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

Abstract

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 partial 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.