Chapter IV Extension to materially non linear problems

Chapter IV  Page 73 

Chapter IV

Extension to materially non linear problems

Content

Summary 74

IV.1. Introduction 76

IV.2. Nonlinear theory 77

IV.3. Domain decomposition 79

IV.4. Discretization 80

IV.5. Equations deduced from the FdV variational principle 83

IV.6. Matrix notation 84

IV.7. Solution to the equations 87

IV.8. Applications 91

IV.8.1. Plasticity and strain hardening in solids 91

IV.8.2. Elasto-plastic material with von Mises linear hardening 92

IV.8.3. Patch tests 94

IV.8.4. Pure bending of a beam 98

IV.8.5. Square membrane with a circular hole 102

IV.9. Conclusions 106

Chapter IV  Page 74 

Summary

In this chapter, the natural neighbours method (NEM) based on the FRAEIJS de VEUBEKE (FdV) variational principle is extended to materially non linear solids in 2D.

Considering a solid of unit thickness in plane strain state, the material has a non linear constitutive equation but the displacements of the solid are assumed to be very small.

Hence, the problem considered in this chapter is geometrically linear and materially non linear.

The FdV variational principle for linear elasticity (chapter II) is extended to the elasto- plastic case in which the assumed velocity, stresses, strain rates and surface support reactions are discretized separately.

The domain is decomposed into N Voronoi cells corresponding to the N nodes distributed inside the domain and on its boundary. Since the displacements are assumed to be infinitesimal, there is no need to update this decomposition as the solid deforms.

The following discretization hypotheses are admitted:

1. The assumed velocities are interpolated between the nodes with the Laplace

interpolation function 2. The assumed strain rates are constant over each Voronoi cell I

3. The assumed stresses are constant over each Voronoi cell I

4. The assumed support reactions are constant over each edge K of Voronoi cells on which displacements are imposed.

Introducing these hypotheses in the extension of the FdV variational principle produces the set of equations governing the discretized solid.

The advantages of this method applied in linear elasticity remain valid in the extension to the elasto-plasticity problems:

• The equations do not require the calculation of the derivatives of the Laplace interpolation functions.

• In the absence of body forces, the equations only involve numerical integrations on the edges of the Voronoi cells.

It is possible to impose displacements u~_i on any edge of any Voronoi cell in a weighted average sense.

Recasting the equations in matrix form, it is shown that the discretization parameters associated with the assumptions on the stresses and on the strain rates can be eliminated at the Voronoi cell level so that the final system of equations only involves the nodal velocities and the assumed support reactions.

Chapter IV  Page 75  If the displacements are only imposed as constant, the support reactions can be further eliminated from the final equation system.

This equation system can be solved step by step by integration on time and Newton- Raphson iterations at the level of the different time steps.

Using the von Mises elasto-plastic model with linear hardening, some applications are presented to evaluate the method.

Some patch tests are performed and show that this approach can pass the patch test successfully.

The method is then applied to the pure bending of a beam to study the convergence. The numerical solution is also compared to the results of another method based on the direct integration of the corresponding differential equations.

A square membrane with a hole is also used for convergence evaluation and for comparison with the finite element solution.

Chapter IV  Page 76 

IV.1. Introduction

We consider a solid of unit thickness in plane strain state. The main notations are summarized in figure IV.1.

The material has non linear constitutive equation but the displacements of the solid are assumed to be very small.

Hence, the problem considered here is geometrically linear and materially non linear.

Notations Initial

configuration

Deformed configuration

Area of the domain A a

Contour of the domain S s

Outside normal to the contour N n

Coordinates of material points X x

Body force (per unit volume) F f

Surface traction (per unit area) T t

Figure IV.1. Conventions for the non linear theory e1

e2

Γ

γ X

u x

N

n

Γ γ

Chapter IV  Page 77  In chapter II, section II.2, the classical approach of the natural neighbours method (NEM) has been introduced and developed in chapter III for linear elastic problems.

