Showing papers in "Mathematics and Mechanics of Solids in 2015"

At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola

Abstract: Gabrio Piola’s scientific papers have been underestimated in mathematical physics literature. Indeed, a careful reading of them proves that they are original, deep and far-reaching. Actually, even ...

Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids

Abstract: In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materia...

The relaxed linear micromorphic continuum: Existence, uniqueness and continuous dependence in dynamics

Abstract: We study well-posedness for the relaxed linear elastic micromorphic continuum model with symmetric Cauchy force-stresses and curvature contribution depending only on the micro-dislocation tensor. In contrast to classical micromorphic models our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. Another interesting feature concerns the prescription of boundary values for the micro-distortion field: only tangential traces may be determined which are weaker than the usual strong anchoring boundary condition. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes.

A two-dimensional peridynamic model for thin plates

Abstract: A general plate model based on the peridynamic theory of solid mechanics is presented. The model is derived as a two-dimensional approximation of the three-dimensional bond-based theory of peridynamics via an asymptotic analysis. The resulting plate theory is demonstrated using a specially designed peridynamics code to simulate the fracture of a brittle plate with a central crack under tensile loading.

On compressible versions of the incompressible neo-Hookean material

Abstract: We consider three different compressible versions of the conventional incompressible neo-Hookean material model. The different versions are not new and have been used in various model studies. They...

Localized standing waves in a hyperelastic membrane tube and their stabilization by a mean flow

Abstract: We first give a complete analysis of the dispersion relation for travelling waves propagating in a pre-stressed hyperelastic membrane tube containing a uniform flow. We present an exact formula for the so-called pulse wave velocity, and demonstrate that as any pre-stress parameter is increased gradually, localized bulging would always occur before a superimposed small-amplitude travelling wave starts to grow exponentially. We then study the stability of weakly and fully nonlinear localized bulging solutions that may exist in such a fluid-filled hyperelastic membrane tube. Previous studies have shown that such localized standing waves are unstable under pressure control in the absence of a mean-flow, whether the fluid inertia is taken into account or not. Stability of such localized aneurysm-type solutions is desired when aneurysm formation in human arteries is modelled as a bifurcation phenomenon. It is shown that in the near-critical regime axisymmetric perturbations are governed by the Korteweg–de Vries...

Remodelling in statistically oriented fibre-reinforced materials and biological tissues:

Abstract: We present a mathematical model of structural reorganisation in a fibre-reinforced composite material in which the fibres are oriented statistically, ie obey a probability distribution of orientation Such a composite material exemplifies a biological tissue (eg articular cartilage or a blood vessel) whose soft matrix is reinforced by collagen fibres The structural reorganisation of the composite takes place as fibres reorient, in response to mechanical stimuli, in order to optimise the stress distribution in the tissue Our mathematical model is based on the Principle of Virtual Powers and the study of dissipation Besides incompressibility, our main hypothesis is that the composite is characterised by a probability density distribution that measures the probability of finding a family of fibres aligned along a given direction at a fixed material point Under this assumption, we describe the reorientation of fibres as the evolution of the most probable direction along which the fibres are aligned T

Multiscale modelling for composites with energetic interfaces at the micro- or nanoscale:

Abstract: A homogenization framework is developed that accounts for the effect of size at the micro- or nanoscale. This is achieved by endowing the interfaces of the micro- or nanoscopic features with their ...

Electromechanical instabilities in fiber-constrained, dielectric-elastomer composites subjected to all-around dead-loading

Abstract: In this work we investigate the possible development of instabilities in a certain class of dielectric-elastomer composites (DECs) subjected to all-around dead electromechanical loading. The DECs consist of a dielectric elastomer matrix phase constrained to plane strain deformations by means of aligned, long, rigid dielectric fibers of elliptical cross-section that are also aligned but randomly distributed in the transverse plane. Two types of instabilities are considered: loss of positive definiteness (LPD), and loss of strong ellipticity (LE). LPD simply corresponds to the loss of local convexity of the homogenized electroelastic stored-energy function for the DECs and can be of two types depending on the resulting instability modes. When the modes are aligned with the ‘principal’ solution, the instability corresponds to a maximum in the nominal electric field, possibly followed by snapping behavior. Alternatively, when the modes are orthogonal to the principal solution, the instability corresponds to a...

