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.
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Notes
- 1.
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
- 2.
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:
$$\begin{aligned} e_{ijk} e_{ijk} = 6\,. \end{aligned}$$ - 3.
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.
- 4.
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),
$$\begin{aligned} (\mathrm{curl \,}(\nabla {{\varvec{u}}}^T))_{ij}=e_{ipq}(u_{q,j}),_p=e_{ipq}u_{q,jp}=(e_{ipq}u_{q,p}),_{j}=2\,w_{i,j}. \end{aligned}$$ - 5.
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.
- 6.
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}.
- 7.
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].
- 8.
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.
- 9.
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, \((\mathrm{div \,}{{\varvec{V}}})_i=V_{ij,j}\).
- 10.
Here, \(\, {{\varvec{S}}}({{\varvec{u}}}):=2\mu {{\varvec{E}}}({{\varvec{u}}})+\lambda (\mathrm{tr \,}{{\varvec{E}}}({{\varvec{u}}})){{\varvec{I}}}\,.\)
- 11.
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,
$$\begin{aligned} E_{11}=\frac{1+\nu }{E} S_{11}-\frac{\nu }{1-2\nu } \frac{(1+\nu )(1-2\nu )}{E}(S_{11}+S_{22})=\frac{1+\nu }{E} \big ( S_{11}-\nu (S_{11}+S_{22})\big )\,\,\mathrm etc. \end{aligned}$$
References
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
Meriam JL, Kraige LG (2003) Engineering mechanics, vol 1. Wiley, New York
Podio-Guidugli P (2000) A primer in elasticity. Kluwer, Dordrecht
Podio-Guidugli P (2004) Examples of concentrated contact interactions in simple bodies. J Elasticity 75:167–186
Podio-Guidugli P (2005) On concentrated interactions. In: Dal Maso G, De Simone A, Tomarelli F (eds) Variational Problems in Materials Science. Birkhäuser, Basel
Podio-Guidugli P, Schuricht F (2012) Concentrated actions on plane cuspidate bodies. J Elasticity 106:107–114
Schuricht F (2007) A new mathematical foundation for contact interactions in continuum physics. Arch Rat Mech Anal 184:495–551
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
Sokolnikoff IS (1956) Mathematical theory of elasticity. McGraw-Hill, New York
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Podio-Guidugli, P., Favata, A. (2014). Elements of Linear Elasticity. In: Elasticity for Geotechnicians. Solid Mechanics and Its Applications, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-01258-2_2
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