In this chapter, we will start from a rate form of the FRAEIJS de VEUBEKE (FdV) variational principle (chapter II, section II.3) to develop an extension of the NEM to materially non linear problems, i.e. problems in which the displacements remain very small but the material constitutive equation is non linear, for example elasto-platic or elasto-visco-plastic.

Hence, referring to figure IV.1, the initial configuration Γ and the deformed configuration γ are very close to each other.

The domain is decomposed into N Voronoi cells corresponding to N nodes distributed inside the domain and on its boundary.

Since the problem is geometrically linear, the domain decomposition remains valid through the deformation process.

After discretization of the velocity field, the stress field, the strain rate field and the support reactions field, we get an equation system that can be solved with the help of Newton-Raphson iterations at each time step.

As an example of non linear constitutive equation, the elasto-plastic von Mises linear hardening model is used to perform some applications to evaluate the method.

In the NEM for linear elastic problems (chapter III), we have seen that:

• the derivatives of the Laplace interpolation functions are not necessary,

• only numerical integration on the edges of the Voronoi cells are required,

• incompressibility locking is avoided.

In this chapter, we will see that these properties remain valid for non linear problems.

IV.2. Non linear theory

The following variational equations constitute the starting point of the approach. They are an extension of the variational approach proposed by FRAEIJS de VEUBEKE for linear elasticity.

∫

= Π

a ij ijδε da σ

δ _{1 &} (IV.1)

da x )

u x

( u

a ij

i j j

ij i

∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂δ δ δε

2 Σ 1 Π

δ ₂ & & & (IV.2)

da x )

u x ( u

a ij

j i j ij i

∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂ ε

2 Σ 1 δ Π

δ ₃ & & & (IV.3)

Chapter IV  Page 78  da

u δ f δ

a i

∫

i

−

= &

Π4 (IV.4)

−∫

=

sttiδu&ids

Π

δ ₅ (IV.5)

ds u r ds ) u u~

(

r _i

s i

s i i i

u u

&

&

& δ

δ Π

δ ₆ = ∫ − −∫ (IV.6)

0 Π δ Π δ Π δ Π δ Π δ Π δ Σ

ε Π

δ (u&_i,&_ij, _ij,r_i )= ₁+ ₂+ ₃+ ₄ + ₅ + ₆ = (IV.7)

with

u&i the assumed velocity field,

ε&ij the assumed strain rate field,

Σijthe assumed stress field,

ri the assumed support reaction field.

In (IV.1), σ_ijare the constitutive stresses at the considered material point in the deformed configuration.

For inelastic materials, these stresses are usually obtained by integration, along the strain path of the considered material point, of a system of equations of the type:

) , , ( _{ij ij ij}

ij fij σ q ε

σ^∇ = & (IV.8)

) , ( _{ij ij}

ij

ij h σ q

q^∇ = (IV.9)

with q_ij a set of internal variables and where the superscript ^∇ is the symbol for an objective derivative (in the particular case of a geometrically linear problem, a simple time derivative is sufficient) .

In (IV.5) and (IV.6), some integrals are computed along the domain contour. This contour is the union of some of the edges of some Voronoi cells. These edges are denoted by s_K and we have

∑

=

= ^M

K sK

s

1

; = ∑

= Mu

K K

u s

s

1

; = ∑

= Mt

K K

t s

s

1

; s=s_u ∪s_t ⇒M =M_u +M_t where M is the number of edges composing the contour, Mu the number of edges on which displacements u~ are imposed and _i M_t the number of edges on which surface tractions t_i are imposed.

The Euler equations corresponding to the variational equation (IV.7) are summarized in table IV.1.

