave speed with the pre-stress and the propagation angle.

An analytical investigation into the non-linear elastic response of helical lattice structures coupled with an elastic medium is presented. Novel composite templates are then obtained to produce bespoke material characteristics by exploiting tuned hierarchy. System behaviour is approximated as a combination of three non-linear “springs”, representing the helical lattice, and the axial and circumferential components of the elastic medium via an energy based approach. Non-dimensional parameters governing each component’s non-linear load–displacement behaviour are identified, demonstrating tailoring potential. Further tunable parameters that govern the combined system response, involving form factor, geometric and stiffness ratios are identified. In particular, pseudo-ductile responses are observed. The feasible region of pseudo-ductility, as determined by these non-dimensional parameters, is obtained, allowing discussion of viable materials and geometries. Finally, load–displacement behaviour is utilised to obtain indicative effective stress–strain curves, thus showing promise as a model for future material development.

Morphoelastic theories have demonstrated that elastic instabilities can occur during the growth of soft materials, initiating the transition toward complex patterns. Within the framework of non-linear elasticity, the theory of incremental elastic deformations is classically employed for solving stability problems with finite strains. In this work, we define a variational method to study the bifurcation of growing cylinders with circular section. Accounting for a constant axial pre-stretch, we define a set of canonical transformations in mixed polar coordinates, providing a locally isochoric mapping. Introducing a generating function to derive an implicit gradient form of the mixed variables, the incompressibility constraint for the elastic deformation is solved exactly. The canonical representation allows to transform a generic boundary value problem, characterized by conservative body forces and surface traction loads, into a completely variational formulation. The proposed variational method gives a straightforward derivation of the linear stability analysis, which would otherwise require lengthy manipulations on the governing incremental equations. The definition of a generating function can also account for the presence of local singularities in the elastic solution. Bifurcation analysis is performed for few constrained growth problems of biomechanical interests, such as the mucosal folding of tubular tissues and surface instabilities in tumor growth. In a concluding section, the theoretical results are discussed for clarifying how anisotropy, residual strains and external constraints can affect the stability properties of soft tissues in growth and remodeling processes.

Journal article An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power¿law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability. Marie Curie Fellowship, Grant Agreement number 219800 peer-reviewed

Mass growth on cylindrical and spherical substrates is a phenomenon which can be related to the biochemical creation of an elastic actin gel shell by polymerization and cross linking of actin filaments either in vivo on bacteria cylindrical surfaces as a means for their motility or in vitro on spherical beads as a means for experimentally studying the previous in vivo case. Such mass growth is strongly effected by the developed stress field. The objective of this paper is to accurately determine this stress field assuming elasticity of the growing mass and symmetrical growth. Based on the special kinematics of mass growing on spherical and cylindrical substrates, inwards or outwards from them, and various isotropic constitutive laws for both small and finite elastic strains, it is possible to obtain the solution for the stress field in closed analytical form for all cases considered. This expands very significantly recent findings [Dafalias, Y.F., Pitouras, Z., 2008. Stress field in actin gel growing on spherical substrate. Biomechanics and Modeling in Mechanobiology, doi:10.1007/s10237-007-0113-y .] for some constitutive laws and outwards growth on spherical substrates only. The effect of biomass compressibility is shown to be of cardinal importance for the developed stress field, questioning the validity of the simplifying assumption of a zero value Poisson ratio usually made in the relevant biomechanics literature for simplicity. Few selected graphs of stress variation along the radial direction illustrate the analytical findings. The obtained closed form analytical expressions for stress can be a standard reference tool in this important area of stress-modulated soft tissue growth.

This paper presents projection methods to treat the incompressibility constraint in small- and large-deformation elasticity and plasticity within the framework of Isogeometric Analysis. After reviewing some fundamentals of isogeometric analysis, we investigate the use of higher-order Non-Uniform Rational B-Splines (NURBS) within the B ¯ projection method. The higher-continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results. A new non-linear F ¯ projection method, based on a modified minimum potential energy principle and inspired by the B ¯ method is proposed for the large-deformation case. It leads to a symmetric formulation for which the consistent linearized operator for fully non-linear elasticity is derived and used in a Newton–Raphson iterative procedure. The performance of the methods is assessed on several numerical examples, and results obtained are shown to compare favorably with other published techniques.

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions.

