Materials Science Forum Vols. 539-543 (27) pp. 2551-2556 online at http://www.scientific.net (27) Trans Tech Publications, Switzerland A multiscale approach for composite materials as multifield continua Patrizia Trovalusci 1,a, Vittorio Sansalone 2,b and Fabrizio Cleri 2,c 1 Dipartimento di Ingegneria Strutturale e Geotecnica, Università "La Sapienza, via Gramsci 53, I-197 Roma (Italy) 2 Ente Nuove Tecnologie, Energia e Ambiente, Unità Materiali e Nuove Tecnologie, Centro Ricerche Casaccia, C.P. 24, I-1 Roma A. D. (Italy) a patrizia.trovalusci@uniroma1.it, b vittorio.sansalone@casaccia.enea.it, c cleri@casaccia.enea.it Keywords: fibre-reinforced composites, micro-mechanical modelling, constitutive models. Abstract. A continuum model for composite materials made of short, stiff and tough fibres embedded in a more deformable matrix with distributed microflaws is proposed. Based on the kinematics of a lattice system made of fibres, perceived as rigid inclusions, and of microflaws, represented by slit microcracks, the stress-strain relations of an equivalent multifield continuum is obtained. These relations account for the shape and the orientation of the internal phases and include internal scale parameters, which allow taking into account size effects. Some numerical analyses effected on a sample fibre-reinforced composite pointed out the influence of the size and orientation of the fibres on the gross behaviour of the material. Introduction The growing demand for high-performance structural materials in several domains of engineering and technology has spurred the research in the field of complex, composite materials, such as: polyphase metallic alloy systems, polymer blends, polycrystalline, porous or textured media, fibrematrix composites, up to masonry-like materials and biomaterials. The ability to design such materials, and to derive their macroscopic properties relies, in turn, on the ability to take into account the (possibly evolving) internal structure, size, shape, spatial distribution of the microstructural constituents by multi-scale and multi-level computational methods [1]. Various approaches have been used to describe the mechanical behaviour of complex materials, most of them based on the homogenization theory [2]. The conventional homogenized models are based on the standard Cauchy continuum and present two major disadvantages. First, they cannot predict the effect of the size and orientation of the heterogeneities, since they deal with only the volume fraction and, in some cases, the morphology of the internal phases distribution. Second, they intrinsically assume the macroscopic uniformity of the stress-strain fields in each representative volume, which is likely to be unappropriate in critical regions of high gradients, e.g. in the vicinity of a macroscopic discontinuity such as a joint or a hole, or close to the point of application of a localized load. A way to retain memory of the fine organization of a material, without renouncing to the advantages of the continuum modelling and avoiding the above mentioned drawbacks, is to resort to the multifield theory [3]. The models developed in this framework have to be understood as continua with different material levels: the macro-structural level of the matrix and one or more micro-structural levels, characterized by the presence of descriptors additional to the standard ones. Therefore, non-standard strain and stress measures can be defined in a rational way, and suitable scale parameters naturally surface, which allow distinguishing the behaviour of media with heterogeneities of different size and orientation. The applicability of such models relies on the possibility to define constitutive functions for all the stress measures introduced. In this work a multiscale approach to derive these functions is proposed starting from the description of the material at the scale of the internal phases (micro-model), the fibres and the All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of the publisher: Trans Tech Publications Ltd, Switzerland, www.ttp.net. (ID: 82.55.228.3-11/1/7,2:44:16)
