Bounds and extremal configurations for the torsional rigidity of coated fiber reinforced shafts

In this paper we derive bounds on the torsional rigidity for coated fiber reinforced shafts. The bounds are used to assess the optimality or suboptimality of fiber reinforcement configurations. This investigation focuses on coated fiber reinforcements with circular cross section. It is shown how the effective antiplane shear modulus and torsional rigidity of each coated fiber are used to determine whether the configuration provides reinforcement above or below that of a homogeneous shaft containing no coated fibers. Simply connected shaft cross sections of arbitrary shape reinforced with any configuration of coated fibers are considered. Precise conditions on the effective antiplane shear modulus and torsional rigidity of each coated fiber are given under which the circular shaft reinforced with a single centered circular coated fiber is either optimal or suboptimal. © 2004 Society for Industrial and Applied Mathematics

Solids containing spherical nano-inclusions with interface stresses: Effective properties and thermal–mechanical connections

AbstractThis work examines the overall thermoelastic behavior of solids containing spherical inclusions with surface effects. Elastic response is evaluated as a superposition of separate solutions for isotropic and deviatoric overall loads. Using a variational approach, we construct the Euler–Lagrange equation together with the natural transition (jump) conditions at the interface. The overall bulk modulus is derived in a simple form, based on the construction of neutral composite sphere. The transverse shear modulus estimate is derived using the generalized self-consistent method. Further, we show that there exists an exact connection between effective thermal expansion and bulk modulus. This connection is valid not only for a composite sphere, but also for a matrix-based composite reinforced by many randomly distributed spheres of the same size, and can be viewed as an analog of Levin’s formula for composites with surface effects

Bounds for the torsional rigidity of shafts with arbitrary cross-sections containing cylindrically orthotropic fibres or coated fibres

In this paper we derive bounds for the torsional rigidity of a cylindrical shaft with arbitrary transverse cross-section containing a number of cylindrically orthotropic fibres or coated fibres. The exact upper and lower bounds depend on the constituent shear rigidities, the area fractions, the locations of the reinforcements as well as the geometric shape of the cross-sections. Specific bounds are derived for circular shafts, elliptical shafts and cross-sections of equilateral triangle. Simplified expressions are also deduced for reinforcements with isotropic constituents. We verify that when additional constraints between the constituent properties of the phases are fulfilled, the upper and lower bounds will coincide. In the latter case, the fibres or coated fibres become neutral under torsion and the bounds recover the previously known exact torsional rigidity. © 2007 The Royal Society