Anisotropic meanfield modeling of debonding and matrix damage in SMC composites
Introduction
With their high mass-specific strength and stiffness, fiber reinforced polymers receive considerable attention in mass-reduction strategies in the automotive and aerospace sector. The strongest and stiffest composites are continuous fiber reinforced composites. Discontinuous fiber reinforced polymers, however, are more amenable to high-volume automotive applications because of their geometric freedom and significantly reduced cycle times in manufacturing processes. The focus on this work lies on a specific discontinuous fiber reinforced polymer: sheet molding compound (SMC) composites. SMC composites consist of a thermoset matrix – in the present case, epoxy or unsaturated polyester polyurethane hybrid (UPPH) resin – reinforced with long ( 25 mm) glass fibers. Parts are typically manufactured by compression molding of pre-impregnated fibers, also known as prepregs. A key benefits of the UPPH resin is the possibility of manufacturing SMC composites with local continuous reinforcements. The resulting hybrid material allows for structural applications, such as, a windshield surround [1], a suspension arm [2], or a subfloor structure [3]. The orientation evolution of fibers during the compression molding manufacturing process leads to a moldflow-induced microstructure.
The demand for a precise and efficient prediction of the resulting inhomogeneous, anisotropic, process-dependent mechanical properties arises from industry's interest in shortening development cycles and reducing extensive and costly prototyping. A wide range of research is, therefore, associated to the virtual process chain of discontinuous fiber reinforced composites, e.g., moldfilling and fiber orientation evolution analysis, prediction of warpage and eigenstresses, and structural analysis including damage.
Meraghni and Benzeggagh [4] investigated damage propagation in randomly-oriented, discontinuous, fiber reinforced composites. Their experimental studies involving the amplitude analysis of acoustic emission signals and microscopic observations revealed two dominant damage mechanisms: matrix damage and interface damage. Several other authors (e.g., [[5], [6], [7]]) have confirmed that fiber-matrix interface debonding is the primary and matrix cracking the secondary damage mechanism in SMC composites.
Experimental characterization of the interface strength is a challenging field. Many authors (e.g., [[8], [9], [10], [11]]) have investigated interface strength using pull-out, push-out, and micro-tensile techniques or via fragmentation tests. Hour and Sehitoglu [12] and Dano [13], e.g., performed experimental studies with focus on the development of damage in glass fiber reinforced composites and Fitoussi et al. [14] conducted experiments specifically on the mechanical behavior of SMC composites.
Jendli et al. [5] qualitatively analyzed the influence of the strain rate on damage threshold and accumulation. Performing monotonic and interrupted tensile tests at different strain rates, Jendli et al. [15] showed that both damage onset and kinetics are sensitive to the strain rate, such that the interface failure strength increases with increasing strain rate. Similar findings were obtained by Fitoussi et al. [7] and Shirinbayan et al. [16].
