Skip to main content
main-content

Über dieses Buch

This book illustrates the deep roots of the geometrically nonlinear kinematics of

generalized continuum mechanics in differential geometry. Besides applications to first-

order elasticity and elasto-plasticity an appreciation thereof is particularly illuminating

for generalized models of continuum mechanics such as second-order (gradient-type)

elasticity and elasto-plasticity.

After a motivation that arises from considering geometrically linear first- and second-

order crystal plasticity in Part I several concepts from differential geometry, relevant

for what follows, such as connection, parallel transport, torsion, curvature, and metric

for holonomic and anholonomic coordinate transformations are reiterated in Part II.

Then, in Part III, the kinematics of geometrically nonlinear continuum mechanics

are considered. There various concepts of differential geometry, in particular aspects

related to compatibility, are generically applied to the kinematics of first- and second-

order geometrically nonlinear continuum mechanics. Together with the discussion on

the integrability conditions for the distortions and double-distortions, the concepts

of dislocation, disclination and point-defect density tensors are introduced. For

concreteness, after touching on nonlinear fir

st- and second-order elasticity, a detailed

discussion of the kinematics of (multiplicative) first- and second-order elasto-plasticity

is given. The discussion naturally culminates in a comprehensive set of different types

of dislocation, disclination and point-defect density tensors. It is argued, that these

can potentially be used to model densities of geometrically necessary defects and the

accompanying hardening in crystalline materials. Eventually Part IV summarizes the

above findings on integrability whereby distinction is made between the straightforward

conditions for the distortion and the double-distortion being integrable and the more

involved conditions for the strain (metric) and the double-strain (connection) being

integrable.

The book addresses readers with an interest in continuum modelling of solids from

engineering and the sciences alike, whereby a sound knowledge of tensor calculus and

continuum mechanics is required as a prerequisite.

Inhaltsverzeichnis

Frontmatter

Prologue

Frontmatter

Motivation: Linear Crystal Plasticity

Mono-crystals display various defects that may potentially act as obstacles to further evolution of inelastic deformations, i.e. to further plastic flow. These are translational defects in terms of (primary and secondary) dislocations, rotational defects in terms of disclinations, and (dilatational) point-defects in terms of lattice vacancies or interstitial atoms. Formulations of generalized crystal plasticity incorporate the densities of these defects in order to capture the hardening of the material. In particular the inclusion of the defect densities other than the dislocation density requires to root the formulation in a second-order continuum description.

Paul Steinmann

Differential Geometry

Frontmatter

Preliminaries

The treatment of geometry goes back to the ancient Greek, thereby the findings documented in the 13 volumes of Euclid’s Elements defined the state of affairs for some 2000 years. The advent of differential geometry is associated with the Habilitation lecture of Riemann in 1854. Its further development enabled and cumulated in the formulation of Einstein’s Theory of General Relativity/Gravitation some sixty years later in 1915. However the necessity for a non-Euclidean geometry may be motivated already from simply considering the failure of some of the corner stones in Euclidean geometry, for example the parallel axiom, on a two-dimensional curved manifold such as a sphere. Differential manifolds may be classified in terms of the two fundamental quantities connection and metric that in turn give rise to the three most essential tensors in differential geometry describing torsion, curvature, and non-metricity.

Paul Steinmann

Geometry on Connected Manifolds

The differential geometry on manifolds is considered in an abstract setting without resorting to the concept of a metric. To this end, vectors and covectors are distinguished according to their transformation behavior upon changes of coordinates. Since partial derivatives of vectors and covectors do not transform like tensors, the concept of the covariant derivative of tensors, obeying proper tensor transformation behavior, is motivated. This is achieved by introducing the connection, a third-order non-tensorial object, as one of the most important objects of differential geometry. The covariant derivative also serves to identify what is considered as the parallel transport of tensors. Moreover, the (right) skew symmetric contribution to the connection is denoted the torsion, a third-order tensor that discriminates symmetric from non-symmetric manifolds. Along these lines also the anholonomic object is introduced as a third-order tensor that is related to the concept of the dislocation density in the sequel. Finally, the fourth-order curvature tensor is derived from considering the parallel transport of vectors and covectors along infinitesimal circuits in the manifold. Various aspects of the curvature tensor, in particular the so-called Bianchi identities and the Ricci tensors, are carefully discussed. For the sake of transparency the exposition follows mainly an index notation, however, in order to relate to more modern representations, the main concepts are also given in a coordinate-free invariant formulation and in terms of elements of exterior calculus.