Analysis and approximation of a strain-limiting nonlinear elastic model:

Abstract: Elastic solids with strain-limiting response to external loading represent an interesting class of material models, capable of describing stress concentration at strains with small magnitude. A theoretical justification of this class of models comes naturally from implicit constitutive theory. We investigate mathematical properties of static deformations for such strain-limiting nonlinear models. Focusing on the spatially periodic setting, we obtain results concerning existence, uniqueness and regularity of weak solutions, and existence of renormalized solutions for the full range of the positive scalar parameter featuring in the model. These solutions are constructed via a Fourier spectral method. We formulate a sufficient condition for ensuring that a renormalized solution is in fact a weak solution, and we comment on the extension of the analysis to nonperiodic boundary-value problems.

A one-dimensional fractional order thermoelastic problem for a spherical cavity:

Abstract: We apply the fractional order theory of thermoelasticity to a one-dimensional problem for a spherical cavity subjected to a thermal shock. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity.

The linear elasticity tensor of incompressible materials

Abstract: With a universally accepted abuse of terminology, materials having much larger stiffness for volumetric than for shear deformations are called incompressible. This work proposes two approaches for ...

The Rayleigh and Courant variational principles in the six-parameter shell theory

Abstract: Courant’s minimax variational principle is considered in application to the six-parameter theory of prestressed shells. The equations of a prestressed micropolar shell are deduced in detail. Courant’s principle is used to study the dependence of the least and higher eigenfrequencies on shell parameters and boundary conditions. Cases involving boundary reinforcements and shell junctions are also treated.

On Timoshenko thin elastic inclusions inside elastic bodies

Abstract: The paper concerns the analysis of equilibrium problems for 2D elastic bodies with thin inclusions modeled in the framework of Timoshenko beams. The first focus is on the well-posedness of the model problem in a variational setting. Then delaminations of the inclusions are considered, forming a crack between the elastic body and the inclusion. Nonlinear boundary conditions at the crack faces are considered to prevent a mutual penetration between the faces. The corresponding variational formulations together with weak and strong solutions are discussed. The model contains various physical parameters characterizing the mechanical properties of the inclusion, such as flexural and shear stiffness. The paper provides an asymptotic analysis of such parameters. It is proved that in the limit cases corresponding to infinite and zero rigidity, we obtain rigid inclusions and cracks with the non-penetration conditions, respectively. Finally, exemplary networks of Timoshenko beams are considered as inclusions as well.

Preferred fiber orientations in healthy arteries and veins understood from netting analysis

Abstract: In this paper we analyze the reinforcement of blood vessels by collagen fibers using the concept of netting analysis from composite theory. To this end, we interpret preferred fiber reinforcement as the solution to a weighted optimization problem having two competing targets, minimal pulse pressure and minimal material usage, under the constraint of mechanical stability against buckling in a physiologically reasonable range of blood pressures. We demonstrate that the solution to this optimization problem for large arteries is a helical reinforcement with fibers oriented diagonally and for small arteries and veins an orthogonal reinforcement consisting of axial and circumferential fibers. Both findings agree well with experimental data reported by others, which suggests the existence of an underlying mechanical principle for the establishment and maintenance of vascular fiber orientations under normal conditions.

Wave propagation and non-local effects in periodic frame materials: Generalized continuum mechanics:

Abstract: Through the analysis of the wave propagation in infinite two-dimensional periodic frame materials, this paper illustrates the complexity of their dynamic behavior. Assuming the frame size is small compared to the wavelength, the homogenization method of periodic discrete media coupled with normalization is used to identify the macroscopic behavior at the leading order. The method is applied on a frame material with the vertical elements stiffer than the horizontal elements. Such a material is highly anisotropic and presents a large contrast between the rigidities of the possible mechanisms. Thus the waves associated with different kinematics appear in different frequency ranges. Moreover, the stiffer elements can deform in bending at the macroscopic scale. The equivalent continuum is a second-grade medium at the leading order and shear waves can be dispersive. A criterion is proposed to easily determine when this bending effect has to be taken into account. Second-grade media, generalized media

Comparison of various linear plate theories in the light of a consistent second-order approximation

Abstract: The uniform-approximation technique in combination with the pseudo-reduction technique is applied in order to derive consistent theories for isotropic and anisotropic plates. The approach has already been used to assess and validate theories established in the literature, e.g., the theories of Reissner and Zhilin. In this contribution, we also present a comparison with the theories of Vekua, Ambartsumyan, Steigmann and Reddy’s third-order theory. The current paper is a corrected and extended version of the one originally published in Kienzler and Schneider (Comparison of various linear plate theories in the light of a consistent second-order approximation. In Pietraszkiewicz, W and Gorski, J (eds), Shell Structures: Theory and Applications. London: Taylor & Francis Group, 2014, pp. 109–112). In addition we put special emphasis on the derivation and validation of Ambartsumyan’s general and simplified theories.

