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The potential of a finite-element technique based on an egalitarian meshing of the space–time domain D of physical problems described by parabolic or hyperbolic differential equations is explored. A least-squares minimization technique is applied in the meshed domain D to obtain stiffness-like matrices associated with various linear differential operators. Applications discussed include problems of boundary growth, and diffusive coalescence, in which D cannot be regarded as the Cartesian product of two independent domains in space and time. [ABSTRACT FROM AUTHOR]

Although it is often asserted that, in view of their reduced length, axially compressible beams have a higher buckling load than their inextensible counterparts, a detailed analysis demonstrates that this is not necessarily the case. The argument to arrive at this conclusion is made in terms of relatively straightforward concepts of elasticity and structural mechanics. It is shown that for certain classes of materials, the reduced prebuckling length is more than compensated for by a softening of the elastic response, leading to a reduction of the Euler critical load. [ABSTRACT FROM AUTHOR]

Under given prestress conditions, solitary waves in fluid-filled elastic tubes are confined to a rather narrow set of combinations of the background fluid velocity and the wave speed. This set, which we call the domain of existence, is bounded by several curves that represent various physical and mathematical restrictions. Remarkably, these restrictions can be cast as purely algebraic conditions to be imposed upon the governing system of differential equations. Paramount among the physical restrictions are the avoidance of wrinkles and the self-impenetrability of the wave profile. In particular, the existence of a critical wave speed of impending wrinkling, independent of the background fluid velocity, is established rigorously. [ABSTRACT FROM AUTHOR]

In the original paper, we erroneously used the total potential energy (Helmholtz free energy of the material plus potential energy of the external forces) in the entropy inequality. [ABSTRACT FROM AUTHOR]

A hybrid quantum–classical model is proposed whereby a micro-structured (Cosserat-type) continuum is construed as a principal Hilbert bundle. A numerical example demonstrates the possible applicability of the theory. [ABSTRACT FROM AUTHOR]

The theory of thin-walled beams proposed in 1940 by Vlasov is shown to emerge naturally within the framework of multivector statics. This circumstance is used as the basis for possible extensions of the theory to media with complex microstructures. [ABSTRACT FROM AUTHOR]

The equations of balance of a continuum under conditions of growth and mass diffusion are derived from a principle of invariance under general observer transformations. The resulting equations are invariant under inertial transformations. Apparent body forces stemming from the mass transport phenomenon are identified and shown to be associated with non-inertial observers. [ABSTRACT FROM AUTHOR]

Inspired by the seminal works of Eshelby (Philos Trans R Soc A 244A:87–112, 1951, J Elast 5:321–335, 1975) on configurational forces and of Noll (Arch Ration Mech Anal 27:1–32, 1967) on material uniformity, we study a thermoelastic continuum undergoing volumetric growth and in a dynamical setting, in which we call the divergence of the Eshelby stress the Eshelby force. In the classical statical case, the Eshelby force coincides with the negative of the configurational force. We obtain a differential identity for the modified Eshelby stress, involving the torsion of the connection induced by the material isomorphism of a uniform body, which includes, as a particular case, that found by Epstein and Maugin (Acta Mech 83:127–133, 1990). In this identity, the divergence of the modified Eshelby stress with respect to this connection of the material isomorphism takes the name of modified Eshelby force. Moreover, we show that Eshelby's variational approach (1975) can be used to formulate not only the balance of material momentum, but also the balance of energy. In this case, we find that what we call Eshelby power is the temporal analogue of the Eshelby force, and we obtain a differential identity for the modified Eshelby power. This leads to concluding that the driving force for the process of growth–remodelling is the Mandel stress. Eventually, we find that the relation between the differential identities for the modified Eshelby force and modified Eshelby power represents the mechanical power expended in a uniform body to make the inhomogeneities evolve. [ABSTRACT FROM AUTHOR]

Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, continuum mechanics and differential geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination of the existence of material defects, such as dislocations. This paper presents a rather intuitive treatment of these ideas. [ABSTRACT FROM AUTHOR]

Highlights • The theory of screws is undergoing a revival in machines and robotics. • The theory is significantly generalized and extended in various directions. • Use is made of exterior algebra of multi-vectors. • A formulation in terms of tensor densities is also provided. Abstract The theory of screws is generalized to yield a formulation of Statics in higher-dimensional spaces in the absence of any distinguished metric structure. This level of generality necessitates the use of the tools of the exterior algebra of multi-vectors. The minimal structure retained serves to bring into relief the essential difference between screws and counter-screws as well as a weak version of the notion of rigid-body velocity field. An essential feature of the theory is the introduction of higher-order forces and displacements, whose relevance in the context of structured media can be determined in specific potential applications. [ABSTRACT FROM AUTHOR]

