Mostly typical simple arrangements like square or S S V , (1) - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS AN EXTENDED NUMERICAL HOMOGENIZATION APPROACH FOR COMPOSITES WITH RHOMBIC FIBER ARRANGEMENTS H. Berger * , M. Würkner, U. Gabbert Institute of Mechanics, University of Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany * Corresponding author (harald.berger@ovgu.de) Keywords : Composite, Rhombic fiber arrangement, Homogenization, FEM 1 Introduction 2 Algorithm and Models Fiber reinforced composites get an increasing The numerical algorithm is based on a micro- attention in the development of new materials. By mechanical unit cell model which contains the real controlling the manufacturing process, it is possible distribution of inclusions. The unit cells represent a to get the desired material properties. With the periodic array of the global structure. To ensure recent advances in numerical modeling of periodicity also after deformation appropriate composites, it is possible to predict the effective periodic boundary conditions must be applied. material properties of the composites. The basic idea for calculating effective material properties is that the strain energy stored in the A number of numerical and analytical methods have heterogeneous system must be approximately the been developed to estimate the effective coefficients same like in the homogeneous system. With FEM using homogenization methods. By micro- S and for elastic case the averaged element strains mechanical models based on unit cells the problem ij can be reduced on investigation of a periodic part of T are calculated and summed over all stresses ij an infinite structure. But existing approaches are elements k of the unit cell often restricted to certain types of arrangements.  1  Mostly typical simple arrangements like square or S S V , (1) ij hexagonal pattern have been investigated which ij k V k result in an overall transverse isotropic behavior of  1  the composite. An interesting goal is to create T T V , (2) ij ij k composites with orthotropic behavior in the V k transverse plane which can be achieved by rhombic V is the element volume and V is the where k fiber arrangements in connection with high volume volume of the unit cell. Then from the following fraction for the fibers. But nearly no results are constitutive equations for such orthotropic case published in literature for such patterns of fibers. Jiang [1] and Guinovart-Díaz [2] calculated with       eff C S T  11   11  11   analytical methods effective shear coefficients for eff eff      T  C C symm. S selected rhombic angles. 22 21 22 22       eff eff eff T C C C S        (3) At our institute a general numerical homogenization 33 31 32 33 33       technique for calculating effective material ef f T 0 0 0 C S     23  44 23  properties of composites with various fiber       eff eff T 0 0 0 C C S 31 54 55 31 distributions has been developed [3,4]. Special       eff eff eff eff  T     C C C 0 0 C  S procedures were used to create a comprehensive, 12 61 62 63 66 12 highly automatic homogenization tool which the effective elastic constants can be calculated by combines pre-processing steps for geometrical constructing six different load cases in this sense modeling and applying of boundary conditions with that only one particular strain component is non-zero finite element solution process. This paper is and all others are zero. This can be achieved by focused on special considerations for models with applying appropriate boundary conditions which rhombic fiber pattern for elastic composites.

produce pure tension and pure shear. E.g., for the To produce the non-zero strains for every load case eff displacement differences are applied between C 11 only S calculation of may be non-zero. Then 11 opposite surfaces of the cell. eff C 11 can be calculated from first row of constitutive In particular to produce pure normal strains we eff T / S and C apply: equations by the ratio of from the 11 11 21 load case 1: non-zero strain S : T / S second row by the ratio of and analogous 11 22 11            X X X X eff u u u and u u u a cos ( ) 1 1 2 2 C etc. Because the effective coefficients are 1 1 1 1 31     calculated in the global coordinate system x 1 -x 2 , X X load case 2: non-zero strain S : u u u , 2 2 22 2 2 which is not identical with the principal axes of the     X X load case 3: non-zero strain S : u u u 3 3 rhombus we get also non-zero coefficients 33 3 3 eff eff eff eff and for pure shear strains: C , C , C , C . 54 61 62 63     X X S u u u , All calculations have been made with FE package load case 4: non-zero strain : 3 3 23 2 2 ANSYS which provides with the included ANSYS     X X S u u u , load case 5: non-zero strain : 3 3 Parametric Design Language (APDL) a convenient 31 1 1     open interface for user specified input scripts. X X S u u u . load case 6: non-zero strain : 2 2 12 1 1 In our approach we extract a unit cell like shown in Here u is an arbitrary non-zero value. For Fig. 1. To calculate all coefficients for the three   X 1 dimensional case a 3D FE model is used with one simplicity u is chosen. The values u and k i element in third direction (Fig. 2).  X u are the i -th displacement components on unit k i   X X like shown in cell boundary surfaces and x 2 k k Fig. 3. a and  which appear in load case 1 are the base length of the unit cell and the rhombic angle, respectively. + x 1 X - 2 X 3 - X Fig. 1. Rhombic fiber arrangement and unit cell 1 x 2 x 1 + X 1 - x 3 X + 2 X 3 Fig. 3. Notation of unit cell surfaces To ensure the full periodicity in every load case all Fig. 2. 3D finite element model of unit cell remaining displacement differences are set to zero. To apply these displacement differences opposite The problem lies in the non rectangular geometry of nodal pairs are coupled by appropriate constraint the cell which arises problems in applying equations. For that a special meshing procedure appropriate loads and periodic boundary conditions. ensures identical mesh configurations on opposite Especially for the case of pure tension in x 1 direction surfaces. To avoid rigid body movement one applying only traction forces results in additional arbitrary node must be fixed in all directions. We shear strain. To overcome this problem modified used the corner node at origin of coordinate system. loads are applied for this case which include a shear With the extension for load case 1 the numerical part to compensate the unwanted shear strains. homogenization algorithm can also be used for composites with parallelogram fiber arrangement.

