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We consider the optimization of the vector of gripping forces that support a known generalized force acting on the gripped object—a rigid body or a mechanism. Working in the framework of finite dimensional normed vector spaces and their dual spaces, the cost function to be minimized is assumed to be a norm on the space of gripping forces. We present an expression for the optimum which depends on the external force and the kinematics of the gripping system. Next, assuming that optimal gripping forces are applied using force control, and assuming that there is a bound on the norm of the admissible gripping forces, we characterize the largest norm of an external force that the gripping system may support, that is, the norm of the worst case loading that may be applied and still be supported. A few simple examples are given for the sake of illustration.

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.

For high-order continuum mechanics and classical field theories configurations are modeled as sections of general fiber bundles and generalized velocities are modeled as variations thereof. Smooth stress fields are considered and it is shown that three distinct mathematical stress objects play the roles of the traditional stress tensor of continuum mechanics in Euclidean spaces. These objects are referred to as the variational hyper-stress, the traction hyper-stress and the non-holonomic stress. The properties of these three stress objects and the relations between them are studied.

A geometric analysis of high order stresses in continuum mechanics is presented. Virtual velocity fields take their values in a vector bundle W over the n -dimensional space manifold. A stress field of order k is represented mathematically by an n -form valued in the dual of the vector bundle of k -jets of W . While only limited analysis can be performed on high order stresses as such, they may be represented by non-holonomic hyper-stresses, n -forms valued in the duals of iterated jet bundles. For non-holonomic hyper-stresses, the analysis that applies to first order stresses may be iterated. In order to determine a unique value for the tangent surface stress field on the boundary of a body and the corresponding edge interactions, additional geometric structure should be specified, that of a vector field transversal to the boundary.

The fundamental ideas and tools of the global geometric formulation of stress and hyper-stress theory of continuum mechanics are introduced. The proposed framework is the infinite dimensional counterpart of the statics of a system having a finite number of degrees of freedom, as viewed in the geometric approach to analytical mechanics. For continuum mechanics, the configuration space is the manifold of embeddings of a body manifold into the space manifold. Generalized velocity fields are viewed as elements of the tangent bundle of the configuration space and forces are continuous linear functionals defined on tangent vectors, elements of the cotangent bundle. It is shown, in particular, that a natural choice of topology on the configuration space implies that force functionals may be represented by objects that generalize the stresses of traditional continuum mechanics.

We construct generalizations of boundary forms of Kupferman, Olami and Segev and of the De Donder form to higher order Lagrangians. In principle, the resulting forms may depend on the coordinate system used. Nevertheless, it leads to many invariant conclusions. In some cases, like second order gravity, the construction leads to a unique De Donder form on the jet bundle that is independent of the choice of coordinates.

The theory of continuous distributions of dislocations and other material defects, when formulated in terms of differential forms, is shown to comprise also the discrete, or singular, counterpart, in which defects are concentrated on lower dimensional regions, such as surfaces, lines, and points. The mathematical tool involved in this natural transition is the theory of de Rham currents, which plays in regard to differential forms the same role as the theory of Schwartz distributions plays with respect to ordinary functions. After a review of the main mathematical aspects, the theory is illustrated with a profusion of examples and applications.

In a global, non-linear, geometric formulation of force and stress theory for continuum mechanics, a body is modeled as a compact manifold with corners and the configuration space consists of all C1-embeddings of the body in a physical space manifold. A Force at a given configuration is a linear functional on the space of C1-generalized velocities, that is, a linear functional on a space of sections of a vector bundle over the body. Stresses are introduced through a representation theorem as measures, valued on the dual of the first jet bundle, which represent forces. This paper considers various properties of such stress measures. In particular, we discuss regularization of forces and stresses, and study the relation between forces and stresses in the non-smooth case from a new point of view.

Various issues raised by the mechanics of an octopus’s arm are presented from the point of view of continuum mechanics. First, we study the conditions for a mixture of a number of fiber bundles to span the space of stress tensors. The geometry of a fiber bundle, as reflected in the direction and density of the fibers, is described by a vector field, or, more precisely, by a differential two-form. For the stress analysis, we propose a three-dimensional continuum model of the arm that is based on constitutive data available for muscle fibers. An analysis of small deformations superimposed on an activated configuration of the muscle bundles reveals the stiffening mechanism which enables the arm to support various external loadings, even if the geometry and the number of the fiber groups does not allow the tensions in them to span the space of stress tensors.