Highlights • Laminates and bundles are important cases of functionally graded materials. • Classical inhomogeneity theory is not able to treat defects in these materials. • A novel definition of inhomogeneity in laminates and bundles is introduced. Abstract Laminates and bundles are common particular cases of functionally graded materials (FGMs). Within this class of non-uniform media, laminates and bundles are distinguished by a property of partial uniformity along surfaces or lines. The question addressed in this paper is whether this property can be exploited to define rigorously a notion of homogeneity eventually leading to a quantitative measure of a density of continuous defects. [ABSTRACT FROM AUTHOR]

A thin-walled structure is homogeneously embeddable if it can be obtained by carving it out of a homogeneous material block or, in other words, if it is materially defect-free. Explicit analytic and geometric conditions for the embedded homogeneity of plane linearly elastic beams are derived and discussed. In the geometric setting, a prominent role is played by the properties of the hodograph of uniaxial material tensor fields defined on the beam axis. [ABSTRACT FROM AUTHOR]

The notion of embedded homogeneity of thin-walled structures is introduced as the property characterizing the provenance of such a structure from a homogeneous material. This property needs to be distinguished from other definitions of homogeneity formulated exclusively in terms of a purely structural constitutive equation. Necessary conditions for embedded homogeneity are derived for planar beams and their geometric interpretation is expressed as the condition for the elastic hodograph to lie on a hypersphere containing the origin of a six-dimensional space of tensors. [ABSTRACT FROM AUTHOR]

A unified theory of material defects, incorporating both the smooth and the singular descriptions, is presented based upon the theory of currents of Georges de Rham. The fundamental geometric entity of discourse is assumed to be represented by a single differential form or current, whose boundary is identified with the defect itself. The possibility of defining a less restrictive dislocation structure is explored in terms of a plausible weak formulation of the theorem of Frobenius. Several examples are presented and discussed. [ABSTRACT FROM AUTHOR]

The theory of singular dislocations is placed within the framework of the theory of continuous dislocations using de Rham currents. For a general n-dimensional manifold, an (n − 1)-current describes a local layering structure and its boundary in the sense of currents represents the structure of the dislocations. Frank’s rules for dislocations follow naturally from the nilpotency of the boundary operator. [ABSTRACT FROM PUBLISHER]

Abstract: An easy-to-apply algorithm is proposed to determine the correct set(s) of boundary conditions for hyperbolic systems of partial differential equations. The proposed approach is based on the idea of the incoming/outgoing characteristics and is validated by considering two problems. The first one is the well-known Euler system of equations in gas dynamics and it proved to yield set(s) of boundary conditions consistent with the literature. The second test case corresponds to the system of equations governing the flow of viscoelastic liquids. [Copyright &y& Elsevier]

For any body-time manifold R × B there exists a groupoid, called the material groupoid, encoding all the material properties of the material evolution. A smooth distribution, the material distribution, is constructed to deal with the case in which the material groupoid is not a Lie groupoid. This new tool provides a unified framework to deal with general non-uniform material evolution. [ABSTRACT FROM AUTHOR]

Self-diffusion, or the flux of mass of a single species within itself, is viewed as an independent phenomenon amenable to treatment by the introduction of an auxiliary field of diffusion velocities. The theory is shown to be heuristically derivable as a limiting case of that of an ordinary binary mixture. [ABSTRACT FROM AUTHOR]

The theory of material evolution is used to model anisotropic plasticity. As suggested by the physics of plasticity, whereby the motion of dislocations yields relative slipping along a particular material plane, a shear-like isochoric material deformation is assumed as the admissible kinematic evolution variable, which is constitutively driven by the thermodynamically dual Mandel stress. A simple and intuitive anisotropic evolution law is considered, where slipping is permitted along a fixed material plane which is prescribed as part of the evolution law. An additional yield criterion is proposed to complete the model. It is demonstrated that the anisotropy of the evolution law renders perfect plasticity, hardening or softening and is rate dependent. [ABSTRACT FROM PUBLISHER]

Studies of solitary waves commonly apply the long-wave approximation, which unnecessarily rigidifies the behavior of the tube, but permits the problem to be solved using certain approximate numerical techniques. In this study, an approach was developed where approximating the contribution of the axial strain as a linear function of the radial strain reduced the system of exact governing differential equations to a single equation of a single dependent variable. The approximated solution was found to agree with the exact solution to within 3%. This approach would be useful for considering more complex problems where the exact solution technique may not be reasonably applied. [ABSTRACT FROM AUTHOR]