AN EXTENDED NUMERICAL HOMOGENIZATION APPROACH FOR COMPOSITES WITH RHOMBIC FIBER ARRANGEMENTS 3 Results For testing the algorithm and comparison with 18.0 results from literature isotropic material properties C 11 were used with a high stiffness ratio of 120 between C 22 16.0 fiber and matrix [1]. In particular the material constants listed in Table 1 were chosen. C 11 , C 22 [GPa] 14.0 12.0 Table 1: Material constants for the components Young's modulus Poisson's ratio 10.0 Matrix 2.6 GPa 0.3 Fiber 312 GPa 0.3 8.0 For general verification of the algorithm the results 6.0 20 30 40 50 60 70 80 90 100 were compared with Jiang [1] who presented values Rhombic angle [°] for rhombic angles of 45 and 75 degrees and different volume fractions (see Tables 2 and 3). A 6.0 C 44 very good agreement was found. 5.5 C 55 5.0 Table 3: Comparison of shear coefficients with Jiang C 44 , C 55 [GPa] for rhombic angle of 45 degrees 4.5 Vol. C 44 C 44 C 54 C 54 C 55 C 55 4.0 frac. Jiang FEM Jiang FEM Jiang FEM 3.5 0.1 1.223 1.227 ‐ 0.005 ‐ 0.005 1.214 1.217 3.0 0.2 1.516 1.526 ‐ 0.024 ‐ 0.025 1.468 1.476 2.5 0.3 1.922 1.937 ‐ 0.071 ‐ 0.071 1.780 1.791 2.0 0.4 2.533 2.547 ‐ 0.177 ‐ 0.174 2.180 2.185 20 30 40 50 60 70 80 90 100 Rhombic angle [°] 0.5 3.621 3.641 ‐ 0.441 ‐ 0.446 2.738 2.749 1.0 Table 3: Comparison of shear coefficients with Jiang for rhombic angle of 75 degrees 0.0 Vol. C 44 C 44 C 54 C 54 C 55 C 55 frac. Jiang FEM Jiang FEM Jiang FEM C 61 , C 62 [GPa] -1.0 0.1 1.218 1.220 0.001 0.001 1.219 1.220 0.2 1.488 1.493 0.007 0.007 1.492 1.497 -2.0 0.3 1.834 1.846 0.020 0.020 1.844 1.857 0.4 2.295 2.318 0.047 0.049 2.320 2.344 -3.0 C 61 0.5 2.952 2.994 0.106 0.109 3.009 3.051 C 62 0.6 4.001 4.087 0.238 0.251 4.129 4.224 -4.0 20 30 40 50 60 70 80 90 100 Rhombic angle [°] To study the overall behavior of such composites all coefficients where calculated for a rhombic angle Fig. 4. Orthotropic behavior in transverse plane for range from 30 to 90 degrees and for various volume selected effective elastic coefficients vs. change of fractions. rhombic angle with fixed fiber volume Fig. 4 shows pairs of selected effective elastic fraction of 0.4 coefficients over change of rhombic angle. It can clearly be seen that for low rhombic angles a typical 3