Chapter IV  Page 79  Table IV.1. Euler equations deduced from the Fraeijs de Veubeke variational principle

Variation Equation Comments

εij

δ& in a σ^ij =Σ^ij The assumed stresses are identified as the

constitutive stresses inside the domain Σij

δ in a ( )

2 1

i j j i

ij x

u x ε u

∂ +∂

∂

= ∂& &

& Compatibility between the assumed strain

rates and the assumed velocities inside the domain

ri

δ on su u&i =u~&i Compatibility between the assumed

velocities and the velocities imposed on the part su of the domain contour

u&i

δ in a Σ 0

=

∂ +

∂

i j

ji f

x

Equilibrium inside the domain between the assumed stresses and the body forces

u&i

δ on st n^jΣ^ji =tⁱ Equilibrium on the part s_t of the domain contour where surface tractions are imposed

IV.3. Domain decomposition

The domain contains N nodes (including nodes on the domain contour) and is decomposed into N Voronoi cells, each cell corresponding to a node.

Since the problem is geometrically linear, the domain decomposition remains valid throughout the deformation process.

The area of the domain is:

∑

=

= ^N

I

aI

a

1

(IV.10) with a_I the area of cell I.

We denote c_I the contour of Voronoi cell n° I.

Then

= ∑ ∫

= N

I a ij ij I

I

da

1 1 σ δε

Π

δ & (IV.11)

∑ ∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂

= N

I I

a ij

i j j

ij i ) da

x u x

( u

1I

2 δ δ δε

2 Σ 1 Π

δ & & & (IV.12)

∑ ∫ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ −

∂ +∂

∂

= ∂

= N

I I

a ij

i j j

ij i ) da

x u x ( u

1 I

3 ε

2 Σ 1 δ Π

δ & & & (IV.13)

Chapter IV  Page 80 

−∑ ∫

= = N

I I

a fi ui da

1 I

4 δ

Π

δ & (IV.14)

∑ ∫

−

= = _K

t

s i i K

M K

ds u

t δ&

Π δ

5 1 (IV.15)

∑ ⎥

⎦

⎢ ⎤

⎣

⎡∫ − − ∫

= =

u

K K

M

K i K

s i s ri(u~i ui )dsK r u ds

6 1 δ δ

Π

δ & & & (IV.16)

IV.4. Discretization

We make the following discretization hypotheses:

1. The assumed strain rates ε&_ij are constant over each Voronoi cell I:

I ij

ij ε

ε_& = _& (IV.17)

2. The assumed stresses Σ_ij are constant over each Voronoi cell I:

I ij ij =Σ

Σ (IV.18)

3. The assumed support reactions r_i are constant over each edge K of Voronoi cells on which displacements are imposed

iK

i r

r = (IV.19)

4. The assumed velocities u&_iare interpolated by Laplace interpolation functions:

∑

=

Φ

= ^N

J

J i J

i u

u

1

&

& (IV.20)

where u&_i^J is the velocity of node J (corresponding to the Voronoi cell J).

As a consequence of (IV.8, IV.9 and IV.17), the stresses σ_ij are constant over each Voronoi cell I:

I ij

ij σ

σ = (IV.21)

The variations of the independent variables are:

Iij ij δε ε

δ& = & (IV.22)

Iij ij δΣ Σ

δ = (IV.23)

iK

i r

r δ

δ = (IV.24)

Chapter IV  Page 81 

= ∑

= N I

iI I

i u

u

1Φ δ

δ& & (IV.25)

Introducing these assumptions in (IV.11) to (IV.16), and integrating by parts, we get:

= ∑

= N

I I I

I ij

ij a

1 1σ δε Π

δ & (IV.26)

∑ =

∑ ∫ −

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂ +∂

∂

= ∂

=

=

N

I I I

I ij ij N

I I

A i

j j

I i

ij ) da a

X u X

( u

I 1

2 1 δ δ Σ δε

2 Σ 1 Π

δ & & &

−∑

∑ ∫

=

=

N

I I I

I ij ij I

N

I c I i

I j

ij n u dc a

I 1

1Σ δ& Σ δε& (IV.27)

∑ =

∑ ∫ −

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂ +∂

∂

= ∂

=

= I I

ij N I

ijI N

I I

A i

j j i

ijI ) da a

X u X ( u

I

ε Σ 2 δ

Σ 1 δ Π

δ

1

3 1 & & &

−∑

∑ ∫

=

=

N

I I I

I ij ij I

N

I c I i

I j

ij n u dc a

I 1

1

ε Σ δ Σ

δ & & (IV.28)

∑ ⎥

⎦

⎢ ⎤

⎣

⎡ ∫ − − ∫

= =

u

K K

M

K Ks i K

s i i K i

iK (u~ u )ds r u ds

r

6 1 δ δ

Π

δ & & & (IV.29)

where n^I_jis the outward normal to the contour of Voronoi cell I.

