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Handbook of Référence
R5.04 booklet: Nonlocal modelings
Document: R5.04.02
Nonlocal modeling with gradient of deformation

Summary

This document presents a model of delocalization of the laws of behavior per regularization of
deformation. It introduces an additional nodal variable: regularized deformation, dependant on the deformation
local by an equation of regularization of the least type square with penalization of the gradient than one solves
at the same time with the traditional equilibrium equation. The regularized deformations are used for calculation
evolution of the internal variables (and not for the calculation of the constraints!). This method makes it possible to avoid
certain problems involved in the digital processing of the local problems like the dependence with the grid.
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1
Nature of the formulation

In the presence of damage (softening), the local laws of behavior lead to one
badly posed problem which results numerically in a localization of the deformations in one
bandage thickness a mesh: in extreme cases, one breaks without dissipating energy.
There are several extensions to the local models which make it possible to mitigate this problem of localization
(relieving of the potential energy, enrichment of kinematics, theories with gradient, models not
buildings). This document deals with nonlocal model with gradient of deformations, modeling
* _GRAD_EPSI, drifting of the model with gradient of equivalent deformation proposed by Peerlings and Al
(1995). One introduces interactions between the material point and his space vicinity by regularizing them
deformations thanks to an operator of delocalization. The regularized deformations are then used
to evaluate the evolution of the internal variable.

It should be noted however that the constraints are calculated starting from the local deformations bus
the use of the deformations regularized in the calculation of the constraints would return to “too
to regularize “the problem, which would call into question the existence even of solutions. One is convinced some
easily thanks to the following example:
Let us consider a bar made up of 2 different materials which have different Young moduli. One
exert on this bar a simple traction. The 2 elements being gone up in series, the constraint is equal
in the two elements:

= 1 1
E = 2 2
E =



E2
E1


With the interface between the two elements, the discontinuity of modulus Young thus imposes one
discontinuity of the deformation. Let us consider now either the local deformation but one
delocalized deformation. The traditional operators of delocalization cause to make continuous
deformation in the structure, which then generates obligatorily a discontinuity of constraint
with the interface because of the difference of Young modulus, and this goes against the equation
of balance.

The regularization of the deformations leads us to introduce a characteristic length definite by
operator DEFI_MATERIAU under the key word factor NON LOCAL which conditions the width of the tapes
of localization. The scales thus are not defined any more by the digital processing of the problem but
by a parameter material.
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2
Limits of the local models

One initially proposes to illustrate the phenomenon of localization in the simple case of one
bar subjected to a uniaxial traction.
One thus considers an assembly of identical elements assembled in series subjected to a traction
as represented on [Figure 2-a].


, U
i-1 I
i+1

Appear 2-a: Assemblage of identical elements assembled in series
subjected to a tensile test

Each element obeys the same law of behavior of the elastic type endommageable with
softening [Figure 2-b]. The state of material is described by two variables which are the deformation
and the damage characterized by scalar variable D. variable Cette is worth 0 when the material
is healthy and grows up to 1 when it is completely damaged.
We will not enter here in detail of the equations governing such a behavior of material.
Let us specify simply that these equations make it possible to describe the behavior completely of
material. They indeed give us access to the constraints and the damage according to the rate
of deformation, to see for example [R5.03.18].


peak
E0 peak



Appear 2-b: Loi of behavior of material in uniaxial simple traction
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The elements of the studied bar are assembled in series, which implies, because of the equation
of balance of the structure, equality of the constraint in all the elements:

I =

One can consequently study the total response of the assembly to a simple tensile test. This
answer breaks up into two phases. In a first phase, the behavior of all them
elements is elastic and the damage remains null. The response of the structure thus exists and is
single. The deformation is identical for all the elements and is worth:


I =

0
E

This phase perdure as long as the peak of constraint is not reached. Microcomputer-heterogeneities of
material imply light fluctuations of the field of elasticity between the various elements, which
will involve the damage of a fastener before the others. The second phase
start when one of the elements which one notes A damages. The constraint in the whole of
structure reached its maximum. While continuing traction, the constraint supported by the structure is
to decrease. Element A having passed the peak, it is in the lenitive phase of the behavior of
material, which means that it will continue to damage itself during traction. The other elements do not have
not reached the critical point, they thus simply will undergo an elastic discharge at the time of
decrease of the constraint. This phase finishes when element A is completely
damaged. Finally, the damage as well as the deformation thus concentrated in one
only element.

One then includes/understands easily the numerical consequences of the localization. The phenomenon described
previously on a simple sample will occur whatever the structure with a grid by elements
finished. For reasons of stability, the localized solution tends to being selected. The damage and
the deformation will concentrate in a tape thickness an element and any refinement of
grid then will modify the total response of the structure. One includes/understands then well that it is
impossible to describe the scale of the tapes of localization, the length of the damaged tape
coming from the grid and not from a physical principle. Moreover, one obtains a result physically
inadmissible from an energy point of view. Indeed, the energy dissipated at the time of the damage goes
to depend on the refinement of the grid, and one can even imagine the total rupture of a structure without
expenditure of energy if an extremely fine grid is considered.
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3
Formulation with regularized deformations

3.1 Principle

One considers the state of material defined locally by the deformation and of the internal variables.
given free potential energy (,) makes it possible to define the constraint.
In manner General, the law of behavior is given by the expression of the constraint and the law
of evolution of the internal variables:

(,)


& = G (&,)

The principle of the method of delocalization of the deformations is to use the deformations
regularized in the law devolution of the internal variables:

(,)

& = G (&,)

One thus includes/understands the general information of the method which does not force to reconsider the integration of the law of
behavior. It is indeed the same one as for the local model but by replacing par. It is necessary
nevertheless to distinguish well the calculation from the internal variables, which utilizes deformations
regularized, of that of the constraints, which utilizes only the local deformations.