The flux of a certain extensive physical quantity across a surface is often represented by the integral over the surface of the component of a pseudo-vector normal to the surface. A pseudo-vector is in fact a possible representation of a second-order differential form, i.e. a skew-symmetric second-order covariant tensor, which follows the regular transformation laws of tensors. However, because of the skew-symmetry of differential forms, the associated pseudo-vector follows a transformation law that is different from that of proper vectors, and is named after the Italian mathematical physicist Gabrio Piola (1794–1850). In this work, we employ the methods of Differential Geometry and the representation in terms of differential forms to demonstrate how the flux of an extensive quantity transforms from the spatial to the material point of view. After an introduction to the theory of differential forms, their transformation laws, and their role in integration theory, we apply them to the case of first-order transport laws such as Darcy’s law and Ohm’s law.

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.

The fundamental phenomenological equations of radiative transfer, e.g., Lambert’s cosine rule and the radiant transport equation, are derived from an analysis based on the Cauchy flux theory of continuum mechanics. For the classical case, where the radiance is distributed regularly over the unit sphere, it is shown that Lambert’s rule follows from a balance law for the transfer of radiative power in each direction u on the sphere, together with the appropriate Cauchy postulates and the additional assumption that the corresponding flux vector field ju be parallel to u. The standard radiant transport equation follows from the additional assumption that radiant flux is given as the advection of radiant energy density. A theory is also presented for the singular limit of isolated rays, where the distribution of radiance on the sphere reduces to a Borel measure.

The paper considers the formulation of higher-order continuum mechanics on differentiable manifolds devoid of any metric or parallelism structure. For generalized velocities modeled as sections of some vector bundle, a variational kth order hyper-stress is an object that acts on jets of generalized velocities to produce power densities. The traction hyper-stress is introduced as an object that induces hyper-traction fields on the boundaries of subbodies. Additional aspects of multilinear algebra relevant to the analysis of these objects are reviewed., Comment: 23 pages

The Eulerian approach to continuum mechanics does not make use of a body manifold. Rather, all fields considered are defined on the space, or the space–time, manifolds. Sections of some vector bundle represent generalized velocities which need not be associated with the motion of material points. Using the theories of de Rham currents and generalized sections of vector bundles, we formulate a weak theory of forces and stresses represented by vector-valued currents. Considering generalized velocities represented by differential forms and interpreting such a form as a generalized potential field, we present a weak formulation of pre-metric, p -form electrodynamics as a natural example of the foregoing theory. Finally, it is shown that the assumptions leading to p -form electrodynamics may be replaced by the condition that the force functional is continuous with respect to the flat topology of forms.

A metric independent geometric framework for some fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, balance principles for extensive properties are formulated and Cauchy's theorem for fluxes is proved. Fluxes in an n-dimensional space are represented as differential (n − 1)-forms. In an analogous formulation of stress theory, a distinction is made between the traction stress, enabling the evaluation of the traction on the boundaries of the various regions, and the variational stress, which acts on the derivative of a virtual velocity field to produce the virtual power density. The relation between the two stress fields is examined as well as the resulting differential balance law. As an application, metric-invariant aspects of electromagnetic theory are presented within the framework of the foregoing flux and stress theory.

In the context of stress theory of the mechanics of continuous media, a generalization of the boundary operator for de Rham currents---the co-divergence operator---is introduced. While the boundary operator of de Rham's theory applies to real valued currents, the co-divergence operator acts on vector valued currents, i.e., functionals dual to differential forms valued in a vector bundle. From the point of view of continuum mechanics, the framework presented here allows for the formulation of the principal notions of continuum mechanics on a manifold that does not have a Riemannian metric or a connection while at the same time allowing irregular bodies and velocity fields.

