The Finite Element Method for the Analysis of Non … and Dynamic Systems: Computational Plasticity...

The Finite Element Method for the Analysis ofNon-Linear and Dynamic Systems: Computational

Plasticity Part II

Prof. Dr. Eleni ChatziDr. Giuseppe Abbiati, Dr. Konstantinos Agathos

Lecture 3 - 5 October, 2017

Institute of Structural Engineering Method of Finite Elements II 1

Learning Goals

To recall the basics of linear elasticity and the importance ofVoigt notation for representing tensors.

To understand basic rate-independent plasticity modelsformulated in terms of stress and strain fields.

To derive displacement-based finite elements based on suchconstitutive models.

References:

Ren de Borst, Mike A. Crisfield, Joris J. C. Remmers, Clemens V.Verhoosel, Nonlinear Finite Element Analysis of Solids andStructures, 2nd Edition, Wiley, 2012.

Example: Forming of a metal profile

Lumped vs. Continuous Plasticity Models

Lumped parameter model:

Finite dimensional stateexpressed in terms of thescalar r

Described by a set ofOrdinary DifferentialEquations (ODE)

Continuous parameter model:

Infinite dimensional stateexpressed in terms of thefield σ (x)

Described by a set of PartialDifferential Equations (PDE)

Voigt Notation

Stresses and strains are second order tensors related by a fourthorder tensor describing the elastic properties of the continuum.

σij = Deijklεkl

i , j , k, l → 1, 2, 3↓

σ6×1

= [De ]6×6ε6×1

However, in order to facilitate the implementation of computerprograms -when possible- it is more convenient to work with vectorsand matrices. A clear description of Voigt notation is reported in:

Belytschko, T., Wing Kam L., Brian M., and Khalil E.. Nonlinearfinite elements for continua and structures, Appendix 1, John wiley& sons, 2013.

Voigt Notation

Graphical representation of the Cauchy stress tensor.

σxx σxy σxzσyy σyz

sym σzz

σxxσyyσzzσyzσxzσxy

Voigt Notation

Graphical representation of the Green-Lagrange (small) strain tensor.

εxx εxy εxzεyy εyz

sym εzz

εxx =

∂x, εxy =

∂y+∂v

)εyy =

∂y, εxz =

∂z+∂w

)εzz =

∂z, εyz =

∂z+∂w

)Institute of Structural Engineering Method of Finite Elements II 6

Voigt Notation

Graphical representation of the Green-Lagrange (small) strain tensor.

εxx εxy εxzεyy εyz

sym εzz

εxxεyyεzz

2εyz2εxz2εxy

εxxεyyεzzγyzγxzγxy

Voigt Notation

Cauchy stress tensor. Green-Lagrange (small) strain tensor.

δw int =3∑

3∑j=1

δεijσij = δεijσij = δε : σ = δεTσ

Voigt Notation

Cauchy stress tensor. Green-Lagrange (small) strain tensor.

δw int =3∑

3∑j=1

δεijσij = δεijσij = δε : σ = δεTσ

Principle of virtual displacement !!!

Voigt Notation

Isotropic elastic compliance from tensor:

εij = C eijklσkl or ε = Ce : σ

to Voigt notation:

ε = [Ce ] σ

1 −ν −ν 0 0 0−ν 1 −ν 0 0 0−ν −ν 1 0 0 00 0 0 2 (1 + ν) 0 00 0 0 0 2 (1 + ν) 00 0 0 0 0 2 (1 + ν)

E : Young modulus, ν : Poisson ratio.

Voigt Notation

Isotropic elastic stiffness from tensor:

σij = Deijklεkl or σ = De : ε

to Voigt notation:

σ = [De ] ε

(1 + ν) (1− 2ν)

1− ν ν ν 0 0 0ν 1− ν ν 0 0 0ν ν 1− ν 0 0 00 0 0 1−2ν

2 0 00 0 0 0 1−2ν

2 00 0 0 0 0 1−2ν

E : Young modulus, ν : Poisson ratio.

