Multi-Scale Material and Product Modeling Using DIGIMAT and ABAQUS

2004 ABAQUS Users’ Conference 65

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: +32 (0) 495 52 56 52

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.

66 2004 ABAQUS Users’ Conference

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.


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: J2 elasto-plasticity and Chaboche's


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 [tn, tn+1] and each 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 to it by ABAQUS are the total macro strains nε and ε (as well as material constants and history information at tn). The DIGIMAT code returns the macro stress σ and macro tangent moduli c at tn+1. The micro-structure is not “seen” by ABAQUS but only by DIGIMAT, which considers 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 (EL, αL) and transversal (ET, αT) directions to the test sample. The charts presented from Figure 2 to Figure 7 summarize DIGIMAT prediction of EL, ET, αL, αT.

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.


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)).


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.


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 110 GPa 0.25 300 MPa Hardening model Hardening modulus (k) Hardening exponent (m)

Matrix : Titanium (Ti)

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)


8. Figures

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.

PMC Microstructure


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.


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.


Figure 7. The thermal expansion coefficient of the PMC in the longitudinal and transverse direction as a function of the fibers’ aspect ratio.

Figure 8. Coupled ABAQUS-DIGIMAT analysis. Definition of the centrifugal cyclic loading.

0.5 1

Maximal centrifugal velocity

time


Figure 9. DIGIMAT standalone parametric analysis. Influence of the inclusion’s volume fraction, aspect ratio and orientation on the average macro stress.

Figure 10. Coupled ABAQUS-DIGIMAT analysis. Prediction of the performance of the blade as a function of the parameters defining the MMC micro-structure.


Figure 11. Coupled ABAQUS-DIGIMAT analysis. von Mises equivalent macro stress response. Homogeneous material (Titanium).

Figure 12. Coupled ABAQUS-DIGIMAT analysis. von Mises equivalent macro stress response. Titanium matrix with transverse SiC fibers.


Figure 13. Coupled ABAQUS-DIGIMAT analysis. von Mises equivalent macro stress response. Titanium matrix with axial SiC fibers.

Figure 14. Coupled ABAQUS-DIGIMAT analysis. Equivalent macro strain response. Titanium matrix with transverse SiC fibers.


Figure 15. Coupled ABAQUS-DIGIMAT analysis. Equivalent matrix strain response. Titanium matrix with transverse SiC fibers.

Figure 16. Coupled ABAQUS-DIGIMAT analysis. Equivalent inclusion strain response. Titanium matrix with transverse SiC fibers.


9. Acknowledgment

This work was done with the financial support of the Walloon Government/D.G.T.R.E under the program FIRST-SPINOFF and FIRST-ENTREPRISE.

Documents

Multi-Scale Material and Product Modeling Using DIGIMAT and ABAQUS