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保持架的注塑模具設(shè)計(jì),保持,維持,注塑,模具設(shè)計(jì) 1 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Multi Scale Modeling of Composite Materials and Structures with DIGIMAT to ANSYS Document Version 1 0 February 2009 Copyright e Xstream engineering 2009 info e X www e X Materials Engineering Plastics Reinforced Plastics e Xstream Technology DIGIMAT Digimat MF Digimat FE Digimat to ANSYS MAP Complementary CAE Technology Moldflow Moldex3D SigmaSoft ANSYS Industry Material Suppliers Automotive Aerospace Consumer see Figure 2 In practice it would be computationally impossible to solve the mechanical problem at the fine micro scale Therefore we consider the macro scale and assume that each material point is the center of a representative volume element RVE which contains the underlying heterogeneous microstructure Classical solid mechanics analysis is carried out at the macro scale except that at each computation point strain or stress values are transmitted as BCs to the underlying RVE In other words a numerical zoom is realized at each macro point The RVE problems are solved and each of them returns stress and stiffness values which are used at the macro scale 3 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Figure 2 Illustration of the multi scale material modeling approach after Nemat Nasser and Hori 1 Now the only difficulty in this two scales and more generally multi scale approach is to solve the RVE problems It can be shown that for a RVE under classical BCs the macro strains and stresses are equal to the volume averages over the RVE of the unknown micro strain and stress fields inside the RVE In linear elasticity relating those two mean values gives the effective or overall stiffness of the composite at the macro scale In order to solve the RVE problem one can use the well known finite element FE method see Figures 7 to 10 This method offers the advantages of being very general and extremely accurate However it has two major drawbacks which are serious meshing difficulties for realistic microstructures and a large CPU time for nonlinear problems such as for inelastic material behaviour Another completely different method is mean field homogenization MFH which is based on assumed relations between volume averages of stress or strain fields in each phase of a RVE see Figure 3 Compared to the direct FE method and actually to all other existing scale transition methods MFH is both the easiest to use and the fastest in terms of CPU time However two shortcomings of MFH are that it is unable to give detailed strain and stress fields in each phase and it is restricted to ellipsoidal inclusion shapes Figure 3 Mean field homogenization process i local strains are computed based on the macro strains ii local stresses are computed based on the local strains and according to each phase constitutive model and iii macro stresses are computed by averaging the local stresses 4 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 A typical example of MFH is the Mori Tanaka model 2 which is successfully applicable to two phase composites with identical and aligned ellipsoidal inclusions The model assumes that each inclusion of the RVE behaves as if it were alone in an infinite body made of the real matrix material The BCs in the single inclusion problem correspond to the volume average of the strain field in the matrix phase of the real RVE The single inclusion problem was solved analytically by J D Eshelby 3 in a landmark paper which is the cornerstone of MFH models Figure 4 Schematic of the Mori Tanaka homogenization procedure Mori Tanaka and other MFH models were generalized to other cases such as thermoelastic coupling two phase composites with misaligned fibers using a multi step approach or multi phase composites using a multi level method The predictions have been extensively verified against direct FE simulation of RVEs or validated against experimental results As a general conclusion it was found that in linear thermo elasticity MFH can give extremely accurate predictions of effective properties although for distributed orientations progress in closure approximation will be welcomed Note also that MFH can be used for UD and for each yarn in woven composites An important and still ongoing effort both in theoretical modeling and in computational methods is the generalization of MFH to the material or geometric nonlinear realms Such extension involves some major difficulties The first one is linearization where constitutive equations at microscale need to be linearized onto linear elastic or thermoelastic like format The second issue is the definition of so called comparison materials which are fictitious materials designed to possess uniform instantaneous stiffness operators in each phase The next problem to be solved is first order vs second order homogenization In first order homogenization comparison materials are computed with real constitutive models but volume averages of strain or stress fields per phase In a second order formulation richer statistical information namely the variance of strain or stress fields per phase