[Paper Review] Correct traction boundary conditions in the indeterminate couple stress model

Patrizio Neff, Ionel‐Dumitrel Ghiba+2 more|arXiv (Cornell University)|Apr 2, 2015

Elasticity and Material Modeling14 citations

TL;DR

This paper identifies and corrects the long-recognized inconsistency in traction boundary conditions for the indeterminate couple stress model, demonstrating that the widely used Mindlin-Tiersten boundary conditions are incomplete. By rigorously deriving the full set of consistent traction conditions through variational principles and surface divergence theorems, the authors establish a new, complete framework that ensures energetic conjugacy and resolves longstanding ambiguities in higher-gradient elasticity models.

ABSTRACT

In this paper we consider the Grioli-Koiter-Mindlin-Toupin indeterminatecouple stress model. The main aim is to show that the traction boundary conditions were not yet completely deduced. As it turns out, and to our own surprise, restricting the boundary condition framework from the strain gradient models to the couple stress model does not reduce to Mindlin's set of accepted boundary conditions. We present therefore, for the first time the complete, consistent set of traction boundary conditions.

Motivation & Objective

• To identify the fundamental flaw in the classical traction boundary conditions of the indeterminate couple stress model as formulated by Mindlin and Tiersten.
• To resolve the inconsistency in the energetic conjugacy between kinematic and traction boundary conditions in higher-gradient elasticity.
• To derive a complete, consistent, and energetically conjugate set of traction boundary conditions using variational principles and surface divergence theorems.
• To clarify the physical and mathematical role of the couple stress tensor, particularly its non-necessarily skew-symmetric nature, in contrast to claims in recent controversial literature.
• To provide a rigorous foundation for the application of the indeterminate couple stress model in micro- and nano-scale structural mechanics.

Proposed method

• Derives the correct traction boundary conditions using the principle of virtual work and the surface divergence theorem, ensuring proper energy conjugacy.
• Applies a decomposition of the displacement gradient into normal and tangential components relative to the boundary, enabling independent prescription of kinematic and traction conditions.
• Identifies the missing term in the Mindlin-Tiersten formulation: $-\frac{1}{2}\nabla\left[\operatorname{anti}((1\!\!1-n\otimes n)\cdot\widetilde{m}\cdot n)\cdot(1\!\!1-n\otimes n)\right]:(1\!\!1-n\otimes n)$, which is essential for energetic consistency.
• Validates the new boundary conditions by showing their equivalence to the traction conditions derived from a more general second gradient elasticity model with third-order moment tensors.
• Uses index notation and tensor calculus, including the Levi-Civita symbol and the anti-operator, to rigorously express the curvature energy and stress tensors.
• Analyzes the role of the boundary curve $\partial\Gamma$ and introduces a jump condition for the anti-symmetric part of the moment tensor across discontinuities, ensuring consistency in non-smooth boundaries.

Experimental results

Research questions

• RQ1 Why are the classical traction boundary conditions in the indeterminate couple stress model physically inconsistent despite being widely used?
• RQ2 What is the complete and energetically conjugate set of traction boundary conditions for the indeterminate couple stress model?
• RQ3 How does the correct boundary condition formulation differ from the Mindlin-Tiersten formulation, and what is the physical significance of the missing term?
• RQ4 Can the new traction boundary conditions be derived from a more general second gradient elasticity framework?
• RQ5 What is the true nature of the couple stress tensor $\widetilde{m}$, and why is it not necessarily skew-symmetric?

Key findings

• The classical traction boundary conditions proposed by Mindlin and Tiersten are incomplete and fail to ensure energetic conjugacy, as they omit a critical surface term.
• The correct traction boundary conditions include an additional term involving the gradient of the anti-symmetric part of the moment tensor projected onto the tangent plane, which performs work against the displacement gradient.
• The complete set of traction conditions consists of two parts: a vector traction condition on the smooth boundary $\partial\Omega\setminus\overline{\Gamma}$ and a jump condition on the boundary curve $\partial\Gamma$, ensuring continuity of moment effects.
• The new boundary conditions are fully consistent with the variational formulation of second gradient elasticity and match those derived from a third-order moment tensor formulation.
• The couple stress tensor $\widetilde{m}$ is not required to be skew-symmetric; it can be symmetric, contradicting claims in recent literature based on the flawed Mindlin-Tiersten formulation.
• The derivation resolves the controversy in recent papers claiming that $\widetilde{m}$ must be anti-symmetric, showing that such conclusions stem from an incomplete boundary condition framework.