2552 THERMEC 26 microcracks [4, 5]. Using an integral equivalence criterion, the expression for the strain energy of the multifield continuum (macro-model) is given and the constitutive functions for all the stress measures are derived. The numerical solution of the multifield problem has been obtained using a specific object-oriented code [6] allowing for Finite Element analysis of plane elasticity problems (both in linear and non-linear regime) in the presence of additional scalar or vectorial fields with respect to the standard ones. As sample test a fibre-reinforced material without microcracks has been examined. The various numerical analyses effected pointed out the influence of the size and the orientation of the fibres on the gross behaviour of the material. Multiscale-multifield material description The gross description of the particle composite as multifield continuum (macro-model) is obtained using a multiscale strategy based on an integral equivalence principle linking the stored energy functions defined for different material scales [4]. At the level of the heterogeneities, the fine description of the material is made of two interacting lattice systems: a lattice made of rigid particles of given shape connected in pairs by linear elastic beams, representing the matrix with the fibres, and another lattice made of interacting slits of arbitrary shape with a predominant dimension, representing the microcracks. Considering materials with periodic (at least centro-symmetric) microstructure, a representative volume element (module) with appropriate geometry can be defined describing various internal material textures. The procedure governing the scale transition between the micromodel and the macromodel is based on two key assumptions: (a) the admissible deformations for the module are considered homogeneous (in a non-standard sense); (b) the volume average of the strain energy of the module is equated to the strain energy density of the macromodel. These assumptions are standard in the classical molecular theory of elasticity (e.g. [7]). In particular, the latter corresponds to the energy-averaging theorem known in the literature as the Hill-Mandel condition [8]. Briefly, the linearised strain measures of the module are: the relative displacement between two fibres, A and B, centred at the positions a and b, represented by the vector u ab u a u b + W a ( p a a) W b ( p b b), where u a (u b ) is the displacement vector of A (B), W a (W b ) is the skew-symmetric tensor of the fibre rotation, and p a and p b are the points on A and B connected by the elastic beam; the crack opening displacement of a slit H, represented by the vector d h ; the displacement jump difference between two interacting slits, H and K, represented by the hk h k vector d d d ; a displacement vector field, ω ah, depending on the position of A and H. Considering the interaction force and couple between the particles, A and B, represented by the vector t ab and the skew-symmetric tensor C ab respectively; the force due to the displacement jump on a slit H, represented by the vector z h ; the interaction force between two slits H and K, represented by the vector z hk ; and the particle-slit interaction force, represented by the vector q ah, the mean strain energy over the module of volume V reads: 1 1 + + + + 2V 2 { } ab ab a a b ab ab h h hk hk ah ah { t u W ( p p ) C W } z d z d q ab h hk ah ε ω. (1) Homogeneous deformations for the module are such that [ ] u a u( x) + u( x) a x
Materials Science Forum Vols. 539-543 2553 [ ] [ ] W a W( x) + W( x) a x (2) d h d( x) + d( x) h x, where u, W, and d are regular fields defined on a continuum neighbourhood centred at x. These fields correspond, respectively, to the standard displacement vector field, the microrotation tensor field (skew-symmetric), and a microdisplacement vector field. The additional microscopic fields respectively account for the rotations of the individual fibres and the distributed displacement jump due to the presence of microflaws in the matrix. After defining linear elastic laws for the particle-particle interactions, t ab and C ab, and for the h opening force on a microcrack, z ; non-linear elastic laws for the microcrack-microcrack force, z hk, and the particle-microcrack force, q ah [5]; and tacking into account Eqs. 2, the mean strain energy of the module can be written in terms of the continuum deformation fields u, W, and d. Finally, considering the energy equivalence (b), the strain energy density for the continuum can be derived 1 1 e { S ( u- W) + S W+ z d+ Z d}. (3) 2 2 The quantities {S, S, z, Z} depend on the constants of the matrix and the fabric vectors of the microstructure. They have the meaning of generalized stress fields: S is a generalization of the standard Cauchy stress tensor; S is the