Introducing in (IV.7), we get:

0 Π δ Π δ Π δ Π δ Π

δ = _VA + _VC + _DC + _EF = (IV.30)

with

−∑

∑ −

= = =

N

I I I

ij I ij N

I I I

ij I ij I ij

VA ( ) a a

1 1

ε Σ δ ε

δ Σ σ Π

δ & & (IV.31)

I N

I c I i

j I ij I

N

I c I i

j I ij

VC n u dc n u dc

I I

∑ ∫

∑ ∫ +

= =1 =1

Σ δ δ

Σ Π

δ & & (IV.32)

∑ ⎥

⎦

⎢ ⎤

⎣

⎡ ∫ − − ∫

= =

u

K K

M

K Ks i K

s i i K i

iK

DC r (u~ u )ds r u ds

1

δ δ

Π

δ & & & (IV.33)

− ∑ ∫

−∑ ∫

= = =

t

K I

M

K s K i K

i N

I I

a i i

EF f u da t u ds

1

1 δ δ

Π

δ & & (IV.34)

In (IV.32) to (IV.34), the velocities and virtual velocities are interpolated by (IV.20) and (IV.25) respectively. Substituting in (IV.32), we get:

Chapter IV  Page 82 

∑ ⎥

⎦

⎢ ⎤

⎣

⎡∑ ∫

= = =

N I

N

J c J I

i J I j I

ij VC

I

dc u n

1 1

δ Φ Σ

Π

δ & ∑ ⎥

⎦

⎢ ⎤

⎣

⎡∑ ∫

+ = =

N I

N

J c J I

i J I j I

ij

I

dc u n

1 1

Φ Σ

δ & (IV.35)

Finally, since the edges of the Voronoi cell are straight lines, the outside normal n_j to edge s_K is constant along this edge and is denoted n^K_j

Now, using the discretization (IV.20), we get:

∑ ⎭⎬⎫

⎩⎨

⎧∫ − ∑ ∫

= = =

u

K K

M K

N

J Js J K

s i K i iK

DC r u~ds u ds

1 1

Φ δ

Π

δ & & ∑

⎭⎬

⎫

⎩⎨

⎧∑ ∫

− = =

u

K

M K

N

J Js J K

K i

i u ds

r

1 1

Φ

δ& (IV.36)

Similarly, (IV.34) becomes:

∑ ∑ ′ ∫

∑ ∑ ∫ −

−

= = = = =

t

K I

M

K s i J K

N J

iJ N

I I

a i J

N J

iJ

EF u f da u t ds

1 1

1 1δ Φ δ Φ

Π

δ & (IV.37)

Collecting all the results, we obtain the discretized rate form of the FdV variational principle.

−∑

∑ −

= = =

N

I I I

I ij ij N

I I I

I ij I ij

ij ) a a

(

1 1

ε Σ δ ε

δ Σ σ Π

δ & &

∑

⎭⎬

⎫

⎩⎨

⎧∑ ∫ +

= =

N I

N

J c I J I

J j I i

ij

I

dc n u

1 1

Φ δ

Σ &

∑

⎭⎬

⎫

⎩⎨

⎧∑ ∫

+ = =

N I

N

J c I J I

J j I i ij

I

dc n u

1δΣ 1& Φ

+ ∑

⎭⎬

⎫

⎩⎨

⎧∫ − ∑ ∫

= =

u

K K

M K

N

J Js J K

s i K i

iK u~ds u ds

r

1δ & 1& Φ

∑

⎭⎬

⎫

⎩⎨

⎧∑ ∫

− = =

u

K

M K

N

J Js J K

K i

i u ds

r

1 1

Φ

δ&

−∑∑ ∫ − ∑ ∑ ∫

= =

= =

t

K I

M

K s i J K

N J

iJ N

I I

a i J

N J

iJ f da u t ds

u

1 1 1 1

Φ δ

Φ

δ& & = 0 (IV.38)