Intended mainly for continuum mechanicists, Epstein's book introduces modern geometry and some of its applications to theoretical continuum mechanics. Thus, examples for the mathematical objects introduced are chosen from the realm of mechanics. In particular, differentiable manifolds, tangent and cotangent bundles, Riemannian manifolds, Lie derivatives, Lie groups, Lie algebras, differential forms and integration theory are presented in the main part of the book. Once the reader's familiarity with continuum mechanics is used for the introduction of basic geometry, geometry is used in order to generalize notions of continuum mechanics. Integration of differential forms is used to formulate flux theory on manifolds devoid of a Riemannian structure. More specialized topics, namely, Whitney's geometric integration theory and Sikorski's differential spaces are used to relax smoothness assumptions for bodies and fields defined on them. Finally, an overview is given of the work that Epstein and co-workers carried out in recent years where the theory of inhomogeneity of constitutive relations is developed using the geometry of principal fiber bundles, G-structures and connections.

For a given structure made of a perfectly plastic material with a yield stress s Y , we consider the load capacity ratio of the structure: the largest positive number C, depending only on the geometry of the structure, which satisfies the following property. For any loading distribution f on the structure whose maximum is f max , the structure will not undergo plastic collapse as long as f max ⩽ s Y C , independently of the distribution of the load. The paper presents the mathematical aspects, related mechanical notions, algorithms and examples corresponding to load capacity ratios of structures.

This paper is concerned with the dynamics of continua on differentiable manifolds. We present a covariant derivation of equations of motion, viewing motion as a curve in an infinite-dimensional Banach space of embeddings of a body manifold in a space manifold. Our main application is the motion of residually-stressed elastic bodies; residual stress results from a geometric incompatibility between body and space manifolds. We then study a particular example of elastic vibrations of a two- dimensional curved annulus embedded in a sphere., Comment: To appear in the Journal of elasticity

In this note, we consider the application of a velocity-free controller for attitude regulation of a rigid spacecraft with gas jet actuators when the effects of time-delays in the feedback loop are taken into consideration. Simple sufficient conditions for exponential stability are presented. Some structural properties of the resulting closed-loop system are studied, and relevant design tools are demonstrated.

A generalized transport theorem for convecting irregular domains is presented in the setting of Federer's geometric measure theory. A prototypical $r$-dimensional domain is viewed as a flat $r$-chain of finite mass in an open set of an $n$-dimensional Euclidean space. The evolution of such a generalized domain in time is assumed to be in accordance to a bi-Lipschitz type map. The induced curve is shown to be continuous with respect to the flat norm and differential with respect to the sharp norm on currents in $\mathbb{R}^{n}$. A time dependent property is naturally assigned to the evolving region via the action of an $r$-cochain on the current associated with the domain. Applying a representation theorem for cochains the properties are shown to be locally represented by an $r$-form. Using these notions a generalized transport theorem is presented., Comment: 27 pages

A continuum mechanical framework for the description of the geometry and kinematics of defects in material structure is proposed. The setting applies to a body manifold of any dimension which is devoid of a Riemannian or a parallelism structure. In addition, both continuous distributions of defects as well as singular distributions are encompassed by the theory. In the general case, the material structure is specified by a de Rham current \(T\) and the associated defects are given by its boundary \(\partial T\). For a motion of defects associated with a family of diffeomorphisms of a material body, it is shown that the rate of change of the distribution of defects is given by the dual of the Lie derivative operator.

The stress-energy tensor of field theory is defined and analyzed in a geometric setting where a metric is not available. The stress is a linear mapping that transforms the three-form representing the flux of any given property, e.g., charge-current density, to the three-form representing the flux of energy. The example of the electromagnetic stress-energy tensor is given with the additional structure of a volume element.

A theory of surface interactions on manifolds is presented as a framework for force and stress theory pertaining to materials with micro-structure. The dependence of surface forces on the normal in the case of Euclidean geometry is replaced by dependence on the tangent hyperplane and its orientation. Odd quantities are used to model interactions that satisfy Newton's law of action and reaction. Stresses are defined as vector-valued forms of order m−1, where m is the dimension of the body manifold. It is shown how a stress field induces surface forces for the various bodies. The special case of a scalar-valued balance law is presented with some additional detail. Finally, we consider the way material structures are induced by balance laws.

