Sommario
Si postula una corrispondenza biunivoca per due continui coordinati. Un continuo viene considerato come lo stato inizialmente non deformato di un continuo attualmente deformato.
I due continui sono trasformazioni ortogonali uno dell'altro. La radice quadrata della metrica quadratica, quando la relativa diade di dilatazione autoconiugata venga espressa nella sua forma principale dà la forma matematicamente lineare alla analisi.
La diade coniugata\(\mathop T\limits^ = \) viene espresse come\(\mathop T\limits^ = \)=grad grad P in termini di una funzione potenziale scalare P. La continuità fisica e matematica della diade di deformazione è assicurata da rot\(\mathop T\limits^ = \)=0.
Una analisi tetravettoriale viene sviluppata da analisi di quinternioni tetravettoriali. L'analisi tetravettoriale è della stessa forma dell'analisi trivettoriale che viene vsiluppata dalla analisi quaternionica trivettoriale di Hamilton.
Summary
One-one correspondence is postulated for two coordinate continua. One continuum is regarded as the initially undeformed state of a currently deformed continuum. The two continua are orthogonal trasformations each to the other. The square root of the quadratic metric, when the appropriate self-conjugate stretch dyadic is expressed in its principal form, gives the mathematically linear form to the analysis. The self-conjugate dyadic\(\mathop T\limits^ = \) is expressed as\(\mathop T\limits^ = \)=grad grad P in terms of a scalar potential function P. The physical and mathematical continuity of the strain dyadic is ensured by curl\(\mathop T\limits^ = \)=0.
A 4-vector analysis is evolved from a 4-vector quinternion analysis. The 4-vector analysis is of the same form as the usual 3-vector analysis that evolved from Hamilton's quaternion analysis.
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Abbreviations
- \(\bar X, \bar Y\) :
-
Position vectors of corresponding typical points in corresponding continua
- \(\bar X, \bar R\) :
-
Position vectors of corresponding points\(\bar X\) and what was\(\bar Y\) when whole-continuum rotation is zero
- \(\mathop \nabla \limits^ - \) :
-
Vector gradient operator (or grad).\(\mathop \nabla \limits^ - X,\mathop \nabla \limits^ - Y,\mathop \nabla \limits^ - R\) when defined relative to the appropriate continua
- \(\bar \nabla ^c \) :
-
Conjugate vector gradient operator (or gradc)
- \(\mathop A\limits^ = , \mathop B\limits^ = , \mathop G\limits^ = \) :
-
Self-conjugate deformation dyadics of corresponding points relative to\(\bar X\)-continuum,\(\bar Y\)-continuum,\(\bar R\)-continuum respectively
- \(\left. \begin{array}{l} d\bar X = \bar e_X dX \\ d\bar Y = \bar e_Y dY \\ d\bar R = \bar e_R dR \\ \end{array} \right\}\) :
-
Differential position vectors, directions and moduli from corresponding points in the corresponding continua
- \(\bar c_x \) :
-
General directions for orthogonal coordinates in a continuumly-fixed reference
- \(\bar x_t ,\bar y_t ,\bar r_t \) :
-
Principal directions of\(\mathop A\limits^ = , \mathop B\limits^ = , \mathop G\limits^ = \) respectively
- \(\left. \begin{array}{l} \bar Z^x = d\bar X/dY \\ \bar Z^v = d\bar Y/dX \\ \end{array} \right\}\) :
-
Convenient abbreviations
- \(\bar U\) :
-
Displacement of a typical point from\(\bar X\) to\(\bar Y\) relative to a continuumly-fixed reference external to the corresponding continua
- \(\bar O_X , \bar O_Y \) :
-
Particular corresponding points in the corresponding continua. Denoted by\(\bar O\) when the two points are brought to coincidence for zero whole-continuum translation
- \(\bar E_X , \bar E_Y ,\bar E_R \) :
-
Denote briefly corresponding deformation ellipsoids at typical corresponding points in the corresponding continua.\(\bar E_X (\bar O), \bar E_Y (\bar O),\bar E_R (\bar O)\) particularises the ellipsoids to be at\(\bar O\)
- \(\bar D\) :
-
Straining-displacement of a typical point\(\bar X\) to its corresponding position\(\bar R\)