On the algebraic structure of isotropic generalized elasticity theories

Abstract: In this paper the algebraic structure of the isotropic nth-order gradient elasticity is investigated. In the classical isotropic elasticity it is well known that the constitutive relation can be broken down into two uncoupled relations between the elementary part of the strain and the stress tensors (deviatoric and spherical). In this paper we demonstrate that this result can not be generalized because in 2nd-order isotropic elasticity there exist couplings between elementary parts of higher-order strain and stress tensors. Therefore, and in certain way, nth-order isotropic elasticity have the same kind of algebraic structure as anisotropic classical elasticity. This structure is investigated in the case of 2nd-order isotropic elasticity, and moduli characterizing the behavior are provided.

Some remarks on generalized continuum mechanics

Abstract: In this essay, a critical review of the bases and main characters of the theories of generalized continua is given within the general context of complex systems. After a brief reminder of the various extensions from standard to so-called “generalized” continua, the emphasis is placed on various disputable essential points. These are: the theoretically irreconcilable concepts of material point and continuum, the opposing notions of “discretization” and “continualization”, the important role played by symmetries and invariances in a definitely modern view of the subject matter, and the inevitable introduction of complexity in the mechanics of real materials as illustrated by several examples.

On Strong Ellipticity for Implicit and Strain-Limiting Theories of Elasticity

Abstract: There has been increasing interest in the archival literature devoted to the study of implicit constitutive theories for non-dissipative materials generalizing the classical Green and Cauchy notions of elasticity, and for the special case of strain-limiting models for which strains remain bounded, even infinitesimal, while stresses can become arbitrarily large. This paper addresses the question of strong ellipticity for several classes of these models. A general approach for studying strong ellipticity for implicit theories is introduced and it is noted that there is a close connection between the questions of strong ellipticity and the existence of an equivalent Cauchy elastic formulation. For most of the models studied to date, it is shown that strong ellipticity holds if the Green–St. Venant strain is small enough, whereas it fails to hold for large strain. The large strain failure of strong ellipticity is generally associated with extreme compression.

Direct determination of stresses from the stress equations of motion and wave propagation for a new class of elastic bodies

Abstract: For a new class of elastic bodies, where the linearized strain tensor is given as a function of the Cauchy stress tensor, the problem of considering unsteady motions is studied. A system of partial differential equations that only depends on the stress tensor is found from the equation of motion, which is a system of six partial differential equations for the six components of the stress tensor. A simple boundary value problem is solved for a 1D bar using exact and numerical methods.

Quasi-static buckling simulation of single-layer graphene sheets by the molecular mechanics method

Abstract: This paper presents a quasi-static nonlinear buckling analysis of compressed single-layer graphene sheets (SLGSs) using the molecular mechanics method. Bonded interactions between carbon atoms are ...

Helical buckling of thick-walled, pre-stressed, cylindrical tubes under a finite torsion:

Abstract: We study the occurrence of torsional instabilities in soft, incompressible, thick-walled tubes with both circumferential and axial pre-stretches. Assuming a neo-Hookean strain energy function, we investigate the helical buckling under a finite torsion in three different classes of boundary conditions: (a) no applied loads at the internal and external surfaces of the cylindrical tube, (b) a pressure load P acting on the external surface or (c) on the internal surface. We perform a linear stability analysis on the axisymmetric solutions using the method of small deformations superposed on finite strains. Applying a helical perturbation, we derive the Stroh formulation of the incremental boundary value problems and we solve it using a numerical procedure based on the surface impedance method. The threshold values of the torsion rate and the associated critical circumferential and longitudinal modes at the onset of the instability are examined in terms of the circumferential and axial pre-stretches, and of th...

Quasistatic adhesive contact delaminating in mixed mode and its numerical treatment

Abstract: An adhesive unilateral contact between visco-elastic bodies at small strains and in a Kelvin–Voigt rheology is scrutinized, neglecting inertia. The flow-rule for debonding the adhesive is considered rate independent, unidirectional, and non-associative due to dependence on the mixity of modes of delamination, namely Mode I (opening) needs (= dissipates) less energy than Mode II (shearing). Such mode-mixity dependence of delamination is a very pronounced (and experimentally confirmed) phenomenon typically considered in engineering models. An efficient semi-implicit-in-time FEM discretization leading to recursive quadratic mathematical programs is devised. Its convergence and thus the existence of weak solutions is proved. Computational experiments implemented by BEM illustrate the modeling aspects and the numerical efficiency of the discretization.

