On conservation laws in elastodynamics

• Published in: International journal of solids and structures Vol. 41; no. 13; pp. 3595 - 3606
• Main Authors: Kienzler, R., Herrmann, G.
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.06.2004
• Subjects: 4 × 4 Formalism; Balance law; Conservation law; Elastic bar; Elastodynamics; Eulerian description; Lagrangian description; Lagrangian description; Conservation law; 4 × 4 Formalism; Eulerian description; Balance law; Elastodynamics; Elastic bar
• ISSN: 0020-7683; 1879-2146
• DOI: 10.1016/j.ijsolstr.2004.01.018

The objective of this investigation is the establishment of governing balance and conservation laws in elastodynamics. The feature of the approach employed here consists in placing time on the same level as the space coordinates, as is done in the theory of relativity, i.e., pursuing a 4 × 4 formalism. Both the Lagrangian and the Eulerian descriptions of the postulated Lagrangian function are formulated. The Euler–Lagrange equations in each of the two descriptions are discussed, as well as the results of the application of the gradient, divergence and curl. The latter two operations are made to act on the product of coordinates and the Lagrangian function, i.e., a four-vector. In this manner a variety of balance and conservation laws are derived, partly well known and partly seemingly novel. In each case the general results for elastodynamics are illustrated for the simple case of a linearly elastic bar.