Chapter 5: Constitutive Equations and Linear Elasticity
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MCEN 5023/ASEN 5012 Chapter 5 Constitutive Equations and Linear Elasticity Fall, 2006 1 Linear Elasticity Constitutive Equations Stresses Strains Forces Displacements 2 Linear Elasticity Constitutive Equations Constitutive equations characterize material properties: Stress – Strain Voltage – Current Temperature – Heat flux … 3 Linear Elasticity Hooke’s Law f = kx σ = Ee Generalized Hooke’s Law For infinitesimal small strain: 4 Linear Elasticity Generalized Hooke’s Law Reduction of total number of constants in Dijkl 1. Symmetry of stress and strain tensors σ ij = σ ji ekl = elk 5 Linear Elasticity Generalized Hooke’s Law Reduction of total number of constants in Dijkl 2. Energy Potential W W = ∫ σ ij d eij 6 Linear Elasticity Generalized Hooke’s Law – Matrix Form 7 Linear Elasticity Generalized Hooke’s Law Reduction of total number of constants in Dijkl 3. Orthotropic Materials The material that has three orthogonal planes of symmetry Collagen Molecule ( ~1nm) Minerals in Holes Minerals Between Molecules (~10nm) 8 Linear Elasticity Generalized Hooke’s Law 3. Orthotropic Materials 9 Linear Elasticity Generalized Hook’s Law 4. Transversely Isotropic Materials 2 1 10 Linear Elasticity Generalized Hook’s Law 5. Isotropic Materials – 2 components E Young’s Modulus ν Poisson’s Ratio 11 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials E ν E G= 2(1 + ν ) 1 e11 = [σ 11 −ν (σ 22 + σ 33 )] E 1 e22 = [σ 22 − ν (σ 11 + σ 33 )] E 1 e33 = [σ 33 − ν (σ 11 + σ 22 )] E 1 σ 12 2G 1 e23 = σ 23 2G 1 e13 = σ 13 2G e12 = 1 ⎞ ⎛ γ τ = ⎜ xy xy ⎟ G ⎠ ⎝ 1 ⎞ ⎛ γ τ = ⎜ yz yz ⎟ G ⎠ ⎝ 1 ⎞ ⎛ γ τ xz ⎟ = ⎜ xz G ⎠ ⎝ 12 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials ν 1 +ν σ ij − σ kk δ ij eij = E E 13 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials 14 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials 15 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials ν 1 +ν eij = σ ij − σ kk δ ij E E σ ij = 2Geij + λ= νE (1 + ν )(1 − 2ν ) ekk δ ij νE (1 + ν )(1 − 2ν ) λ, G, are called Lamé Constants. 16 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials σ ij = 2Geij + νE (1 + ν )(1 − 2ν ) ekk δ ij 17 Linear Elasticity Constitutive Model for Linear Elastic Isotropic Materials Five elastic constants: E ν λ G K Only two of them are independent 18 Linear Elasticity Linear Elasticity What is linear elasticity about? P P deformed undeformed X2 X2 X1 X1 Question: If we apply a force on a material, what are the stresses, strains and displacements? Object: Linear Elastic Body (Mr. Potato; Machine elements; Human hard tissue……) (E and v are given) Input Boundary conditions (Applied force; Applied displacement …) Output Stresses, strains, displacements, at each material point (x1,x2,x3) ??? 19 Linear Elasticity Linear Elasticity Things we want: Stresses, strains, displacements, at each material point (x1,x2,x3) 20 Linear Elasticity Linear Elasticity If we take displacements as basic unknowns: 21 Linear Elasticity Linear Elasticity Displacements can be obtained by integration of strains. Things we want: Stresses, strains, at each material point (x1,x2,x3) 22 Linear Elasticity Linear Elasticity If we take stresses as basic unknowns: 23 Linear Elasticity v Linear Elasticity T Boundary conditions: 1 sσ Prescribed displacements Prescribed tractions 1 u s X2 2 u s X1 24 Linear Elasticity Linear Elasticity Boundary conditions: Example T c b a X2 X1 25 Linear Elasticity Linear Elasticity Boundary conditions: Example T c b a X2 X1 26 Linear Elasticity Linear Elasticity Boundary conditions: Boundary conditions due to symmetry 27 Linear Elasticity Linear Elasticity Boundary conditions: Boundary conditions due to symmetry 28 Linear Elasticity Boundary conditions: Boundary conditions due to symmetry Nanoindentation: three-sided pyramidal tips are most often used. Berkovich tip (included angle, 142.3º) Cube corner tip (included angle, 90º) Indentation Impression 29 Linear Elasticity Nanoindentation: 1000 900 Force 800 600 ng i ad o L 500 400 Un loa din g Time Force (uN) 700 300 200 100 0 0 20 40 60 80 Indentation Depth (nm) Loading: Elastic + Plastic Unloading: Elastic 30 Linear Elasticity Boundary conditions: Boundary conditions due to symmetry Top Views 3D perspective 31 Linear Elasticity Summary of Equations of Linear Elasticity 1 (ui, j + u j ,i ) 2 Kinematics: Eij = eij = Strain Compatibility: eij ,kl + ekl ,ij − eik , jl − e jl ,ik = 0 Navier’s: Gui , jj + (λ + G )u j , ji + f i = 0 Constitutive: 1 +ν ν σ ij − σ kk δ ij eij = E E Equilibrium: σ ij , j + f i = 0 Stress Compatibility (B-M) ∇ 2σ ij + σ ij = 2Geij + λekk δ ij ν 1 θ ,ij = − δ ij X k ,k − (X i , j + X j ,i ) 1 +ν 1 −ν 32 Linear Elasticity Methods for solving linear elasticity problem Compatibility Displacement Kinematics Strain Boundary Conditions Constitutive Stress S-SF Rel. Stress Function Equilibrium 33 Linear Elasticity Two Principles 1. Principle of Superposition T1 T1 T2 T2 = X2 + X2 X1 X2 X1 X1 34 Linear Elasticity Two Principles 2. St. Venant’s Principles: Statically equivalent systems of forces produce the same stresses and strains within a body except in the immediate region where the loads are applied. P = qL q L L If characteristic length of the area where a force is acting on is L, the dimension of immediate region is ~L. 35 Linear Elasticity Example: Simple Tension Gauge Length 36 Linear Elasticity x2 Example: Simple Tension x3 x1 Gauge Length, L0 37 Linear Elasticity x2 Example: Simple Tension x3 x1 σ ij = 2Geij + νE (1 + ν )(1 − 2ν ) ekk δ ij 38 Linear Elasticity x2 Example: Simple Tension x3 x1 σ ij = 2Geij + νE (1 + ν )(1 − 2ν ) ekk δ ij 39 Linear Elasticity x2 Example: Simple Tension x3 x1 1 +ν ν σ ij − σ kk δ ij eij = E E 40 Linear Elasticity x2 Example: Simple Tension x3 x1 1 +ν ν σ ij − σ kk δ ij eij = E E 41
