Nonlinear finite elements/Lagrangian finite elements

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Lagrangian Finite Elements[edit]

Two types of approaches are usually taken when formulating Lagrangian finite elements:

Total Lagrangian:
- The stress and strain measures are Lagrangian, i.e.,they are defined with respect to the original configuration.
- Derivatives and integrals are computed with respect to the Lagrangian (or material) coordinates $(\mathbf{X})$ .
Updated Lagrangian:
- The stress and strain measures are Eulerian, i.e.,they are defined with respect to the current configuration.
- Derivatives and integrals are computed with respect to the Eulerian (or spatial) coordinates $(\mathbf{x})$ .

The following 1-D examples illustrate what these approaches entail.

Consider the axially loaded bar shown in Figure 1.

Figure 1. Axially loaded bar

In the reference (or initial) configuration, the bar has a length

L 0

, an area

A 0 (X)

, and density

ρ 0 (X)

. A tensile force

T

is applied at the free end. In the current (or deformed) configuration at time

t

, the length of the bar increases to

L

, the area decreases to

A (X, t)

, and the density changes to

ρ(X, t)

Motion in Lagrangian Form[edit]

The motion is given by

${ x = \varphi(X, t) = x(X, t) ~, ~~\qquad X \in [0, L_0]~. }$

For the reference configuration,

$X = \varphi(X, 0) = x(X, 0) ~.$

The displacement is

${ u(X, t) = \varphi(X, t) - X = x - X ~. }$

For the reference configuration,

$u_0 = u(X, 0) = \varphi(X, 0) - X = X - X = 0 ~.$

The deformation gradient is

${ F(X, t) = \frac{\partial }{\partial X}[\varphi(X,t)] = \frac{\partial x}{\partial X} ~. }$

For the reference configuration,

$F_0 = F(X, 0) = \frac{\partial }{\partial X}[\varphi(X,0)] = \frac{\partial X}{\partial X} = 1 ~.$

The Jacobian determinant of the motion is

${ J = \cfrac{A}{A_0} F ~. }$

For the reference configuration,

$J_0 = \cfrac{A_0}{A_0} F_0 = 1 ~.$

Hingga nonlinier unsur / elemen Lagrangian terbatas

< Nonlinier terbatas elemen

Lagrangian Elemen Hingga[ sunting ]

Dua jenis pendekatan biasanya diambil ketika merumuskan Lagrangian elemen hingga:

Jumlah Lagrangian:
- Tegangan dan regangan adalah tindakan Lagrangian, yaitu, mereka didefinisikan sehubungan dengan konfigurasi asli.
- Derivatif dan integral dihitung sehubungan dengan Lagrangian (atau bahan) koordinat $(\ Mathbf {X})$ .
Diperbarui Lagrangian:
- Tegangan dan regangan adalah tindakan Eulerian, yaitu, mereka didefinisikan sehubungan dengan konfigurasi saat ini.
- Derivatif dan integral dihitung sehubungan dengan Eulerian (atau spasial) koordinat $(\ Mathbf {x})$ .

Contoh 1-D berikut menggambarkan apa pendekatan ini memerlukan.

Pertimbangkan bar dimuat secara aksial ditunjukkan pada Gambar 1.

Gambar 1. Secara aksial dimuat bar

Dalam referensi (atau awal) konfigurasi, bar memiliki panjang

L 0,

area

A ρ 0 (X),

dan kepadatan

0 (X).

Sebuah gaya tarik $T$ diterapkan pada akhir gratis. Dalam konfigurasi (atau cacat) saat ini pada waktu $t,$ panjang bar meningkat menjadi $L,$ daerah tersebut menurun ke $A (X, t),$ dan perubahan densitas untuk

ρ (X, t).

Gerak dalam Formulir Lagrangian[ edit ]

Gerakan ini diberikan oleh

${X = \ varphi (X, t) = x (X, t) ~, ~ ~ \ qquad X \ pada [0, L_0] ~. }$

Untuk konfigurasi referensi,

$X = \ varphi (X, 0) = x (X, 0) ~.$

Perpindahan dilakukan

${U (X, t) = \ varphi (X, t) - X = x - X ~. }$

Untuk konfigurasi referensi,

$u_0 = u (X, 0) = \ varphi (X, 0) - X = X - X = 0 ~.$

Gradien deformasi adalah

${F (X, t) = \ frac {\ partial} {\ partial x} [\ varphi (X, t)] = \ frac {\ partial x} {\ partial x} ~. }$

Untuk konfigurasi referensi,

$F_0 = F (X, 0) = \ frac {\ partial} {\ partial x} [\ varphi (X, 0)] = \ frac {\ partial x} {\ partial x} = 1 ~.$

Determinan Jacobian gerak adalah

${J = \ cfrac {A} {} a_0 F ~. }$

Untuk konfigurasi referensi,

$J_0 = \ {cfrac a_0} {} a_0 F_0 = 1 ~.$

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Hingga nonlinier elemen

Category:

Nonlinear finite elements

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Jumat, 02 Maret 2012