18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A COMPARISON OF IMPLICIT AND EXPLICIT FINITE ELEMENT METHODS FOR THE MESO-LEVEL SIMULATION OF DRY WOVEN FABRIC COMPOSITES M. Komeili, A. S. Milani School of Engineering, University of British Columbia, Kelowna, Canada mojtaba.komeili@ubc.ca; abbas.milani@ubc.ca Keywords : multi-scale modeling , woven fabrics , meso-level unit cell , finite element method 1. Abstract of dry yarns. Small shear stiffness, nonlinear Accurate and fast modeling of unit cells is a crucial transverse stiffness, which highly depends on the step in multi-scale analysis of woven fabric transverse and axial strains, significantly high axial composites. Due to complex fibrous nature of yarns, stiffness under tension, and the incapability of fibers the conventional finite element models are not in bearing compression are among the most capable of capturing the accurate behavior of these important properties which should be included in the materials and modeling modifications are necessary material models of unit cell yarns [3]. to enhance the definition of their unit cells. These In particular, the substantial difference between modifications primarily include postulating suitable stiffness values in the axial direction compared to meso-level material constitutive laws that can be other directions implies that the material properties incorporated into commercial finite element should be defined in a frame attached to the fibers. packages via user-defined subroutines. Secondly, the However, a given finite element software (here material stiffness matrix in fabric yarns should be ABAQUS) may be working with a different frame. defined with respect to fiber directions. However, Therefore, modifications in stress updates of the the frame of a finite element solver may be different finite element package should be applied. There are from the frame of fiber yarns under large shear two different approaches in dealing with such deformation. The latter problem is normally handled material frame rotations. using either the TSM (Transforming Stiffness The first approach, known as TSM (Transforming Matrix) or TSS (Transforming Stress and Strain Stiffness Matrix) is conducted via rotating the vectors) method in user-subroutines. Finally, there is material stiffness matrix defined in fiber direction to the possibility of selecting explicit or implicit the working frame of the software and updating the integrator to simulate the unit cell behavior, which in new stress. Mathematically, turn can affect the definition of a material subroutine T' (1) [ ] C [ ][ ] [ T C ] = due to different parameters required in each e f integrator. The aim of this article is to conduct a where [ ] C is the material stiffness matrix, [ ] T and thorough study on the advantages and disadvantages [ T' are the transformation matrices, and e and f ] of the TSS and TSM methods under both explicit and implicit modeling frameworks in axial and shear subscripts refers to the working frame of the loading of a fabric unit cell. software and the fiber frame, respectively. The second approach is referred to as TSS (Transforming Stress and Strain vectors) in which 2. Introduction: Modeling fibrous materials the stress and strain increments are rotated to the frame of fiber and stress updates are calculated at the 2.1. Material model frame of fiber. Then, the new stress state is One of the biggest challenges in the finite element transformed back to the working fame of the modeling of dry woven fabrics is their multi-scale software. This can be expressed mathematically as: material behaviour nature [1,2]. To assure that the (2) n n � � n � � n { } [ T' ]{ } ; { } [ T' ]{ } effect of micro-scale fibers are included in a fabric σ σ σ σ = σ σ σ σ = f e f e unit cell model, particular material behaviors should be defined for fiber yarns. Such material behaviors are postulated by considering the physical behavior
u y = 4 (mm) (3) n 1 n � � n 1/2 + + { } { } [ ] { C } = + σ σ σ σ σ σ σ σ f f f f u x = 4 (mm) (4) n 1 n 1 + + { } = T [ ]{ } σ σ σ σ σ σ σ σ e f The main axial z y � � where { } and { } are the stress and strain σ σ σ σ stiffness direction increments and the superscript refers to the step x number. The current fiber direction can be tracked using the deformation gradient tensor and the initial direction of fiber [2,3]. Fig. 1. The schematic of a cube element and applied Apart from choosing a stress update method and loading mode applying transformations for stress calculations, the integrator of finite element solver can play an Results of this finite element model on a single important role in defining a correct unit cell model. element with the above mentioned stress update It can also affect the accuracy of simulation results models (TSM or TSS) along with two different and the run time. In this paper, two different integrators (implicit or explicit) are compared in integrators (implicit/explicit) and two Table 1. The table also shows the percentage of error aforementioned approaches for handling the specific compared to the analytical solution. Results confirm material behavior of yarns are studied under the the superior advantage of TSM and TSS compared axial tension and pure shear deformation modes of a to the default method by the software. plain weave. Comparison of the methods in terms of their convergence and computational cost is Table 1. Obtained axial stresses ( � 1 ) using different presented. combinations of integrator/material models and their error percentages compared to the analytical solution of the problem shown in Fig. 1 2.2. A simple geometry example case Explicit Implicit First, to show the difference between using the Method � 1 (GPa) Error � 1 (GPa) Error default material models (stress update) available in Default 0.994 48.3% 1.013 47.3% built-in modules of Abaqus and the modifications TSM 1.983 3.1% 1.961 2.0% proposed using the TSM or TSS algorithms, a TSS 1.983 3.1% 1.961 2.0% simple case study is illustrated in this section. The geometry is made of a simple 20×20×20 mm unit cube. The assigned material properties contain a 3. Modeling woven fabrics fictitious stiffness, which is relatively high in the 3.1. Geometry axial direction- z compared to other non-zero stiffness components (namely, 50 GPa for the axial To study the computational effectiveness of each component and 5 MPa for the shear and lateral stress update method (TSM or TSS) along with components). The direction of the main axial different integrators, a plain woven fabric is selected stiffness and the assigned loading condition on the for the analysis (Figure 2-a). The first step of the cube are illustrated in Figure 1. A lateral analysis is defining a representative element (also displacement of 4 units (mm) is applied on the top known as the unit cell) which can describe the surface. This induces a stretch ratio of � 1 =1.0392 in overall behaviour of the fabric [1, 3]. The unit cell the main axial direction (along the cube height) should be as small as possible to reduce the which corresponds to an axial stress of � 1 =1.923 modeling and computation costs, yet it should not be GPa. Nearly zero stress components are obtained in too small not to reflect the actual behaviour of the the other directions. It should be noted that the fabric. There are different types of unit cells strains are calculated using the logarithmic strain suggested for balanced woven fabrics, based on the which is the default strain measure used by the applied loading mode and different assumptions on software. the behaviour of a given fabric [3]. Figure 2-b shows
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