Paul Steinmann

Geometry on Metric Manifolds

The

metric

is an important tensorial object that introduces more structure into a (differential) manifold. The metric coefficients allow for example to determine the length of parameter curves in the manifold and make it possible to relate corresponding co- and contravariant objects defined on the manifold. Thus the

inner product

of either co- or contravariant quantities and in particular the angle between two vectors or (covectors) may be computed. Adopting the Ricci postulate of vanishing covariant derivative of the metric coefficients results in a decomposition of the fully covariant connection into its Riemann part, that depends exclusively on the metric, and the contortion, that depends exclusively on the torsion. Moreover, in this case the length of vectors and their angle with respect to geodesics are preserved upon parallel transport. Based on the metric coefficients the fully covariant curvature tensor, displaying left and right skew symmetry, is introduced and the corresponding modifications of the Bianchi identities are highlighted. Likewise the metric allows to introduce the mixed-variant Ricci tensor together with corresponding identities. Finally the metric enables to compute the Ricci scalar from the previous curvature tensors. The particular case of a symmetric, metrically connected manifold represents a Riemann geometry with associated Riemann curvature tensor as fundamental for example in the Einstein Theory of General Relativity/Gravitation. A generalization is obtained by allowing for metric manifolds with non-vanishing covariant derivative of its metric coefficients. The presence of the non-metricity is then subsequently reflected by extra terms in the decomposition of the connection and in the explicit representation of the curvature tensor, the Ricci tensor, the Ricci scalar, and the corresponding Bianchi identities for the various curvature quantities.

Paul Steinmann

Representations in Four-, Three-, Two-Space

The main results for metric and metrically connected manifolds of arbitrary dimension as presented in the previous chapter shall now be specialized and detailed to the cases of four-, three- and two-dimensional manifolds. (The case of one-dimensional manifolds, i.e. curves is not of relevance here, since the curvature tensor, that measures solely the internal geometry of a manifold, degenerates to zero in one dimension.) The above manifolds will be denoted as four-, three- and two-space, respectively, in the sequel. Four-, three- and two-spaces have most relevant applications such as gravitation, continuum physics (mechanics/electromagnetism) and surfaces. It is interesting to observe the similarities of the different cases that are only fully appreciated by such a deductive approach. For these concrete cases it proves convenient to exploit skew symmetries in the torsion tensor, the contortion tensor and the curvature tensor by introducing the dual torsion tensor, the dual contortion tensor, and the double-dual curvature tensor along with the corresponding double-dual Ricci tensor and scalar. Moreover, the relation between the dual torsion and contortion tensors as well as between the doubledual curvature tensor and the Einstein tensor will be highlighted. Finally, the Bianchi identities will be re-examined in terms of the dual quantities.

Paul Steinmann

Nonlinear Continuum Mechanics

Frontmatter

Continuum Kinematics

A continuum body is always embedded in three-dimensional Euclidean space, however its kinematics may be described in either rectilinear (Cartesian) or curvilinear coordinates. Expressed in curvilinear coordinates the differential geometry of flat Euclidean space is captured by the Christoffel symbols that take the role of a symmetric and integrable metric connection with associated zero curvature tensor. The position of a physical point together with the distortion, the double-distortion, and the triple-distortion are essential quantities to describe the continuum kinematics. Thus their representation is carefully elaborated. The kinematics of a continuum body are further characterized by an embedded non-Euclidean manifold. The embedded manifold is represented by a connection that additively decomposes into an integrable and a non-integrable contribution. The integrability conditions for the distortion and the double-distortion prove to be governed by the anholonomic object, the torsion, the curvature, and the non-metricity of the embedded manifold. To describe the deviation from integrability four defect density tensors, i.e. the primary and the secondary dislocation density tensors, the disclination density tensor, and the point-defect density tensor are introduced. Various types of continua with defects may be classified based on these defect density tensors.

