### Monday, 11 October 2004 - 1:50 PM

### This presentation is part of : Horgan Symposium

### Homogenization and Global Responses of Inhomogeneous Spherical Nonlinear Elastic Shells

**Yi-chao Chen**^{1}, K. Raj Rajagopal^{2}, and Lewis Wheeler^{1}. (1) University of Houston, Department of Mechanical Engineering, Houston, TX 77204, (2) Texas A&M University, Department of Mechanical Engineering, College Station, TX 77843
Homogenization of radially inhomogeneous spherical nonlinear elastic shells subject to internal pressure is studied. The equivalent homogeneous material is defined in such a way that it gives rise to exactly the same global response to the pressure load as that of the inhomogeneous shell. This leads to an integro-differential equation. For an inhomogeneous shell with general strain-energy function and inhomogeniety, this integro-differential equation is solved, yielding the strain-energy function of the equivalent homogeneous material. The resulting formula, which is in the form of a series, is used to study layered composite shells. It is found that the constitutive function of the equivalent homogeneous material is in general of different type from that of the composite shell. For example, the constitutive function of the homogeneous material equivalent to a two-layer neo-Hookean shell is not neo-Hookean. For an infinitely fine layered composite shell of general nonlinear elastic materials, the constitutive function of the equivalent homogeneous material is determined explicitly and is found to assume an extremely simple form. Furthermore, it is found that such a homogeneous material gives not only the same global response, but also the same average stress field as the composite shell does.

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