Appunti Computational Mechanics and inelastic structural analysis

1. THE STRUCTURAL PROBLEM

• Given:
  • 3D continuous body w/ volume V and surface S
  • Body forces F
  • Surface forces f
  • Assigned displacement on the constrained surface

• We wanna define
  • Displacement field
  • Stress field
  • Strain field

• Hypothesis
  • Small strains & displacements
  • Isothermal process
  • No dynamics
  • Linear elastic (isotropic) material behavior

• Governing equations

1. Equilibrium
   • \(\sigma_{ij,j} + F_i = 0 \quad \forall x \in V\)
   • \(\sigma_{ij} n_j = f_i \quad \forall x \in S_f\)
   • \(\frac{\partial \sigma_{xx}}{\partial x} + \frac{\partial \sigma_{xy}}{\partial y} + \frac{\partial \sigma_{xz}}{\partial z} + F_x = 0\)
   • \(\sigma_{Mx} + \tau_{xy} M_y + \tau_{xz} M = 0\)

2

2) KINEMATIC COMPATIBILITY

\( \{ \begin{array}{ll} \varepsilon_{ij} = \frac{1}{2} (s_{ij} + s_{ji}) \ \ \ & \text{on } V \\ s \cdot = \dot{s_i} \ \ \ & \text{on } Su \end{array} \)}

3) CONSTITUTIVE LAW \(\{ \begin{array}{ll} \sigma_{ij} = d_{ijkl} \varepsilon_{kl} \ \rightarrow \ \underline{\underline{\sigma}} = \underline{\underline{d}} \ \underline{\underline{\varepsilon}} \\ \varepsilon_{ij} = c_{ijkl} \sigma_{ij} \ \rightarrow \ \underline{\underline{\varepsilon}} = \underline{\underline{c}} \ \underline{\underline{\sigma}} \end{array}\)

\(\cdot\) If the POTENTIAL ELASTIC ENERGY exists, then \(\underline{\underline{s}}\) and \(\underline{\underline{\varepsilon}}\) must be symmetric & positive definite.

\(\cdot\) In the case of an ELASTIC & ISOTROPIC material

it is possible to express \(\underline{\underline{\sigma}}\) and \(\underline{\underline{\varepsilon}}\) by

using the Lame's constants

• \(\lambda = \frac{\nu E}{(1+\nu)(1-2\nu)} \ , \ E \left[ \frac{F}{A} \right] \)
• \(\mu = G = \frac{E}{2(1+\nu)}\)

\(-1 < \nu < \frac{1}{2}\)

• \(\underline{\underline{d}} = \begin{bmatrix} \lambda + 2\mu & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda + 2\mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda + 2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix} \ , \ \underline{\underline{c}} = \frac{1}{E}\begin{bmatrix} 1 & -\nu & -\nu & 0 & 0 & 0 \\ -\nu & 1 & -\nu & 0 & 0 & 0 \\ -\nu & -\nu & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{2(1+\nu)}{2(1+\nu)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{2(1+\nu)}{2(1+\nu)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{2(1+\nu)}{2(1+\nu)} \end{bmatrix}\)

\(\ast\) Meaning of \(\nu < 0\)

• \(\rightarrow\) First 2 equations of \(\underline{\epsilon} = \underline{\underline{d}} \cdot \epsilon\)
• \(\varepsilon_x = \frac{1}{E} \{ \sigma_x - \nu(\sigma_y + \sigma_z)\}\)
• \(\varepsilon_y = \frac{1}{E} \{ \sigma_y - \nu(\sigma_x + \sigma_z)\}\)

Proof of (3):

• Internal work definition + kinematic compatibility condition

⇒ _V∂_j^*ŷ_j dV

ŷ_j = 1/2 (s_j + s_j,c)⇒ _t = _V∂_ij^* 1/2 (S_j + S_j,c) dV

• due to the symmetry of ∂_ij and the implicit summation we can write

⇒ ℓ_i = _V∂_ij^* S̅_ij dV

• Divergence theorem:

V (∂_ij,jSi + ∂_ijS_ij) dV = S ∂_ijm_js_i dS

⇒ ℓ_i = -_V∂_ij,js_idV + S ∂_ijm_js_idS

• Equilibrium condition

{∂_ij,j = -f_i^* on V∂_ijm_j = f_i^* on S_u,se

⇒ ℓ_i = V f_i^*s_idV + ∂_Sf f^*_is_idS + S_u,se∂_ijm_js̅_cdS - L_e

PRINCIPLE OF VIRTUAL DISPLACEMENT (PVD)

• Let's consider a REAL STATIC SET of a REAL BODY and a VIRTUAL KINEMATIC SET.

• The VIRTUAL KIN SET must satisfy the following and:
• Si = Si_i
• Si + δSi = Si

⇒ δSi = 0 on Su

_Eij + δE_ij = 1/2 ((Si_i − δSi_i)_,j + (Si_j + δSj_j)_,i)

due to linearity

⇒ |δE_ij = 1/2 (δSi_,j + δSj_,j) on V

E^* set of kinematic compatible field with zero data on the boundary

Principle:

The condition δti - δte = 0 implies EQUILIBRIUM on V and St

∀ δS, δE_ij, δSi

• ∫_v δti ∙ τij δE_ij dV
• ∫_su δte ∙ F_i δS_i ds dV
• + ∫_v fi δSi dV

Proof:

δti - δte ∙ ∫_v τij δE_ij dV - ∫_su Fi δSi ds dV

•Recalling the kinematic compatibility δE_ij = 1/2 (δS_i,j + δSj_,i)

3D ELASTIC BEHAVIOR

1) ELASTIC POTENTIAL

w(ε) = ∫ δt dε

dw = δt dε → δt = dw/dε

2) COMPLEMENTARY ELASTIC POTENTIAL

w_c(δ) = ∫ εt dδ

dw_c = εt dδ → εt = dw_c/dδ

By considering a body with volume V, we can define:

• TOTAL ELASTIC POTENTIAL Ω(ε) = ∫_V w(ε) dV

In case of linear elastic behavior w = ½εt Eε

Ω(ε) = ½ ∫_V εt dε Eε dV

• TOTAL COMPLEMENTARY ELASTIC POTENTIAL Ω_C(δ) = ∫_V w_c(δ) dV

In case of linear elastic behavior Ω_C(δ) = ½ ∫_V δt E^-1 δ dV

OSSERVAZIONE:

ε = E^-1 δ → equivalent of δ = Eε in 1D

δ = dw/dε = dεt dε → d = d²w/dε²t

d is a matrix containing the 2^nd derivatives of w

• Recalling the Schwartz’s theorem

∂²f(x,y)/∂y∂x = ∂²f(x,y)/∂x∂y → d = M_sym

Proof

• Let consider a stress field σ* and a variation δσ* + dσ*
  • σ* = ξ
  • δσ* st ξ ∈ ξ_n
  The variations δσ* and δt* must satisfy also
  • δσ* = 0 on V
  • δt* = 0 on Sf
• The VARIATION of the TCE is
  • δTCE = TCE(σ* + δσ*) - TCE(σ*)

• = -^½_V (σ* + δσ*) ξ (σ* + δσ*) dV = ∫_Su (σ* + δt*)^T ξ s dS
• -^½_V σ*^T ξ σ* dV - ∫_Sc t*^T ξ s dS

• = ^½_V σ^τ ξ σ* dV ±¹_2Vf σ^τ ξ δσ* dV ±¹₂ δσ*^τ ξ σ* dV +
• + ^½_Vf δσ*^τ ξ δσ* dV - ∫_Su δξ*^τ s dS ± ^½_2V* ξ σ* dV

• = ∫_V ξ*^τ ξ δσ* dV - ∫_Su δξ*^τ s dS + ^½_2V ν δσ*^τ ξ δσ* dV