Abstract: Boundary growth is defined as a particular case of surface growth. Within this restricted context, it is shown that the kinetics of boundary growth is not essentially distinguishable from that of volumetric growth and that, consequently, the a-priori notion of material particle may be devoid of an intrinsic meaning. [ABSTRACT FROM AUTHOR]

Abstract: We present a unified purely hyperbolic model for compressible and incompressible viscoelastic liquids. The proposed model is then solved numerically using the least-squares finite element method (LSFEM). [Copyright &y& Elsevier]

The possible role of the Eshelby stress in the general context of theories of material evolution is examined with particular emphasis on evolution laws that are not necessarily of the anelastic type. [ABSTRACT FROM AUTHOR]

Abstract: After drawing a distinction between anelastic evolution and aging, it is shown that for certain solid classes a canonical decomposition of any symmetry-preserving evolutive process into an anelastic-like component and an aging component can be achieved. The theory thus obtained can be regarded as the time-like counterpart of the theory of inhomogeneities in functionally graded bodies. [Copyright &y& Elsevier]

Abstract: A method to derive the stiffness of self-similar elastic fractals is presented based on structural mechanics principles and a physically motivated similarity criterion, which is assumed as a postulate. Using this method, the stiffnesses of both the Von Koch curve and the Sierpiński gasket in the small-deformation regime are derived. For these fractal structures, it is shown that the stiffness matrix is completely determined by a single elastic constant. The procedure to tile a planar domain with Sierpiński gaskets is explored and shown to require the consideration of hexagonal-shaped combinations of gaskets joined continuously along their edges. This continuity leads to a phenomenon of geometrically induced inextensibility along the common edges. After deriving the stiffness matrix for the basic hexagon, the analog of the Boussinesq–Flamant problem for a tiled half-plane is solved numerically to demonstrate the potential of the method in modeling of solid mechanics applications. [Copyright &y& Elsevier]

The notions of uniformity and homogeneity of elastic materials are reviewed in terms of Lie groupoids and frame bundles. This framework is also extended to consider the case of Functionally Graded Media, which allows us to obtain some homogeneity conditions. [ABSTRACT FROM AUTHOR]

Abstract: A systematic procedure to extract a projected first-grade constitutive equation from a given second-grade one is proposed and proven to be independent of the reference configuration used. The symmetry group of the projected material is shown to contain the first-grade projection of the symmetry group of the original material. The proposed method is illustrated by means of an example. [Copyright &y& Elsevier]

For evolutionary processes of material remodelling and growth, a comparison is drawn between a conventional formulation and one that postulates the existence of additional balance laws for the configurational forces. [ABSTRACT FROM AUTHOR]

Abstract: Fibres, aponeuroses, and tendons are often considered mechanically “in series” in skeletal muscles. This notion has led to oversimplified calculations of fibre forces from tendon forces, to incorrect derivations of constitutive laws for aponeuroses, and to misinterpretations of the recovery of elastic energy in stretch-shortening cycles of muscles. Here, we demonstrate theoretically, using examples of increasing complexity, that tendon and aponeurosis are not in series in a muscle fibre–aponeurosis–tendon complex. We then demonstrate that assuming the tendon and aponeurosis to be in series can lead to the appearance of mechanical work creation in these passive viscoelastic structures, a result that is mechanically impossible. Finally, we explain the mechanical role of the incompressible muscle matrix in force transmission from fibres to aponeuroses and tendon, and emphasize that incompressibility necessitates the introduction of extra forces necessary to maintain this constraint. Unfortunately, this requirement eliminates, for all but the simplest cases, a theoretical approach of muscle modeling based on intuitive free-body diagrams. [Copyright &y& Elsevier]

Abstract: A mechanical theory of fractals and of non-smooth objects in general is developed on the basis of the theory of differential spaces of Sikorski. Once the (generally infinite dimensional) configuration space is identified, an extended form of the principle of virtual work is used to define the concept of generalized force and stress. For the case of self-similar fractals, an appropriate integration based on the Hausdorff measure is introduced and applied to the numerical formulation of stiffness matrices of some common fractals, which can be used in a finite element implementation. [Copyright &y& Elsevier]

A pseudo-rigid coin is a thin disk that can deform only to the extent of undergoing an arbitrary affine deformation in its own plane. The coupling of the classical rolling problem with this deformability, albeit limited, may shed light on such phenomena as the production of noise by a twirling dish. From the point of view of analytical dynamics, one of the interesting features of this problem is that the rolling constraint turns out to be nonholonomic even in the case of motion on a straight line in a vertical plane. After the analytical formulation of the general problem, explicit solutions are obtained for special shape-preserving motions. For more general motions, numerical studies are carried out for various initial conditions. [ABSTRACT FROM AUTHOR]