Of interest here is the influence of loading rate on the stability of structures where inertia is taken into account. The approach currently used in the literature to analyze these stability problems, is the method of modal analysis that determines the structure’s fastest growing wavelength, which is meaningful only for cases where the velocity of the perfect structure is significantly lower than the associated characteristic wave propagation speeds. The novel idea here is to analyze the time-dependent response to perturbations of the transient (high strain rates) states of these structures, in order to understand the initiation of the corresponding failure mechanisms. We are motivated by the recent experimental studies of Zhang and Ravi-Chandar (2006) on the high strain rate extension of thin rings that show no evidence of a dominant wavelength in their failure mode and no influence of strain-rate sensitivity on the necking strains. In the interest of analytical tractability, we study the extension of an incompressible, nonlinearly elastic bar at different strain rates. The dynamic stability of these bars is studied by following the evolution of localized small perturbations introduced at different times. It is shown that these structures are stable until the static necking strain is reached at some point. Moreover their failure pattern is dictated by the distribution of defects, the minimum distance between necks diminishes with increasing strain rate and there is no dominant wavelength mode, exactly as observed experimentally in Zhang and Ravi-Chandar (2006).

The present paper concerns the stable multiphase isochoric deformations for an isotropic elastic body subject to a surface traction of uniform Piola stress with two equal principal forces which are opposite to the third. To model the occurrence of such deformations, we consider a strain energy density function which depends on the first principal invariant of deformation through a non-convex function and which has an added linear dependence on the second invariant. We establish existence conditions for equilibrium multiphase deformations which give restrictions on the morphology of the connecting phases as well as on the orientation of the flat interfaces between the phases. Finally, by considering a special, but representative, form for the non-convex strain energy function, we show that there exists a “critical” value of the external load which allows for the emergence of stable coexistent deformation fields.

In this work finite element simulations are conducted based on the micro structure of polymers in order to transfer the information of the micro level to the macro level. The micro structure of polymers is characterized by chain-like macromolecules linked together at certain points. In this way an irregular three-dimensional network is formed. Many authors use the tool of statistical mechanics to describe the deformation behaviour of the entire network. Most of these concepts can be reformulated as traditional continuum mechanical formulations. They are, however, restricted to affine deformation, regular chain arrangements and purely elastic material behaviour. For this reason, in the present contribution, we propose a new finite element-based simulation method for polymer networks which enables us to include non-affinity and arbitrary chain configurations. It can be easily extended to include chain breakage and reconnection. The polymer structure to be investigated, e.g. a rubber boot or a seal, is discretized by means of tetrahedral elements. To each edge of a tetrahedral element one truss element is attached which models the force–stretch behaviour of a bundle of polymer chains. Each of these tetrahedral unit cells represents the micro mechanical material behaviour in a certain point of the network. The proposed method provides the possibility to observe how changes at the microscopic level influence the macroscopic material behaviour. Such information is especially valuable for the polymer industry.

This article deals with a family of non-linear hyperelastic materials M (e) depending on a parameter e varying from 0 to 1; M (0) is a masonry-like material and M (1) is linear elastic. Some properties of the function T ( E ;e) delivering the Cauchy stress corresponding to the infinitesimal strain E, are proved; in particular, it is shown that T ( E ;e) is strongly monotone for e >0 and monotone for e=0. Moreover, denoting by [ u (·;e), E (·;e), T (·;e)] the solution to the equilibrium problem for solids made of a material M (e), the convergence of [ u (·;e), E (·;e), T (·;e)] for e going to 0 and 1, is investigated.

The constraint of in-plane rigidity is examined within the general framework of the theory of internally constrained materials. It is shown that, for in-plane rigid materials, local strain and active stress are both defined by vectorial quantities. Representation formulae for the elastic response mapping are established in the cases of transverse isotropy and maximal symmetry, compatible with the constraint manifold. The equilibrium problem for an elastic body reinforced with parallel inextensible planes is also considered. In particular, universal solutions for bodies with maximal material symmetry are determined within the class of deformations which leave rigid every reinforcing plane.

