Soutis / Beaumont Multi-Scale Modelling of Composite Material Systems

The Art of Predictive Damage Modelling
1. Auflage 2005
ISBN: 978-1-84569-084-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 528 Seiten

Reihe: Woodhead Publishing Series in Composites Science and Engineering

One of the most important and exciting areas of composites research is the development of modelling techniques to predict the response of composite materials to different types of stress. Predictive modelling provides the opportunity both to understand better how composites behave in different conditions and to develop materials with enhanced performance for particular industrial applications. Multi-scale modelling of composite material systems summarises the key research in this area and its implications for industry.The book covers modelling approaches ranging from the micron to the metre in scale, and from the single fibre to complete composite structures. Individual chapters discuss a variety of material types from laminates and fibre-reinforced composites to monolithic and sandwich composites. They also analyse a range of types of stress and stress response from fracture and impact to wear and fatigue. Authors also discuss the strengths and weaknesses of particular models.With its distinguished editors and international team of contributors, Multi-scale modelling of composite material systems is a standard reference for both academics and manufacturers in such areas as aerospace, automotive and civil engineering. - Extensive coverage of this important and exciting area of composites research - Understand how composites behave in different circumstances - Compiled by an expert panel of authors and editors

Autoren/Hrsg.

Soutis, C.

Beaumont, P W R

Leseproben

2

Interfacial damage modelling of composites

C. Galiotis    University of Patras, Greece

A. Paipetis    Hellenic Naval Academy, Greece

2.1 Introduction: definition of the interface

The inherent notion of a composite material is that it comprises more than one phase. It is desirable to make the optimum use of each distinct phase, something that may only be achieved if the phases are joined together to make a contiguous or integral unit. Upon joining, the phases are separated by an interface which, thus, becomes a dominant feature of the composite. It is the existence of the interface that makes a composite differ physically from its components (Ebert, 1965). The distinct phases in the composite possess different properties, which constitute the reason for joining them. Any loading situation results in an interaction between the two constituents, which plays a crucial role in the composite properties.

In the case of fibre reinforced composite materials, the interface is the entire ‘shared’ surface between the fibre and the matrix. There have been a number of basic approaches to studying the nature and the role of the interface: the (global) macroscopic approach, and the microscopic approaches, which in turn, may differ in scale (i.e. microscopic versus atomic). The former is based on classical mechanics and the theory of elasticity. The latter takes into account not only the mechanical but also the chemical and physical nature of the interface.

A schematic illustration of the model upon which all various macroscopic approach studies are based is shown in Fig. 2.1(a) (Vlattas 1995). According to this model, there is no difference between the material properties in the vicinity of the interface and those of the bulk material away from the interface. Bonding is assumed to be perfect (Hull 1996). If, on the other hand, we take into account crucial physico-chemical phenomena occurring between the fibre and the matrix in the vicinity of the interface, a more realistic model representing the interface is obtained (Fig. 2.1(b)). Various parameters such as wetting, material absorption and interdiffusion arise during the composite fabrication but they are completely neglected from a macroscopic point of view. Wetting and absorption represent the compatibility between the (solid) fibre and the (liquid) matrix in terms of the thermodynamic work of adhesion when the surfaces are brought close to each other. Interdiffusion is the formation of bonding between two surfaces by the entanglement of the polymer molecules on one surface with the molecular network of the other surface.

2.1 The different scales of interfacial adhesion: from the macroscopic to the atomic scale.

If we focus even further on the area between fibre and matrix (Fig. 2.1(c)), at the atomic scale level, we may consider the electrostatic attractions and chemical bonding. Electrostatic attractions occur between two surfaces due to opposite charges. The strength of the interface will depend on the charge density. A chemical bond is formed between a chemical grouping of the fibre surface and a compatible chemical group in the matrix. The strength of the bond depends on the number and type of bonds. Bond rupture occurs during interfacial failure.

