Determination of the critical size of a statistical representative volume element (SRVE) for carbon reinforced polymers☆
Introduction
Fibre-reinforced polymeric composites are widely used in structural applications because of their good specific stiffness and strength. However, the use of these materials is limited by the lack of efficient tools to predict their degradation and lifetime under service loads and environment. The inhomogeneity and anisotropy of their microstructure leads to complex damage mechanisms (basically: fibre breakage, matrix cracking and yielding, fibre–matrix debonding and delamination).
The development of specific design tools for composites is being pursued since the early stages of their application in aircraft structures. Two approaches may be distinguished: a phenomenological approach and a mechanistic approach. Phenomenological approaches are based on the empirical laws of mechanical behavior obtained from experimental tests. These models require a heavy experimental background and are not general in the sense that the behavior of a particular material and ply sequence cannot be inferred from the behavior of a simpler configuration, that is, each laminate requires a complete experimental characterization. The simplest, and more extensively used, design tools assume a complete elastic behavior of the material until a failure criterion is satisfied. Once this happens, it may be considered as either a complete breakage of the structural element or a stiffness reduction by an arbitrary factor. However, there is no agreement concerning the failure criteria to be used in the design of composite structures [1].
On the other hand, mechanistic approaches aim to simulate the occurrence of damage on the constituent scale and to reproduce the interaction of the different damage mechanisms. Moreover, the final objective of these models is to establish the degraded mechanical properties of the composite resulting from the damaged microstructure. Although this is a complex and computationally expensive task, the powerfulness of such a model as a design tool motivates the research activity in this field. Moreover, they can certainly contribute to clarify the physics of the damage processes.
For a more realistic simulation of the microstructure of these materials, a representative volume element (RVE) may be used. Because of its ability in reproducing the real stress and strain evolution, the simulation through a RVE may provide the understanding of damage mechanisms and the identification of the possible sources and scenarios, which cause their initiation. This understanding is a need for the proposal of macroscopical failure criteria and failure laws. On the other hand, the simulation of a RVE can also be employed in a two-scale method and then, for the simulation of failure and fracture in real structural components.
In the analysis of the microstructure, the periodicity hypothesis of the fibre within the composite has been traditionally employed. This hypothesis reduces the analysis of the microstructure to the analysis of a single unit cell (the simpler RVE) and may lead to analytical solutions. Although these unit cells can be useful for some purposes and can be employed successfully in two-scale methods to reproduce macroscopical behavior [2], [3], they do not reflect the reality of composite materials, in which the fibre is randomly distributed, and consequently, they are not usable to simulate some of the complex mechanisms which take place in long fibre reinforced polymers and which may cause microscopic failure [4].
One of the most revealing simulations which can be performed with a RVE is the mechanical analysis of the plane which is perpendicular to the fibres in long fibre reinforced polymers (FRPs), the transverse plane. In this plane, the mechanical behavior is dominated by the matrix properties, both in the elastic and in the failure regime.
Due to the complex geometries and the extended use of laminates with multiple ply orientations, usual laminates in structural components may have plies which work mainly in the direction perpendicular to the fibres. Also because of this design approach, transverse failure of composites is normally not critical but, of course, it has to be verified in the design process. Matrix cracking also contributes to other degradation phenomena like in stiffness degradation [5], damage [6], [7] and fatigue [8] and are related to delamination [9] and even fibre breakage [10]. Furthermore, matrix cracking is crucial for some applications such as hydrogen tanks and aerospatial vehicles in which this phenomena must be avoided.
The simulation through RVEs of the transverse plane can also provide useful information on other damage agents like residual thermal stresses, the role of voids and defects, or the influence of fibre–matrix interface and it be applied to quantify reliability of carbon fibre reinforced polymers (CFRPs).
The phenomena of matrix cracking has been in depth analyzed by Asp et al. [11], [12], [13], [14]. The main conclusions of this analysis are that the large mismatch in the elastic properties between usual fibres (glass or carbon fibres) and matrix (epoxy), under these load circumstances, submits the matrix to a triaxial stress state [11], [12] and that the cracking phenomena is closely related to the dilatational energy density [12], [13], [14] in the matrix, given by:where E and ν are the Young’s modulus and Poisson’s ratio of the matrix and σi are the principal stresses.
More recently, Fiedler et al. [15] have performed finite element simulations using periodic unit cells which also confirm the importance of the triaxial stress state in matrix cracking of CFRPs loaded transversely and additionally consider a parabolic criterion for its prediction.
The present work is a part of a deeper research project whose main objective is to model the random microstructure in carbon fibre reinforced polymers and simulate the probability of failure associated to matrix cracking. For this purpose, the employed model must precisely reproduce the stress maps present in the microstructure of real materials. These real material stress maps are caused by the real distribution of the fibre within the composite and they differ from those stress maps obtained with the periodic models which have been classically employed [16].