International audience; This paper is concerned with numerical simulations of three-dimensional finite deformation of a thick-walled circular elastic tube subject to internal or external pressure and zero displacement on its ends. We formulate the system of equations that can accommodate large strain and displacement for the incompressible isotropic neo-Hookean material. The fully non-linear governing equations are solved using the Cþþ based object-oriented finite element library libMesh. A Lagrangian mesh is used to discretize the governing equations, and a weighted residual Galerkin method and Newton iteration solver are used in the numerical scheme. To overcome the sensitivity of the fully non-linear system to small changes in the iterations, the analytical form of the Jacobian matrix is derived, which ensures a fast and better numerical convergence than using a numerically approximated Jacobian matrix. Results are presented for different parameters in terms of wall thickness/radius ratio, and length/radius ratio, as well as internal/external pressure. Validation of the model is achieved by the excellent agreement with the results obtained using the commercial package Abaqus. Comparison is also made with the previous work on axisymmetric version of the same system (Zhu et al., 2008 [34]; Zhu et al. 2010 [43]), and interesting fully three-dimensional post-buckling deformations are highlighted. The success of the current approach paves the way for fluid–structure interaction studies with potential application to collapsible tube flows and modeling of complex physiological systems.

The mechanical behavior of soft matter is characterized by large shape changes, often accompanied by small changes of elastic energy; non-linear elasticity, with large, inhomogeneous, and anisotropic distortions, that may evolve in time, proved to be an effective modeling tool for many of such soft materials. Here, we deal with the definition of an appropriate strain measure, called the elastic metric, upon which the elastic energy density can be defined. Moreover, we discuss two key issues about distortions: one deals with the notion of compatible distortions, that is, distortion fields yielding a global configuration without any change of the elastic energy; the other concerns the symmetries of the material responses. We also present few selected examples of non-linear, anisotropic, elastic response where distortions play a key role.

Strain measures consistent with the classical, infinitesimal form of the strain-energy function are obtained within the context of isotropic, homogeneous, compressible, non-linear elasticity. It will be shown that there are two distinct families of such measures. One family has already been much studied in the literature, the most important member being the strains whose principal values are a function only of the corresponding principal stretches. The second family of strains appears new. The motivation for studying such strains is the intuitive expectation that, for at least moderate deformations, a good fit with experimental data from material characterisation tests will be obtained with the corresponding strain-energy functions. In particular, there is the expectation that such models could prove useful for the modelling of biological soft tissue, whose physiological response is characterised by moderate strains. It will be shown that this is indeed the case for simple tension tests on porcine brain tissue. Deposited by bulk import

A general implicit solution for determining volume-preserving transformations in the n -dimensional Euclidean space is obtained in terms of a set of 2 n generating functions in mixed coordinates. For n =2, the proposed representation corresponds to the classical definition of a potential stream function in a canonical transformation. For n =3, the given solution defines a more general class of isochoric transformations, when compared to existing methods based on multiple potentials. Illustrative examples are discussed both in rectangular and in cylindrical coordinates for applications in mechanical problems of incompressible continua. Solving exactly the incompressibility constraint, the proposed representation method is suitable for determining three-dimensional isochoric perturbations to be used in bifurcation theory. Applications in non-linear elasticity are envisaged for determining the occurrence of complex instability patterns for soft elastic materials.

We present a simplified heart model with the aim of introducing a mechanical point of view in the interpretation of the pressure–volume loops. In particular, we propose a mechanical description of muscle contraction, and discuss its implications with reference to a specific pressure–volume loop representative of a normal human patient.

International audience; A study of direction-based models for the representation of isotropic and anisotropic hyperelastic behaviour of rubber-like materials is proposed. The interest in such models is sustained by their ability to account for the Mullins effect induced anisotropy. For such a purpose, the directional models should be initially isotropic and representative of the hyperelastic behaviour of rubber-like materials. Various models were defined according to different sets of directions. Models were tested in terms of their initial anisotropy and their ability to reproduce the classic full-network hyperelastic behaviour. Various models were proved to perform very well.

International audience; We discuss three different ways of defining the strain measures in the non-linear micropolar continuum: (a) by a direct geometric approach, (b) considering the strain measures as the fields required by the structure of local equilibrium conditions, and (c) requiring the strain energy density of the polar-elastic body to satisfy the principle of invariance under superposed rigid-body deformations. The geometric approach (a) generates several two-point deformation measures as well as some Lagrangian and Eulerian strain measures. The ways (b) and (c) allow one to choose those Lagrangian strain measures which satisfy the additional mechanical requirements. These uniquely selected relative strain measures are called the natural ones. All the strain measures discussed here are formulated in the general coordinate-free form. They are valid for unrestricted translations, stretches and changes of orientations of the micropolar body, and are required to identically vanish in the absence of deformation. The relation of the Lagrangian stretch and wryness tensors derived here to the ones proposed in the literature is thoroughly discussed.