Chapter IV  Page 83 

IV.5. Equations deduced from the FdV variational principle

Let us reorganize the terms of (IV.38)

{ }

∑ −

= = N

I I I

I ij I ij

ij ( )a

1

Σ σ ε δ Π

δ &

∑

⎭⎬

⎫

⎩⎨

⎧− + ∑

+ = =

N I

N J

IJj iJ I I

I ij

ij a u A

1 1

ε Σ

δ & &

+ ∑

⎭⎬

⎫

⎩⎨

⎧ − ∑

= =

Mu

K

N J

KJ iJ iK

iK U~ u B

r

1 1

δ & &

+ ∑

⎭⎬

⎫

⎩⎨

⎧∑ − − ∑ − ∑

= = = =

N J

N I

M K

iKJ M

K

KJ iK iIJ

IJj ijI iJ

t

ur B t~

)

~f A ( u

1δ& 1 Σ 1 1 =0 (IV.39)

In this result, the following notations have been used.

∫

^Φ

=

cI

I J I j IJ

j n dc

A (IV.40)

∫

^Φ

=

sK

K J

KJ ds

B (IV.41)

∫

=

sK

K i K

i u~ds

U~& & (IV.42)

I a

J i IJ

i f da

~f

I

∫

^Φ

= (IV.43)

K s

J i KJ

i t ds

t~

K

∫

^Φ

= (IV.44)

Equation (IV.40) involves the integration on the contour c_I of Voronoi cell I.

Equations (IV.41) and (IV.42) involve the integration on the edge s_K (belonging to the domain contour) of a Voronoi cell.

We are now able to deduce the discretized Euler equations.

1. In all the Voronoi cells I

I ij I ij =Σ

σ for I =1,N (IV.45)

These equations identify the assumed stresses Σ_ij^I as the constitutive stresses σ_ij^I deduced from the constitutive equations (IV.8) and (IV.9) and from the assumed strains ratesε&_ij^I in each Voronoi cell.

2. In all the Voronoi cells I

Chapter IV  Page 84 

∑

=

= ^N

J

IJ j J i I

I

ija u a

1

&

&

ε (IV.46)

This is a compatibility equation linking the assumed strain rate ε_&ij^I in Voronoi cell I with the assumed nodal velocity u&_i^J.

3. On the edges K =1,M_uof Voronoi cells submitted to imposed velocities

K i N

J

KJ J

i B U~

u& = &

∑

=1

(IV.47)

These are also compatibility equations taking account of the imposed velocities u~&_i on the part s_u of the domain contour.

4. In all the Voronoi cells J

0

1

1 1

=

−

−

−

Σ

∑

∑ ∑

=

= =

u

t M

K

KJ K i N

I

M

K KJ i IJ

i IJ j I

ij f~ ) t~ r B

A

( (IV.48)

These are equilibrium equations taking account of the body forces f_i, the surface tractions t_i and the support reactions r_i.

We note that, in the developments above, the only term that implies an integration over the area of the Voronoi cells is _I

a J i IJ

i f da

f~

I

∫

^Φ

= .

Hence, if there are no body forces, the problem of choosing integration points is simplified: there are only integrations along the straight edges of Voronoi cells. A classical Gauss integration scheme can be used. In chapter III, some tests have shown that 2 integration points give enough precision.

Furthermore, this formulation does not require the derivatives of the shape functions.

In the linear elastic problem (Chapter III), using the FdV functional as starting point, we obtained the same advantages as with the stabilized confofming integration [CHEN J. S.

et al. (2001), YOO J. et al. (2004)]. This property remains valid in non linear problem.