The chapter also contains a comprehensive account on tensor calculus in Euclidean space in an extended supplement (which also introduces the symbolic notation used extensively throughout Part III).

Paul Steinmann

Elasticity

The previous concepts of differential geometry, in particular aspects related to compatibility, shall be applied to the kinematics of first- and second-order (nonlinear) elasticity. Thereby it shall be noted that both the material and spatial configurations of first- and second-order elasticity ought to be compatible, see Fig. 7.1. Then, two cases may be considered: firstly the material configuration is assumed compatible and the conditions on the deformation for the spatial configuration to also be compatible are sought; secondly the situation is reversed: the spatial configuration is assumed compatible and the conditions on the (inverse) deformation for the material configuration to be compatible are sought. Thereby distinction can be made between the straightforward conditions for the distortion (and double-distortion) being integrable and the more involved conditions for the metric (and double-metric) being integrable. The latter leads to nonlinear (and extended) versions of the famous St-Venant compatibility conditions. Comprehensive accounts on first- and second-order elasticity in Euclidean space are provided for the sake of reference at the end of the chapter in two extended supplements.

Paul Steinmann

Elasto-Plasticity

The previous concepts of differential geometry, in particular aspects related to incompatibility, shall be applied to the kinematics of firstand second-order (nonlinear) elasto-plasticity. Thereby it shall be noted that the intermediate configuration of first- and second-order elasto-plasticity is incompatible, see Fig. 8.1. Then two cases may be considered: firstly the incompatibility of the intermediate configuration is measured based on the nonintegrability of plastic tensorial quantities; secondly the situation is reversed: the incompatibility of the intermediate configuration is measured based on the non-integrability of elastic tensorial quantities. Thereby distinction can be made between straightforward measures in terms of the non-integrable plastic or elastic distortions (and double-distortions) and more involved measures in terms of the non-integrable plastic or elastic (strain) metrics (and doublemetrics). The former leads to various dislocation density tensors, whereas the latter results in various incompatibility density tensors.

Comprehensive accounts on first- and second-order elasto-plasticity in Euclidean space are provided for the sake of reference at the end of the chapter in two extended supplements.

Paul Steinmann

Epilogue

Frontmatter

Integrability and Non-Integrability in a Nutshell

The integrability conditions for the spatial and material configurations in first- and second-order elasticity may be stated either in terms of the distortions and double-distortions or, likewise, in terms of the (strain) metrics and connections (corresponding to the double-metrics/strains). In analogy, the lack of integrability for the intermediate configuration in firstand second-order elasto-plasticity is captured by non-integrability measures either in terms of the plastic or elastic distortions and double-distortions or, likewise, in terms of the plastic or elastic (strain) metrics and connections. It is emphasized that the integrability conditions and non-integrability measures have clear counterparts in differential geometry. This chapter aims in a concise summary of the previously derived relations and in a juxtaposition of the various cases considered.

Paul Steinmann

Backmatter

Weitere Informationen

Premium Partner

in-adhesivesMKVSNeuer Inhalt

BranchenIndex Online

Die B2B-Firmensuche für Industrie und Wirtschaft: Kostenfrei in Firmenprofilen nach Lieferanten, Herstellern, Dienstleistern und Händlern recherchieren.

Whitepaper

- ANZEIGE -

Technisches Interface Design - Beispiele aus der Praxis

Eine gute Theorie besticht nur darin, dass am Ende einer Entwicklung sinnvolle und nutzergerechte Produkte herauskommen. Das Forschungs- und Lehrgebiet Technisches Design legt einen starken Wert auf die direkte Anwendung der am Institut generierten wissenschaftlichen Erkenntnisse. Die grundlegenden und trendunabhängigen Erkenntnisse sind Grundlage und werden in der Produktentwicklung angewendet. Nutzen Sie die Erkenntnisse aus den hier ausführlich dargestellten Praxisbespielen jetzt auch für Ihr Unternehmen.
Jetzt gratis downloaden!

Marktübersichten

Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen. 

Bildnachweise