Second-Order Computational Homogenization Scheme Preserving Microlevel C1
Continuity
Tomislav Lesičar, Zdenko Tonković, Jurica Sorić
Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb
Ivana Lučića 5, 10000 Zagreb, Croatia
tomislav.lesicar@fsb.hr, zdenko.tonkovic@fsb.hr, jurica.soric@fsb.hr
Keywords: heterogeneous materials, C1 finite element, C1 continuity microlevel, second-order
computational homogenization, gradient generalized periodic boundary conditions
Abstract. The paper deals with a new second-order computational homogenization procedure for modeling
of heterogeneous materials at small strains, where C1 continuity is preserved at the microlevel. The multiscale
model is based on the Aifantis theory of gradient elasticity. The C1 two dimensional triangular finite element
used for the discretization of macro- and microlevel is described. Contrary to the C1 - C0 transition, here besides
the displacements, the displacement gradients are included into the boundary conditions on the representative
volume element (RVE). According to the second order continuum at microlevel, the relevant homogenization
relations are derived. Finally, the performance of the algorithms derived is investigated. Dependency of
homogenized stresses on mesh density and microstructural parameter l are examined in simple loading cases.
Introduction
It is well known that classical continuum mechanics cannot consider structural effects in the material
at the microlevel and therefore, multiscale techniques for modeling on multiple levels have been
developed [1]. Several homogenization methods have been proposed, where the second-order
computational homogenization scheme has been proven as most versatile tool [1, 2, 3]. The multiscale
algorithm comprising second-order homogenization procedure requires C1 continuity at the
macrolevel and employment of nonlocal continuum theory. The discretization of RVE at the
microlevel is usually performed preserving standard C0 continuity. However, due to the C1 - C0
transition, some inconveniences arise. Firstly, the microlevel second-order gradient cannot be related
to the macrolevel as volume average. Besides, to prescribe the full macrolevel second-order gradient,
the microfluctuation boundary integral is needed. As the second, in the Hill-Mandel energy condition,
a modified definition of the second-order stress should be derived.
To overcome these shortcomings, the present contribution deals with a multiscale second-order
computational homogenization algorithm keeping C1 continuity at both levels under assumptions of
small strains and linear elastic material behavior. Accordingly, the computational models at both
levels are discretized by the C1 continuity finite element (FE) reformulated for multiscale analysis in
the authors’ previous papers [4, 5]. Herein, a new scale transition methodology is derived in which
the volume average of every macrolevel variable prescribed at the microlevel is explicitly satisfied.
Furthermore, employing the Hill-Mandel energy condition, the macrolevel stress tensors are extracted
as volume average of microlevel stress tensors. The macroscopic consistent constitutive matrix is
computed from the RVE global stiffness matrix using the standard procedures. A special attention is
directed to the application of the gradient displacement- and generalized gradient periodic boundary
conditions on the RVE. The algorithms derived are implemented into FE software ABAQUS [6] via
user subroutines. Finally, the performance of the proposed homogenization approach, as well as the
finite element are verified.
C1 continuity triangular finite element
The finite element used for the discretization of macro- and microlevel is based on the Aifantis theory,
as a special case of the Form II Mindlin’s theory. The basic relations of strain gradient elasticity
theories of Mindlin as well as Aifantis are compiled in [4]. Fig. 1 represents the element which has
three nodes with twelve nodal degrees of freedom, corresponding to the two displacements and their
first and second order derivatives with respect to Cartesian coordinates. It describes the full fifth order
polynomial displacement field with plane strain under small displacement assumptions. More detailed
element derivation is presented in [4, 5].
C1 - C1 computational homogenization algorithm
As mentioned before and presented in Fig. 2, the multiscale algorithm consists of two levels obeying
C1 continuity. In the following, macro quantities are denoted by the subscript “M”, while
microvariables are labeled by “m”. Accordingly, the RVE displacement field is defined as
1
u m = ε M x + x  ε M  x + r ,
2
(1)
ε
where x is the spatial coordinate on the RVE boundary, M denotes the macrostrain tensor and r
represents the microfluctuation field. It is important to note that in Eq. (1) displacement vector um
comprises displacements as well as their first and second derivatives, according to the finite element
formulation. In the macro to micro transition, volume average of the microstrains must be equal to
macrostrains. Exploiting this requirement with account to Eq. (1) and quadratic RVE geometry, the
following relations arise:
1
1
1
1
ε m dV  ε M   nr dA  0,
ε m dV  ε M   n  r  dA  0.