From Lumped to Continuous Plasticity Models

Lumped plasticity modelr, u, Ke

Continuous plasticity modelσ, ε, [De ]

Elastic regimeif f (r) < 0

↓r = Ke u

if f (σ) < 0

↓σ = [De ] ε

Elastoplastic regimeif f (r) = 0

↓r = Ke (u− up)

with up = λm

if f (σ) = 0

↓σ = [De ] (ε − εp)f = 0

with εp = λm

if f (r) = 0

↓r = Ke (u− up)

with up = λm

if f (σ) = 0

↓σ = [De ] (ε − εp)f = 0

with εp = λm

Yield criterion : this is a scalar function that determines theboundary of the elastic domain.

if f (r) = 0

↓r = Ke (u− up)

with up = λm

if f (σ) = 0

↓σ = [De ] (ε − εp)f = 0

with εp = λm

Flow rule : this is a vector function that determines the direction ofthe plastic strain flow.

if f (r) = 0

↓r = Ke (u− up)

with up = λ∂f

if f (σ) = 0

↓σ = [De ] (ε − εp)f = 0

with εp = λ∂f

In the case of associated plasticity, the same function f defines bothyield criterion and flow rule i.e. the plastic displacement/strain flow

is co-linear with the yielding surface normal.

Invariants of the Stress Tensor

Invariants of stress tensor σ are used to formulate yielding criteria.

sym σzz

det (σ − λI) = det

σxx − λ σxy σxzσyy − λ σyz

sym σzz − λ

λ3 − I1λ2 − I2λ− I3 = 0

where I1, I2 and I3 are the invariants of the stress tensor andλ = σ11, σ22, σ33 are the eigenvalues of the stress tensor alsocalled principal stresses.

Invariants of the Stress Tensor

Invariants of stress tensor σ are used to formulate yielding criteria.

λ3 − I1λ2 − I2λ− I3 = 0

I1 = σxx + σyy + σzz

I2 = σ2xy + σ2

yz + σ2zx − σxxσyy − σyyσzz − σzzσxx

I3 = σxxσyyσzz + 2σxyσyzσzx − σxxσ2yz − σyyσ2

zx − σzzσ2xy

2σT [Ce ] σ =

(I 21 + 2I2 (1 + ν)

)where Ψ is the elastic energy potential.

Invariants of the Deviatoric Stress Tensor

Invariants of deviatoric stress tensor s are used to formulate yieldingcriteria.

sym σzz

p =σxx + σyy + σzz

s = σ − pI =

σxx − p σxy σxzσyy − p σyz

sym σzz − p

sxx sxy sxzsyy syz

sym szz

where p is the hydrostatic pressure.

sxx sxy sxzsyy syz

sym szz

det (s− λI) = det

sxx − λ sxy sxzsyy − λ syz

sym szz − λ

λ3 − J1λ2 − J2λ− J3 = 0

where J1, J2 and J3 are the invariants of the deviatoric stress tensor.

λ3 − J1λ2 − J2λ− J3 = 0

J1 = sxx + syy + szz

J2 = s2xy + s2

yz + s2zx − sxxsyy − syy szz − szzsxx

J3 = sxxsyy szz + 2sxy syzszx − sxxs2yz − syy s

2zx − szzs

Ψd =1

2sT [Ce ] s =

1 + 2J2 (1 + ν))

where Ψd is the deviatoric elastic energy potential.

λ3 − J1λ2 − J2λ− J3 = 0

J1 = sxx + syy + szz = 0

J2 = s2xy + s2

yz + s2zx − sxxsyy − syy szz − szzsxx =

J3 = sxxsyy szz + 2sxy syzszx − sxxs2yz − syy s

2zx − szzs

Ψd =1

2sT [Ce ] s =

J2 (1 + ν)

)(1 + ν)

where Ψd is the deviatoric elastic energy potential.Institute of Structural Engineering Method of Finite Elements II 16

Von Mises Yield Function

The J2 invariant of the deviatoric stress tensor is used to define theVon Mises yield function:

fVM (σ) = q − σ = 0

where σ is the pure uniaxial yielding stress and,

q =√

√(σxx − σyy )2 + (σyy − σzz)2 + (σzz − σxx)2

2+ 3σ2

xy + 3σ2xz + 3σ2

√(σ11 − σ22)2 + (σ22 − σ33)2 + (σ33 − σ11)2

Von Mises Yield Function

The J2 invariant of the deviatoric stress tensor is used to define theVon Mises yield function:

fVM (σ) = q − σ = 0

q =√

2σTPσ

2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0

0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

Drucker-Prager Yield Function

The J2 invariant of the deviatoric stress tensor is used to define theDrucker-Prager yield function that accounts for hydrostatic pressuredependency:

fDP (σ) = q + αp − k = 0

where α and k are material parameters and,

q =√

2σTPσ, p = πTσ

2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0

0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

, π =

1/31/31/3

Tresca Yield Function

The Tresca yield function reads,

fTR (σ) =

σ11−σ222 − τmax = 0

σ22−σ112 − τmax = 0

σ11−σ332 − τmax = 0

σ33−σ112 − τmax = 0

σ22−σ332 − τmax = 0

σ33−σ222 − τmax = 0

where τmax = σ/2 is used to approximate the Von Mises yieldfunction.

Coulomb Yield Function

The Coulomb yield function reads,

fCL (σ) =

σ11−σ222 + σ11+σ22

2 sin (ϕ)− c · cos (ϕ) = 0σ22−σ11

2 + σ11+σ222 sin (ϕ)− c · cos (ϕ) = 0

σ11−σ332 + σ11+σ33

2 sin (ϕ)− c · cos (ϕ) = 0σ33−σ11

2 + σ11+σ332 sin (ϕ)− c · cos (ϕ) = 0

σ22−σ332 + σ22+σ33

2 sin (ϕ)− c · cos (ϕ) = 0σ33−σ22

2 + σ22+σ332 sin (ϕ)− c · cos (ϕ) = 0

where α = 6sin(ϕ)3−sin(ϕ) and k = 6c·cos(ϕ)

3−sin(ϕ) are used to approximate theDrucker-Prager yield function.

Continuous Plasticity Problem

Stress-strain response of an elastic perfectly-plastic material.

Let’s imagine to turn this into a computer program:

1: function [σj+1] = material (εj+1)2: ...3: end

Return Mapping Algorithm: (σ,ε) vs. (r,u)

The return mapping algorithm if form of residual minimizationproblem is reported for a generic continuous plasticity model:

σj+1, ∆λj+1 :

εσ = σj+1 − σe+ Dem∆λj+1

εf = f (σj+1)

For the sake of comparison, the return mapping algorithm is reportedalso for a generic lumped plasticity model (e.g. spring-slider):

rj+1, ∆λj+1 :

εr = rj+1 − re + Dem∆λj+1

εf = f (rj+1)

Return Mapping Algorithm: (σ,ε) vs. (r,u)

The corresponding Newton-Raphson algorithm is reported for ageneric continuous plasticity model:[

σk+1j+1

∆λk+1j+1

[σkj+1

∆λkj+1

]−[∂εσ∂σ

∂εσ∂∆λ

∂εf∂σ

∂εf∂∆λ

]−1 [εkσεkf

]The Newton-Raphson algorithm is reported also for a generic lumpedplasticity model (e.g. spring-slider):[

rk+1j+1

∆λk+1j+1

[rkj+1

∆λkj+1

]−[∂εr∂r

∂εr∂∆λ

∂εf∂r

∂εf∂∆λ

]−1 [εkrεkf

Von Mises Plasticity with Associated Flow Rule

The gradient of the Von Mises yield surface is function of σ:

fVM (σ) =

2σTPσ − σ = 0

nVM = mVM =∂fVM∂σ

32σTPσ

2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0

0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

Drucker-Prager Plasticity with Associated Flow Rule

The gradient of the Drucker-Prager yield surface is function of σ:

fDP (σ) =

2σTPσ+ απTσ − k

nDP = mDP =∂fVM∂σ

32σTPσ

+ απ

where α and k are material parameters and,

2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0

0 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

1/31/31/3

Return Mapping Algorithm with Curved Yield Surfaces

In order to guarantee convergence of the return mapping algorithmwhen the yield surface is curved, the strain increment has to besmall.

e.g. spring-slider return mapping.

re = rj + Ke∆uj+1

e.g. Von Mises return mapping.

