Historic development of elasticity laws with emphasis on the logarithmic strain tensor

Armand Imbert (1850-1922) seems to be the first to use a logarithmic force-elongation law. Assuming proportionality between current length and length increment due to applied incremental load, on page 53 Imbert arrives at the conclusion (in modern notation) t_1 = E log λ, where λ is the stretch ratio in uniaxial tension of a vulcanized rubber sheet and t_1 is the corresponding nominal traction (with respect to the reference area). Here E is the initial Young’s modulus.
The developpment of Imbert is restricted to the one-dimensional setting in tension.

  • A. Imbert. Recherches théoriques et expérimentales sur l'élasticité du caoutchouc / par M. A. Imbert / Lyon : impr. de P. Goyard , (1880) pdf

Karl Ernst Hartig (1836-1900) uses Imbert's uniaxial logarithmic law to fit vulcanized rubber data in tension and compression.

  • E. Hartig. Der Elastizitätsmodul des geraden Stabes als Funktion der spezifischen Beanspruchung. Z. Civilingenieur (1893) p.120 pdf

George Ferdinand Becker (1847-1919) was a famous geologist. In 1893 he axiomatically deduced a constitutive law for isotropic elastic substances connecting Biot stress (the initial loads) and logarithmic strain based on a superposition principle for principle forces.

  • G. F. Becker. Finite homogeneous strain, flow and rupture of rocks. Bulletin of the Geological Society of America 4 (1893), pp. 13-90 pdf

  • G. F. Becker. The Finite Elastic Stress-Strain Funktion. American Journal of Science (1893) Nr. 46, pp. 337-356 original newlatexversion

  • G. F. Becker. Die mathematische Beziehung zwischen endlichen elastischen Deformationen und Kräften. German review by Dr. Lübeck. Beiblätter zu den Wiedemannschen Annalen (Annalen der Physik) (1894), p. 515 pdf

  • R. Mehmke. Zum Gesetz der elastischen Dehnungen. Zeitschrift für Mathematik u. Physik 42 (1897), Nr. 6, pp. 327-338 pdf

  • G. F. Becker. Experiments on Schistosity and Slaty Cleavage (1904) pdf

In many textbooks and articles one encounters statements like the following: "The concept of natural strain has been introduced (first suggested) by P. Ludwik, (1909)."
In that small booklet, mention of the natural strain log λ is made but at one place (p. 17) with no further explanation or motivation given.

  • P. Ludwik. Elemente der technologischen Mechanik. (1909) pdf

Heinrich Hencky (1885-1951) introduced the logarithmic strain tensor in 1928 in an axiomatic way based finally on a principle of superposition for coaxial Kirchhoff stresses.

  • H. Hencky. Über die Beziehungen der Philosophie des Als Ob zur mathematischen Naturbeschreibung. Annalen der Philosophie (1921/1923), Vol. 3, Issue 1, pp. 236-245 pdf

  • P. Neff, B. Eidel, R. Martin (translators). The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky.  pdf

  •  L. Prandtl. Elastisch bestimmte und elastisch unbestimmte Systeme. Festschrift zum 70. Geburtstag August Föppls (1924);  pp. 52-61, Berlin, Springer (1924) pdf

  • H. Hencky. Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen. Zeitschrift für technische Physik (1928), Nr. 6 pdf

  • H. Hencky. Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern? Zeitschrift für technische Physik (1929), Nr. 9 pdf

  • H. Hencky. Das Superpositionsgesetz eines endlich deformierten relaxationsfähigen elastischen Kontinuums und seine Bedeutung für eine exakte Ableitung der Gleichungen für die zähe Flüssigkeit in der Eulerschen Form. Annalen der Physik 5.Folge (1929) Band 2, Heft 6 pdf

  • H. Hencky. The law of elasticity for isotropic and quasi-isotropic substances by finite deformations. Journal of Rheology (1931) pdf

  • H. Hencky. The elastic behavior of vulcanized rubber. Rubber Chem. Technol. 6 (1933), pp. 217-224 pdf

  • F. D. Murnaghan. The compressibility of solids under extreme pressures. Theodore von Karman Anniversary Volume (1941) (log-strain p.127). pdf

  • F. D. Murnaghan. The compressibility of media under extreme pressures. Nat. acad. Sci. 30 (1944), pp.244-247 pdf

