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Elasticity problem formulation

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- 1. Lecture 5 Elasticity problem formulation The Game Print version Lecture on Theory of Elasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACEG 5.1 Contents 1 Field equations review 1 2 Boundary conditions 2 3 Fundamental problem formulation 3 3.1 Displacement formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3.2 Stress formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.3 Principle of superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.4 Saint-Venant’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4 General solution strategies 5 4.1 Direct method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.2 Inverse method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 4.3 Semi-inverse method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4.4 Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.2 1 Field equations review Field equations review Boundary Value Problem • The ﬁeld equations are differential and algebraic relations between strain, stresses and displacements within the body • We need of appropriate boundary conditions (BCs) to solve the elasticity problem. These BCs specify the loading and supports that create stress, strain and displacement ﬁelds • Field equations are same but the BCs are different • Field Equations + Boundary Conditions = Boundary Value Problem Note • It is very important to imply proper BCs for a problem solution 5.3 Field equations review Field Equations • Strain-displacement relations- 6 pieces, 9 variables (6 strains+3 displacements) εij = 1 2 (ui,j +uj,i) or ε = 1 2 ∇u+(∇u)T 1
- 2. • Compatibility equations- 6 pieces (to check only) εmn,ij +εij,mn = εim,jn +εjn,im or ∇×(∇×ε)T = 0 • Equilibrium equations- 3 pieces, 6 variables (6 stresses) σij,i + fj = 0 or ∇·σ +f = 0 5.4 Field equations review Field Equations • Constitutive equations- 6 pieces, 12 variables (6 stresses + 6 strains) σij = 2µεij +λεkkδij or σ = 2µε +λtr(ε)I • Recapitulation – 15 unknowns (6 stresses + 6 strains + 3 displacements) – 15 equations (6 → (ε −u) + 6 → (σ −ε) + 3 → (σ − f)) • Everything is OK. The elasticity problem can be solved!?! 5.5 2 Boundary conditions Boundary conditions Loading and supports • The BCs specify how the body is loaded or supported 5.6 Boundary conditions Loading and supports • Stress BCs- specify the tractions at boundary ti = njσji = ¯ti or t = σ ·n = ¯t 5.7 Boundary conditions Loading and supports • Displacement BCs- specify the displacements at boundary ui = ¯ui or u = ¯u 5.8 2
- 3. Boundary conditions Loading and supports • Mixed BCs- specify the displacements or tractions at boundary t = σ ·n = ¯t or u = ¯u Note • Don’t specify stress and displacements BCs at the same boundary simultaneously! 5.9 Boundary conditions Example u = ¯u σ ·n = ¯t • Using the above relations determine the displacement and stress BCs at the body’s bound- aries 5.10 3 Fundamental problem formulation 3.1 Displacement formulation Fundamental problem formulation Displacement formulation • We try to develop a reduced set of ﬁeld equations with displacement unknowns • Replacement of kinematic relations into Hooke’s law gives σij = µ (ui,j +uj,i)+λuk,kδij • And next into the equilibrium equations leads to µui,j j +(µ +λ)uj,ji + fi = 0 5.11 Fundamental problem formulation Displacement formulation • In tensor form µ∇2 u+(µ +λ)∇(∇·u)+f = 0 • The above expressions are known as Lamé-Navier equations Note • Method is useful with displacement BCs • Avoid compatibility equations • Mostly for 3D problems 5.12 3
- 4. Fundamental problem formulation Displacement formulation • Claude-Louis Navier (1785-1836) 5.13 3.2 Stress formulation Fundamental problem formulation Stress formulation • Stress-strain relations are replaced into compatibility equations σij,kk + 1 1+ν σkk,,ij = − ν 1−ν fk,kδij −(fj,i + fi,j) • In tensor form ∇2 σ + 1 1+ν ∇[∇(tr∇)] = − ν 1−ν (∇·f)I− ∇f+(∇f)T • The above expressions are known as Beltrami-Michell compatibility equations Note • Method is effective with stress BCs • Need to work with compatibility eqns. • Mostly for 2D problems 5.14 Fundamental problem formulation Stress formulation • Eugenio Beltrami (1835-1900) 5.15 4
- 5. 3.3 Principle of superposition Principle of superposition Superposition • Linear ﬁeld equations → Method of superposition t = t(1) +t(2) , f = f(1) +f(2) σ = σ(1) +σ(2) , ε = ε(1) +ε(2) , u = u(1) +u(2) 5.16 3.4 Saint-Venant’s principle Saint-Venant’s principle Deﬁnition • The stress, strain, and displacement ﬁelds caused by two different statically equivalent force distributions on parts of the body far away from the loading points are approximately the same 5.17 4 General solution strategies 4.1 Direct method Direct method Direct integration • Solution by direct integration of the ﬁeld equations (stress or displacement formulation) • Exactly satisﬁed BCs • A lot of mathematics • The method is applicable on problems with simple geometry 5.18 4.2 Inverse method Inverse method Reverse solution • Select particular displacements or stresses that satisfy the basic ﬁeld equations • Seek a speciﬁc problem that ﬁt to the solution ﬁeld, i.e. ﬁnd the appropriate problem geometry, BCs and tractions • Difﬁcult to ﬁnd practical problems to apply a given solution 5.19 5
- 6. 4.3 Semi-inverse method Semi-inverse method Direct + inverse method • Part of the displacement or stress ﬁelds is speciﬁed • The remaining portion - by direct integration of the basic ﬁeld equations and BCs • Start with approximate MoM solutions • Enhancement by using the St-Venant’s principle 5.20 4.4 Solution methods Solution methods Mathematical methods for solutions • Analytical solutions – Power series method – Fourier method – Integral transform method etc. • Approximate solution procedures – Ritz method – Bubnov-Galerkin method etc. • Numerical solutions – Finite difference method – Finite element method – Boundary element method 5.21 Solution methods The End • Any questions, opinions, discussions? 5.22 6

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