A material body with smoothly distributed microstructure can be seen geometrically as a fibration or, when the symmetry group is specified, as a fiber bundle. Within this very general framework, we present a geometric description of such material bodies in terms of fiber jets. We introduce the notion of fiber frame and construct the corresponding Lie groupoid and fiber G-structure. Then, physical properties of a material body with microstructure as uniformity and homogeneity can be translated in geometrical terms as transitivity for the Lie groupoid or integrability for the fiber G-structure. [Copyright &y& Elsevier]

The purpose of this note is to reaffirm the fact that there exists a natural connection between Noll’s theory of inhomogeneities and the Eshelby tensor. One way to expose this connection consists in allowing the inhomogeneity pattern to evolve in time and then exploring the thermodynamic implications. [Copyright &y& Elsevier]

Studies a complete theory of saturated nonlinear elasticity which was formulated in terms of a relaxed function. Basic notions of saturated elasticity; Information on the relaxed energy function; Applications of saturated elasticity.

The constitutive characterization of the uniformity and homogeneity of binary elastic composites is presented in terms of a combination of the material groupoids of the individual constituents. The incorporation of these two groupoids within a single double groupoid is proposed as a viable mathematical framework for a unified formulation of this and similar kinds of problems in continuum mechanics. [ABSTRACT FROM AUTHOR]

• We apply a difference of potential to the inner/outer boundaries of the tube. • For a larger difference of potential, the hoop deformation solitary wave decreases. • The period of the hoop solitary wave increases for a larger difference of potential. [Display omitted] The influence of an electric field on the propagation of axially symmetric solitary waves in a fluid-filled thin-walled tube made of a dielectric elastomer is analysed. The electric field is applied by means of soft flexible electrodes lining the internal and external surfaces of the tube and connected to a power source with a given constant difference of potential. For a specific hyperelastic constitutive law, numerical results are obtained and discussed. [ABSTRACT FROM AUTHOR]

Provides information on a study which explained the evolution of the material structure for materials of first and second order in terms of first-order differential equations for a set of `transplant' maps. Bodies and configurations; Second-order material points; Explanation on evolution laws.

• A new model for coupled diffusive processes with a TLP bonding application. • Model framed within the theory of partial differential equations of a mixed order. • Comparison between single interlayer, separate and nanocomposite interlayers. • Spatial distribution variance of concentration as a measure of the diffusive process. A model for coupled diffusive processes is framed within the theory of partial differential equations of mixed order. Its numerical implementation is achieved by a technique that does not require additional tools beyond the standard procedures for the solution of the diffusion equation. The interface between the diffusant and the substrate retains its sharpness during the bulk of the diffusive process. The results demonstrate that, as expected in pharmaceutical and diffusion bonding applications, the amount of material made available for diffusion is controlled by the area of the interface and, therefore, the use of nanoparticle layers may greatly accelerate the rate of the overall process. [ABSTRACT FROM AUTHOR]

• Background on differential geometric concepts is provided in the short space available. • The theory of G-structures is shown to be relevant to both approaches mentioned in title. • A non-trivial example is presented to illustrate the theory. • Some epistemological questions are addressed, if very briefly. An attempt is made to bring into harmony two of the paradigms commonly used in the theory of continuous distributions of defects. It is shown that the common differential geometric apparatus is provided neatly by the theory of G-structures. In the case of a structural model, based on putative experimental observations at the microscopic level, a G-structure can be shown to emerge from the group of linear transformations that preserve a tensorial quantity. For the phenomenological (macroscopic) constitutive model, the G-structure arises from the notion of material isomorphism and the underlying local symmetry group of the constitutive law. A comparative example is presented in the framework of certain smectic liquid crystals. [ABSTRACT FROM AUTHOR]

Discusses differential equation for the amplitude of wrinkles. Derivation of the equation; Relationship between wrinkling strain, curvature, amplitude and wavelength; Obtaining a nonlinear ordinary differential equation governing the amplitude of the wrinkles.

The concept of material distribution is introduced as describing the geometric material structure of a general non-uniform body. Any smooth constitutive law is shown to give rise to a unique smooth integrable singular distribution. Ultimately, the material distribution and its associated singular foliation result in a rigorous and unique subdivision of the material body into strictly smoothly uniform components. Thus, the constitutive law induces a unique partition of the body into smoothly uniform sub-bodies, laminates, filaments and isolated points. [ABSTRACT FROM AUTHOR]

A Lie groupoid, called material Lie groupoid, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called material algebroid, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of G−structures is discussed in detail. An illustrative example is presented as an application of the theory. [ABSTRACT FROM AUTHOR]