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Theory of elasticity and plasticity (Equations sheet part 01) Att 8676

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Theory of elasticity and plasticity (Equations sheet part 01)

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Theory of elasticity and plasticity (Equations sheet part 01) Att 8676

  1. 1. Theory of elasticity and plasticity Equations sheet, Part 1 Theory of stress Cauchy stress formula σT · n = t, or   σxx σyx σzx σxy σyy σzy σxz σyz σzz     nx ny nz   =   tx ty tz   where tx, ty and tz are the traction components; nx = cos(n, x), ny = cos(n, y) and nz = cos(n, z) are the direction cosines of the outward normal vector. Normal and shear components of the stress (traction) vector tn = t · ˆn = tini, ts = |t|2 − t2 n Hint: Unite normal to the plane ˆn = φ | φ| Principal stresses (σ − λI)n = 0 Characteristic equation λ3 − I1λ2 + I2λ − I3 = 0 Stress invariants I1 = σxx + σyy + σzz, I2 = σxx σxy σyx σyy + σxx σxz σzx σzz + σyy σyz σzy σzz , I3 = σxx σxy σxz σyx σyy σyz σzx σzy σzz Principal directions (σ − λiI)ni = 0 1
  2. 2. Equilibrium equations ∂σxx ∂x + ∂σyx ∂y + ∂σzx ∂z + fx = 0 ∂σxy ∂x + ∂σyy ∂y + ∂σzy ∂z + fy = 0 ∂σxz ∂x + ∂σyz ∂y + ∂σzz ∂z + fz = 0 Displacements and strains Displacement gradient tensor u =    ∂u ∂x ∂u ∂y ∂u ∂z ∂v ∂x ∂v ∂y ∂v ∂z ∂w ∂x ∂w ∂y ∂w ∂z    where = e1 ∂ ∂x + e2 ∂ ∂y + e3 ∂ ∂z and u is the displacement vector Strain tensor ε = 1 2 u + ( u) T Strain components εxx = ∂u ∂x , εyy = ∂v ∂y , εzz = ∂w ∂z εxy = 1 2 ∂u ∂y + ∂v ∂x , εxz = 1 2 ∂u ∂z + ∂w ∂x , εyz = 1 2 ∂v ∂z + ∂w ∂y Saint-Venant compatibility equations εxx,yy + εyy,xx = 2εxy,xy, εyy,zz + εzz,yy = 2εyz,yz, εzz,xx + εxx,zz = 2εzx,zx εxy,xz + εxz,xy = εxx,yz + εyz,xx, εyz,yx + εyx,yz = εyy,zx + εzx,yy, εzx,zy + εzy,zx = εzz,xy + εxy,zz 2
  3. 3. Constitutive equations- Hooke’s law Generalized Hooke’s law- isotropic material         εxx εyy εzz 2εxy 2εyz 2εzx         = 1 E         1 −ν −ν 0 0 0 1 −ν 0 0 0 1 0 0 0 2(1 − ν) 0 0 sym. 2(1 − ν) 0 2(1 − ν)                 σxx σyy σzz σxy σyz σzx                 σxx σyy σzz σxy σyz σzx         = E (1 + ν)(1 − 2ν)         1 − ν ν ν 0 0 0 1 − ν ν 0 0 0 1 − ν 0 0 0 1−2ν 2 0 0 sym. 1−2ν 2 0 1−2ν 2                 εxx εyy εzz 2εxy 2εyz 2εzx         • Representation by the Lamé coefficients         σxx σyy σzz σxy σyz σzx         =         2µ + λ λ λ 0 0 0 2µ + λ λ 0 0 0 2µ + λ 0 0 0 2µ 0 0 sym. 2µ 0 2µ                 εxx εyy εzz εxy εyz εzx         where µ = G = E 2(1+ν) and λ = νE (1+ν)(1−2ν) are Lamé constants. Boundary conditions Stress and displacement BCs (see Cauchy stress formula) σT · n = ¯t, u = ¯u Energy principles Total potential energy functional ΠT P E = 1 2 V σ : εdV − V f · udV − S t · udS Principle of minimum potential energy δΠT P E(u) = 0 3
  4. 4. Classical beam and bar theories TPE functionals for bars and beams ΠT P E(u) = 0 EA 2 du dx 2 dx − 0 qxudx ΠT P E(w) = 0 EI 2 d2 w dx2 2 dx − 0 qwdx BCs for beams • Fixed end- w = 0 and dw dx = 0 • Free end- M = −EI d2 w dx2 = M0 and Q = −EI d3 w dx3 = Q0 • Pinned end- w = 0 and M = −EI d2 w dx2 = 0 Ritz method Displacement approximation functions u = u0 + n i=1 aiui, v = v0 + n i=1 bivi, w = w0 + n i=1 ciwi where the terms of u0, v0 and w0 are chosen to satisfy any non-homogeneous displacement BCs and ui, vi and wi satisfy the corresponding homogeneous displacement BCs. The TPE functional ΠT P E = ΠT P E(ai, bi, ci) Minimum conditions ∂ΠT P E ∂ai = 0, ∂ΠT P E ∂bi = 0, ∂ΠT P E ∂ci = 0 4

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