Multi-Scale Material and Product Modeling Using DIGIMAT and ABAQUS

16 Pages·2004·0.48 MB·English

Multi-Scale Material and Product Modeling Using
DIGIMAT and ABAQUS
Roger A. Assaker*, Laurent Tinel, Laurent Adam and Issam Doghri
e-Xstream engineering SA
4, Avenue Georges Lemaître, B-1348 Louvain-la-Neuve, Belgium
e-mail : [email protected]
Phone: [phone]
Abstract: The objective of this paper is to describe and demonstrate the capability to perform
accurate nonlinear multi-scale material and product modeling using ABAQUS and DIGIMAT (i.e.
The Linear and nonlinear multi-scale material modeling software from e-Xstream engineering).
To this end we will start by presenting the semi-analytical micro-macro homogenization
techniques implemented in DIGIMAT to predict the nonlinear anisotropic thermo-elasto-plastic
material behavior of a Representative Volume Element (RVE) of a Polymer Matrix Composite
(PMC) and a Metal Matrix Composite (MMC) subjected to a non monotonic loading (e.g.
cycling).
Next, DIGIMAT to ABAQUS/CAE Graphical User Interface and to ABAQUS/Standard UMAT
interface (i.e. DIGI2ABA) will be presented and used to provide a detailed multi-scale description
of the composite material (PMC and MMC) behavior within an ABAQUS finite element model.
Two applications will be shown to illustrate the use of DIGIMAT and DIGIMAT to ABAQUS
interface. The first is a parametric study of the thermoelastic material behavior of a PMC. The
second will focus on the modeling of the elasto-plastic behavior of a Ti-SiC MMC Turbojet fan
blade subjected to a cyclic loading in order to illustrate the link between the parameters
describing the material microstructure taken in account by DIGIMAT (i.e. inclusions’ volume
fraction, aspect ratio, orientation …) and the product performance indicators predicted by
ABAQUS. The use of the material microstructure as a design parameter and the influence of a
micro-structural change on the macroscopic material and product performance level will notably
be shown.
Keywords: Composites, Constitutive Model, Homogenization, Muti-Scale, Plasticity, MMC, PMC.
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1. Introduction
The high level of end-performances (e.g. stiffness to weight ratio, miniaturization, etc.) that are
required from high-tech products very often translate into a set of equally high performance targets
at the constituent material level. Composite materials, where a matrix phase is reinforced with one
or more phases, are a major way of achieving these performance specifications.
A large variety of composites exists; Polymer Matrix Composites (PMC) and Metal Matrix
Composites (MMC) are just two families of composite materials where a polymer matrix or a
metal matrix is reinforced with glass or carbon inclusions or any other type of appropriate
inclusions, respectively. Some of the basic questions facing a material design engineer are:
What kind of material to use for the matrix and reinforcing phases?
What kind of reinforcement shape to use? How much? How to distribute them within the matrix ?
The challenge is to find the right answer to all these questions in order to find the optimal balance
between the material performance and cost and, more importantly, the optimal set of material
parameters needed to achieve the final product performance targets.
To design composite materials one can use the traditional prototyping and testing approach to
develop a discrete set of material grades that are then non-optimally used in the end-product. The
material grade can be physically used within an actual prototype of the final product, or the
material test results are used to compute the parameters of a phenomenological material law that
approximates, to a certain extent, the material constitutive behavior. This standard approach
suffers from several shortcomings. First, it is costly and time consuming (physical prototyping
and testing). Second, it is non-predictive at the material level (the material prototype has to exist
and to be tested). Third, it does not give an insight into the material microstructure (it is the final
behavior of the composite that is measured). Fourth, a phenomenological constitutive behavior
has to be postulated in order to approximate the material behavior. This last point is especially
difficult when the composite behavior is nonlinear (e.g. elasto-plastic) and anisotropic.
Multi-scale material modeling techniques, which consist of predicting the final (i.e. macroscopic)
behavior of a composite material based on the modeling of its microstructure, relax, to a certain
extent, each of the previously listed shortcomings. In fact, multi-scale material modeling allows
the material design to limit the number of material prototyping and testing cycles by a predictive
numerical simulation of the composite material response that can be directly used inside a finite
element model of the product.
There are two major approaches to multi-scale modeling. The first is based on a detailed
modeling, through finite element discretization, of the microstructure. The second is semi-
analytical based on mean field, Eshelby-based, homogenization methods. It is this second set of
methods that are implemented in DIGIMAT and interfaced to ABAQUS for the linear and
nonlinear modeling of composite materials and structures using these materials.
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In the next section, we will describe the semi-analytical homogenization methods implemented in
DIGIMAT. Next, we will describe DIGIMAT’s UMAT interface to ABAQUS. Finally, two
applications will be described and conclusions will be drawn.
2. DIGIMAT: Multi-Scale Material Modeling
The DIGIMAT (for "Digital Materials") software is based on mean-field Eshelby-based
homogenization models and robust numerical algorithms for the micro/macro simulation of linear
and nonlinear inclusion-reinforced materials.
We use the generic term of "inclusions" to designate the reinforcing phase. The inclusions can be
particles, fibers (long or short) or platelets. Actually, all those shapes can be generated by setting
appropriate values to the aspect ratio of a spheroid (i.e., an ellipsoid with an axis of revolution).
There are many industrial examples for those composites, as illustrated hereafter.
• Polymer matrix composites (PMCs) reinforced with ceramic, glass or Kevlar fibers.
Objective: improve stiffness and strength. Examples: boat hulls, aircraft wings, cars
(body frames, hood and door panels), sporting equipment.
• Polymer matrix with low modulus rubber particles. Objective: improve toughness and
impact resistance. Example: car bumpers.
• Rubber matrix with carbon-black particles. Objective: improve toughness and stiffness.
Example: tires.
• Metal matrix composites (MMCs) with ceramic particles or short fibers. Objective:
