1. Introduction
Continuous fiber reinforced ceramic matrix composites (CMC) are being used in many fields, such as aeronautical, aerospace and automobile, because of their excellent thermomechanical properties at high temperatures and relatively low density when compared to their metallic counterparts [
1,
2,
3,
4,
5]. Filament winding technology is one of the preforming methods which is employed to manufacture CMCs with rotational symmetry axis and application-tailored fiber orientation [
6,
7]. Two such CMCs that are manufactured at German Aerospace Centre (DLR) are C/C–SiC (Institute of Structures and Design) and WHIPOX™ (Institute of Materials Research) via filament winding.
Despite the presence of a brittle ceramic matrix in CMCs, this material class exhibits inelastic behavior because of energy dissipating mechanisms, such as matrix cracking and fiber–matrix interphase debonding [
8,
9]. Many damage models have been proposed in the literature to describe the inelastic behavior of CMCs. The micromechanics-based approach proposed by Lamon [
10] delivers accurate results in predicting damage but requires model parameters which can only be obtained from the individual constituents of a CMC, namely fiber and matrix. However, in a complex matrix system like C–SiC, the material under consideration, it is challenging to evaluate the effective properties of a representative matrix material. Even the bulk material properties of a monolithic matrix material cannot be used because the matrix undergoes massive changes when employed for a fiber-reinforced material due, to effects such as hindered matrix shrinkage during the manufacturing process [
11]. Baranger has summarized damage models and the rupture criterion at different microscopic scales in his work where stress and strain are used as limits to design a CMC component [
12]. A thermodynamical formulation of the anisotropic damage model is discussed by Wulfinghoff et al. by linking crack channels and pores with degradation in properties [
13]. Models describing the damage of a homogenous CMC material at the macro-level have also been a topic of investigation for many researchers [
14,
15,
16,
17,
18]. These models describe damage as the degradation of stiffness in principal material directions as the load increases. A model with a similar theoretical background of continuum damage mechanics is proposed by Barbero [
19] and is implemented in the current work because of the lower number of tests required in parameter-identification for the model. Apart from that, testing norms for CMCs already exist for these required tests.
The above-mentioned continuum damage models successfully describe non-linearity in the material but do not explicitly predict the final failure of composite. A failure criterion is required in order to predict the failure stress or strain of a laminate. Several physics-based and empirical failure criteria for fiber-reinforced composites have been reported in the literature and are summarized in the World Wide Failure Exercise (WWFE) [
20]. Failure criteria based on micromechanics are not appropriate in the case of some CMCs because of the same above-mentioned reasons as for damage models. Apart from that, phenomenological criteria such as Cuntze and Puck require material parameters based on tests conducted on unidirectional (UD) ply, which is not possible in the case of the wound materials under consideration. For these reasons, the Tsai-Wu failure criterion is found to be the most appropriate as the parameters required for its implementation can be obtained from standardized tests in case of CMCs. The availability of standardized tests for CMCs is not trivial. For example, there are no standardized combined shear–compression tests for CMCs to obtain the required friction parameter for the Cuntze criterion [
21]. Based on the strength of tensile samples with different fiber orientations, failure stresses and strains for a virtual UD ply are evaluated. However, Tsai-Wu failure has its shortcomings as it is not able to differentiate between failure modes. The Tsai-Wu failure criterion is extended by Paepegem in order to determine the stress component responsible for the failure of a ply [
22]. A quantitative comparison of direction-dependent failure modes is, though, not demonstrated in his work. Tushtev et al. implemented a damage model with a failure criterion based on thermodynamic forces for a 2D-woven C/C composite [
23]. A similar damage-dependent quadratic failure criterion is proposed by Yang et al. as well, where strength predictions are made for a 2D-woven C/SiC material [
24]. The proposed model is applicable not only on laminates with woven fiber architecture, but also on CMCs with wound fiber architecture.
In the current work, an attempt has been made to combine continuum damage mechanics with a damage-based quadratic failure criterion. Although only tensile and shear tests were considered for the materials under consideration, the proposed failure criterion allows the integration of damage under compression if the damage behavior varies under tension and compression. A damage-based criterion has the advantage over the Tsai-Wu failure criterion in stress- or strain-space that it can give an insight into the fracture mechanics of the laminate because of its direction dependency. It can be used for designing CMC structures where damage, i.e., loss of stiffness in different directions, can be considered as a design limit instead of strength or strain. In this way, an empirical damage-based failure criterion based on macro-mechanics can deliver more information about phenomenon at the micro-level through damage variables which can be correlated to properties like crack density within a material. The proposed model exhibits an advantage over other empirical failure criteria as it can determine the responsible component in the damage-space, and thus can predict the failure mode in laminates with varying fiber orientations.
4. Conclusions
In this work, an anisotropic damage model based on continuum mechanics is integrated with a damage-based failure criterion in order to predict the inelastic behavior of two ceramic matrix composites manufactured via filament winding technology. Damage in the material is defined as continuous stiffness reduction in the laminate with increasing load. As there were no test results available for unidirectional plies, the elastic properties of a virtual unidirectional ply were evaluated from the tensile tests carried out on laminates with different fiber orientations. The parameters required for damage models were derived from the stress–strain curves obtained from tensile tests. The damage model was implemented in a commercial finite element software, ANSYS Workbench. It was observed that laminates with a fiber orientation where the fiber carries most of the load, e.g., ±15°, exhibit a relatively linear stress–strain curve compared to the matrix-dominant direction, e.g., ±60°. In this way, the inelastic behavior of laminates with varying fiber orientations can be predicted with a single parameter set for damage description.
In order to predict the failure stress and strain of laminates, a damage-based failure criterion inspired by the Tsai-Wu failure criterion was proposed. The failure criterion considers the coupling of damage variables in different directions, which makes it appropriate to be used for anisotropic CMCs. The predictions made by the damage-based criterion were closer to the experimental results in comparison to the stress- or strain-based Tsai-Wu failure criterion. Moreover, a damage-based criterion can be directly related to the physical attributes of a material, such as crack density, and has potential to give an insight into the micromechanics of the material despite its macroscopic nature.
The application of the proposed model on Iosipescu shear tests showed discrepancy when compared with the experimental results. All three in-plane stress components (tension, compression and shear) are present in the sample based on its fiber orientation. This discrepancy is attributed to the assumption that the material behaves linearly elastically under compression, i.e., no damage occurs under compression. The further investigation of the fracture surfaces of materials after failure led to the conclusion that the Iosipescu shear test can determine the shear modulus of the material, but is not an appropriate test to evaluate the shear strength of the material because of the interdependence of notches and the fiber orientation of the laminate.
The proposed damage-based failure criterion considers loss of stiffness in principal material directions, which can be used as design limits for a CMC component under thermomechanical loading and can differentiate between damage mechanisms under tension and compression. Loss of stiffness can be measured via non-destructive testing (NDT) methods, such as acoustic emissions, and can be used as a quality assurance criterion in order to determine the life cycle of a CMC component after being in operation for a definite period of time.