16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity
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1 6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz = ε 33 γ yz = ε 4 = 2 ε yz = 2 ε 23 γ xz = ε 5 = 2 ε xz = 2 ε 3 γ xy = ε 6 = 2 ε xy = 2 ε 2 EQUATIONS OF ELASTICITY y 3, z y 2, y y, x Right-handed rectangular Cartesian 5 equations/5 unknowns coordinate system. Equilibrium (3) σ + σ 2 + σ 3 + f = 0 y y 2 y 3 σ 2 + σ 22 + σ 32 + f 2 = 0 σ mn + f n = 0 y y 2 y 3 y m σ 3 + σ 23 + σ 33 + f 3 = 0 y y 2 y 3 Paul A. Lagace 2002 Handout 2-
2 2. Strain-Displacement (6) ε = ε 22 = u ε 2 = ε u 2 = + u 2 y 2 y 2 y u 2 ε 3 = ε u 3 = + u 3 ε y 2 2 y 3 y mn = u m u + n 2 y n y m ε 33 = u 3 y 3 ε 32 = ε 23 = u 2 + u 3 2 y 3 y 2 3. Stress-Strain (6) Anisotropic: Generalized Hooke s Law: σ mn = E mnpq ε pq σ E E 22 E 33 2E 23 2E 3 2E 2 ε E 22 E 2222 E E E 223 2E 222 ε 22 σ 22 σ 33 E 33 E 2233 E E E 333 2E 332 ε 33 = σ 23 E 23 E 2223 E E E 323 2E 223 ε 23 σ 3 E 3 E 223 E 333 2E 323 2E 33 2E 23 ε 3 σ 2 E 2 E 222 E 332 2E 223 2E 23 2E 22 ε 2 Orthotropic: σ E E 22 E ε σ 22 E 22 E 2222 E ε 22 σ 33 E 33 E 2233 E ε 33 = σ E ε 23 σ E 33 0 ε 3 σ E 22 ε 2 Compliance Form: ε mn = S mnpq σ pq where: E - = S ~ ~ Paul A. Lagace 2002 Handout 2-2
3 DEFINITION OF ENGINEERING CONSTANTS. Longitudinal (Young s) (Extensional) Moduli: E mm = σ mm ε mm due to σ mm applied only (no summation on m) 2. Poisson s Ratios: ε ν nm = mm ε nn due to σ nn applied only (for n m) Reciprocity: ν nm E m = ν mn E n (no sum) (m n) 3. Shear Moduli: G mn = σ mn 2ε mn due to σ mn applied only (for m = 4, 5, 6) (for n m) 4. Coefficients of Mutual Influence: (using contracted notation) η mn = ε n ε m for σ m applied only (for m, n, =, 2, 3, 4, 5, 6, m n) (Note: one strain extensional, one strain shear) Reciprocity here as well 5. Chentsov Coefficients: (using contracted notation) η mn = ε n ε m for σ m applied only (for m, n, = 4, 5, 6, m n) Paul A. Lagace 2002 Handout 2-3
4 ENGINEERING STRESS-STRAIN EQUATIONS (using contracted notation) ε = [σ ν 2 σ 2 ν 3 σ 3 η4σ 4 η 5 σ 5 η 6 σ 6 E ] ε 2 = E2 [ ν 2 σ + σ 2 ν 23 σ 3 η σ 4 η σ 5 η σ 6 ] ε 3 = E3 [ ν 3 σ ν σ 2 + σ 3 η 34 σ 4 η 35 σ 5 η 36 σ 6 ] ν 32 γ 4 = ε 4 = γ 5 = ε 5 = γ 6 = ε 6 = [ η 4 σ η 42 σ 2 η 43 σ 3 + σ 4 η 45 σ 5 η 46 σ 6 ] G 4 [ η 5 σ η 52 σ 2 η 53 σ 3 η 54 σ 4 + σ 5 η 56 σ 6 ] G 5 [ η6 σ η 62 σ 2 η 63 σ 3 η 64 σ 4 η 65 σ 5 + σ 6 ] G 6 In general: ε n = E 6 n m= ν nm σ m Note: ν nn = - and η s --> ν s Orthotropic form In terms of ENGINEERING CONSTANTS (using contracted notation): ε ν 2 ν σ E E E ε 2 ν 2 ν σ 2 E 2 E 2 E 2 ε 3 ν 3 ν 32 σ E = 3 E 3 E 3 ε σ 4 G 4 ε G σ ε G 6 σ 6 6 Paul A. Lagace 2002 Handout 2-4
5 Isotropic form ε / E ν / E ν / E σ ε 2 ν / E / E ν / E σ 2 ε 3 ν / E ν / E / E σ 3 = ε / G 0 0 σ 4 ε / G 0 σ 5 ε / G σ 6 with: G = E 2 ( + ν) PLANE STRESS h << a, b σ zz, σ yz, σ xz = 0 = 0 z Anisotropic stress-strain equations ε = ε 2 = ε 6 = [σ ν 2 σ 2 η 6 σ 6 ] E [ ν 2 σ + σ 2 η 26 σ 6 ] Primary E 2 [ η 6 σ η 62 σ 2 + σ 6 ] G 6 Paul A. Lagace 2002 Handout 2-5
6 ε 3 = E3 [ ν 3 σ ν 32 σ 2 η 36 σ 6 ] ε 4 = ε 5 = [ η 4 σ η 42 σ 2 η 46 σ 6 ] G 4 [ η 5 σ η 52 σ 2 η 56 σ 6 ] G 5 Secondary PLANE STRAIN L >> x, y = 0 z ε 3 = ε 23 = ε 33 = 0 Paul A. Lagace 2002 Handout 2-6
