Micromechanics - notes of the course

MICROMECHANICS

Toughness: capability to stop propagation of a crack. It is related to energy.

Strength: maximum allowable stress.

Bone: hydroxyapatite (ceramic-like) + collagen (rubber-like).

High strength of composite / biomimetic materials. Different orders of magnitude for example between tension and compression, parallel and orthogonal direction. Relative sliding between layers under tension; buckling failure mechanism under compression.

TENSOR ANALYSIS

v tensor (on bold letter) on indicia notation v_i (i-th component of the vector)

Displacement: u , u_i is a component, Cartesian coordinates

=_[ u_1]

          u₂

          u₃

A.B = A^TB = A_iB_i + A₂B₂ + A₃B₃ = A_iB_i

         A

         └&bhsl;¨ℬ&kcirc;

         екк

A∧B          

C = _[ j  î  κ_]

          A₁  A₂  A₃

          B₁  B₂  B₃

Nⁿ => M_ij matrix

P_3×1 = ^N⊂3×3⊃; V_3x1 (row three column...)

Lb individual notation: P_i = M_ikV_k   P₄ = M₄V₄ + M₂V₂ + N₁₃V₃ (k=1) (k=2) (k=3)

N¹ + N² = N^jikM_ij

N³ = NⁱN²

N³    

Unit second order tensor I = δ_ij Kronecker delta         

i = j

ó

i ≠ j

M_ij: δ⊂ρ⊂ı_{∑sub>⊃⊂sub>⊂33&thrissa∑&msup;&macesⁱ′iiiVn}

M_ii = Xi

C_ijkl     strain

σ_ij = C_ijklE_kl   -> 9 equations in total, but we have to

σ_ij:symmetric tensor -> 6 linear independent equations (

Θ stress              y stress

differentigma_ij^a →  9 arithmetic angles in total

Continuum:    all physical points are occupied by solid matter

we are neglecting interspace between molecules and atoms, even with crystalline materials

         for each point (characterized by {x vector})

we can define a displacement vector "u".

We can define ε that is a tensor and a field E(x).

We can define the stress σ(x) that is a Cauchy tensor.

The body is not Φε, it is fixed.

ε -> displacement

+ + 6 linear independent equation of the second order tensor E(x)

5: unknown fields, -> 15 equations needed

Body forces b (N/m³) and surface loadings γ (sources).

If we know the geometry (V, Ω), the material properties (E, μ),

MICROMECHANICS

Toughness: capability to stop propagation of a crack. It is related to energy.

Strength: maximum allowable stress.

Bone: hydroxyapatite (ceramic-like) + collagen (rubber-like).

Mechanism of composite laminate materials. Different orders of magnitudefor example between tension and compression parallel and orthogonal direction. Relative sliding between layers under tension, buckling failure mechanism under compression.

TENSOR ANALYSIS

v tensor (on bold letter) on indicial notation v_i: i-th component of the vector.

Displacement: u_i, i is a component, Cartesian coordinates [u₁, u₂, u₃]

a·b = a_ib_i, a₁b₁ + a₂b₂ + a₃b₃ = a_ib_iEinstein notation saturated indexes. Σ is omitted.

a,b: vectors, c: vector

C = i j k A₁ A₂ A₃ B₁ B₂ B₃

• ^j