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