7 SUMMARY Plane Stress Plane Strain Geometry: thickness (y 3 ) << in-plane dimensions (y, y 2 ) length (y 3 ) >> in-plane dimensions (y, y 2 ) Loading: Resulting Assumptions: Primary Variables: Secondary Variable(s): Note: σ 33 << σ αβ σ i3 = 0 ε αβ, σ αβ, u α ε 33, u 3 Eliminate ε 33 from eq. set by using σ 33 = 0 σ - ε eq. and expressing ε 33 in terms of ε αβ σ αβ only / y 3 = 0 ε i3 = 0 ε αβ, σ αβ, u α σ 33 Eliminate σ 33 from eq. Set by using σ 33 σ - ε eq. and expressing σ 33 in terms of ε αβ TRANSFORMATIONS σ mn = l mp l nq σ pq ε mn = l mp l nq ε pq x m = l mp x p ũ m = l mp u p Ẽ mnpq = lmr lns l pt l qu Erstu where: l~ ~ mn = cosine of angle from y m to y n Paul A. Lagace 2002 Handout 2-7
8 OTHER COORDINATE SYSTEMS F ( y, y 2, y 3 ) = ξ F2 ( y, y 2, y 3 ) = η F 3 ( y, y 2, y 3 ) = ζ Example - Cylindrical Coordinates 2 y ξ = r F ( y, y 2, y 3 ) = + y 2 η = θ F2 ( y, y 2, y 3 ) = tan - (y 2 / y ) ζ = z F3 ( y, y 2, y 3 ) = y 3 Equilibrium: r : σ rr + σ θr + σ zr + σ rr σ θθ + f r = 0 r r θ z r 2 θ : σ rθ + σ θθ + σ zθ + 2σ rθ + f θ = 0 r r θ z r z : σ rz + σ θz + σ zz + σ rz + f z = 0 r r θ z r (Engineering) Strain-Displacement: ur ε rr = r ε θθ = u θ r θ ε = zz ε rθ = u 3 z u θ + u r u θ r r θ r ε θz = u 3 r θ + u θ z ε zr = u r + u 3 z r Paul A. Lagace 2002 Handout 2-8
9 (Isotopic) Stress-Strain: ε rr = E [σ rr ν(σ θθ + σ zz )] ε θθ = E [σ θθ ν(σ rr + σ zz )] ε zz = E [σ zz ν(σ rr + σ θθ )] ε rθ = ε θz = ε = zr 2 ( + ν) σrθ E 2 ( + ν) σθz ( E 2 + ν) σ E zr STRESS FUNCTIONS 4 φ = Eα 2 ( T) ( ν) 2 V (isotropic) 2 2 where: 2 = + x 2 y 2 2 φ σ xx = 2 + V y σ = yy σ xy 2 φ x 2 2 φ = xy + V Paul A. Lagace 2002 Handout 2-9
10 EFFECTS OF THE ENVIRONMENT Temperature Thermal Strain: ε T = α T α = Coefficient of Thermal Expansion (C.T.E.) T general form: ε ij = α ij T Total Strain = Mechanical Strain + Thermal Strain M ε ij = ε ij + ε ij T M ε ij = S ijkl σ kl σ kl = E ijkl ε ij E ijkl α ij T Transformation of α ij : α = cos 2 θ α + sin 2 θ α 22 α *, α * 22 are C.T.E. s in α 22 = sin 2 θ α + cos 2 θ α 22 α 2 = cos θ sinθ (α 22 α ) principal material axes Sources of temperature differential Ambient environment Convection Aerodynamic heating specific Mach heat ratio number Adiabatic wall temp = T AW = + γ r M T 2 heat flux: q = h (T AW - T s ) recovery factor 2 T = ambient temperature ( K) heat transfer coefficient surface temperature of body Paul A. Lagace 2002 Handout 2-0
11 Radiation Emissivity Absorptivity Conduction q = - ε σ T s 4 q = α I s λ q = heat flux e = emissivity σ = Stefan-Boltzman constant T s = surface temperature q = heat flux α = absorptivity I s = intensity of source λ = angle factor q i T T T T q = k i = heat flux ij T x j k ij = thermal conductivity Fourier s equation: kz T 2 T T = ρc z 2 t thermal conductivity Degradation of material properties Glass transition temperature E(T), σ ult (T), σ y (T) Creep Paul A. Lagace 2002 Handout 2-
12 Other Environmental Effects M ε ij = ε ij + Σ ε ε ij total = mechanical + environmental Moisture: General: s ε ij = β ij c S ε ij = swelling strain β ij = swelling coefficient c = moisture concentration E ε ij = environmental strain E ε ij = χ ij χ χ ij = environmental operator χ = environmental scalar Piezoelectricity Piezoelectric strain: ε ij p = d ijk E k Coupled equations: E k = electric field d ijk = piezoelectric constant σ mn = E mnij ε ij E mnij d ijk E k D i = e ik E k + d inm σ mn e ik = dielectric constant D i = electrical charge Paul A. Lagace 2002 Handout 2-2
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