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Derivation of Full Elastic Fields for Force Doublets Near an Interface Between Two Materials
Paperback
Technology & EngineeringGeneral Mathematics
ISBN13: 9781792796722
Publisher: Independently Published
Published: Dec 28 2018
Pages: 124
Weight: 0.67
Height: 0.26 Width: 8.50 Depth: 11.00
Language: English
Richard Yegian's research expands 2D plane elasticity solutions into 3D Airy Stress functions by applying boundary conditions at the (z=0) interface, ensuring continuity of displacement and stress vectors. The work derives exact, closed-form formulas in Cartesian coordinates for force doublets near a welded bimaterial interface, optimizing the equations for direct programmatic calculation and digital simulation.
This is a workbook second edition with formulas and code for easy calculation and manipulation. Plane elasticity solutions for concentrated forces near a welded interface between two materials can be used for a variety of purposes, among them, the derivation of elastic fields for force doublets (dipoles) or nuclei of strain. The objective of this work is to derive and record the full elastic fields for the force doublets near an interface between two materials. The solutions are exact within the classical theory of elasticity and are in closed form in terms of elementary functions. Explicit formulae are given for the derivation of 3-D (three dimensional) Airy stress functions expanded from 2-D, the components of stress and displacement, and the tractions on the interface (in Cartesian co-ordinates).Richard Yegian's research focuses on resolving discontinuous boundaries-the exact point where two different materials are welded together. At the plane interface where z = 0, the displacement vectors (u) and the normal/shear stress vectors (σ) of both materials must match perfectly across the interface so the materials do not rip apart. To enforce this continuous bond mathematically, Yegian applies the following system of boundary conditions. Yegian enforces continuous bonding at the interface (z=0) by matching displacement (u) and stress (sigma) vectors:
(u_x)_1 = (u_x)_2, (u_y)_1 = (u_y)_2, (u_z)_1 = (u_z)_2 [at z = 0](σ_zz)_1 = (σ_zz)_2, (σ_zx)_1 = (σ_zx)_2, (σ_zy)_1 = (σ_zy)_2 [at z = 0] To solve these complex biharmonic equations, Yegian derives exact, closed-form 3D Airy Stress functions expanded from traditional 2D plane elasticity solutions. He evaluates the governing field mechanics across an infinite three-dimensional continuum using biharmonic operators. The 3D Airy Stress functions, derived from the biharmonic operator (Δ 4 Φ = 0), are structured into code-ready, isolated sub-expressions for efficient, direct computational simulation, into a general form. The resulting formulations completely isolate the coordinate variables (x, y, z), components of stress, displacement, and interface tractions in standard Cartesian coordinates.
Unlike purely theoretical frameworks, this algebraic layout is structured into discrete, isolated sub-expressions. By computing distinct spatial radius vectors relative to the source depth (h) along the z-axis, the equations are optimized to avoid redundant processing strain. This code-ready architecture culminates in explicit closed-form field equations designed for immediate computational execution. This structural format allows engineers to seamlessly loop over a dense 3D coordinate mesh to map and simulate 3D stress fields in computing software.
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