is also taken into account Finally a very technical difficulty concerns the computation of Eshelby s or Hill s tensors and is related to the anisotropy of the comparison instantaneous stiffness operator Within a coupled multi scale analysis FE method is used at macro scale while at each Gauss integration point MFH computation is carried out either in the linear or nonlinear regime This is the most feasible approach in practice See Figure 5 Each inclusion RVE homogenization 5 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Figure 5 Comparison between the classical FE and the coupled FE Digimat MF approaches Extensive verification and validation results show that MFH can be used in practice for nonlinear problems and leads to good predictions in general while work continues on improving accuracy in some situations and reducing CPU time for coupled multi scale analysis FE Homogenization an application to nanocomposites Most likely will nanomaterials be the materials of tomorrow as they offer new horizons of applications in a wide variety of fields e g nanoelectronics bio nanotechnology and nanomedicine As such more and more effort is put in understanding and modeling their behavior as well as acquiring know how about nanoeffects While new tools are being developed to tackle this engineering challenge some are already available to the engineer of today Among them Finite Element Homogenization FEH Modeling Filler Clustering a typical nanoeffect Material scientists face several challenges related to the design and the processing of nanocomposites as at the nano scale new physics and phenomena that are negligible at the macro scale enter the picture For instance uniform dispersion of the nanofiller inside the composite matrix is sought to improve the material mechanical properties while clustering and percolation are desired when the conductivity of a base material thermal or electrical needs to be increased see Figure 6 Achieving one or the other nowadays constitutes a challenge in terms of both material processing and study 6 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Dispersed Ag nanoparticles 4 Clustered and percolated silica nanofiller in a polymer matrix 5 Figure 6 Nanofiller dispersion FEH as it requires the studied geometry to be explicitly generated and meshed allows an accurate modeling of percolation and clustering effects As an illustration we present the effect of clustering on the elastic mechanical properties of a macroscopic material point Figure 7 presents two periodic nanostructures also referred to as Representative Volume Element RVE that have been generated using Digimat FE Clustering parameters have been introduced to generate the rightmost geometry whose inclusions are concentrated around 2 distinct clustering points Volume fraction of the inclusion phase is 5 and the inclusions are spherical Once meshed these geometries will be subjected to uniaxial tensile conditions in the RVE x y and z directions and the finite element problem will be solved using the ANSYS finite element solver Figure 7 Microstructures with uniformly distributed inclusions left and clustered inclusions right Clustering Percolation 7 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Result Comparison Figure 8 S11 stress distribution in the inclusions left and in the matrix right for randomly placed inclusions Figure 8 to 10 illustrate the stress distribution in the matrix and inclusion phases in the case of the x axis uniaxial tensile test Due to the proximity of the inclusions around the clustering centers stress concentrations appear As such up to 30 higher tensile stresses are observed for the clustered case under x direction uniaxial tensile loading conditions see Figure 10 Figure 11 plots the S33 stress and E33 strain distribution in the inclusion and matrix phases as well as in the RVE One clearly observes the higher stress levels in the inclusion phase Such higher stress concentrations that are not observed for randomly or uniformly placed inclusions could lead to debonding during loading Figure 9 S11 stress distribution in the inclusions left and in the matrix right for clustered inclusions 8 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Figure 10 2D section view of clustered left and random right RVEs Tensile stress distribution Figure 11 S33 stress left and E33 strain right distributions in the nano phases and in the RVE for both cases for a z direction uniaxial loading At this low volume fraction of inclusions we see that clustering does not significantly alter the macroscopic mechanical properties of the material see Table 1 Such a placement of nanoinclusions is thus preferably avoided by the material scientists when trying to increase the stiffness of a base material Ematrix 2195 MPa by combination with a nanofiller Efiller 7000 MPa Random MPa Clustered MPa E1 2319 2322 E2 2318 2324 E3 2317 2328 Average 2318 2325 Rel Diff 0 3 Table 1 Young s moduli for both geometries obtained by FEH 9 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 FE MFH Coupled Computation an application to an industrial part For many reasons manufacturing costs and flexibility