mainly high-temperature applications. Example: fossil-fuel engine components (e.g.,
turbochargers).
• Concrete matrix with: metallic fibers (strength in tension or bending), polymer or natural
fibers (better ductility, lower density), rubber inclusions (impact resistance, acoustic
isolation).
In all those cases, we wish to predict the influence of the micro-structure on the overall properties
of the material or the product. An elegant solution is provided by a micro/macro or two-scale
approach with a macro scale (that of the body) and a micro scale (the heterogeneous micro-
structure). Transition between the two scales is made possible by average field theories, also
known as homogenization models.
The current version of DIGIMAT is able to simulate within reasonable accuracy, computer time
and memory: (1) any nonlinear rate-independent model for any phase; (2) cyclic and otherwise
non-proportional loadings; (3) any multi-axial stress state; (4) structures (real products) made of
composite materials.
In the present version of DIGIMAT, we implemented two homogenization schemes: Mori-Tanaka
(M-T) and Double Inclusion (D-I) and two material models: J elasto-plasticity and Chaboche's
2
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cyclic plasticity. Those models can be used for any phase of a composite material. The inclusions
can be spheres, fibers or spheroids with any aspect ratio. For validation purposes, we validated our
numerical simulations against experimental data or direct FE computations on unit cells.
3. DIGI2ABA: DIGIMAT’s UMAT interface to ABAQUS
We also integrated our homogenization code DIGIMAT into the FE program ABAQUS through
its user-defined material interface UMAT. We developed the following two-scale approach. A
classical FE analysis is carried out at macro-scale, and for each time interval [t , t ] and each
n n+1
iteration of the global equilibrium equations at macro scale, and at each quadrature point of the
macro FE mesh, the homogenization module UMAT/DIGIMAT is called. The data that are passed
ε ε
to it by ABAQUS are the total macro strains and (as well as material constants and history
n
σ
information at t ). The DIGIMAT code returns the macro stress and macro tangent moduli
n
c at t . The micro-structure is not “seen” by ABAQUS but only by DIGIMAT, which considers
n+1
each quadrature point to be the center of a representative volume element (RVE) which contains
the heterogeneous micro-structure.
As we shall see, this two-scale procedure allows to compute structures (products) made of
nonlinear composite materials within reasonable CPU time and memory usage.
4. Applications
4.1 Thermoelastic behavior of a PMC: A parametric study
The objective of this section is to predict the macroscopic thermo-elastic material properties of a
series of PMCs where the polymer matrix is reinforced with glass fibers (Figure 1). The problem
definition and the material’s properties are given in Table 1.
DIGIMAT analyses were successfully performed to predict the influence of the fibers volume
fraction, orientation and aspect ratio on the composite’s thermo-elastic properties (i.e. Young’s
Modulus (E) and Thermal Expansion Coefficient (α)) in the longitudinal (E , α ) and transversal
L L
(E , α ) directions to the test sample. The charts presented from Figure 2 to Figure 7 summarize
T T
DIGIMAT prediction of E , E , α , α .
L T L T
Each data point corresponds to a DIGIMAT analysis which requires a small fraction of a CPU
second to complete on a regular PC.
The predicted values compare very favorably to experimental results.
4.2 Elasto-plastic behavior of a MMC: A Ti-SiC fan blade under cyclic loading
The objective of this section is to predict the nonlinear macroscopic elasto-plastic material and
structural response of a jet engine turbine blade subject to a cyclic centrifugal loading. The blade
is made of titanium material reinforced with Silicon Carbide fibers.
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First, a parametric study was performed using DIGIMAT multi-scale material modeling approach
to predict the influence of the fibers volume fraction, orientation and aspect ratio on the elasto-
plastic properties of a virtual material sample (or Representative Volume Element (RVE)) of the
composite subject to a virtual tension test.
Next, a series of coupled DIGIMAT-ABAQUS multi-scale Finite Element Analyses (FE) were
done in order to predict the performance (e.g. Elasto-Plastic stress-strain level) of the blade as a
function of the parameters defining the MMC micro-structure (i.e. Volume fraction, Aspect ratio,
Orientation).
The problem definition and the material’s properties are given in Table 2 and Table 3. The
ABAQUS model is submitted to a centrifugal cyclic loading (Figure 8).
Material Parametric Study: DIGIMAT Multi-Scale Analyses
The curves in Figure 9 summarize a small parametric study where the influence of the SiC fibers’
Aspect Ratio (Ar), Volume Fraction (VF) and Orientation (Φ) on the elasto-plastic response of the
material sample (or RVE) is studied. Each curve is the result of a DIGIMAT standalone analysis
where the RVE is submitted to a strain load that was monotonically increased up to 6%. Each
DIGIMAT analysis, using Mori-Tanaka homogenization method, runs in a fraction of a second on
a regular PC.
Multi-Scale Material & Structure Modeling: Coupled ABAQUS-DIGIMAT Analyses
In this section we present the results of 3 different ABAQUS-DIGIMAT analyses where the
structure is modeled, at the macroscopic scale, by an ABAQUS FE model and the composite
material is described using DIGIMAT’s semi-analytical homogenization approach. Three
materials were considered:
• Homogeneous material (Titanium only)
• Fiber reinforced Titanium where the fibers are aligned with the direction of traction (i.e.
Axial);
• Fiber reinforced Titanium where the fibers are aligned in a direction perpendicular to the
traction (Transverse).
The curves in Figure 10 depict the Stress–Strain along the blade axis (axis 3) at an integration
point located in a critical region (i.e. highest plastic deformation) of the ABAQUS Finite Element
model.
The pictures from Figure 11 to Figure 13 show the contour plots of von Mises equivalent stress
across the blade at maximum velocity (900 rad/s) for the 3 FE analyses. Each analysis
corresponds to a different material. Using ABAQUS-DIGIMAT coupled multi-scale material and
structure modeling approach one can also access the average stress and strain in each phase of the
composite material being modeled by DIGIMAT. Figures 14 to 16 present a contour plot of the
strain field at the macroscopic scale (homogenized results, left) and, at the microscopic scale, the
average strain in each phase (matrix (center) and fibers (right)).

