On the origins of the idea of the multiplicative decomposition of the deformation gradient

The objective of this short note is to trace back the major contributions that led to the multiplicative decomposition of the deformation gradient in finite plasticity, nonlinear thermoelasticity, and growth mechanics. In the 1940s, Eckart in the US and Kondo in Japan independently paved the road to the formulation of a nonlinear theory capable of modeling anelastic phenomena. As opposed to assuming, for a given body, the existence of a global stress-free configuration (the “principle of relaxability-in-the-large” according to Eckart) that the body takes whenever it is completely relaxed, Eckart [1] suggested an alternative framework for anelasticity based on what he called “relaxability-in-the-small”. He conceptually constructed a local stress-free “fragmented” state following a local relaxation of the reference configuration by “cutting out” a “small bit of matter” around every material point and letting it relax independently of the remainder of the body. He also asserted that such a construction should be accompanied by an elastic deformation to ensure that the body keeps its structural integrity. This is nothing but the decomposition of the deformation gradient into an anelastic relaxation, leading to the so-called “intermediate” configuration, followed by the elastic portion of the deformation gradient. Independently of Eckart’s work, Kondo [2] observed that due to plastic deformations, the relaxed state of a body has a non-trivial geometry that is not compatible with that of the Euclidean space. This observation first led him to construct a stress-free configuration as a Riemannian manifold in which a non-vanishing curvature is a measure of the incompatibility of the plastic deformation. Inspired by the works of Cartan [3, 4] on non-trivial holonomy groups, Kondo [5–7] extended his framework to consider the material body as a non-Riemannian space with a non-zero torsion. He used this geometric framework in the context of crystals with geometrical imperfections, e.g. dislocations, and introduced the idea of considering the stress-free state as “an amorphous aggregation” of small pieces of relaxed perfect “crystalline pieces” that he modeled as a non-Riemannian manifold. Further, he interpreted the torsion tensor as a measure of the density of dislocations and initiated the development of a geometric theory of dislocation mechanics. Soon after, further contributions to the nonlinear theory of dislocation mechanics were introduced by Kröner [8, 9] and Bilby et al. [10]. For a review of the interactions between the Japanese (led by Kondo), the British (led by Bilby), and the German (led by Kröner) schools and their contributions, see [11]. It is worth mentioning that Sedov [12] independently realized that a body in plastic deformation can be relaxed in a stress-free intermediate configuration, which he called “a new starting position”, with a changing metric that is generally non-Euclidean. Following the original idea of local relaxation inspired by the pioneering works cited above, the first formal introduction of the multiplicative decomposition of the deformation gradient in finite plasticity appeared in the