σe = σj + [De ] ∆εj+1

Return Mapping Algorithm

A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models, is reported in form ofresidual minimization problem:

σj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1) ∆λj+1 = 0

εf = f (σj+1) = 0

↓[σk+1

∆λk+1j+1

[σkj+1

∆λkj+1

]−[∂εσ∂σ

∂εσ∂∆λ

∂εf∂σ

∂εf∂∆λ

]−1 [εkσεkf

Return Mapping Algorithm

A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models, is reported in form ofresidual minimization problem:

σj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1) ∆λj+1 = 0

εf = f (σj+1) = 0

↓[σk+1

∆λk+1j+1

[σkj+1

∆λkj+1

]−[I + [De ] ∂m∂σ∆λkj+1 [De ]m

∂f∂σ 0

]−1 [εkσεkf

]where m and f and their partial derivatives are functions of σkj+1.

Consistent Tangent Stiffness

A formulation of the consistent tangent operator, which iscompatible with both Von Mises and Drucker-Prager plasticitymodels, is reported:

σj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1) ∆λj+1 = 0

εf = f (σj+1) = 0

↓[σk+1

∆λk+1j+1

[σkj+1

∆λkj+1

[∂σ∂εσ

∂σ∂εf

∂∆λ∂εσ

∂∆λ∂εf

] [εkσεkf

[D]j+1 =∂σj+1

∂εj+1= −

∂σj+1

∂εσ

∂εσ∂εj+1

∂ (∆εj+1) = ∂ (εj+1 − εj) = ∂εj+1 −*constant

∂εj = ∂εj+1

Hardening Behaviour

So far we assumed that the yield function f depends only on thestress tensor σ and material parameters are constant. However,this is almost never the case:

Cyclic loading in metals: Bauschinger effect.

Hardening Behaviour

We can identify two complementary hardening phenomena:

Isotropic hardening: expansion ofthe yield surface.

f = f (σ, κ)

κ is a scalar variable.

Kinematic hardening: translationof the yield surface.

f = f (σ, α)

α is a tensor variable.

Isotropic Hardening

The Von Mises yield function modified by the linear isotropichardening rule reads,

fVM (σ) = q (σ)− (σ0 + hκ)

where the evolution of κ, which accounts for the expansion of theyield surface, reads,

κ = λp (σ, κ)→ κ =

∫κdt

with σ0 is the initial yield strength, h is the hardening modulus andp (σ, κ) is a scalar function depending on the hardeninghypothesis. It is noteworthy that the gradient of the yield functiondoes not depend on the isotropic hardening variable κ in this case:

∂fVM∂σ

32σTPσ

Isotropic Hardening

These are some examples of isotropic hardening hypothesis:

σTεp = λ

), work-hardening√

23εpTQεp = λ

√23m

TQm, strain-hardening

−3πT εp = −λ(3πTm

), volumetric-hardening

2/3 −1/3 −1/3 0 0 0−1/3 2/3 −1/3 0 0 0−1/3 −1/3 2/3 0 0 0

0 0 0 1/2 0 00 0 0 0 1/2 00 0 0 0 0 1/2

, π =

1/31/31/3

, εp = mλ

Return Mapping Algorithm with Isotropic Hardening

A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with isotropic hardening,is reported in form of residual minimization problem:

σj+1, κj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1, κj+1) ∆λj+1

εκ = κj+1 − κj −∆λj+1p (σj+1, κj+1)

εf = f (σj+1, κj+1)

↓σk+1j+1

κk+1j+1

∆λk+1j+1

σkj+1

κkj+1

∆λkj+1

−∂εσ∂σ ∂εσ

∂κ∂εσ∂∆λ

∂εκ∂σ

∂εκ∂κ

∂εκ∂∆λ

∂εf∂σ

∂εf∂κ

∂εf∂∆λ

−1 εkσεkκεkf

Return Mapping Algorithm with Isotropic Hardening

A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with isotropic hardening,is reported in form of residual minimization problem:

σj+1, κj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1, κj+1) ∆λj+1

εκ = κj+1 − κj −∆λj+1p (σj+1, κj+1)

εf = f (σj+1, κj+1)

↓σk+1j+1

κk+1j+1

∆λk+1j+1

σkj+1

κkj+1

∆λkj+1

−I + [De ] ∂m∂σ∆λkj+1 [De ] ∂m∂κ∆λkj+1 [De ]m

− ∂p∂σ∆λkj+1 1− ∂p

∂κ∆λkj+1 −p∂f∂σ

∂f∂κ 0

−1 εkσεkκεkf

where m, p and f and their partial derivatives are functions ofσkj+1 and κkj+1.