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5. Conclusions
In this paper we have briefly described the semi-analytical homogenization methods that are
implemented in DIGIMAT and in DIGI2ABA, DIGIMAT’s UMAT interface to ABAQUS, for the
linear and nonlinear multi-scale modeling of composite materials and their use with ABAQUS
structural FEA.
Two applications were presented to illustrate the industrial applicability and advantages of this
technology using DIGIMAT and ABAQUS. It is quite clear from this paper that:
1- The semi-analytical mutli-scale modeling techniques implemented in DIGIMAT are an
accurate and efficient approache for performing “what if ?” analyses and numerical
DOEs for the optimal design of linear and nonlinear composite materials.
2- Using this approach in conjunction with ABAQUS enables accurate, efficient and
seamless multi-scale material and structure modeling.
3- Mutli-Scale material modeling in DIGIMAT and ABAQUS is a powerful tool for the
optimal design of a composite material within its final application and as a result from the
manufacturing process.
6. References
1. Doghri, I., Mechanics of deformable solids- Linear, nonlinear,analytical and
computational aspects, Springer-Verlag, Berlin, 2000, XVIII+579 pages, ISBN 3-540-
66960-4.
2. I. Doghri & A. Ouaar, “Homogenization of two-phase elasto-plastic composite materials
and structures. Study of tangent operators, cyclic plasticity and numerical algorithms”,
International Journal of Solids and Structures, 40 (2003) 1681-1712.
3. I. Doghri & C. Friebel, “Effective elasto-plastic properties of inclusion-reinforced
composites. Study of shape, orientation and cyclic response", Mechanics of Materials,
(2004) in press.
4. O. Pierard, C. Friebel & I. Doghri, “Mean-field homogenization of multi-phase thermo-
elastic composites: a general framework and its validation”, Composites Science and
Technology, 64 (2004) in press.

