nach oben

Erschienen in:

2014 | OriginalPaper | Buchkapitel

2. Elements of Linear Elasticity

verfasst von : P. Podio-Guidugli, A. Favata

Erschienen in: Elasticity for Geotechnicians

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

In this chapter we give a short and yet fairly complete exposition of the elemental features of classic elasticity having relevance to our subject matters.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel One-Dimensional Paradigms

Nächstes Kapitel Geometric and Analytic Tools

For more information about the role of ${{\varvec{E}}}$ and, more generally, about the local analysis, both exact and approximate, of a deformation see [11], Chap. I

In terms of the vectors composing the orthonormal Cartesian basis we chose, Kronecker’s symbol $\delta _{ij}$ is given by

$$\begin{aligned} \delta _{ij}:={{\varvec{e}}}_i \cdot {{\varvec{e}}}_j\!, \end{aligned}$$

whence

$$\begin{aligned} \delta _{ij} =\left\{ \begin{array}{lr} 1 \,\mathrm \ if\ \;&{}i=j \\ 0\, \mathrm \ if \ \;&{}i\ne j \end{array}\right. \!\!; \end{aligned}$$

moreover, relation

$$\begin{aligned} e_{ijk}:={{\varvec{e}}}_i\times {{\varvec{e}}}_j\cdot {{\varvec{e}}}_k \end{aligned}$$

defines Ricci’s symbol, so that

$$\begin{aligned} e_{ijk} = \left\{ \begin{array}{l} +1 \quad \text {if all indices i, j, k are different and, in addition,}\\ \qquad \;\;\text {their sequence is an even-class permutation of 1, 2, 3};\\ \;\mathrm \; \,0 \quad \,\text {if at least two of the indices i, j, k are equal;}\\ -1 \quad \text {if all indices i, j, k are different and, in addition,}\\ \qquad \;\;\text {their sequence is an odd-class permutation of 1, 2, 3}. \end{array} \right. \end{aligned}$$

Ricci’s and Kronecker’s symbols are linked by the following relation:

$$\begin{aligned} e_{ijk} e_{lmk} = \delta _{il}\delta _{jm} - \delta _{im}\delta _{jl}. \end{aligned}$$

(2.5)

By repeated saturation of pairs of free indices, two easy and often useful consequences of (2.5) are obtained:

(i) formal multiplication of both sides by $\delta _{jm}$ yields:

$$\begin{aligned} e_{ijk} e_{ljk} = 2\,\delta _{il}\,; \end{aligned}$$

(ii) one more saturation gives:

$$\begin{aligned} e_{ijk} e_{ijk} = 6\,. \end{aligned}$$

Recall that the symbol $\otimes $ signifies dyadic product, a notion introduced in the first footnote of Sect. 1.3; the second-order tensor ${{\varvec{a}}}\otimes {{\varvec{b}}}$ is defined by specifying its linear action on vectors.

That $\mathrm{curl \,}(\nabla {{\varvec{u}}})= {{\varvec{0}}}$ follows from the definitions of (the two involved operators and) Ricci symbol:

$$\begin{aligned} (\mathrm{curl \,}(\nabla {{\varvec{u}}}))_{ij}=e_{ipq}(\nabla {{\varvec{u}}})_{jq,p}=e_{ipq}(u_{j,q}),_p=e_{ipq}u_{j,qp}=0. \end{aligned}$$

Furthermore, in view of (2.4),

$$\begin{aligned} (\mathrm{curl \,}(\nabla {{\varvec{u}}}^T))_{ij}=e_{ipq}(u_{q,j}),_p=e_{ipq}u_{q,jp}=(e_{ipq}u_{q,p}),_{j}=2\,w_{i,j}. \end{aligned}$$

For a proof of this result, which is due to the great Italian elasticist Eugenio Beltrami (1835–1900), who established it in 1889, see [6], Sect. 14, where various other results included in this section are also proved.

When Greek indices are used, it is understood that they take the values 1 and 2; the range of Latin indices is the set {1, 2, 3}.

It appears that the concept of diffused contact interactions between internal adjacent body parts begun to condensate in Cauchy’s mind on the basis of a similarity with standard examples of diffused contact loads exerted on a body by an environment of a different nature, such as the hydrostatic pressure of a fluid on an immersed solid [3]. Cauchy’s model of internal contact interactions has been applied without changes to contact interactions of a body with its exterior, with the stress-vector mapping accounting for both. An implicit drawback of this practice is that no difference is made between geometrical surfaces obtained by ideal cuttings and fabricated surfaces obtained by actual cuttings [4]; moreover, the issue of boundary compatibility of a (body,environment) pair is completely overlooked [1, 2].