couple-stress tensor; z is the internal volume force related to the presence of the microcraks, and Z is the micro-stress tensor. More generally, z is an auto-force that plays the role of a configurational force responsible of the internal changes of the system configuration [1]. The explicit functions for {S, S, z, Z} can be expressed as S S z Z A E I O W) W) W) + B W + C d + D d + F W + G d + H d + L W + M d 2 2 W) + P W + Q d + R d + Ψ ( d, d d, d ) + N d (4) The elastic tensors A-R depend on the geometry of the microstructure and the orientation of the internal phases, other than on the elastic constants of the matrix. Moreover, the tensors B, C, E, F, H, I, M, N, P and Q have components depending on the size of the heterogeneities, allowing to take properly into account size effects. Finally, Ψ is a constitutive function accounting for the microcracks interactions. Fibre-reinforced materials, numerical simulations In order to show the effectiveness of the multifield model in capturing the main features of heterogeneous materials, for the sake of simplicity and without loss of generality, we consider undamaged fibre-reinforced materials. We particularly refer to a specific class of polymer-matrix fibre composites, such as epoxy/glass representing a wide class of composite materials, to show that the model is able to evaluate the macroscopic elastic energy and stress distribution as functions of fibre size and orientation. At the microscopic scale, we assume infinitely rigid fibres embedded in a deformable matrix and a perfect adhesion between matrix and fibres, Fig. 1. (See [9] for a more detailed discussion about this assumption). The matrix is assumed to be linear-elastic and isotropic, i.e., characterized by the Young modulus E and the shear modulus G. In this case, tacking into account the central symmetry, characteristic of each periodic assembly, Eqs. (4) become
2554 THERMEC 26 S A W) S D W (5) The non-null components of the constitutive tensors A and D are [9]: A 1111 Eμρ, A 2222 2 E, A 1212 2 Eμ, A 2121 Eμρ /2, and D 121121 E (μ /2 + αρ 2 ) w 2, D 122122 E (1/4 + 2 μ /ρ 3 ) w 2, where μ G/E, ρ w/t is the aspect ratio of the fibre, and α a constant depending on the fibres arrangement. This model has been implemented in a FE computer code [6]. Some numerical simulations have been performed, in order to investigate the capabilities of this approach. As a reference material we will adopt the experimental values of E3.5 GPa and G/E.4, common to epoxy and polyester. Both such thermosets have an experimental value of Poisson s ratio of ν.25, i.e., identical to the value ν iso E/2G-1.25 corresponding to an ideally isotropic material. Fig. 1. Schematic of the module. The fibre is represented by the shaded area. The matrix is the embedding volume. Fig. 2. Schematic of the geometry of the system and of the loading modes: (a) traction test, (b) shear test, (c) 4-point bending test. Our 2-dimensional reference system, shown in Fig. 2, is a beam of length L (parallel to the x Cartesian axis) and height HL/1 (parallel to the y Cartesian axis), in plane-strain, constant-force loading condition. Three types of simulated loading tests will be considered: (1) traction, represented by a symmetric tensile loading at both ends of the beam, (Fig. 2a); (2) shear, represented by two equal and opposite loadings at the two ends of the beam (Fig. 2b); (3) four-point bending, represented by two point loads applied on the top side, at a distance L/3 and 2L/3 along the beam, while the two ends are held fixed at the bottom side (Fig. 2c). In all cases, the applied force is such that the maximum displacement is within 2-3% of L. All the results will be presented as a function of the scale parameter λw/l, representing the size of the module compared with the system size, and of the initial fibre orientation ϕ, with ϕ indicating fibres parallel to the x-axis of the system. In Fig. 3 and 4 the strain energy of the system E (i.e., the strain energy density, Eq. 3, integrated over the volume of the beam) is plotted as a function of the initial fibre orientation ϕ and of the scale parameter λ, respectively. In the same figures, dashed curves represent the corresponding anisotropic Cauchy (a-c) model, obtained by setting the microrotation equal to the skew-symmetric part of the displacement gradient, Wskw( u), and D in Eq.(5) [4]. The E values are scaled by the corresponding isotropic-elasticity value, E C (matrix without fibres).