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7. Tables
Table 1 : Material’s properties and Design parameters of the PMC parametric study
Material’s Properties
Phase Young’s modulus Poisson’s ratio Thermal expansion coefficient
Matrix : Polymer 4 000 MPa 0.36 6.0E-05 K-1
Reinforcements : Glass Fibers 74 000 MPa 0.25 5.0E-06 K-1
Design Parameters
Phase Volume fraction Aspect ratio Orientation (φ)
Reinforcements : Glass Fibers 0% to 50 % 1 to 50 0° to 45°

Table 2 : Material’s properties and Design parameters of the Ti-SiC study
Material’s Properties
Phase Young’s modulus Poisson’s ratio Yield stress
Matrix : Titanium (Ti) 110 GPa 0.25 300 MPa
Hardening model Hardening modulus (k) Hardening exponent (m)
R(p) = k pm 331.98 MPa 0.1827
Phase Young’s modulus Poisson’s ratio
Reinforcements : Glass Fibers 410 GPa 0.17
Design Parameters
Phase Volume fraction Aspect ratio Orientation (φ)
Reinforcements : Glass Fibers 8% to 12 % 10 to 50 0° to 90°

Table 3 : Definition of the ABQUS model
ABAQUS Model
Number of nodes Number of elements Number of DOF Element Type
6 344 24 205 19 032 Tetrahedron (C3D4)

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8. Figures
PMC
Microstructure

Figure 1. Parametric study on a PMC. Polymer matrix reinforced with glass fibers
under an uniaxial macro strain loading.

Figure 2. Young’s modulus of the PMC in the longitudinal and transverse direction
as a function of the fibers’ volume fraction.

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Figure 3. The thermal expansion coefficient of the PMC in the longitudinal and
transverse direction as a function of the fibers’ volume fraction.

Figure 4. Young’s modulus of the PMC in the longitudinal and transverse direction
as a function of the fibers’ orientation.

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Figure 5. The thermal expansion coefficient of the PMC in the longitudinal and
transverse direction as a function of the fibers’ orientation.

Figure 6. Young’s modulus of the PMC in the longitudinal and transverse direction
as a function of the fibers’ aspect ratio.

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