[Paper Review] The axiomatic deduction of the quadratic Hencky strain energy by Heinrich Hencky

Patrizio Neff, Bernhard Eidel+1 more|arXiv (Cornell University)|Feb 17, 2014

Control and Stability of Dynamical Systems7 references19 citations

TL;DR

This paper presents a faithful English translation and annotation of Heinrich Hencky's original 1928 work on the axiomatic derivation of the quadratic Hencky strain energy. Using a variational approach based on the logarithmic strain tensor and the principle of superposition for finite deformations, Hencky derives a strain energy function that is quadratic in the logarithmic strains, establishing a foundational framework for nonlinear elasticity theory with rotational invariance and consistent stress-strain relations.

ABSTRACT

The introduction of the quadratic Hencky strain energy based on the logarithmic strain tensor log V is a milestone in the development of nonlinear elasticity theory in the first half of the 20th century. Since the original manuscripts are written in German, they are not easily accessible today. However, we believe that the deductive approach taken by Hencky deserves to be rediscovered today. In this work we have gathered parts of the original contributions "Über die Form des Elastizitätsgesetzes bei ideal elastischen Stoffen", "Welche Umstände bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen Körpern?" and "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" which center around this deductive approach. We tried to provide, for the first time, a faithful translation into English. All footnotes are our addition.

Motivation & Objective

• To recover and translate Hencky's original axiomatic derivation of the quadratic Hencky strain energy, which remains inaccessible in German.
• To clarify the physical and mathematical foundations of the quadratic Hencky energy model in nonlinear elasticity.
• To demonstrate the consistency of the strain energy function with the principle of superposition under finite deformations.
• To provide a rigorous, invariant-based derivation of the strain energy using the logarithmic strain tensor and deformation velocity fields.
• To establish the theoretical basis for the quadratic Hencky energy as a minimal, physically consistent model for finite elasticity.

Proposed method

• Derives the strain energy function using a variational principle applied to the deformation velocity tensor and stress rate, ensuring rotational invariance.
• Introduces the logarithmic strain tensor via the principal stretches, defining the strain measure as $ e_i = 1 - 1/λ_i $, where $ \lambda_i $ are the principal stretches.
• Applies the principle of superposition to finite deformations by considering the time derivative of the stress tensor in a rotating material frame, incorporating Coriolis-like terms via the spin tensor.
• Uses a power series expansion of the stress tensor in terms of the strain tensor to derive the linearized law of superposition for small perturbations.
• Derives the key equation (8b) relating the rate of change of stress to the deformation velocity tensor and the strain tensor, valid for arbitrary finite deformations.
• Eliminates the strain tensor from the stress evolution equation using a series expansion (8c), yielding a closed-form expression for the strain energy in terms of stress invariants.

Experimental results

Research questions

• RQ1 How can the quadratic Hencky strain energy be derived from first principles without relying on empirical fitting?
• RQ2 What is the correct mathematical form of the strain energy function that ensures rotational invariance and consistency with the superposition principle under finite deformations?
• RQ3 Why is the logarithmic strain tensor the appropriate measure for finite elasticity, and how does it relate to the physical behavior of ideal elastic materials?
• RQ4 Under what conditions does a linear superposition law for stress and deformation hold in the context of finite elasticity?
• RQ5 How can the strain energy function be expressed as a power series in the stress tensor, and what is the significance of the quadratic term?

Key findings

• The strain energy function is derived as a quadratic function of the logarithmic strain tensor, ensuring rotational invariance and consistency with the superposition principle.
• The law of superposition for finite deformations is derived in the form of a power series (equation 8b), valid for arbitrary deformation sizes.
• The strain tensor $ e_{m n} $ is expressed as a series in the stress tensor $ \sigma'_{m n} $, with the quadratic term being the dominant contribution for small deformations.
• The derivation shows that the quadratic Hencky energy model is the minimal consistent model that satisfies the superposition principle and rotational invariance.
• The stress rate equation (9a) and (9b) reduce to the classical linearized form under small perturbations, confirming consistency with infinitesimal elasticity.
• The method eliminates the displacement field entirely from the final equations, showing that the stress-strain relation depends only on the deformation velocity and rotation fields.