Deformation

Huamin Wang

Deformation Mode

• A deformation mode specifies how an object is deformed. Cloth and paper:

Compression Tension Shearing Bending

Planar DeformationOut-of-Plane Deformation

Deformation Mode

• A deformation mode describes how an object is deformed. 3D soft bodies:

Three stages in Deformation

Deformation

Forc

e

Elastic DeformationPlastic Deformation

Fracture

If there is no force, the object returns to its rest shape

The object cannot return to its rest shape. (The shape change becomes permanent…)

Linear Elastic Deformation

DeformationFo

rce

Elastic DeformationForce is linearly proportional to deformation.

The deformation-force curve is a straight line.

Commonly known as Hooke’s law!

A 1D Spring

L0 L0 d

Rest Deformed

F = −k(L − L0 )

F = −C LL0

−1

Constant Strain

What is Strain?• A strain is the description of the deformation.• When strain is zero, no deformation (L=L0).• When strain gets bigger, larger deformation.• There are many ways to describe the deformation. • So the strain definition is not unique.

LL0

−1

Cauchy Strain

lnLL0

Log Strain

L2

L02 −1

Green Strain

1−L0

2

L2

Euler-Almansi Strain

Ideal Metal Spring Commonly used in 2D/3D

2D Strain• How to define strain for 2D deformation?• Any point x in the rest shape is deformed into the new

position y: y=Fx+b.• F is a matrix (Deformation Gradient).• b is a translation vector.• If F is orthonormal, what deformation is it?

x0 x1

x2

y0

y1

y2

Basic Assumption:y=Fx + b

2D Strain• The goal is to get the deformation gradient matrix F.

x0 x1

x2

y0

y1

y2

Basic Assumption:y=Fx + b

Since y0=Fx0 + by1=Fx1 + by2=Fx2 + b

y1-y0=F(x1-x0), y2-y0=F(x2-x0)

[y1-y0, y2-y0]=F[x1-x0, x2-x0]

F=[y1-y0, y2-y0] [x1-x0, x2-x0]-1

2D Strain• Once we get F, we can get the Green strain as:

• Intuitively, it is the 1D green strain in u, v, and uv direction.

x0 x1

x2

y0

y1

y2

Basic Assumption:y=Fx + b

G =12

FTF − I( )= εuu εuv

εuv εvv

u

v

uv

2D Stress• Question: what about force?• Force exists in every direction and it varies.• A stress S is a matrix that describes of forces per

unit area.

y0

y1

y2

f f

f

f

Strain-Stress Conversion

• 2D Hooke’s Law!

G =εuu εuv

εuv εvv

S =σ uu σ uv

σ uv σ vv

Strain Stress

σ uu

σ vv

σ uv

=

2µ + λ λλ 2µ + λ

µ

εuu

εvv

εuv

μ and λ are called Lamé constants.

From Stress to Force

x0 x1

x2

y0

y1

y2

y=Fx + b

d02

d10

d21

• Define dij as a vector perpendicular to edge ij:

• The force applied on vertex 0 is: dij =

−(yi − yj )

xi − x j

f0 = −12

FS d10 + d02( )

Cloth Animation

• A cloth piece is made of triangles.• In the plane of each triangle k, we

assume y=Fkx+bk.• After computing the force in the

2D plane, covert it back into the 3D space.

• This idea is known as Finite Element method.

• A triangle is a 2D element.

3D Soft Body Simulation• The same idea, except for everything is now

in 3D.

• Deformation gradient F, strain G, and stress Sare all 3-by-3 matrices!

x0 x1

x3

x2

y0

y1

y3

y2y=Fx + b

F=[y1-y0, y2-y0, y3-y0] [x1-x0, x2-x0, x3-x0]-1

From Stress to Force

• Define dijk as a vector perpendicular to triangle ijk:

• The force applied on vertex 0 is: dijk =

12

x j − xi( )× xk − xi( )

f0 = −13

FS d021 + d013 + d032( )

x0 x1

x3

x2

y0

y1

y3

y2y=Fx + b

2D Algorithm• For every vertex i

– fi=mg;• For every triangle ijk

– F=[y1-y0, y2-y0] [x1-x0, x2-x0]-1 ……… Deformation Gradient

– G=(FTF-I)/2; ……… Green Strain

– S=Strain_To_Stress(G) ……… Stress

– Fi+=-FS(dij+dki)/2 ……… get the force!

– Fj+=-FS(dij+djk)/2 ……… get the force!

– Fk+=-FS(djk+dki)/2 ……… get the force!

• For every vertex i– vi = vi *damping;– vi = vi + t * Force / mi;– xi = xi + t *vi;

3D Algorithm• For every vertex i

– fi=mg;• For every tetrahedron ijkl

– F=[y1-y0, y2-y0, y3-y0] [x1-x0, x2-x0, x3-x0]-1

– G=(FTF-I)/2; – S=Strain_To_Stress(G) ; – Fi+=-FS(dikj+dijl+dilk)/3;– …

• For every vertex i– vi = vi *damping;– vi = vi + t * Force / mi;– xi = xi + t *vi;

More details in http://physbam.stanford.edu/~fedkiw/papers/stanford2003-07.pdf