Unidirectional epoxy matrix composite with carbon fibers was cured by hot pressing the commercial prepreg Hexcel M39 in a mold. Macromechanical characteristics of tested coupons with different fiber content and density, were determined by standard tensile, flexure, interlaminar shear strength, and in-plane shear strength tests. The results obtained from different mechanical tests have been correlated, and have been compared with literature data. The reliability of the results for mechanical properties of tested composites as constructional material have been evaluated.

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 2004, Thesis (M.Sc.) -- İstanbul Technical University, Institute of Science and Technology, 2004, Kauçuk konstrüksüyon geliştirmede en büyük problem, kauçuğun doğrusal olmayan, zamana ve yük tekrar sayısına, pişme yöntem ve değişkenlerine bağlı mekanik özelliklerine ampirik ve analitik olarak hakim olmanın güçlüğüdür, özellikle de ortada kesinleşmiş bir konstrüksüyon yokken. Bu çalışmada bitmiş konstrüksüyona ihtiyaç duyulmaksızın, sabit kalınlıktaki arıplakalardan standart ölçülerde kesilmiş denekler üzerinde yapılan mekanik test sonuçlarının, sonlu elemanlar yazılımına girilmesi, ve yazılımda bir defa doğru malzeme modeli elde edildikten sonra, bu malzemenin atandığı karmaşık konstrüksüyonların çeşitlilik arz eden testlerinde dahi, sanal ortamda tasarlananla, bu doğrultuda imal edilen bitmiş parçanın test sonuçlarının çakışacağını göstermektir. Böylelikle , istenen mekanik davranış çerçevesindeki nihai konstrüksüyon yazılım üzerinde oluşturulup, bitmiş parça için prototip kalıp bir defa üretilmiş olur, The most difficult problem in developping a rubber construction is the difficulty of amprical and analytic control over the non-linear and time dependent mechanical behaviour of the rubber being highely effected by the number of repetitive loading cycles , as well as by the type and parameters of curing. Particularly if there isn’t any certain completed construction in use. In this work , it’s aimed to show , that final-construction-independant rubber material properties for an FEA software can be obtained , using the experimental resulsts of simple lab-slabs only. And once the appropriate material model is acquired, it’s quite possible to obtain succesfully coinciding load-displacement curves for the final shape of the real part and its corresponding FEM model , even for different sorts of mechanical tests of varying complexity. Hence the final construction shape of the part can be completed thoroughly on a software platform, within any mechanical limitations requested by the client, asiding the necessity of manifacturing several moulds for provisional trial constructions before reaching the appropriate final one., Yüksek Lisans, M.Sc.

The problem of free shape consists in finding the form that an elastic body must have in a natural stare in order that it shall assume a given form in an equilibrium configuration under the action of assigned loads. The problem, that is of interest in itself, arises in some practical applications and can constitute a preliminary step in the study of some mechanical properties of classes of equilibrium configurations that are not natural states. This paper examines the problem of free shape for inextensible elastic rods which in equilibrium are subject only to the action of forces and couples applied to the ends. and whose deformations can be described by the theory of finite displacements of thin rods due to Kirchhoff. After the general equations governing the problem have been deduced, they are employed to give a classification of the free shapes of rods that in equilibrium are circular rings.

This paper shows that the FEM implementation of Koiter's asymptotic method [W.T. Koiter, On the stability of elastic equilibrium, 1970, Ph.D. Thesis, Delft, 1945. English transl. NASA TT-F10, 883, 1967, AFFDL-TR70-25] outlined by Casciaro et al. [Finite element asymptotic analysis of slender elastic structures: a simple approach, Int. J. Num. Meth. Eng. 35 (1992) 1397-1426] provides accurate and reliable results in the critical and post-critical analysis of non-linear elastic structures. Care, however, does have to be taken in implementing (apparently) minor details to avoid locking effects which adversely affect accuracy and which can destroy the method reliability. As the effects related to the finite element interpolation have been discussed before this paper focuses on the nonlinear locking due to the use, implicit in the method, of finite distance extrapolations. Within this scope, it is shown that perturbation algorithms based on compatible formulations can imply a strong critical and post-critical locking when analysing structures characterized by high stiffness ratios in the presence of moderate pre-critical rotations. On the contrary, perturbation algorithms based on independent extrapolations of displacements and stresses furnish reliable results in excellent agreement with those provided by step-by-step analysis, at a small fraction of its computational cost. (C) 1999 Elsevier Science S.A. All rights reserved.