Finally, in the approach developed here, it is possible to impose displacements

u~

_ion any edge of any Voronoi cell. From (IV.42) and (IV.47), it is clear that the imposed displacements are respected in a weighted average sense.

IV.6. Matrix notation

We introduce the following matrix notations.

{ }

⎭⎬⎫

⎩⎨

=⎧ _I^I

I

u u u

2 1

&

&

& ;

{ }

⎭⎬

⎫

⎩⎨

=⎧ _IJ^IJ

IJ

~ff

~

~f

2

1 ;

{ }

⎭⎬

⎫

⎩⎨

=⎧ _KJ^KJ

KJ

t~

t~ t~

2

1 ; (IV.49)

Chapter IV  Page 85 

{ } _∑ { }

=

= ^N

I IJ

J ~f

~f

1

;

{ } ∑ { }

=

= ^Mt

K KJ

J t~

t~

1

(IV.50)

{ }

⎪

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

I I I I

12 22 11

2ε ε ε ε

&

&

&

& ;

{ }

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

I I I I

12 22 11

σ σ σ

σ ;

{ }

⎪⎪

⎭

⎪⎪⎬

⎫

⎪⎪

⎩

⎪⎪⎨

⎧

=

I I I I

12 22 11

Σ Σ Σ

Σ (IV.51)

[ ]

_⎥

⎦

⎢ ⎤

⎣

=⎡ ^IJ _{IJ IJ}^IJ

IJ

A A

A A A

1 2

2 1

0

0 ;

{ }

⎭⎬

⎫

⎩⎨

=⎧ _K^K

K

r r r

2

1 ;

{ }

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

=⎧

K K K

U~ U~ U~

1

&2

&

& (IV.52)

Then, we get successively:

{ } { }

^{I I}

I ij I

ij =Σ ⇒ σ = Σ

σ (IV.53)

[ ]

^IJ

{ }

^I

IJ j I

ijA ⇒ A Σ

Σ ; ^{~f IJ ⇒}

{ }

^{~f IJ; t~IJ ⇒}

{ }

^{t~ IJ} (IV.54)

[ ] { }

^M

{ }

^K

{ }

^J

{ }

^J

K I KJ N

I M IJ

K KJ i N

I IJ i N

I

M

K

KJ K i IJ j I

ij ~f t~

r B A

t~

f~ B

r

A − ^u = + ^t ⇒ Σ − ^u = +

Σ

∑ ∑ ∑ ∑

∑ ∑

=

=

=

=

=1 =1 1 1 1 1

(IV.55) The term

[ ] { }

^M

{ }

^K

K I KJ N

I

IJ B r

A

∑

^u

∑

= =

− Σ

1 1

is the interior nodal force at node J, i.e. in cell J.

It is the sum of the contributions

[ ] { }

^{AIJ Σ I} of the stresses Σ_ij^I existing in all the Voronoi cells I and of the contributions B^KJ

{ }

r ^K of the support reactions r_i^K existing on the contour edges K where velocities are imposed.

The term

{ }

^{~f J +}

^{{ }}

^{t~ J} is the exterior nodal force at node J, i.e. in cell J. It is the sum of :

• the contributions

{ }

^{~f IJ}of the body forces F_i existing in all the Voronoi cells I

• the contributions

{ }

^{t~ IJ}of the surface tractions t_i applied on the part st of the domain contour.

Now, consider equation (IV.46). It can be written

{ }

^N

[ ] { }

^J

J

T , IJ

I I A u

a &

∑

&

=

= ε

1

(IV.56)

where

[ ]

A^IJ,T is the transpose of

[ ]

A^IJ.

Note that in

{ }

ε_& ^I, the third component is 2ε_&12^I .

Chapter IV  Page 86  The compatibility equation (IV.56) defines the strain rate

{ }

ε_& ^Iin a Voronoi cell I as the sum of the contributions

[ ]

A^IJ,T

{ }

u& ^J of all the nodes J.

On the edges K submitted to imposed velocities, we must consider (IV.47) that becomes

{ } { }

^K

N

J

KJ u J U~

B & = &

∑

=1

(IV.57)

The tables IV.2 and IV.3 below collect all the results in matrix form.