The Cauchy postulates are required for the formulation and proof of Cauchy's theorem for the existence of stress. The generalized postulates and theorem in the geometric setting of differentiable manifolds was considered in a previous paper. This note presents an inconsistency in one of the proposed postulates, the boundedness postulate, and corrects it by specifying a weaker requirement.

A formulation of the Cauchy theory for balance laws of scalar valued quantities is considered from a general geometric point of view. It is assumed only that the ambient space is an orientable m-dimensional manifold. The analog of the usual flux vector field is an (m−1)-differential form. Both the Cauchy theorem and differential version of the balance laws are formulated in this context.

A generalization of the Cauchy theory of forces and stresses to the geometry of differentiable manifolds is presented using the language of differential forms. Body forces and surface forces are defined in terms of the power densities they produce when acting on generalized velocity fields. The normal to the boundary is replaced by the tangent space equipped with the outer orientation induced by outward pointing vectors. Assuming that the dimension of the material manifold is m, stresses are modelled as m − 1 covector valued forms. Cauchy's formula is replaced by the restriction of the stress form to the tangent space of the boundary while the outer orientation of the tangent space is taken into account. The special cases of volume manifolds and Riemannian manifolds are discussed.

In the setting of an n-dimensional Euclidean space, the duality between velocity fields on the class of admissible bodies and Cauchy fluxes is studied using tools from geometric measure theory. A generalized Cauchy flux theory is obtained for sets whose measure theoretic boundaries may be as irregular as flat ( n − 1)-chains. Initially, bodies are modeled as normal n -currents induced by sets of finite perimeter. A configuration space comprising Lipschitz embeddings induces virtual velocities given by locally Lipschitz mappings. A Cauchy flux is defined as a real valued function on the Cartesian product of ( n − 1)-currents and locally Lipschitz mappings. A version of Cauchy’s postulates implies that a Cauchy flux may be uniquely extended to an n-tuple of flat ( n − 1)-cochains. Thus, the class of admissible bodies is extended to include flat n-chains and a generalized form of the principle of virtual power is presented. Wolfe’s representation theorem for flat cochains enables the identification of stress as an n-tuple of flat ( n − 1)-forms representing the flat ( n − 1)-cochains associated with the Cauchy flux.

A numerical algorithm for increasing the accuracy of the approximated solutions of equilibrium problems is presented. The proposed method is capable of reaching approximate solutions which are more accurate than other finite difference methods, when the same number of nodal points participate in the local scheme and the mesh is arbitrary. The basic idea of the proposed method is to reduce the required information from the surrounding grid nodes of the finite difference template by taking more information at the local level and applying differentiation on the governing partial differential equations and their boundary conditions. As a result the global characteristic matrix of the problem is considerably simplified. Calculations based on the proposed method and on another finite difference method are compared to analytical results. The comparison clearly illustrates the superiority of the proposed method.

The Reynolds transport theorem for the rate of change of an integral over an evolving domain is generalized. For a manifold $B$, a differentiable motion $m$ of $B$ in the manifold $\mathcal{S}$, an $r$-current $T$ in $B$, and the sequence of images $m(t)_{\sharp}T$ of the current under the motion, we consider the rate of change of the action of the images on a smooth $r$-form in $\mathcal{S}$. The essence of the resulting computations is that the derivative operator is represented by the dual of the Lie derivative operation on smooth forms., uncertainty has risen on the changes made in Version 2

This paper considers a control problem that requires the design of an observed-based compensator in order to guarantee the asymptotic convergence of the output (namely, the joint generalized coordinates of ann-degree of freedom robot) to a corresponding output produced by a linear reference model. It is shown that, for a model having a relative degree vector {r 1, ...,r n}, withr i ≥2, it is possible to determine a controller-observer yielding a robot output that converges exponentially to the desired model output. Moreover, the combined system comprising the plant and the controller is internally stable in the semiglobal sense. The solution to the standard trajectory-following problem in robotics is obtained as a simplified case of the underlying model matching problem. While the controller-observer is nonlinear, its dynamic part is linear time-varying; in addition, the construction does not assume a priori the knowledge or even the existence of an upper bound to the model state vector.