This work establishes some criteria which a microstructure model (a statistical representative volume element) must satisfy for the probabilistic simulation of matrix cracking failure in long fibre reinforced polymers and according to them, determines the critical size for the SRVE. First, some definitions and some criteria (including both mechanically and statistically based) from the scientific literature are reviewed. The criteria which, determine the validity of the micro-model, are then selected and analyzed in a scale-dependent analysis. Each criterion is satisfied by a critical RVE size. The criteria which needs a larger size to be satisfied determines the minimal size for a valid RVE.
Section snippets
Mechanical definitions and criteria for a RVE
In the mechanics and thermodynamics of solids, the definition of a RVE is of paramount importance. According to the very first definition by Hill [17] a RVE is:
“a sample that (a) is structurally entirely typical of the whole mixture on average, and (b) contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the surface values of traction and displacement, so long as these values are “macroscopically uniform”. That is, they fluctuate about a
Random geometry criteria for the RVE size
A quite different statistical approach is to analyze the point pattern formed by the distribution of fibres or inclusions centers in the matrix, that is to focus on the representativeness of the random geometry. The statistical functions which, describe these point patterns, are well established. For the long-range interaction, Ripley’s K-function (K(r), or second-order intensity function) and the pair distribution function (g(r)) are useful [27]. Ripley’s K-function can be defined as the
Methodology
For the determination of the finite size of the statistical representative volume element, models of increasing size are constructed and the evolution of some variables or functions versus the size of the SRVE are analyzed.
First, let us define the dimensionless variable δ, which relates the side length of the SRVE L and the fibre radius R:From the literature review values of δ between 4 and 100 are chosen. The upper limit for the analysis is chosen from both preexisting literature and
Effective properties
Following Hill’s very first definition of the RVE, a valid RVE has to be typical of the whole material in average. That means that the effective properties of the RVE are the same of those of the material. Although the elastic properties of a unidirectional lamina can be modelled as a linear function of the fibre content [38], in a microscopic scale they can show some deviation from this relation. Consequently, an analysis of scale-dependency of the elastic properties is performed in this
Hill condition
The so-called Hill condition of Eq. (2) should be satisfied by any RVE. To verify the satisfaction of this criteria, both sides of equality (2) have been computed for each SRVE candidate. Results (plotted in Fig. 4) show as the bigger the SRVE size is, the closer are the measures of the energy. Fig. 4 also shows the relative difference between energy bounds is plotted. In the same figure, a tendency line with a exponential fit is shown. It can be verified that for δ > 15 the relative difference
Convergence of stress and strain fields
In a probabilistic mechanical analysis, mechanical properties as seen as random variables, which can be fitted with a probability density function, characterized by some statistics (for instance, the mean and the variance fully characterize a normal distribution). According to the criterion established by Asp and co-workers for matrix cracking [14], each polymer has a constant critical value of the dilatational energy density Uv of Eq. (1), . When the value of Uv at a point in the matrix
Probability density functions of stress and strain fields in the matrix
Most of the failure criteria for the composite transverse direction consider that the failure is caused by cracks in the matrix [13], [41]. One of the objectives of this work is to find probability distribution functions for the failure in the transverse direction. For this reason, when trying to develop a statistical representative volume element, the stress and strain probability distribution functions in the matrix have to be analyzed.
The probability distribution function for ε22 in the
Distance distributions
The clustering or homogeneity of fibre (particle) distribution may affect strongly the damage behavior of a composite [42]. Moreover, the random distribution of fibres within the composite gives place to different statistical stress and strain distributions than an ideal periodical distribution of fibres [16]. For this reason, the SRVE has to represent the real statistical distance distribution of the fibres in the bulk material. As described in Section 2, a useful way to analyze inclusions or
Concluding remarks
This paper has reviewed the criteria which can be used to define the minimal required size for a SRVE. The SRVE defined this way reproduces the mechanical and the statistical behavior of the material. The criteria which have been analyzed are: the effective properties, the Hill condition, the mean and the variance of the stress and strain components in the fibre, the matrix and the composite, the probability density function of the stress and strain components in the matrix and the typical
Acknowledgements
This work has been enriched by some discussions at the Congress “Micromechanics and Microstructure Evolution: Modeling, Simulation and Experiments” held in Madrid, specially those by professor Helmut Böhm.
The present work has been partially funded by the Spanish Government under research project MAT2003-09768-C03-001. The first author express gratitude to the University of Girona for research Grant BRAE00/02.
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This manuscript was presented at the “Micromechanics and Microstructure Evolution: Modeling, Simulation and Experiments” held in Madrid/Spain, September 11–16, 2005.