review of continuum mechanics: kinematics fem objectives of chapters 7-8 review fundamentals of 3d...

Nonlinear FEM

Review ofContinuum Mechanics:

Kinematics

7

NFEM Ch 7 – Slide 1

Nonlinear FEM

Objectives of Chapters 7-8

Review fundamentals of 3D continuum mechanicsas needed for developments in geometrically nonlinear finite element methods in Chapters 9 and beyond.

Students already familiar with continuum mechanics should simply skim it to absorb notation.

NFEM Ch 7 – Slide 2

Nonlinear FEM

Notational Systems in Mechanics

Indicial: powerful, general, obscures physics ("index soup")

Direct: compact, physically transparent, limited

Matrix: oriented to FEM, occasionally confusing

Full: ambiguity free but verbose; sometimes helps in programming

NFEM Ch 7 – Slide 3

Nonlinear FEM

Notational System Examples

Dot product of two vectors:

Differential equilibrium equations:

Total force residual as gradient of Total Potential Energy function:

ai bi︸︷︷︸indicial

= a.b︸︷︷︸direct

= aT b︸︷︷︸matrix

= a1b1 + a2b2 + a3b3︸ ︷︷ ︸full

.

σi j, j + bi = 0︸ ︷︷ ︸indicial

, ∇σ + b = 0︸ ︷︷ ︸direct

, DT σv + b = 0︸ ︷︷ ︸matrix

,

∂σ11

∂x1+ ∂σ12

∂x2+ ∂σ13

∂x3+ b1 = 0, plus 2 more︸ ︷︷ ︸

full

.

ri = ∂�

∂ui

def= �,i︸ ︷︷ ︸indicial

, r = ∇�︸ ︷︷ ︸direct

, r = ∂�

∂u︸ ︷︷ ︸matrix

, r1 = ∂�

∂u1, r2 = ∂�

∂u2, . . .︸ ︷︷ ︸

full

.

NFEM Ch 7 – Slide 4

Nonlinear FEM

Particle Trajectory, Body Motion

(to be completed)

NFEM Ch 7 – Slide 5

Nonlinear FEM

Expression of Motion in Control-State Space

(to be completed)

NFEM Ch 7 – Slide 6

Nonlinear FEM

Configuration as "Snapshot" ofBody Moving in Control-State Space

(to be completed)

NFEM Ch 7 – Slide 7

Nonlinear FEM

Name Alias Definition Equilibrium Identification Required?

Admissible A kinematically admissible configuration No

Perturbed Kinematically admissible variation of No of an admissible configuration

Current Deformed Any admissible configuration taken during the No or Spatial analysis process. Contains all others as special cases

Base Initial The configuration defined as the origin of Yes , or Undeformed displacements. Strain free but not necessarily Material stress free

Reference Configuration to which stepping computations TL,UL: yes TL: UL: in an incremental solution process are referred CR: no, yes CR: and

Iterated Configuration taken at the kth iteration No of the nth increment step

Target Equilibrium configuration accepted Yes after completing the nth increment step

Corotated Shadow Body- or element-attached configuration obtained No Ghost from through a RBM (CR description only)

Aligned Preferred A fictitious body ot element configuration aligned No Directed with a particular set of axes (usually global axes)

Distinguished Configurations (Table in Ch 7)

Blue: used only in theoretical & applied mechanics. Yellow: used only in computational mechanics. Green: used in both

NFEM Ch 7 – Slide 8

Nonlinear FEM

Three Important Configurations for Geometrically Nonlinear Analysis

Current Configuration

Reference Configuration(identifier depends on

kinematic description chosen)

Base Configuration , or

or

NFEM Ch 7 – Slide 9

Nonlinear FEM

Kinematic Descriptions Used in Geometrically Nonlinear Analysis in Computational Solid and Structural Mechanics

Name Acronym Definition Primary applications

Total Lagrangian TL Base and reference configurations Solid and structural mechanics with finite coalesce and remain fixed throughout but moderate displacements and strains. the solution process Primarily used for elastic material. Unreliable for flow-like behavior or topology changes Updated Lagrangian UL Base configuration remains fixed but Solid and structural mechanics with finite reference configuration is periodically. displacements and possibly large strains. updated. Most common update strategy Handles material flow-like behavior well, is to set reference configuration to last (e.g., forming processes) as well as topology converged solution changes (fracture) Corotational CR Reference configuration is split into base Solid and structural mechanics with arbitrarily and corotated. Strains and stresses are large finite motions, but small strains and measured from corotated to current, while elastic material behavior. Extendible to nonlinear base configuration is maintained as materials if inelasticity is localized so most of reference to measure rigid body motions structure stays elastic.

All three descriptions are Lagrangian: computations are always referred to a previous configuration (base and/or reference)

Eulerian formulations, which are common in fluid mechanics, are not popular in solid and structural mechanics

NFEM Ch 7 – Slide 10

Nonlinear FEM

Total Lagrangian (TL) Kinematic Description

Current Configuration

Base and Reference Configuration =

TOTAL LAGRANGIAN (TL)Kinematic Description

NFEM Ch 7 – Slide 11

Nonlinear FEM

Updated Lagrangian (UL) Kinematic Description

Current Configuration

Base Configuration

UPDATED LAGRANGIAN (UL)Kinematic Description

B

nReference Configurationupdated after each incremental step

NFEM Ch 7 – Slide 12

Nonlinear FEM

Corotational (CR) Kinematic Description

Current Configuration

Base Configuration

COROTATIONAL (CR)Kinematic Description

0

RCorotated Configurationa rigid motion of the base configuration

NFEM Ch 7 – Slide 13

Nonlinear FEM

Global Coordinate Systems

V

u = x − X

X, x Y, y

Z , z

X ≡ x0

Base configuration (for drawing simplicity, assumed tocoalesce with reference, as in TL)

