Soft materials such as rubberlike and biological tissues are usually modeled as incompressible. Mechanical properties of polymeric materials can be controlled either by exposing them to ultraviolet light for different time durations or by changing their molecular structure. A challenge is to find the spatial variation of the moduli to fully utilize the material. One way to achieve this is to have a uniform distribution of the stress component likely to cause their failure. To achieve this, we analytically analyze 3-dimensional infinitesimal flexural deformations of a functionally graded (FG) and linearly elastic beam of rectangular cross-section with Young's modulus a continuous function of the thickness coordinate. The problem is first studied for an incompressible material and then for a compressible material for which Poisson's ratio is assumed to be a constant. It is found that when Young's modulus at a point is inversely proportional to its distance from the neutral axis, then the magnitude of the bending stress is uniform over beam's cross-section, the beam is the lightest possible and its deflections are 2/3rd of that of the corresponding beam of a homogeneous material that has the same maximum bending stress as the FG beam. Noting that Young's modulus cannot be infinity at the neutral axis, we avoid this by assuming that it is a constant over a small region around the neutral axis. For a transversely isotropic incompressible material beam, it is shown that (2 mu(1) + mu(2)) determines beam's flexural stiffness where mu(1) and mu(2) are, respectively, the shear moduli along and perpendicular to the axis of transverse isotropy. Beams of homogeneous materials having the same geometry as the FG beam have different shear moduli depending upon whether the FG and the homogeneous material beams have the same maximum deflection, the same maximum bending stress or the same total strain energy of deformation.
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