3.2
Choice of the operator of delocalization

The choice of the operator of regularization is purely arbitrary and is not based on any reasoning
physics. One however may find it beneficial to choose an operator who is integrated easily and directly in
STAT_NON_LINE by the finite element method. Thus, the use of an integral formulation, where
the coupling between the finite elements on the level of the integration of the laws of behavior causes
to increase considerably the bandwidth of the tangent matrix and to thus increase it
a many operations to be carried out, are not judicious. The operator of regularization retained, proposed
by Peerlings and Al (1995), a delocalization by least squares with penalization employs of
gradient:
1
1
R () = min
(
2
2


-) + (L) D
C
2
2


The term in gradient introduces the interaction between the material point and its vicinity and makes it possible to limit
strong concentration of gradient of deformations. To minimize such an integral amounts solving
the following differential equation:
- L2

C =

One sees appearing a major interest of the choice of this operator of regularization. The differential equation
can be integrated classically by the finite element method, and this without introducing news not
linearities. It is enough for that to introduce new nodal variables representing the deformations
generalized.
There is moreover a tangent matrix of reasonable bandwidth (compared to a formulation
integral) but it should be noted that the tangent matrix is not symmetrical, as it further will be seen
by clarifying the tangent matrix.
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3.3 Formulation
variational

In the model, two equations control the process of deformation, on the one hand the equation
of traditional balance and in addition the differential equation characterizing the regularization of
deformations. The integral formulation of our problem is as follows:

v Vad
(v) C: D = v.T D + F FD





V AD
: space acceptable displacements

T: forces imposed on the edge

E [
H1 ()]6

(E + E
.L2
)

D = E
D
C




The limiting conditions for the generalized deformations are the natural conditions rising from
the equation of regularization. They are of Neumann type:

.n = 0

One indeed imposes no particular condition on the edge in the equation of regularization.

4 Discretization
4.1 Equations
discretized

The equilibrium equation discretized between the external and interior forces is traditional form
(cf [R5.03.01]):
T
Fint + D = ext.
F

with F
T
int = BT D and Fext = NR T D


(
T
D: cf T
B of [R5.03.01])

where NR are related to forms associated with the field with displacement and B the derivative with
functions of forms.

The differential equation on the regularized deformations is discretized in the same way:

K
F
=



~ ~
2 ~
with K
= (NT
~
NR + L BT B

~T
C
) D and F = NR D


~
~
where NR are related to forms associated with the field with generalized deformations and B them
derived from the functions of forms. It should be noted here that functions of forms associated with
generalized deformations are different from the functions of forms associated with displacements.
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The nodal residues associated these two equations are as follows:

F U = Fint + DT - Fext



F = K - F

The tangent matrix associated the resolution of this system by the method of Newton is as follows:

F U F U


U

K =

F
F


U


The various blocks of the tangent matrix are as follows:

Fu
T
=

B

Data base


U

i-1

Fu
T
=
~
B
NR
D




i-1


F
= (~ ~ 2~ ~
NT NR + L BT B
C
)
D
i-1
F
= - ~
NT

Data base


U i-1

It should be noted that the tangent matrix is not-symmetrical.

4.2
Choice of the finite elements

The introduction of new nodal variables forces to use new compatible elements
with the new formulation. One is in the presence of two nodal unknown factors: displacements
and regularized deformations. Deformation being the space derivative of a displacement, if one
use functions of P2 form for displacement, it is preferable to use functions of form
P1 for the deformations regularized for reasons of homogeneity. Quadratic elements,
TRIA6 and QUAD8 for the 2D, TETRA10, PENTA15 and HEXA20 for the 3D, were developed.
components of displacement are assigned to all the nodes of the element whereas the components
regularized deformations are affected only with the nodes nodes. For more clearness,
element TRIA6 is represented below:

3

6

5
Nodal variables
(U,)
1
(U)
4
2


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Titrate:
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4.3 Modelings
available

These various elements are used in three types of modelings:
Calculation 2D in plane deformations:
D_PLAN_GRAD_EPSI (cf [U3.13.06])
Calculation 2D in plane constraints:
C_PLAN_GRAD_EPSI (cf [U3.13.06])
Calculation 3D:
3d_GRAD_EPSI (cf [U3.14.11])

The axisymmetric mode is not yet available.

5
Interface with the laws of behavior

The use of this method of delocalization requires the calculation of the following terms on the level of
law of behavior:

(,)



,

,
,






The last two terms are necessary only for the calculation of the tangent matrix.

6 Bibliography

[1]
BADEL P.: Contribution to the digital simulation for the reinforced concrete structures. Thesis
of doctorate of the university Paris 6 (2001).
[2]
LORENTZ E.: Laws of behavior to gradients of internal variables: construction,
variational formulation and implementation numerical. Thesis of doctorate of the university
Paris 6 (1999).
[3]
PEERLINGS R.H.J., OF BORST R., BREKELMANS W.A. Mr., OF VREE J.H.P.
:
Computational modelling off gradient-enhanced ramming for fracture and fatigue problems.
Computational Plasticity, Share 1, Pineridge Near, pp.975-986 (1995).
[4]
PEERLINGS R.H.J: Enhanced ramming modelling for fracture and fatigue. PhD Thesis
Eindhoven University off Technology, Faculty off Mechanical Engineering (1999).

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