Saint-Venant End Effects in the Plane Problem for Linearly Elastic Functionally Graded Materials
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Saint-Venant End Effects in the Plane Problem for Linearly Elastic Functionally Graded Materials Aisa Biria1
Received: 17 May 2020 © Springer Nature B.V. 2020
Abstract An isotropic elastic strip, with a continuous inhomogeneity profile for Young’s modulus is considered, subject to a self-equilibrated load on one of its axial ends and free of traction on the remainder of its surface boundaries. By taking advantage of the analytical flexibility of an exponential inhomogeneity profile, the full equations of linear theory of elasticity are employed to find the two-dimensional Saint-Venant decay rate in terms of an inhomogeneity parameter that measures gradation steepness in a Functionally Graded Material (FGM). The results show that softening axial inhomogeneity may be introduced to engineered FGM strips and beams to accelerate the decay of stresses. For a hardening axial inhomogeneity on the other hand, the end effects extend beyond a one-width-size distance from the loaded end. The plane problem end effects are shown to decay faster compared to the anti-plane shear counterparts, consistent with what is observed in homogeneous media. For inhomogeneity in the lateral direction from the core toward the outer edges, the qualitative behaviour changes with the degree of inhomogeneity. Whether the core is softer or harder relative to the outer edges, a steep lateral gradation of elastic modulus can significantly increase the decay length of end effects. Keywords End effects · Saint-Venant’s principle · FGM · 2D elasticity · Non-homogeneous material · Inhomogeneity Mathematics Subject Classification 74B05 · 74E05 · 74E30 · 74G05 · 74G50
1 Introduction Mechanics of inhomogeneous solids has been the subject of interest for decades with the motive developing over time. The mining drilling industry, structural engineers, and geologists are interested in mechanics of sands, soils, and rocks, elastic properties of which
B A. Biria
[email protected]
1
Department of Mechanical, Industrial and Aerospace Engineering, Concordia University, 1455 Maisonneuve W., Montreal, Quebec H3G 1M8, Canada
A. Biria
gradually change by depth beneath the Earth’s surface. Biomedicine and bioengineering followed suit, as many biological tissues tailor their mechanical properties to their functionality through gradations in composition, microstructure, or porosity [5, 30, 34, 36, 39]. Inspired by nature, material scientists also engineered composites known as Functionally Graded Materials (FGM) with properties graded continuously or over fine steps in a spatial direction, enabling desired engineering advantages. FGMs have found various applications as surface coatings and interfaces between dissimilar materials, impeding failures due to thermal and contact stresses. More recently, they were employed in soft robotics, in order to allow smooth transition from the hard electronics to the soft body [2, 14]. Saint-Venant’s principle is commonly invoked in engineering applications to justify the neglect of end and edge effects when determinin
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