Along with their experimental findings, Fitoussi et al. [17] also proposed a micromechanical model based on an equivalent, anisotropic inhomogeneity approach for damaged fibers. Their fiber-matrix interface debonding model is based on a criterion with linear coupling of the local shear and normal stress on the interface [18]. This work was followed by an extension that considered local fluctuations of strain and stress, and a probabilistic interface-strength distribution [19]. Meraghni et al. [20] developed a similar model that combined a microcrack density parameter with fiber-matrix decohesion in order to decrease the fiber strain localization tensor. Derrien et al. [21] further developed the probabilistic interface-damage model introduced by Fitoussi et al. [19]. Here, the approach of replacing fibers having damaged interfaces with an equivalent, anisotropic, undamaged inhomogeneity or matrix material were validated experimentally. Desrumaux et al. [22] extended the statistical representation of failure to each constituent (fibers, matrix, and interface). Here, the interface damage was implemented in such a way that the interface of each fiber is experiencing damage represented by the introduction of directionally-dependent matrix cracks. Later, Desrumaux et al. [23] introduced a two-step homogenization damage model for a randomly-oriented fiber composite based on a numerically determined Eshelby tensor. In the first step, an anisotropic, equivalently-damaged matrix is calculated by considering the undamaged matrix and microcracks. In the second step, the fibers are embedded in the damaged matrix using a numerical Eshelby tensor. A comparable two-step homogenization framework was pursued by Jendli et al. [24] and Kammoun et al. [25], who followed approaches for interfacial decohesion and pseudo-grain sub-regions. Meraghni et al. [26] further developed the probabilistic strength model. Analogously, Guo et al. [27] introduced a damageable-elastic law for randomly-reinforced composites based on a two-scale approach. Nguyen and Khaleel [28] developed a matrix degradation model based on the experimental findings of Meraghni and Benzeggagh [4]. Here, the macroscopic stiffness of the randomly-oriented composite is calculated by an orientation average over aligned fibers, and a thermodynamically consistent damage evolution law predicts the stiffness reduction. Baptiste [29] proposed a model that captures the inelastic behavior of a composite due to plasticity, viscosity, and damage. The damage to matrix, reinforcement, and interface is considered utilizing the probabilistic approach for interface-strength developed by Fitoussi et al. [19]. Lee and Simunovic [30] evolved a model to predict the elasto-plastic-damage behavior of a ductile matrix composite containing aligned fibers. They combined an associative flow rule and a hardening law with evolutionary interfacial debonding. In their model, partially debonded fibers are replaced by equivalent, perfectly-bonded fibers. Additionally, they extended their model to treat random fiber orientations [31]. A similar model was developed by Ju and Lee [32] for a three-phase composite. Here, completely debonded fibers are treated as voids within the three-phase homogenization scheme. Using an effective yield criterion, an associative plastic flow rule, and a hardening law, the macroscopic mechanical behavior predicted by the model fits the experimental observations. Zaïri et al. [33] described interface debonding as nucleation and the growth of voids. They combined a critical void volume criterion with a vanishing element technique to capture damage accumulation and failure. Ben Cheikh Larbi et al. [6] investigated the elastic behavior of SMC composites under cyclic loading. They found appropriate parameters for the two-scale probabilistic damage model by evaluating fatigue tests via scanning electron microscopy. Yang et al. [34] introduced a phenomenological damage model based on two coupled damage variables to capture matrix cracking and interface debonding. A von-Mises type criterion and a cohesive zone model are applied to capture the two damage mechanisms. A transversely isotropic stiffness reduction law based on mesoscale RVE calculations is established. Notta-Cuvier et al. [35] developed a model to describe interface debonding at fiber head surfaces in injection-molded, short-fiber reinforced composites. The model for interface debonding utilizes the accumulation of voids and phenomenological parameters.
Despite the enormous work already conducted within the research of discontinuous fiber reinforced polymers, some deficiencies still remain open. Some of those deficiencies are addressed by the presented model. Few models, e.g., are physically motivated and take the microscale into account, but are still efficiently applicable to calculations of structural components (e.g., parts that are of interest to the industry [[1], [2], [3]]). Hereby, a numerical regularization based on the redistribution of load from fibers with debonded interfaces to undamaged fibers can be helpful. To the authors' knowledge, a Weibull weakest link approach with a heterogeneous stress distribution on each interface was not presented so far. Added value lies in the simulation of non-proportional loading paths and a rigorous visualization of the resulting anisotropic evolution of the effective and the microstructural quantities (see Appendix A). This allows for a better discussion and a better understanding of, e.g., damaged-induced anisotropy within SMC composites.