We present a framework for the study of bodies that contains evolving nonmaterial interfaces. Although analogs of material points and sub-bodies do not exist for such interfaces, we are able to construct a kinematic structure that allows the definition of an interfacial configuration. Equipping the collection of all such configurations, the interfacial configuration space, with the structure of an infinite-dimensional manifold leads to the definition of generalized velocities and forces as elements of the tangent and cotangent bundles of that space. A representation theorem then yields a non-classical force system that arises in recent continuum theories for the study of coherent phase transitions. Associating the underlying body with a particular reference configuration, we find, further, that the elements of that force system are subject to a balance that is also imposed in such theories.

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.

A geometrical framework for the formulations of the invariant mechanical theories of materials with microstructure is presented. The suggested framework is based on the construction of an infinite-dimensional manifold containing the configurations of the body. Each configuration includes both the macro- and the micro-states of the body. The notions of forces, stresses, balance laws and latent microstructure are discussed. Liquid crystals are used as an example.

A mathematical framework for the fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, velocity fields over bodies, modeled as sections of a vector bundle W, are generalized using notions of homological integration theory such as flat chains and cochains. The class of bodies includes fractal sets whose irregular boundaries may have infinite measures. Stresses, initially modeled as smooth differential forms valued in the dual of the jet bundle of W, are generalized to cochains represented by L ∞-sections whose weak divergences are also L ∞. The divergence of a stress field, defined in an earlier work, is generalized to apply to stress cochains. The co-divergence of a velocity field is a weak form of the jet extension mapping and it is the counterpart of the boundary operator for real valued flat chains.

Algorithms for solving partial differential equations which extend previous applications of the nonconforming Taylor discretization method (NTDM) are presented. In one modification the number of interrelated grid points is variable, thus enabling additional geometric flexibility. Another modification is the approximation of the governing differential equation using the method of weighted residuals. A simple one-dimensional test case with a known analytic solution is solved using this code. The results demonstrate that precision is enhanced when using the method of weighted residuals with an increased number of interrelated points. The algorithm is applied as a general purpose two-dimensional code for nonlinear steady state heat-conduction. Two-dimensional examples with complex geometry and boundary conditions are then solved both by the NTDM and by the finite elements method (FEM). The results obtained by the two methods are compared.

A mathematical setting that applies to the analysis of strength of perfectly plastic bodies and structures is presented. Specifically, the worst case analysis under the loads in any vector subspace of the space of all loadings is performed. It is shown that there is a number C, depending only on the geometry of the body, such that the body will not collapse plastically under any applied boundary load t, as long as ‖t‖≤sY|C independently of the distribution of t, where sY is the yield stress of the material. We give examples for computations of C, the load capacity ratio of the body, and examples of computations of worst case loadings.The analysis also shows that perfectly plastic materials are optimal in the following sense. Without specifying a constitutive relation, let ∑f denote the collection of all stress fields σ that are in equilibrium with a given loading f and consider soptf = infσ∈∑f{supx|σ(x)}—the optimal maximal stress. Then, for perfectly plastic materials at the limit state, the optimum opt ...

The kinematics of the octopus’s arm is studied from the point of view of robotics. A continuum three-dimensional kinematic model of the arm, based on a nonlinear rod theory, is proposed. The model enables the calculation of the strains in various muscle fibers that are required in order to produce a given configuration of the arm—a solution to the inverse kinematics problem. The analysis of the forward kinematics problem shows that the strains in the muscle fibers at two distinct points belonging to a cross section of the arm determine the curvature and the twist of the arm at that cross section. The octopus’s arm lacks a rigid skeleton and the role of material incompressibility in enabling the configuration control is studied.

An algorithm for the numerical solution of field problems is presented. The method is based on expanding the unknown function in a Taylor series about some nodal points so that the coefficients of the series for the various nodes are considered as unknowns. Discrepancy between the values resulting from Taylor expansions about distinct nodes is allowed if it is on the same order of magnitude as the estimated error resulting from the discretization. This enables considerable savings in computation effort in addition to the advantage of specifying the accuracy of the obtained solution. Geometrical flexibility, which enables handling complex boundaries for a large variety of field problems, is another advantage of the proposed scheme. The algorithm is applied to nonlinear steady-state heat-conduction. A test case is treated numerically, and for the evaluation of the scheme the results are compared with the analytical solution.