Kinematic Hardening

The Von Mises yield function modified by the Ziegler kinematichardening rule reads,

fVM (σ) = q (σ − α)− σ

where the evolution of α, which represents the position of thecentroid of the yield function, reads

α = λa (σ − α)→ α =

∫αdt

where a is a material parameter. It is noteworthy that the gradientof the yield function depends on the hardening variable α in thiscase:

∂fVM∂σ

=3 (Pσ − α)

32 (Pσ − α)T P (Pσ − α)

Return Mapping Algorithm with Kinematic Hardening

A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with kinematichardening, is reported in form of residual minimization problem:

σj+1, αj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1, αj+1) ∆λj+1

εα = αj+1 −αj −∆λj+1a (σj+1 − αj+1)

εf = f (σj+1, αj+1)

↓σk+1j+1

αk+1j+1

∆λk+1j+1

σkj+1

αkj+1

∆λkj+1

− ∂εσ∂σ ∂εσ

∂α∂εσ∂∆λ

∂εα∂σ

∂εα∂α

∂εα∂∆λ

∂εf∂σ

∂εf∂α

∂εf∂∆λ

−1 εkσεkαεkf

Return Mapping Algorithm with Kinematic Hardening

A return mapping algorithm, which is compatible with both VonMises and Drucker-Prager plasticity models with kinematichardening, is reported in form of residual minimization problem:

σj+1, αj+1, ∆λj+1 :

εσ = σj+1 − σe+ [De ]m (σj+1, αj+1) ∆λj+1

εα = αj+1 −αj −∆λj+1a (σj+1 − αj+1)

εf = f (σj+1, αj+1)

↓σk+1j+1

αk+1j+1

∆λk+1j+1

σkj+1

αkj+1

∆λkj+1

−I + [De ] ∂m∂σ∆λkj+1 [De ] ∂m∂α∆λkj+1 [De ]m

−a∆λkj+1 1 + a∆λkj+1 −a (σj+1 − αj+1)∂f∂σ

∂f∂α 0

−1 εkσεkαεkf

where m, p and f and their partial derivatives are functions ofσkj+1 and αk

Return Mapping Algorithm (σ,ε): Code Template

1: ∆εj+1 ← εj+1 − εj2: σe ← σj + [De ] ∆εj+1

3: if f (σe) ≥ 0 then4: σj+1 ← σe5: ∆λj+1 ← 06: εr ← σj+1 − σe + [De ]m∆λj+1

7: εf ← f (σj+1)8: repeat

[σj+1

∆λj+1

]←[σj+1

∆λj+1

[∂εr∂σ

∂εr∂∆λ

∂εf∂σ

∂εf∂∆λ

]−1 [εrεf

]10: εr ← σj+1 − σe + [De ]m∆λj+1

11: εf ← f (σj+1)12: until ‖ε‖ >= Tol

13: [D]j+1 ← −∂σ∂εr

∂εr∂ε

14: else if f (σe) < 0 then15: σj+1 ← σe16: [D]j+1 ← [De ]17: end if

Finite Element Discretization: from (σ,ε) to (r,u)

So far we derived a procedure for calculating the punctual stressresponse given a punctual strain increment for a generic plasticityconstitutive model ...

... however we want to formulate finite (length, area or volume)elements that relate nodal forces to nodal displacements.

The principle of virtual displacements facilitates their derivation:

r (uj) = f (tj)∫ΩδεTσjdΩ =

∫ΩδuTpvolj dΩ +

∫ΓδuTpsurj ·

−→dΓ

σj : stress state generated by volume pvol and surface psur

loads up to tj .

δu and δε : compatible variations of displacement u andstrain ε fields.

The restoring force is calculated according to the principle of virtualdisplacement for a bar element:

r (uj) =

0δεTσjdx

The two nodal displacements completely characterize displacementand strain fields within the element:

Shape functionsn1 (x) = 1− x

n2 (x) = xL

Shape functions’ derivativesb1 (x) = dn1

dx = − 1L

b2 (x) = dn2dx = 1

r (uj) =

0δεTσjdx

Displacement field Shape functions

u (x) = u1n1 (x) + u2n2 (x) =[n1 (x) n2 (x)

]= N (x)u

r (uj) =

0δεTσjdx

Strain field Shape functions’ derivatives

ε (x) = u1b1 (x) + u2b2 (x) =[b1 (x) b2 (x)

]= B (x)u

Finite Element Discretization: Restoring Force

The restoring force is calculated according to the principle of virtualdisplacement by integrating the stress field:

uj → interp.→ εj → ret. mapping→ σj

r (uj) =

0δεTσjdx

r (uj) =

[b1 (x)b2 (x)

]σj (x) dx

r (uj) ≈∑m

[b1 (xm)b2 (xm)

]σj (xm)

Interpolation works exactly like for linear finite elements. The returnmapping algorithm is formulated for the specific plasticity model.