  • H. Hencky. Über die Berücksichtigung der Schubverzerrung in ebenen Platten. Ingenieur-Archiv (1951) Vol. 16, Issue 1, pp. 72-76 pdf

  • H. Hencky. Neuere Verfahren in der Festigkeitslehre. R. Oldenbourg, München (1947) pdf

  • Heinrich Hencky. Die Bewegungsgleichungen beim nichtstationären Fließen plastischer Massen. Zeitschrift für angewandte Mathematik und Mechanik, Vorträge der Dresdner Tagung, Band 5, Heft 2, 1925 pdf

  • R. Tanner, E. Tanner. Heinrich Hencky: a rheological pioneer. Rheol Acta (2003) Vol. 42: pp.93-101 pdf

Hans Richter (1912-1978) was an applied mathematican, who would later become a professor of probability theory in Munich. In his early years he amply deduced axiomatically the special role played by logarithmic strains. His works are referenced by Clifford Truesdell. In his works he also postulates a superposition principle for coaxial stretches, giving rise to the logarithmic strain tensor. Probably Richter was the first to derive the relation τ = DlogV W(logV). He is also the first to establish the voumetric-isochoric split together with the corresponding properties of the Cauchy stress, 15 years before Flory.

  • H. Richter. Das isotrope Elastizitätsgesetz. Zeitschrift für angewandte Mathematik und Mechanik (1948) Band 28, Heft 7/8: Originaltext ,Text neu gesetzt

  • H. Richter. Verzerrungstensor, Verzerrungsdeviator und Spannungstensor bei endlichen Formänderungen. Zeitschrift für angewandte Mathematik und Mechanik (1949) Band 29, Heft 3 pdf newlatexversion - Review by R. Moufang. Zentralblatt (1949), pp. 426-427 pdf 

  • H. Richter. Zum Logarithmus einer Matrix. (1950) pdf

  • H. Richter. Zur Elastizitätstheorie endlicher Verformungen. (1952) pdf

The reference to Imbert seems to first appear in a survey article of C. Truesdell in 1952, where he introduces the "Hencky's logarithmic measure". Typical for the schoolmaster Truesdell is the remark that: "...while Hencky himself does not give a systematic treatment, we may introduce the tensors H=1/2 log C...". Truesdell certainly never read Imbert, since he refers the reader to Mehmke, who himself refers to Hartig's paper. The reference to Ludwik can be found in Hencky's 1931 paper.

  • Claude Vallée. Lois de comportement élastique isotropes en grandes déformations (Vallée derives the relation τ = DlogV W(logV), as did Richter before him). Int. J. Engng Sci. (1978) Vol. 16, pp. 451-457 pdf

  •  J. Moreau. Lois d'Elasticite en grande Deformation (Moreau rederives the relation τ = DlogV W(logV)) . Seminaire d'Analyse convexe (1979) pdf

  • J. W. Hutchinson, K. W. Neale. Finite strain J_2 deformation theory. (1980) pdf (Loss of ellipticity for |dev log V|^N, 0<N≤1)

  • J. B. Leblond. A constitutive inequality for hyperelastic materials in finite strain. Eur. J. Mech., A/Solids 11 (1992) No.4, pp. 447-466 pdf


Historic Documents

  • W. C. Röntgen. Ueber das Verhältnis der Quercontraction zur Längendilatation bei Kautschuk (Röntgen defines a nonlinear Poisson ratio based on logarithmic elongations). Annalen der Physik, Vol. 235, Issue 12 (1876), pp. 601-616 pdf

  • G. Grioli. Una proprietà di minimo nella cinematica delle deformazioni finite. (1940) pdf

  • M. Reiner. Elasticity Beyond the Elastic Limit. American Journal of Mathematics, Vol. 70, No. 2 (1948), pp. 433-446 pdf

  • Patrizio Neff, Johannes Lankeit, Angela Madeo. On Grioli’s minimum property and its relation to Cauchy’s polar decomposition. International Journal of Engineering Science; 2014 Vol. 80, pp. 209–217, arXiv:1310.7826  abstract pdf


Experimental data

Experimental data for the biaxial deformation of vulcanized rubber, measured in 1944 by L.R.G. Treloar and in 1975 by Treloar and D.F. Jones and provided by courtesy of R. Ogden, in the form of tab-separated ASCII-files.


Prof. Dr. rer. nat. habil.
Patrizio Neff

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