Current configuration

NFEM Ch 7 – Slide 14

Nonlinear FEM

Global Coordinate Systems

V

u = x − X

X, x Y, y

Z , z

X ≡ x0

Base configuration (for drawing simplicity, assumed tocoalesce with reference, as in TL)

Current configuration

X or {X, Y, Z} : material coordinate framex or {x, y, z} : spatial coordinateTaken be identical in this course

NFEM Ch 7 – Slide 15

Nonlinear FEMDisplacement Vector Field

x = X + u

u =[ u X

uYuZ

]=

[ x − Xy − Yz − Z

]= x − X

V

u = x − X

X, x Y, y

Z , z

X ≡ x0

NFEM Ch 7 – Slide 16

Nonlinear FEM

Deformation Gradient and Its Inverse

F = ∂(x, y, z)

∂(x, y, z)

∂(X, Y, Z )

∂(X, Y, Z )

=

∂x∂ X

∂x∂Y

∂x∂ Z

∂y∂ X

∂y∂Y

∂y∂ Z

∂z∂ X

∂z∂Y

∂z∂ Z

F−1 = =

∂ X∂x

∂ X∂y

∂ X∂z

∂Y∂x

∂Y∂y

∂Y∂z

∂ Z∂x

∂ Z∂y

∂ Z∂z

dx =[ dx

dydz

]= F

[ d XdYd Z

]= F dX, dX = F−1 dx

Can be used to relate differentials of spatial and material global frames:

NFEM Ch 7 – Slide 17

Nonlinear FEM

Displacement Gradient and Its Inverse

G = F − I =

∂x∂ X − 1 ∂x

∂Y∂x∂ Z

∂y∂ X

∂y∂Y − 1 ∂y

∂ Z∂z∂ X

∂z∂Y

∂z∂ Z − 1

=

∂u X∂ X

∂u X∂Y

∂u X∂ Z

∂uY∂ X

∂uY∂Y

∂uY∂ Z

∂uZ∂ X

∂uZ∂Y

∂uZ∂ Z

= ∇u

J = I − F−1 =

1 − ∂ X∂x

∂ X∂y

∂ Z∂x

∂Y∂x 1 − ∂Y

∂y∂Y∂z

∂ Z∂x

∂ Z∂y 1 − ∂ Z

∂z

=

∂u X∂x

∂u X∂y

∂u X∂z

∂uY∂x

∂uY∂y

∂uY∂z

∂uZ∂x

∂uZ∂y

∂uZ∂z

NFEM Ch 7 – Slide 18

Nonlinear FEM

Example 1: Simple ExtensionY, y

AA0

BB0

0

Z, z X, x

0L L

Reference-to-current motion:

Displacements:

Deformation and displacement gradients:

x = λ1 X y = λ2 Y z = λ3 Z

u X = x − X = (λ1 − 1)X uY = y − Y = (λ2 − 1)Y uZ = z − Z = (λ1 − 1)Z

F =[

λ1 0 00 λ2 00 0 λ3

]G =

[λ1 − 1 0 0

0 λ2 − 1 00 0 λ3 − 1

]

NFEM Ch 7 – Slide 19

Nonlinear FEMExample 2: Pure Shear

X, x

Y, y

A BA0 B0

γ Y

θγ = tan θ

0

Z, z

x = X + γ Y y = Y z = Z

u X = γ Y uY = uZ = 0

F =[

1 γ 00 1 00 0 1

]G = F − I =

[0 γ 00 0 00 0 0

]

Reference-to-current motion:

in which γ = tan θ is called the amount of shear. Material fibers aligned withX translate horizontally and do not change length, so the motion is isochoric.

Displacements:

Deformation and displacement gradients:

NFEM Ch 7 – Slide 20

Nonlinear FEM

Example 3: Combined Translation, Stretch and Rotation

L

X, x

Y, y

(3) plane rigid rotation by angle ψ (positive CCW)

(1) rigid translation by u , uXC YC

Y0H

H

C

ψ

XCu

YCu

0

Z, zC0

Y

(2) stretching by λ = L /L , λ =H /H , λ = 1

1

2 3

0Y0

Y

0L

NFEM Ch 7 – Slide 21

Nonlinear FEMExample 3: Combined Translation,Stretch and Rotation (Cont'd)

Using multiplicative composition to combine stretching and rigid rotation, the combined motion can be expressed in the matrix form

in which c= cos ψ and s = sin ψ. Expanding gives

Since matrix products do not necessarily commute, the order in which stretch & rotation is applied is important. If the sequence were reversed: rotate-then-stretch, the motionwould be different unless λ = λ . The deformation and displacement gradients are

The rigid translation does not affect these tensors, but the rigid rotation does.

1 2

X, x

Y, y

C

0

Z, zC0

[xyz

]=

[c −s 0s c 00 0 1

][λ1 0 00 λ2 00 0 1

][X − u XC

Y − uY C

Z

]

x = λ1 c (X − u XC ) − λ2 s (Y − uY C )y = λ1 s (X − u XC ) + λ2 c (Y − uY C )z = Z

F =[

λ1 c −λ2 s 0λ1 s λ2 c 0

0 0 1

]G = F − I =

[λ1 c − 1 −λ2 s 0

λ1 s λ2 c − 1 00 0 0

]

NFEM Ch 7 – Slide 22

Nonlinear FEM

Example 4: Non-Homogeneous Bar Extension

X, x

Y, y0

Z, zC =30 0 C 310 120

2

0L0L /2 0L /2

1u3u

2u

1L /2 2L /2

L /2L

L /2

This is worked out in Chapter 7 as Example 7.4

NFEM Ch 7 – Slide 23