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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]

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]

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 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]

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]

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 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]

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]

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]

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.

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.