The interface can be visualised as the boundary between two materials with different properties. It is easy to assume that this boundary has no volume for calculation purposes. However, this is not the case on a smaller scale where phenomena such as surface roughness or chemical interaction are of crucial importance. Moreover, it is much easier to attribute a physical meaning to a zone with a gradient of properties between the two constituents. In this case, the zone is termed the interphase and extends in three dimensions. If the mechanical characteristics of the interface in multi-phase materials are not affected by the typical length scale of the composite, two-dimensional models can be applied (Herrmann, 1996). However, if the bonding layer is of similar order of magnitude to that of a typical macroscopic size, like the reinforcement diameter, then the interphase has to be included in the mathematical modelling. The recent advent of nanocomposites will certainly require the inclusion of the interphase as a third phase in all calculations.

The scale of the interphase may differ dramatically in different composite materials. In the case of fibrous composites it may extend from a few nanometres to several fibre diameters. A typical unsized carbon fibre/epoxy interphase only extends for less than ~ 10 nm (Guigon, 1994). The interphase of SiC fibres in an Al matrix may extend to hundreds of nanometres (Long, 1996). In thermoplastic matrices, the presence of transcrystallinity changes the matrix properties in the interphase region totally. In this case the interphase may be more than one order of magnitude larger in diameter than the reinforcing fibre (Heppenstall-Butler, 1996).

2.2 The interface and composite properties

The simplest assumption predicting composite properties is the rule of mixtures, by which the properties of the constituent materials are added according to their respective volume fraction. In the case of a continuous fibre unidirectional composite loaded in the fibre direction, the rule of mixtures defines the stress s sustained by the fibre and the matrix, through their respective volume fraction V (Jones, 1975):

=sfVf+smVm

  2.1

where the subscripts f and m denote the fibre and the matrix, respectively.

The rule of mixtures is no longer macroscopically valid when shear forces are present due to the presence of local discontinuities, or if the material is not homogeneous, that is, if the material properties change from point to point (Halpin, 1992). Two simple cases for inhomogeneity may be regarded, where shear stresses are of primary importance: (i) The short fibre composites, where the reinforcement is discontinuous, and (ii) the continuous fibre composites, where inhomogeneity is present at the locus of a discontinuity of the reinforcement.

The simplest case is the presence of a fracture in a unidirectional composite loaded in the fibre direction. In that situation, fracture of the fibres is regarded as the primary event which controls the local damage development and accumulation that will lead to the failure of the composite (Reifsnider, 1994). In Fig. 2.2(a), such fractures within the composite zone are shown. The fracture of a fibre causes a local stress perturbation (Fig. 2.2(b)) (Paipetis 2001). The stress is redistributed to the neighbouring fibres through the interface. At some length from the fracture locus, the axial stress is restored on the broken fibre. This length is termed the ineffective (Reifsnider, 1994) or transfer (Galiotis, 1993) length and is dependent on the stress transfer efficiency of the interface. The ineffective length defines the zone of influence of the fracture; small ineffective lengths create large stress concentrations in neighbouring fibres; large ineffective lengths increase the size of the ‘flaw’ within the composite, raising, thus, the possibility of cumulative flaws leading to fracture.

2.2 Fibre fractures (a) and shear perturbation around a fibre fracture (b). (Source: Paipetis, 2001.)

For a single fibre embedded in an epoxy matrix, Drzal and Madhucar (1993) have identified the failure mechanisms with relation to various levels of interfacial adhesion (Fig. 2.3); as interfacial adhesion increases, the axis of failure propagation changes from a mode II crack which propagates along the fibre axis (Fig. 2.3(a)) to a mixed mode (Fig. 2.3(b)) and finally a mode I crack (Fig. 2.3(c)), which propagates transversely to the loading axis. It is worth noting that increased interfacial adhesion may improve the on-axis properties, such as the tensile strength, by 45% (Drzal, 1993).

2.3 Modes of interfacial failure.

2.3 Analytical modelling of the shear transfer

2.3.1 The problem statement

The problem of the stress transfer at the locus of a discontinuity is a typical case of torsionless axisymmetric stress state (Timoshenko, 1988). In cylindrical coordinates r, ?, z with corresponding displacements components u, v, w, the component ? vanishes and u, w, are independent of ?. The same is valid for the stress components with tr? and t?z being zero (Fig. 2.4). Thus, the strain components are reduced to:

r=?u?re?=urez=?w?zGrz=?u?z+?w?r

  2.2

2.4 The stress field at the axisymmetric stress state (Timoshenko, 1996).

and the equilibrium conditions...