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 + collagen (like rubber + rice).

Microstome of composite/bouvrage 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

i^th tensor (on bold letter) on indicio notation v_i (i-th component of the vector)

Displacement: x_i, u_i = a component Cartesian coordinates x = [u₁ u₂ u₃]

A·B = A^TB = A_iB₁ + A₂B₂ + A₃B₃ = A_iB_i

ΔAB

c = j i k A₁ A₂ A₃ B₁ B₂ B₃

N^o ⇒ N_ij metric

P_3x1 = N_3x3V_3x1

N = Nⁱ_ju_iu_k

N³ = Nⁱ N^j N²

Unit second order tensor I = δ_ij Kronecker delta I = 1 (i = j), I = 0 (i ≠ j)

N_ij δ_i^κ = N_ii + N_1δ1κ + N_2δ1κ + N_3δ1κ + N_2δ2κ + N_3δ2κ + N_3δ3κ + N_3δ31 + N_3δ32 + N_3δ3κ + N_3δ37 + N_3δ3 + N_3δ33

C_ijak (stress), E_km strain

σ_ij = C_ijakE_km

• 9 equations in total
• 6 linear independent equations

Continuum...

All physical points are occupied by solid matter. We are neglecting interspace between molecules and atoms, even within crystalline material.

For each point (characterized by the x vector) we can define a displacement vector.

We can define E^t that is a tensor and a field E(x). Cp field

We can define the stress g(x) that is a calling tensor. The body is not floating, it is fixed.

1. Δ displacement
2. P_i
3. σ linear independent equation of the second order tensor E^t

∇E, σ_i unknown fields → 15 equations needed

Body forces b (N/m³) and surface tractions (f sources)

we know the geometry (V, Γ), the materials property (E, λ)

Hypothesis: small strain/small displacement -> some of the governing equations can be linear. Materials are linear-elastic (linear relationship between stress and strain).

1. Equilibrium equations: linear, establish a relationship between stress and displacement, partial differential equations. 3
2. Kinematic equations: establish a relationship between strain and displacements. Always partial differential equation. 6
3. Constitutive equations: establish a relationship between stress and strain depending on the materials. 6

Equilibrium Equations

σ_i,j + b_i = 0

Neglecting the dynamics effect (quasi-static conditions)

i: 1 When you have 3 equations, i is repeated, summation is implied.

σ₁₁ / ∂x₁ + σ₁₂ / ∂x₂ + σ₁₃ / ∂x₃ + b₁ = 0

i: 2.

σ₂₁ / ∂x₁ + σ₂₂ / ∂x₂ + σ₂₃ / ∂x₃ + b₂ = 0

i: 3.

σ₃₁ / ∂x₁ + σ₃₂ / ∂x₂ + σ₃₃ / ∂x₃ + b₃ = 0

Boundary conditions:

^σn = f (3.89)

σ_ijn_j = f_j on Γ

▶ Slide: Basic Solid Mechanics

Linear governing equation → superposition principles ... small displacements, small strains and linear-elastic solids.

Constitutive equation: explicit relationship between stress and strain depending on the kind of materials.

Kinematic Equations and Constitutive Law

ε_ij = ½(∂u_i/∂x_j + ∂u_j/∂x_i)

F - F

(u_i = u_i on the boundary)

Shear strain (E12,...)

Displacement gradient φ_i = δu_i/δx_j ε = ½ (φ + φ^T)

Elastic body:

1. natural state - stresses and strains = 0 (rest). Natural states are not so easy to be recognized, as some objects don't have a natural state. For example, body forces even when they're out of the body without blood on a table.
2. External forces → increase of stresses and strains (in a non-linear way)
3. External forces to zero → same path of loading

Linear mat. can be isotropic or not...

Linear elastic materials: σ_ij = l_ijklε_kl (like 4^th order tensor)

When _Ef = 1 equal mixture _Cr = _Nr

_C⁰ _D⁰ _F⁰

lim the square cosθ _{Ef Df} (transversal direction)

Heterogeneity → disturbance in inter/instrain fields we estimate the average strain within some kind of materials → we can determine macroscopic strain of the system ⟨_Ef⟩_ij, averaged on local domain of every material phase: _Eⁱ = i,j dV All particles are perfectly bonded to the surrounding matrix (no detaching on hyp).

Linearity → governing eqns → linear relationship between _E^f and _F through Ar

_E^f = _E^c + _Cr ⟨_E⟩ C_i: volumetric fraction C_ot + Z_f = 4

matrix contribution: _E^c = _Z + _cfE ⟨_E⟩ = _E + _Zf Cf _Ar +

(the same for stresses)

Effective elastic tensor is a 1^st order tensor

average stiffness tensor

_Lf = ?

LD if no static mismatch between particles and matrix like: L_f → L₀ → trivial selection, the second term has gone to zero.

1) Fictious problem: is the same of the real problem: ideal symmetry as assumption. Considering particles far apart from each other number interaction between them. With this approx

⟨_E^f⟩ = _E + _E + _f → _Ef = _TF e_ar = _dF + If

Shelby method (low Cr: dilute systems) (dual approach with stresses)

ex: ^{knowing geometries and Lo} → [ →] D → A =

Shelby method with spherical particles: isotropic material.

2) Mori-Tanaka method: L₀Lo0 → E₀i same strain in the matrix

weakly coupling interaction between particles → L₁f explains the same

⟨ _E^F ⟩ = _E+ _E2 [ _Co + _{Zf Cf ] ^TF ⟨E⟩}

At strain concentration tensor (dual approach for stresses)

even here with spherical particles we find an isotropic material. The method can be used for every value of _Cr. Quite accurate (much more than Shelby method) used very cautiously.

3) Self-consistent method (the best one). Mechanical properties of the infinite fictitious matrix and the macroscopic elastic properties L = _E^F _f = → but we want to calculate it. The equation depends on the solution itself → iterative method: first attempt L₀ → etc --> Also the Shelby tensor is unknown.

L(_E, _i)

^{1 effective Poisson’s ratio}

even L and ^{1 are implicit equation}

^interactions

Resume: 1) Ar = [^T_S] 2) Ar = IF[^{… []}] 3) Ar = TF

When particles are not an ellipsoidal one we have complex anisotropic properties → complex models: use TEN to answer the strain concentration and tensor → then numerically you can find elastic tensor and so on.

Geometrical shape ratio n=not being ellipsoid →possibility to use analytical method.