[Paper Review] Size-dependent couple stress Timoshenko beam theory

Ali R. Hadjesfandiari, Arezoo Hajesfandiari+2 more|arXiv (Cornell University)|Dec 20, 2017

Nonlocal and gradient elasticity in micro/nano structures55 references4 citations

TL;DR

This paper develops a size-dependent Timoshenko beam theory using consistent couple stress theory to capture microscale effects in small-scale beams. By deriving governing equations and boundary conditions, the authors analytically solve pure bending and cantilever beam problems, showing strong agreement with 2D finite element simulations, thus validating the model's accuracy in predicting size-dependent mechanical behavior at small scales.

ABSTRACT

In this paper, a new size-dependent Timoshenko beam model is developed based on the consistent couple stress theory. In the present formulation, the governing equations and corresponding boundary conditions are obtained. Afterwards, this formulation is used to investigate size-dependency for several elementary beam problems. Analytical solutions are obtained for pure bending of a beam and for a cantilever beam with partially and fully clamped boundary conditions. These analytical results are then compared to the numerical results from a two-dimensional finite element formulation for the corresponding couple stress continuum problem.

Motivation & Objective

• To develop a consistent size-dependent beam theory that captures microscale effects in small beams.
• To formulate governing equations and boundary conditions based on consistent couple stress theory.
• To analyze size-dependent behavior in classic beam problems such as pure bending and clamped beams.
• To validate the analytical model against 2D finite element simulations of the couple stress continuum.

Proposed method

• Formulate a consistent couple stress theory-based Timoshenko beam model with size-dependent effects.
• Derive the governing partial differential equations and associated boundary conditions from variational principles.
• Solve analytically for pure bending and cantilever beam configurations with partially and fully clamped ends.
• Implement a 2D finite element formulation to simulate the same couple stress continuum problem for comparison.
• Use the principle of virtual work and consistent kinematic assumptions to ensure thermodynamic consistency.
• Compare analytical results with FEM simulations to validate the beam model's accuracy and size-dependent predictions.

Experimental results

Research questions

• RQ1 How does the inclusion of microstructure-dependent couple stress effects alter the deflection and stress distribution in Timoshenko beams at small scales?
• RQ2 Can the proposed beam theory accurately predict size-dependent behavior in classic beam configurations like pure bending and cantilevers?
• RQ3 What is the level of agreement between analytical solutions of the beam model and 2D finite element simulations of the underlying couple stress continuum?
• RQ4 How do the derived boundary conditions influence the accuracy of the size-dependent beam model?
• RQ5 To what extent does the beam model capture the scale effects observed in micro- and nano-beams?

Key findings

• The analytical solutions for pure bending and cantilever beams show excellent agreement with 2D finite element results, validating the proposed beam model.
• The size-dependent behavior is clearly captured, with stiffer response at smaller scales due to couple stress effects.
• The derived boundary conditions are consistent with the variational formulation and correctly reflect microstructural influences.
• The model successfully predicts size effects even in simple loading cases, demonstrating its applicability to microscale beam design.
• The comparison confirms that classical Timoshenko beam theory underestimates stiffness at small scales, while the proposed model corrects this deficiency.
• The consistent couple stress formulation ensures thermodynamic consistency and avoids unphysical assumptions in the beam theory.