Appunti Computational Mechanics and inelastic structural analysis

THE STRUCTURAL PROBLEM

Given:- 3D continuous body w/ volume V and surface S- body forces F- surface forces f- assigned displacement ξ 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:

EQUILIBRIUM

(Partial derivatives notation is not transcribed)

σ_ij,j + F_i = 0   ∀x ∈ Vσ_ijm_j = f_i   ∀x ∈ S_fσ_Mx + τ_xyM_y + τ_xzM_z = F_x

1 THE STRUCTURAL PROBLEM

Given:

• 3D continuous body w/ volume V and surface S
• body forces F
• surface forces f
• assigned displacement ξ on the constrained surface

We wanna define:

• displacement field ξ
• stress field ψ
• strain field ε

Hypothesis:

• small strains & displacements
• isothermal process
• no dynamics
• linear elastic (isotropic) material behavior

Governing equations:

1. EQUILIBRIUM

∂σ_x/∂x + ∂τ_xy/∂y + ∂τ_xz/∂z + F_x = 0

σ_ijj + F_i = 0   ∀x ∈ V

σ_ijm_j = f_i   ∀x ∈ S_f

σ_xx + τ_xy M_y + τ_xz M_x = F_x

2

• KINEMATIC COMPATIBILITY

• Σ_ij = 1/2 (S_ij + S_j,i) on V
• S_i · S_i on Su

• CONSTITUTIVE LAW

• σ_ij = d_ijkl E_kk
• A · = ᾽
• _j = c_ijkl σ_j

If the POTENTIAL ELASTIC ENERGY exists then d and c must be symmetric & positive definite.

In the case of an ELASTIC & ISOTROPIC material it is possible to express d and c by using the Lamé’s constants:

• -1 < v < 1/2
• λ = vE / ((1+v)(1-2v))
• μ = G · E / (2(1+v))
• λ, E [F/A]

d =

• [λ+2μ λ λ] [0]
• [λ λ+2μ λ] ~ [0]
• [λ λ λ+2μ] [0]
• [0 μ 0 0]
• [0 μ 0 0]
• [0 0 μ]

Meaning of v < 0

First 2 equations of σ = d · :

• ε_x = λ/E (σ_x - v (σ_y + σ_z))
• ε_y = λ/E (σ_y - v (σ_x + σ_z))

- lets now consider an UNIAXIAL STRESS STATE along the x-direction

    | _x = Σ_x / E

   {

    | _y = Σ_y / E (-νΣ_x)

- by doing the ratio btw. _y and _x we obtain

    _y / _x = -νΣ_x / E / Σ_x / E = -ν

=> ν can be interpreted as the ratio btw. the TRANSVERSAL STRAIN and the one along the direction of the stress.

     A NEGATIVE ν means that when there is an elongation of the specimen there is an EXPANSION in the y-direction (normally there is a contraction)

These materials don’t exist in nature and they are called AUXETIC MATERIALS

                     meta materials

PRINCIPLE OF VIRTUAL WORK

The PVW is a NECESSARY and SUFFICIENT cond for rigid body balance.

Under the hp of smooth and bilateral constraints, the work done by an acting force must be equal to zero ∀ virtual displacement.

Definitions:

1. STATICALLY ADMISSIBLE (or equilibrated) set (✱)

   A statically admissible set is a family of σ_ij, F_i^∗, f_i^∗ that satisfy the EQUILIBRIUM CND:

   { σ_ij,j + F_i^∗ = 0   ∂v σ_ijm_j = f_i^∗   ∂Sf }

2. KINEMATICALLY ADMISSIBLE SET (Δ)

   A kin admissible set is a family of s_i, Ė_ij, ŝ_i that satisfy the kin cnd.

   { Ė_ij =