Diffusion of chemically reacting fluids through nonlinear elastic solids: Mixture model and stabilized methods

Abstract: This paper presents a stabilized mixed finite element method for advection-diffusion-reaction phenomena that involve an anisotropic viscous fluid diffusing and chemically reacting with an anisotropic elastic solid. The reactive fluid–solid mixture theory of Hall and Rajagopal (Diffusion of a fluid through an anisotropically chemically reacting thermoelastic body within the context of mixture theory. Math Mech Solid 2012; 17: 131–164) is employed wherein energy and entropy production relations are captured via an equation describing the Lagrange multiplier that results from imposing the constraint of maximum rate of entropy production. The primary partial differential equations are thus reduced to the balance of mass and balance of linear momentum equations for the fluid and the solid, together with an equation for the Lagrange multiplier. Present implementation considers a simplification of the full system of governing equations in the context of isothermal problems, although anisothermal studies are bein...

Determining the time of bulge formation in an elastomeric tube as it inflates, elongates and alters chemorheologically

Abstract: An elastomeric cylindrical membrane with ends attached to rigid end caps is inflated and extended by a fluid that accesses its interior though one of the end caps. The membrane is at a high enough temperature that it undergoes microstructural changes due to simultaneous scission and re-crosslinking of macromolecular network junctions. Motivated by the known elastic response at lower temperatures, it is assumed that the membrane undergoes time-dependent extension and inflation. The deformation history is axially symmetric and nearly cylindrical for short times. There is then a time when a bulge-like deformation starts to form. This is treated as branching from the cylindrical deformation. For times beyond this ‘branching time’, the governing equations are satisfied by both the continuation of the initial deformation and the branched deformation. Criteria are derived for determining this branching time for two different assumptions about the branched response: when it is another cylindrical deformation and ...

Reviewing the roots of continuum formulations in molecular systems. Part III: Stresses, couple stresses, heat fluxes

Abstract: This contribution is the third part in a series devoted to the fundamental link between discrete particle systems and continuum descriptions. The basis for such a link is the postulation of the primary continuum fields such as density and kinetic energy in terms of atomistic quantities using space and probability averaging.In this part, solutions to the flux quantities (stress, couple stress, and heat flux), which arise in the balance laws of linear and angular momentum, and energy are discussed based on the Noll’s lemma. We show especially that the expression for the stress is not unique. Integrals of all the fluxes over space are derived. It is shown that the integral of both the microscopic Noll–Murdoch and Hardy couple stresses (more precisely their potential part) equates to zero. Space integrals of the Hardy and the Noll–Murdoch Cauchy stress are equal and symmetric even though the local Noll–Murdoch Cauchy stress is not symmetric. Integral expression for the linear momentum flux and the explicit he...

Fractional dissipation generated by hidden wave-fields

Abstract: This paper considers the damping induced on a single degree-of-freedom system when it is coupled at one end of a waveguide in which waves are radiated producing an energy loss in the oscillator mot...

Mechanics of materially uniform thin films

Abstract: A pure-bending model for thin films, accurate to leading order in film thickness, is rigorously established on the basis of the three-dimensional theory of materially uniform elastic bodies. A mode...

Pitchfork and octopus bifurcations in a hyperelastic tube subjected to compression: Analytical post-bifurcation solutions and imperfection sensitivity

Abstract: This paper studies the axisymmetric deformations of a nonlinearly hyperelastic tube subjected to axial compression. We aim at investigating the critical buckling stresses and modes, deriving the analytical solutions for the post-bifurcation deformations and studying the imperfection sensitivity. For a general isotropic hyperelastic tube, a coupled series-asymptotic method is utilized to derive two simplified model equations with specified constraints on the tube geometry. Then, we specialize to the Blatz–Ko material. With greased end conditions, through linear bifurcation analysis, we obtain the critical stress values and the corresponding mode numbers. The analytical solutions for the post-bifurcation states are constructed by the multiple scales method. By examining the solution behavior in the post-bifurcation regime, it is found that a thick tube could be considerably softer than a thin one. The singularities theory is used to consider the imperfection sensitivity, which reveals the mechanism is the e...