On Spencer's displacement function approach for problems in second-order elasticity theory

Area of Science:

• Solid Mechanics
• Continuum Mechanics
• Mathematical Physics

Background:

• Second-order elasticity theory extends classical linear elasticity to account for larger deformations and material nonlinearities.
• Traditional methods for second-order elasticity problems can be complex, involving intricate integrations and potential ambiguities with rigid body terms.

Purpose of the Study:

• To present and apply AJM Spencer's displacement function approach for formulating and solving second-order elasticity problems.
• To demonstrate the efficiency of the displacement function method in handling second-order stress and displacement fields.

Main Methods:

• Utilizing Spencer's displacement function approach to derive a single inhomogeneous partial differential equation for the second-order problem.
• Employing Stokes' and Laplace operators within the derived differential equations.
• Applying the method to classical elasticity problems like Kelvin's concentrated force, Love's doublet, and Boussinesq's problems.

Main Results:

• The displacement function approach simplifies the evaluation of second-order displacement fields through derivatives, avoiding integration-related issues.
• Second-order isotropic stress is governed by an equation involving the Laplace operator, dependent on the classical elasticity solution.
• The method successfully solves established problems in second-order elasticity theory, validating its applicability.

Conclusions:

• Spencer's displacement function approach provides an effective and streamlined framework for second-order elasticity problems.
• The method facilitates the systematic analysis of higher-order effects, potentially aided by symbolic computation.
• This approach offers a robust alternative to traditional formulations, enhancing the solution of complex elasticity problems.