Finite Element Discretization: Consistent Tangent Stiffness

The consistent tangent stiffness is calculated according to theprinciple of virtual displacement:

uj → interp.→ εj → ret. mapping→ [D]j

∂rj∂uj

= Kj =

0δεT

∂σj∂εj

∂εj∂uj

[b1 (x)b2 (x)

][D]j (x)

[b1 (x) b2 (x)

Kj ≈∑m

[b1 (xm)b2 (xm)

][D]j (xm)

[b1 (xm) b2 (xm)

]Interpolation works exactly like for linear finite elements. Theconsistent tangent stiffness is formulated for the specific plasticitymodel.

Nonlinear Static Analysis (r,u)

We derived a procedure for calculating the force response of a singleelement given a displacement trial ...

... but we want to solve the static displacement response of a model,which combines several elements, subjected to an external loadhistory.

The corresponding balance equation reads,

uj : r (uj)− f (tj) = 0

where,

uj : global displacement vector

r (uj) : global restoring force vector

f (tj) : global external load vector

at time step j-th.

Nonlinear Static Analysis (r,u): Code Template

1: for j = 1 to J do2: uj ← uj−1

3: for i = 1 to I do4: ri,j ← elementForce (Ziuj)5: rj ← rj + ZT

i ri,j6: end for7: εr ← rj − f (tj)8: repeat9: for i = 1 to I do

10: Ki,j ← elementStiff (Ziuj)11: Kj ← Kj + ZT

i Ki,jZi

12: end for13: uj ← uj −K−1

j εr14: for i = 1 to I do15: ri,j ← elementForce (Ziuj)16: rj ← rj + ZT

i ri,j17: end for18: εr ← rj − f (tj)19: until ‖εr‖ >= Tol20: end for

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Spike timing dependent plasticity Homeostatic regulation of synaptic plasticity.

A Stochastic Finite Element Method with a Deviatoric ... · A Stochastic Finite Element Method with a Deviatoric-volumetric Split for the Stochastic Linear Isotropic Elasticity Tensor

Introduction to Geometric Knot Theory 1: Knot invariants ... · Introduction to Geometric Knot Theory 1: Knot invariants deﬁned by minimizing curve invariants Jason Cantarella University

Lessons on integral invariants

An introduction to categorifying quantum knot invariants · An introduction to categorifying quantum knot invariants BEN WEBSTER We construct knot invariants categorifying the quantum

Linear Realization Theory, Integer Invariants and Feedback ...mst.ufl.edu/pdf papers/Linear realization theory, integer invariants... · Linear Realization Theory, Integer Invariants

Braids and topological invariants for non-linear ...vdvorst/Thesis-munao.pdf · Braid invariants for non-linear differential equations ISBN: ... Braid invariants for non-linear differential

Topology optimization for minimum weight with compliance ...the invariants of the stress tensor and its deviatoric Maxime Collet Department of Aerospace and Mechanical Engineering

Knot/Link Invariants

Symmetric invariants and cohomology of groupsadem/symmetric-invariants-cohomology.pdf · Symmetric invariants and cohomology of groups Alejandro Adem, 1"* John Maginnis, 2,* and R.

ON JONES KNOT INVARIANTS AND VASSILIEV INVARIANTS...Jones knot invariants and Vassiliev invariants via a substitution t = ex. We show here that this relation can be rearranged into

Eta-Invariants and Molien Series for Unimodular Groupshome.mathematik.uni-freiburg.de/degeratu/thesis.pdf · Eta-Invariants and Molien Series for Unimodular Groups ... Eta-Invariants

Brief Introduction to Continuum Mechanics (Deviatoric Stress and Strain)

Invariants des hypermatrices - IGM