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1/71/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem CONTINUUM MECHANICS (BOUNDARY VALUE PROBLEM - BVP)

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2/72/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem NE CE HE SBC KBC The set of NE+CE+ HE equations consists of 15 linear differential-algebraic equations – and is always the same for any static problem (except of material constants in HE). Individual problems are different only due to different boundary conditions, which define body shape i, loading q i and displacements u i on the body surface (supports). Here is where name Boundary Value Problem of Elasticity comes from.,

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3/73/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem 1. Reduction of unknown functions number in exchange for upgrading the differential equations order a/ Substitution of CE to HE and next to NE; this yields the set of 3 differential equations of the second order for displacements as unknowns (Lamé equations): b/ Elimination of displacements by transforming CE into compatibility equations and substitution HE; this yields the set of 6 differential equations of the second order for stress components (Beltrami-Michell equations): Analytical methods

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4/74/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem 2. Inverse method In this method the full solution compaltible with NE, CE and HE is guessed, then SBC and KBC are checked to comply with a given problem. 3. Semi-inversed methods a/ Displacement approach: 3 functions u i satisfying KBC are guessed, and then the strains are found by differentiation according to CE, and inserted into algebraic HE to obtain stresses which have to satisfy NE and SBC u i + KBC ij σ ij SWB? SubstitutionDifferentiation CEHENE? Analytical methods Substitution CE (Cauchy)NE (Navier)HE (Hooke) SBC KBC

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5/75/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem σ ij + NE + SBC ij uiui KBC? Substitution Integration HECE b/ Stress approach: 6 functions ij satisfying NE and SBC are guessed, and the strains are found by inserting them into HE; then set of Cauchy Equations CE has to be integrated to find displacements u i. The only remaining action left is to check KBC by inserting displacements 3. Semi-inversed methods CE (Cauchy) NE (Navier) HE (Hooke) SBC KBC Analytical methods

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6/76/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem σ ij + NE + SBC ij uiui KBC? Substitution Integration HECE b/ Stress approach: Out of these two semi-inverse methods, the displacement approach seems to be superior as it requires only three displacements to be guessed which are physical quantities and can be measured experimentally. Moreover, only two operations to be performed are insertion and differentiation, the latter being much easier than integration required by stress approach. The price to be paid in displacement approach is a necessity of checking Navier Equation of equilibrium and Static Boundary Condition. u i + KBC ij σ ij SWB? Substitution Differentiation CEHENE? Substituition a/ Displacement approach Comparison of semi-inverse methods

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7/77/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem Numerical Methods features space discretisation and application of one of numerous methods: development in power series, finite differences, finite elements, boundary integrals, meshless methods etc. Numerical methods are discussed in detail as a separate subject of curriculum and will not be dealt with here. However, it is worthwhile to emphasise that numerical methods allow for overcoming of the fundamental problem of theory of elasticity which is solving problems with singular boundary conditions (sharp edges of structures, concentrated loadings etc.) Numerical methods

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8/78/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem stop

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9/79/7 M.Chrzanowski: Strength of Materials SM1-11: Continuum Mechanics: Boundary Value Problem stop

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