- \(\bar D*\) :
-
Lamellar relative-displacement corresponding to straining-displacement\(\bar D\)
- \(\bar X*\) :
-
Relative initial position of point\(\bar R\) corresponding to relative-displacement\(\bar D*\)
- \(\bar C\) :
-
Convection-displacement of point\(\bar R\) to pass from\(\bar X\) to\(\bar X*\) and from\(\bar D\) to\(\bar D*\). Otherwise,\(\bar C\) allows for\(\bar x_i (\bar X)\) rotating into\(\bar r_i (\bar R)\) du to the strain field established between\(\bar O\) and\(\bar R\)
- A Emphasis>2 i ,G Emphasis>2 i :
-
Scalar coefficients of the self-conjugate dyads in the dyadics\(\bar \bar A\),G respectively
- f, g :
-
Unit stretch in the scalar line elementsdX, dR respectively
- \(\bar \bar S, \bar \bar T\) :
-
Corresponding self-conjugate strain dyadics in\(\bar X\)-continuum,\(\bar R\)-continuum respectively
- \(\left. \begin{array}{l} f_t ,g_t \\ S_t ,T_t \\ \end{array} \right\}\) :
-
Scalar coefficients for the principal directions of\(\bar \bar S, \bar \bar T\) respectively
- \(\left. \begin{array}{l} S_{xx} , S_{xv} \\ T_{xx} , T_{xv} \\ \end{array} \right\}\) :
-
Normal and shear strains (scalar) components of\(\bar \bar S, \bar \bar T\) respectively for general directions
- \(\bar \bar \Theta \) :
-
Versor operator to rotate a vector through angle Θ about the direction\(\bar \theta \)
- δ :
-
Increment operator
- \(\bar \nabla \) ≡ curl:
-
Curl operator defined with respect to\(\bar R\)-continuum in art. 7 onwards
- P :
-
Relative-displacement potential function. Strain dyadic potential function
- t :
-
Time\(3 - continuum | 4 - continuum\)
- Λ, ·, ×:
-
Quaternion (& quinternion), scalar, vector product of vectors signs respectively. Operationally Λ op=−·+×
- x, y, z, w :
-
Subscripts denoting orthogonal directions and following the sequences\(\left. \begin{array}{l} x = 1 \\ y = 2 \\ z = 3 \\ \end{array} \right|\left. { \begin{array}{*{20}c} 2 \\ 3 \\ 1 \\\end{array}} \right| \begin{array}{*{20}c} 3 \\ 1 \\ 2 \\\end{array} \left| {\left. {\begin{array}{*{20}c} {x = 1} \\ {y = 2} \\ {z = 3} \\ {w = 4} \\\end{array}} \right|\left. {\begin{array}{*{20}c} 2 \\ 3 \\ 4 \\ 1 \\\end{array}} \right|} \right.\left. {\begin{array}{*{20}c} 3 \\ 4 \\ 1 \\ 2 \\\end{array}} \right|\left. {\begin{array}{*{20}c} 4 \\ 1 \\ 2 \\ 3 \\\end{array}} \right|\)
- [\(\bar f\), g]:
-
Angle between directions\(\bar f\) and\(\bar g\).
References
K. H. Swainger,Analysis of deformation, Chapman & Hall, London, Macmillan, U.S.A., Vol. IMathematical Theory, 1954. 1.1 p. 6; 1.2 pp. 90–93; 1.3 p. 13; 1.4 pp. 234–236; 1.5 p. 257.
K. H. Swainger,Analysis of deformation, Chapman & Hall, London; Macmillan, U.S.A., Vol. V,Relativity, In preparation.
K. H. Swainger,Analysis of deformation, Chapman & Hall, London; Macmillan, U.S.A., Vol. III,Fluidity, 1956. 3.1 pp. 67–68, 83, 85; 3.2 pp. 65, 101.
L. I. Sedov,Foundations of the non-linear mechanics of continua, Translation from the Russian by Rebecca Schoenfeld-Reiner; edited by J. E. Adkins & A. J. M. Spencer, Published by Pergamon Press, Oxford, 1966.
A. E. Green andW. Zerna,Theoretical elasticity, Oxford Univ. Press, 1954.
T. Levi-Civita (Edited by E. Persico),The absolute differential calculus, Blackie & Son., London, 1947.
P. G. Bergmann (Foreword by Albert Einstein),Introduction to the theory of relativity, Prentice-Hall, U.S.A., 1942, Reprint 1960.
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A generalisation that, when particularised, applies to the finite or infinitesimal straining of elastic bodies, incremental straining of plastic or fluidic bodies in 3-dimensional continua and to the space-time continuum as a 4-dimensional continuum.
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Swainger, K.H. The mathematically linear analysis of strain in continua. Meccanica 3, 283–291 (1968). https://doi.org/10.1007/BF02186948
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DOI: https://doi.org/10.1007/BF02186948