This work presents an anisotropic continuum mechanical meanfield damage model taking into account an arbitrary orientation distribution of straight fibers. The outline of the paper is as follows: In Sec. 2, a meanfield Mori-Tanaka homogenization scheme based on an empirical fiber orientation distribution is presented. The model accounts for fiber-matrix interface debonding and matrix damage in SMC composites. In the latter, a maximum principal stress criterion is applied, whereas the former is based on a direction-dependent equivalent interface stress which is related to a Weibull survival probability. Hence, interface debonding is modeled by an anisotropic evolution of load-carrying fibers. Fiber breakage is neglected. Section 3 introduces the discretization procedure and numerical implementation of the damage models. A regularization approach ensures numerical and computational efficiency of the model. Section 4 deals with the identification of the parameters needed to properly describe the material model. Section 5 is devoted to the experimental validation.
A direct tensor notation is followed throughout the text. Components of vectors and tensors refer to the orthonormal basis . Vectors and second-order tensors are denoted by lower case and upper case bold letters, respectively (e.g., and ). Fourth-order tensors are denoted by, e.g., . The composition of second-order or two fourth-order tensors is formulated by and . A linear mapping of vectors by second-order tensors and second-order by a fourth-order tensor is written as . The scalar product is denoted by . The Rayleigh product describes, e.g., the rotation of a fourth-order tensor with the orthogonal rotation tensor . Here, tensor components are expressed by Latin indices, and Einstein's summation convention is applied. The identity on symmetric fourth-order tensors is denoted by . Column vectors and matrices are identified via underscores, e.g., .
Section snippets
Microstructure of SMC composites
Here, the SMC composite is treated as a two-phase composite consisting of a thermoset matrix phase and glass fibers . The matrix is characterized linear elastically by an isotropic matrix stiffness tensor and the corresponding volume fraction . All fibers are modeled linear elastic with an isotropic stiffness . Due to the low shear rates in the manufacturing process, fiber curvature and breakage during the manufacturing process are neglected. As shown, e.g., by Jendli et al. [15],
Implicit time integration
The model is implemented as a UMAT (User Material) in the commercial implicit finite element-software ABAQUS/Standard [50]. Thereby, the algorithmic tangent was calculated numerically by a perturbation of all strain components. A goal of this work was a computationally efficient and robust implementation that allows application to larger structures. Hereafter, the vector lists all fractions of intact fibers
The implicit Euler time integration scheme is applied. Here, a
Matrix damage
In this paper, two matrix materials are considered: epoxy resin (reinforced with 43 vol% and 50 vol% e-glass fibers) and UPPH resin (reinforced with 23 vol% e-glass fibers). Monotonic tensile tests of neat resin bone specimens were performed to characterize the matrix behavior. The epoxy neat resin samples were casted pressureless in a net shape mold. The UPPH neat resin samples were manufactured by a project partner in the International Research Training Group GRK 2078 (Trauth [54]). Fig. 10
Variation of fiber content
The epoxy matrix SMC composite was available with two fiber contents. After fitting the model to the lower fiber content , the model was applied to the higher fiber content under the assumption that the interface strength distribution is not affected by the fiber content and, thus, remains constant. The simulations were performed in one macroscopically homogeneous material point. Fig. 13a shows the simulated and measured stress-strain behavior. The model slightly
Conclusions
The present elasto-damage model for SMC composites captures matrix damage and interface debonding on the microscale. A Mori-Tanaka homogenization scheme is applied to calculate the corresponding macroscopic behavior. The model accounts for an arbitrary, moldflow-induced, inhomogeneous fiber orientation distribution of straight fibers. The complete model can predict the damage behavior of SMC composites for different matrix systems, fiber contents, and stress states. However, the applicability
Acknowledgements
The authors thank Vanja Ugresic, Dan Park and Sebastian Gajek (Fraunhofer Project Center @ Western) for manufacturing and testing the epoxy SMC based composites. The support of David Bücheler (Fraunhofer Institute of Chemical Technology) for the UPPH SMC based composite production, including the SMC composite with the unidirectional reinforcements is appreciated. Anton Helfrich (Institute of Production Science wbk, KIT) was responsible for the milling operations on the biaxial tensile specimen.
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