Appunti di Modeling of space structures notes (Space structures)

[SPACE STRUCTURES]

MODELING ON SPACE STRUCTURES

NOTATION

VECTORS

A vector is associated with a no of components and the set of a base.

V = V₁ e₁ + V₂ e₂ + V₃ e₃ in 3D Space

where i = 1, 2, 3 -> Versor

they build up an orthonormal basis

ORTHONORMAL -> ORTHOGONALe_i ⋅ e_j = 0

• e₁ ⋅ e₂ = 0
• e₁ ⋅ e₃ = 0
• e₂ ⋅ e₃ = 0

-> NORMALe_i ⋅ e_i = 1 since they're properly normalized

V = _Σi=1³ V_i e_i = V_i e_i

Einstein notation

RULE: Whenever an index is repeated then there is a SUMMATION implied

V_i e_i -> No Σ

V_m e_n -> Σ

[SPACE STRUCTURES]

MODELING ON SPACE STRUCTURES

NOTATION

VECTORS

A vector is associated with a number of components and the set of a base.

V = V₁ e₁ + V₂ e₂ + V₃ e₃ in 3D space

• e_i i=1,2,3 → Vectors
• Their buildup is an orthonormal base

ORTHONORMAL → ORTHOGONAL → NORM

• e_i • e_j = 0
• e₁ • e₃ = 0
• e₂ • e₃ = 0
• e_i • e_i = 1 (Since they are properly normalized)

V = Σ_i=1³ V_i e_i = V_i e_i

Einstein Notation

Rmk Whenever an index is repeated then there is a summation implied.

V_i e_i → No Σ

V_m e_m → Σ

Rmk

The letter used as index is not important, it’s the relative position that matters → it’s possible to rename indices but never change their positions.

Rmk

The bounds of the Σ are expressed only if there is ambiguity, else they are not implicit in the context.

V_i → component

V = V_i e_i

The components are the projection of V onto the basis, if we change the orthonormal basis obs the components will be ≠ BUT the vector doesn’t → change and it’s independent from the choice of the reference system.

V ↓ invariant vector depends on

|V| = { V₁ V₂ V₃ } → method vector

depends on the reference system

When we formulate a physical pb we don’t want to be dependent on the reference system

SCALAR PRODUCT

u • V = w

where w = |U| |V| cosθ

Vector ≡ 1st Order Tensor

Scalar product lowers the order of the tetradents (from 1 to 0).

U · V

the SP converts the two tetradents to a pure leg, remains -> 0 order tensor

W = u · v = (u_ie_i) · (v_ie_j) = u_iv_jE - e_j = ×

the letters must be ≠ ->if u and v are defined the sameletter it can have a unique Σ

That's possible to havescalar quantities but notvectorial tensors

then as we need to keep the tone ->cause j = i -> 0so relabel j with i

• × = u_kv_jδ_ij = u_kv_j = u_iv_i + u₂v₂ + u₃v₃ =

e_ic_ij = {1 if i = j0 if i ≠ j

μ_it + v_jv = ϕ*v^j₂i

in matrices

where Kronecker Delta

• δ_ij = {1 if i = j0 if i ≠ j

2nd Order Tensor

Tool that creates a vector from another vector

b = A^α_q

σ = ϕ · u

stress vector normal stress tensor vector direction

RHM In 2nd order tensors we have 2 numbers & 2 basis vectors

= A_ij e_i e_j

⊗ = TENSOR PRODUCT —> we don't need the legs, noted in We have 2 legs and not 0.

RHM (μ⊗ν) · w = (w · ν)μ

2nd order tensor upper has gives the the the magnitude direction

(μ_⟂ ⊗ν_‖) · w = (w · ν_‖)μ_⟂

(w_‖·y) A

dyadic representation of a 2nd order tensor

= A₁₁ e₁ ⊗ e₁ + A₁₂ e_i ⊗ e₂ + A₁₃ e₁ ⊗ e₃ + A₂₁ e₁ e₂ ⊗ e_i + + A₃₅ e₃ e₂ e₃