Two-dimensional elasticity solution of elastic strips and beams made of functionally graded materials under tension and bending

This paper presents a theoretical approach to solve elastic problems of functionally graded materials (FGMs). For FGMs with exponential gradient, based on a two-dimensional theory of elasticity, a governing equation is derived by means of the Airy stress function method together with the strain compatibility equation. Simple uniaxial tension and bending are solved. For an FGM layer with transversely and/or vertically varying material properties, stress distribution and strain field under simple tension are determined according to two different assumptions. The obtained results indicate that for a thin elastic layer of thickness-wise gradient as a transition zone linking two dissimilar materials, there is a horizontal displacement difference across the transition zone due to mismatch of the material properties. In particular, when the thickness of the FGM layer reduces to zero, the horizontal displacement difference has a severe mismatch across the interface of two perfectly bonded dissimilar materials. An FGM beam subjected to a bending moment is also analyzed. The normal stress exhibits a nonlinear distribution and may arrive at its maximum tensile stress inside the beam, not at the surface. The obtained elasticity solution is useful for better understanding of the mechanical behaviors of FGMs subjected to different combined loads.