processing methods high strength vs lightness ratio etc injected parts made up of short glass fiber reinforced plastics have become omnipresent in our daily life But when it gets to model such materials can macroscopic constitutive material models capture effects such as the injection process The answer is no as they do not capture the influence of the fiber orientation which depends on the injection process The following example which consists of a neon light clasp subjected to loading introduces the process of a coupled analysis between Moldex3D DIGIMAT MF and ANSYS This process which is illustrated in Figure 12 consists of the following steps 1 The injection molding process is simulated using Moldex3D Among the available results are the fiber orientation tensors that will serve as input to DIGIMAT in the structural simulation 2 The orientation tensors computed in 1 are mapped from the injection mesh onto the coarser structural one using Map the mapping tool available in DIGIMAT 3 The structural simulation is run using the ANSYS finite element solver coupled with Digimat MF the multi scale material modeler that performs MFH at each integration point of the structural mesh Figure 12 Coupled analysis process DIGIMAT takes the fiber orientation tensor obtained from Moldex3D as input in addition to the material properties and serves as material modeler for the ANSYS finite element simulation Problem Description The light clasp consists of four independent parts see Figure 13 that also illustrates the contacts between the different parts Two of them are made up of 30 glass fiber reinforced polyamide Bergamid and were injected Their injection was simulated in Moldex3D The slide and support block are assumed to be made up of steel 10 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Closure of the clasp is simulated by imposing a displacement to the slide while blocking the support and part of the inner part Symmetry boundary conditions are also applied to limit the study to half the part The goal of the simulation is to evaluate the maximum von Mises stress in the outer part during loading and to compare the response obtained using a linear elastic model of the material and using DIGIMAT MF to perform MFH with elastic glass fibers and an elasto plastic model for the PA Material Modeling To model the PAGF in DIGIMAT MF the following hypotheses are made Glass fibers remain in their linear elastic domain The polyamide behaves elasto plastically The fiber aspect ratio length diameter ratio is 30 See Figure 14 for the tensile response of the material models Figure 13 Representation of the neon light clasp and of the contacts between the four independent parts Courtesy of Trilux and CADFEM GmbH 11 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Figure 14 Modeling of the Bergamid material Tensile response for the isotropic case fixed fiber orientation 1D random 2d orientation 2D and random 3d orientation 3D Courtesy of Trilux and CADFEM GmbH Simulation Results While the FEH approach offers the advantage of yielding an accurate description of the strain stress fields in the RVE MFH only yields the average stresses and strains at the micro level Nonetheless it gives us information we would not have access to if we were to use a macroscopic constitutive model As such the average accumulated plastic strain in the matrix phase can be visualized to observe the plasticity distribution in the plastic parts The largest plastic deformations are to be observed in the outer part See Figure 15 Figure 15 Average accumulated plastic strain distribution in the matrix phase for both the inner and outer parts Range is 0 01 blue to 0 09 red Courtesy of Trilux and CADFEM GmbH 12 P a g e C o p y r i g h t e X s t r e a m e n g i n e e r i n g 2 0 0 9 Figure 16 compares the linear elastic isotropic response classical FE to the nonlinear anisotropic one FE MFH Up to 21 percent difference is observed in the stress magnitude with the stiffer linear elastic model yielding the higher stresses Figure 16 S11 stress MPa distribution in the clasp for the isotropic linear elastic left and nonlinear anisotropic models right Courtesy of Trilux and CADFEM GmbH This case study illustrates the superiority of the multi scale nonlinear approach on the linear elastic homogeneous one to model the material as both accounting for the fiber orientation and the material nonlinearity help predict more accurately the mechanical response of the clasp under loading Bibliography 1 Nemat Nasser S and Hori M Micromechanics Overall Properties of Heterogeneous Solids s l Elsevier Science Publisher 1993 2 Mori T and Tanaka K Average stress in the matrix and average elastic energy of materials with misfitting inclusions Acta Metall Mater 1973 Vol 21 571 574 3 The determination of the elastic field of an ellipsoidal inclusion and related problems Eschelby J D 1226 London Royal Society of London 1957 Vol 241 pp 376 396 4 Polymer nanocomposites prospects of application Chmutin I 5 Nanomechanic Properties of Polymer Based Nanocomposites with Nanosilica by Nanoindentation Guo et Al 2004 Journal of Reinforced Plastics and Composites