The construction of an interaction theory general enough to allow for concentrated contact interactions between adjacent body parts has been undertaken by Schuricht [15, 16]; among the intriguing features of such a theory is the rethinking it involves of the body-part notion. In [14], examples are given of interactions in cuspidate bodies that concentrate at the cusp point, regarded as a body part.

The laplacian of a vector field ${\varvec{v}}$ is the vector field that obtains by taking the divergence of the gradient of ${\varvec{v}}$:

$$\begin{aligned} \varDelta {\varvec{v}}=\mathrm{div \,}(\nabla {\varvec{v}}); \end{aligned}$$

its Cartesian components have the form just shown because $(\nabla {\varvec{v}})_{ij}=v_{i,j}$ and because, for ${{\varvec{V}}}$ a second-order tensor field, $(\mathrm{div \,}{{\varvec{V}}})_i=V_{ij,j}$.

Here, $\, {{\varvec{S}}}({{\varvec{u}}}):=2\mu {{\varvec{E}}}({{\varvec{u}}})+\lambda (\mathrm{tr \,}{{\varvec{E}}}({{\varvec{u}}})){{\varvec{I}}}\,.$

For example, let us show how the first of (2.54) is arrived at: from (2.33)$_{1,2}$ we have that

$$\begin{aligned} S_{11}=\frac{E}{1+\nu } \Big ( E_{11} + \frac{\nu }{1-2\nu }(E_{11}+E_{22} )\Big ), \quad S_{11}+S_{22}= \frac{ E}{(1+\nu )(1-2\nu )}(E_{11}+E_{22}); \end{aligned}$$

consequently,

$$\begin{aligned} E_{11}=\frac{1+\nu }{E} S_{11}-\frac{\nu }{1-2\nu } \frac{(1+\nu )(1-2\nu )}{E}(S_{11}+S_{22})=\frac{1+\nu }{E} \big ( S_{11}-\nu (S_{11}+S_{22})\big )\,\,\mathrm etc. \end{aligned}$$

Capriz G, Podio-Guidugli P (2004) Whence the boundary conditions in modern continuum physics? In: Capriz G, Podio-Guidugli P (eds) Atti dei Convegni Lincei N. 210. Accademia dei Lincei, Roma, pp 19–42.

Carillo S, Podio-Guidugli P, Vergara Caffarelli G (2002) Second-order surface potentials in finite elasticity. In: Brocato M, Podio-Guidugli P (eds) Rational Continua. Classical and New, Springer Italia, Milano, pp 19–38

Cauchy AL (1823) Recherches sur l’équilibre et le mouvement intérieur des corps solides ou fluides, élastiques ou non élastiques. Bull Soc Philom Paris 9–13:300–304

DiCarlo A (2004) Actual surfaces versus virtual cuts. In: Capriz G, Podio-Guidugli P (eds) Atti dei Convegni Lincei N. 210. Accademia dei Lincei, Roma, pp 97–113.

Gladwell GML (1980) Contact problems in the classsical theory of elasticity. Sijthoff and Noordhoff, Alphen aan den Rijn

Gurtin ME (1972) The Linear theory of elasticity. In: Flügge S (ed) Handbuch der Physik, vol. 2. Springer, Berlin

Gurtin ME (1981) An introduction to continuum mechanics. Academic Press, Boston

Love AEH (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge

Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice-Hall, Englewood Cliffs

10.

Meriam JL, Kraige LG (2003) Engineering mechanics, vol 1. Wiley, New York

11.

Podio-Guidugli P (2000) A primer in elasticity. Kluwer, Dordrecht

12.

Podio-Guidugli P (2004) Examples of concentrated contact interactions in simple bodies. J Elasticity 75:167–186MathSciNetCrossRefMATH

13.

Podio-Guidugli P (2005) On concentrated interactions. In: Dal Maso G, De Simone A, Tomarelli F (eds) Variational Problems in Materials Science. Birkhäuser, Basel

14.

Podio-Guidugli P, Schuricht F (2012) Concentrated actions on plane cuspidate bodies. J Elasticity 106:107–114MathSciNetCrossRefMATH

15.

Schuricht F (2007) A new mathematical foundation for contact interactions in continuum physics. Arch Rat Mech Anal 184:495–551MathSciNetCrossRefMATH

16.

Schuricht F (2007) Interactions in continuum physics. In: Silhavý M (ed) Mathematical modeling of bodies with complicated bulk and boundary behavior. Quaderni di Matematica 20:169–196

17.

Sokolnikoff IS (1956) Mathematical theory of elasticity. McGraw-Hill, New York

Titel: Elements of Linear Elasticity
verfasst von: P. Podio-Guidugli
A. Favata
Verlag: Springer International Publishing
Buch: Elasticity for Geotechnicians
Print ISBN: 978-3-319-01257-5

Electronic ISBN: 978-3-319-01258-2

Copyright-Jahr: 2014
DOI: https://doi.org/10.1007/978-3-319-01258-2_2

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"