VV
VV
VV
VV
(2)
For both, gradient displacement- and gradient generalized periodic boundary conditions, Eqs. (2)
are automatically satisfied. As can be noticed, every macrovariable is defined as volume average of
its microconjugate, without additional integral relations, as in [1, 2, 5]. Furthermore, the Hill-Mandel
energy condition in a full C1 continuity boundary value problem is expressed as
1
σ m : δε m  l 2 μm δ  ε m   dV  σ M : δε M  l 2 μ M δ  ε M  
V V 
(3)
Substitution of Eq. (1) into Eq. (3), after some straightforward calculus, yields the following
relationships
1
1
(4)
σ M   σ mdV ,
l 2μ M    l 2μm  σ mx  dV .
VV
VV
Next, using Eq. (3) and the principle of virtual work based on the Aifantis theory, as defined in
Eq. (9) of Ref. [4], it is possible to derive another form of Eq. (4) that can be used to calculate the
macrostress tensors, as follows
1
1
1 1
1

σ M    t x  τ n  dA  σ M  Dfb ,
μ M    x t x  τ n x  dA  μ M  Hfb .
(5)
VV
V
V V2
V

In Eq. (5), t and τ are the traction and the double surface traction, respectively. D and H are the
coordinate matrices of boundary nodes and f b represents the vector of boundary nodal forces. The
tangent stiffness matrices can be obtained through the static condensation procedure, as described in
[1, 5]. In the Aifantis theory, the stress tensor σ depends only on the strain tensor ε , and analogously,
the double stress tensor μ depends only on the strain tensor gradient ε . According to
aforementioned restrictions, the tangent stiffness matrices at the macrolevel are
1
1
C  DK bbDT ,
C  HK bbHT .
(6)
V
V
In Eq. (6) C is the matrix relating stress to strain, C  relates double stress to the energy conjugate
strain gradients and K bb is the condensed RVE stiffness matrix.
Fig. 1 C1 triangular finite element
Fig. 2 Scheme of full C1 continuity
micro-macro algorithm
Numerical example
The C1 - C1 computational homogenization procedure has been tested on the heterogeneous RVE of
side length 0.2 mm shown in Fig. 3. The RVE represents linear elastic porous steel with material data
E  210 GPa and   0.3 , consisting of 13% randomly distributed voids of the average radius of
0.043 mm. Two loading cases have been considered: (a) the tension of the RVE for homogenization
of σ and (b) the RVE bending for homogenization of μ . Firstly, the influence of the mesh density
and the microstructural parameter l on the homogenized stresses has been examined. In this sense,
three different models consisting of 486, 790 and 1281 elements are considered.
Fig. 3 RVE with 790 FEs
(a)
(b)
Fig. 4 Mesh dependency of homogenized stress tensors
Fig 4. shows that in all examples neither stress tensors nor microstructural parameter l are not
significantly influenced by mesh density, which is to understand because of high order of
interpolation polynomials and high integration points number of the FE that are used. Considering
these facts, mesh density is more dependent on qualitative description of the RVE geometry than the
homogenization results.
As the second, influence of the microstructural parameter l on homogenized stress tensors has been
tested, again for the tension and the bending load cases. Several values of l have been chosen:
l = 0, 0.01, 0.03, 0.0577, 0.1, 0.2, 0.3, 0.5 and 1, where l  0.0577 is a specific value, according to
[7, 4], providing the relationship between RVE size and microstructural parameter. In Fig. 5 it is
obvious that after l  0.0577 , the homogenized stress becomes constant and l has no further
influence. On the other hand, the double stress rises constantly with parabolic character, accordingly
to Aifantis principle of virtual work.
(a)
(b)
Fig. 5 Influence of l on homogenized stress tensors
Conclusions
A second-order C1 - C1 computational homogenization procedure for modeling of heterogeneous
materials at small strains is presented. In this case the macro- and microlevel are discretized by the
same C1 two dimensional triangular finite element. The gradient displacement- and gradient
generalized periodic boundary conditions on the RVE have been derived. The dependency of the
homogenized stresses on mesh density and parameter l has been investigated on two simple loading
cases. The finite element has shown very good convergence characteristics, while stress σ shows no
significant influence of l. In the principle of virtual work, double stress μ has a quadratic relation to
l, which is confirmed in computational homogenization algorithm. The C1-C1 computational
homogenization procedure derived is only a first step in development of nonlocal multiscale model
for simulation of the damage localization at microlevel in brittle heterogeneous materials.
Acknowledgements
The investigations described herein are part of the project “Centre of Excellence for Structural
Health” (CEEStructHealth) supported by the EU under contract IPA2007/HR/16IPO/001-040513.
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