When functionally graded materials meet higher order simplified strain gradient elasticity theory

We investigate the constitutive equations when the functionally graded materials (FGMs) are considered under the strain gradient elasticity theory (SGET). The SGET considered includes both the Helmholtz and bi-Helmholtz types. The constitutive equations become more complicated and contain more terms due to the interaction between the material gradation and the nonlocal effect of the strain gradient. We also derive the governing par-tial differential equations (PDEs) by the balance law of linear momentum, and the PDEs also appear to be highly complicated. Under the assumption that material gradation is of the exponential form, a simpler set of governing PDEs can be reached. Solutions to those PDEs under the formulation of different modes of crack problems are discussed.Published by Elsevier Inc.

Automated summary

This study investigates the constitutive equations for functionally graded materials (FGMs) under the strain gradient elasticity theory (SGET). The interaction between material gradation and the nonlocal effect of the strain gradient leads to more complex and intricate constitutive equations. The governing partial differential equations (PDEs) derived from the balance law of linear momentum also appear to be highly complicated. Assuming the material gradation is exponential, a simpler set of governing PDEs can be obtained. Solutions to these PDEs are discussed for different modes of crack problems.