Materials Science Forum Vols. 539-543 2555.25.2 λ4x1-3 4-point bending shear traction.25.2 λ.1.1.4.1.1.1.2.2.16.16 E/E C.15.1 multifield.15.1 E/E C.12 ϕπ/4 multifield 4-point bending Shear Traction.12 anisotropic Cauchy.8 anisotropic Cauchy.8 π/4 π/2 Fig. 3. Plot of the integrated energy density E as a function of the fibre orientation ϕ. ϕ.4 1 1 1 1 1 1/λ.4 Fig. 4. Plot of the integrated energy density E as a function of the scale factor λ. With reference to Fig. 3, it can be observed that in each loading test the behaviour of the multifield solution is close, or even equal, to the a-c solution, at ϕ and ϕπ/2. On the other hand, very large differences are seen at any intermediate value, the maximum discrepancy always occurring around ϕπ/4. In all cases, the multifield model appears to describe a less rigid composite than a corresponding a-c model, since the stored elastic energy is always higher under fixed forces. All the plots refer to a scale factor λ4.e-3. Fig. 4 shows the main difference between the multifield and the a-c model. The latter cannot describe local rotations of the fibres inside the module, and therefore it is unable to describe size effects in the material response. On the contrary, the multifield model can represent different rotations of the fibres inside the module, and explicitly takes into account their size in the constitutive relation, Eqs. 5, thus being able to catch the dependency of the material response on the scale parameter. All the plots refer to an initial fibre orientation ϕπ/4. In Fig. 5 we show a detailed picture of the fibre rotation depicted by both the multifield (λ4.e-3) and the a-c model. The suitable descriptor in the multifield model is the value attained by the additional microrotation field, W, actually representing the rotation of the fibres (Eq. 2), or more precisely by its unique independent component, namely W 21. In the a-c model the fibres are constrained to follow the rotation of the matrix, so the appropriate kinematical descriptor is the unique independent out-of-diagonal component of the skew-symmetric part of the displacement gradient, namely skw( u) 21. Once again, the a-c model comes out to be much stiffer than the multifield one. In fact, it is unable to describe high microrotation gradients, as in the regions close to the points where loads are applied: in fact, the fibre rotation realized in the multifield model is always much higher than in the a-c model (+2% near the loading points, +5% near the ends). It must be pointed out that such results hold for the chosen value of λ, since the multifield response strongly depends on it.
.3.36 2556 THERMEC 26.1.64.7.1.6.94 1 Fig. 5. Contour map of the fibre rotation (initial fibre orientation: ϕ); darker colours indicate higher values of the fibre rotation. On the left: multifield model (λ4.e-3), map of the microrotation field. On the right: a-c model, map of the skew-symmetric part of the displacement gradient (macrorotation). Acknowledgements Financial support from the Italian Ministry of Instruction, University and Research (MIUR) is gratefully acknowledged. References [1] R. Phillips R. (21): Crystals, Defects and Microstructures: Modeling Across Scales, (Cambridge University Press, Cambridge 21) [2] S. Nemat-Nasser and M. Hori, Micromechanics: Overall Properties of Heterogeneous Materials (Elsevier, Amsterdam 1993) [3] G. Capriz, Continua with Microstructure (Springer-Verlag, Berlin 1989) [4] P. Trovalusci and R. Masiani, Int. J. Solids & Struct., Vol. 42, 5578 (25) [5] P. Trovalusci, A multiscale continuum for damaged fibre composites, Mater. Sci. Forum, Vol. 426-432 (23), p. 2133 [6] V. Sansalone, P. Trovalusci an F. Cleri, Int. J. Multiscale Meth. Engn. (in press) [7] J. L. Ericksen, Advances in Applied Mechanics, Vol. 17, Yih C. S., Ed. (Academic Press, London, 1977), 189 [8] R. Hill, J. Mech. Phys. Solids, Vol. 11 (1963), 357 [9] V. Sansalone, P. Trovalusci and F. Cleri, Acta Mater. (in press) [1] G. Maugin, Material Inhomogeneities in Elasticity (Chapman & Hall, London, 1993).