In these tables, taking account of (IV.53),

{ }

Σ ^I is replaced by

{ }

σ ^I

Table IV.2 Matrix notations for the materially non linear case.

Notations and symbols Comments

{ }

⎪

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

I I I I

12 22 11

2ε ε ε ε

&

&

&

& Strains in cell I

{ }

⎪

⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

=

I I I I

12 22 11

σ σ σ

σ Stresses in cell I

{ }

⎭⎬⎫

⎩⎨

=⎧ _I

I I

u u u

2 1

&

&

& Displacements velocity of node I

belonging to cell I

{ }

⎭⎬⎫

⎩⎨

=⎧ _K^K

K

r r r

2

1 Support reactions on edge K

submitted to imposed displacements

aI ; c_I Area and contour of cell I

sK Length of edge K of a cell

ΦJ Interpolant associated with node J

I a

J i IJ

i f da

~f

I

∫

^Φ

= ;

{ }

⎭⎬

⎫

⎩⎨

=⎧ _IJ^IJ

IJ

~ff

~ f~

2

1 ;

{ } _∑ { }

=

= ^N

I IJ

J ~f

f~

1

{ }

^{~f J}is the nodal force at node J equivalent to the body forces f_i

applied to the solid

Chapter IV  Page 87 

K s

J i KJ

i t ds

t~

K

∫

^Φ

= ;

{ }

⎭⎬

⎫

⎩⎨

=⎧ _KJ^KJ

KJ

t~

t~ t~

2

1 ;

{ } ∑ { }

=

= ^Mt

K KJ

J t~

t~

1

{ }

^{t~ J}is the nodal force at node J equivalent to the surface tractions t_i

applied to the contour of the solid

∫

=

sK

K i K

i u~ds

U~& & ;

{ }

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

=⎧

K K K

U~ U~ U~

1

&2

&

&

{ }

^{U~& K}is a generalized displacement

velocity taking account of imposed velocities u~&_i on edge K

∫

^Φ

=

sK

K J

KJ ds

B Integration over the edge K of a cell

∫

^Φ

=

cI

I J I j IJ

j n dc

A ;

[ ]

_⎥

⎦

⎢ ⎤

⎣

=⎡ ^IJ _{IJ IJ}^IJ

IJ

A A

A A A

1 2

2 1

0

0 A^IJ_j can also be computed by

J ) I ( K ) I ( K all

) I ( K j IJ

j n B

A ⁼

∑

Table IV.3. Discretized equations in matrix form for the materially non linear case

Equations Comments

[ ] { }

^M

{ }

^K

{ }

^J

^{{ }}

^J

K I KJ N

I

IJ B r f t

A ^u ~ ~

1 1

+

=

−

∑

∑

= =

σ Equilibrium equation of cell J (IV.58)

[ ]

^J

N

J

T IJ I

I A u

a ⎭⎬⎫

⎩⎨

= ⎧

⎭⎬

⎫

⎩⎨

⎧

∑

=

&

&

1

ε , Compatibility equation for cells I (IV.59)

{ } { }

^K

N

J

KJ J U~

u

B & = &

∑

=1

Compatibility equation on edge K

submitted to imposed velocities (IV.60)

IV.7. Solution of the equations

The equations to solve are:

• The equilibrium equations for all the Voronoi cells J

[ ] { }

^{A M B}

{ }

^{r K}

{ }

^{~f J}

{ }

^t~

K I KJ N

I

IJ Σ −

∑

^u = +

∑

=1 =1

(IV.61)

• The compatibility equations in all the Voronoi cells I

{ }

^N

[ ] { }

^J

J

T , IJ

I I A u

a &

∑

&

=

= ε

1

(IV.62)

Chapter IV  Page 88 

• The compatibility equation on edge K submitted to imposed velocities

{ } { }

^K

N

J

KJ u J U~

B & = &

∑

=1

(IV.63)

Integrating (IV.63) in time, we get

{ } { }

^K

N

J

KJ J U~(t)

) t ( u

B =

∑

=1

(IV.64) With the notations

{ } { } { }

{ }

^⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

= u N

u u

q

. .

2 1

;

{ } { } { }

{ }

^⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

= f N

~. .

~f~f

~f

2 1

;

{ } { } { }

{ }

^⎪⎪

⎪

⎭

⎪⎪

⎪

⎬

⎫

⎪⎪

⎪

⎩

⎪⎪

⎪

⎨

⎧

= t~ N

. . t~

t~

t~

2 1

;

{ } { } { }

^{Q~ = F~ + T~}

equation (IV.64), in matrix form, becomes

[ ] { }

^{BT q(t) =}

{ }

^{U~(t) where}

{ }

^U~(t) is a column matrix containing generalized displacements imposed to some nodes at time t.

This constitutes a set of constraints on the nodal displacements

{ }

q(t) deserving the same remarks as in the linear elastic case.

In particular, if displacements u~_i =0 are imposed at any time t on the segment CD joining 2 nodes C and D of the domain contour, it is easy to show that (IV.64) leads to

=0 ) t (

u_i^C and u_i^D(t)=0 at any time t.

In such a case, the displacements u_i^C and u_i^Dcan be removed from the unknowns

{ }

q . This reasoning can be extended to the case of displacements imposed to zero on any number of similar segments belonging to the contour.

For the solution of the equations, a classical step by step procedure is used.

Assume that (IV.61) and (IV.62) are satisfied at time t_A so that we know exactly the values of:

iJ A iJ A iJ A iJ A iJ A ijJ A ijJ

A f~ , t~ , u~

, u , u ,

, ε

σ & & ,^Ar_i^J

We want to calculate, at time t_B =t_A +Δt,

iJ B ijJ B ijJ

Bσ , ε& , u& ,^Br_i^J

and at time t_B, (IV.61) and (IV.62) must be satisfied.

Let ^Bσ_ij^J=^Aσ_ij^J +Δσ_ij^J (IV.65)

Chapter IV  Page 89  ^Bu_i^J=^Au_i^J +Δu_i^J (IV.66) During the time interval Δt, we assume the evolution of the displacements is a linear function of time:

t u t

u u u

J i J i A J i B J

i Δ

= Δ Δ

= −

& (IV.67)

If ^Bu_i^J is known, we can calculate u&_i^J by (IV.67) and ε&^J_ij by (IV.62).

Then we can calculate the stresses ^Bσ_ij^J in each Voronoi cell by integration of the constitutive equation.

What we have to do is to find the value of ^Bu_i^J such that (IV.61) can be satisfied.

We proceed by iterations.

Let

( )

Δui^J _k be the value of Δu_i^J at iteration k and

( )

^Δui^J _k+1 be the value of Δu_i^J at iteration k+1.

Let

( ) ( )

k i^J J

k i J

i u du

u = Δ +

Δ ₊₁ (IV.68) After some developments, we get:

[ ( )

^σ

] ( ) ( )

⁰

1

≠

=

−

−

∑ −

= J k

k i iJ iJ B iJ N B

I

IJj k ijI

B f~ p R

t~

A (IV.69)

[ ( )

^σ

] ( )

₁ ⁰

1 1 − − − =

∑ ₊

= + J k

J i B i iJ N B

I

IJj k ijI

B ~f p

t~

A (IV.70)

where

{ }

^M

{ }

^K

K

J BKJ r

p

∑

^u

=

=

1

(IV.71) and (R_i^J )_k is the out of balance force at node J for iteration k.

From (IV.69), (IV.70), we get:

( ) ( ) ( )

1 ⁰ 1

=

− +

+ ₊

∑

^N= ^{iJ k iJ k iJ k} I

IJ j I

ijA R p p

dσ (IV.72)

This equation gives the dσ_ij^I that we should have in order to ensure equilibrium at the end of the iteration.

From the constitutive equations of the material and according to the integration scheme used to integrate them, we can write: