Progressive Failure of a Unidirectional Fiber-reinforced Composite Using the Method of Cells: Discretization Objective Computational Results

cells (HFGMC), and finite element method (FEM) to capture the nonlinear response, due to progressive failure, of a unidirectional fiber-reinforced polymer matrix composite (PMC) under a combination of transverse tension/compression and transverse shear loading. The focus is on developing a semi-analytical methodology that is insensitive to changes in the density of the numerical discretization, which is necessary for predicting damage state and failure. With the method of cells (MOC), developed by Ref. 2, a rectangular composite RUC was discretized into four subvolumes, called subcells. One of the subcells was occupied by the fiber material and the rest were occupied by the matrix. Linear displacement fields were assumed in each of the subcells. Displacement and traction continuity conditions were enforced, in an average integral sense, at the subcell interfaces, along with periodic boundary conditions at the RUC boundaries to derive a set of equations that yielded a strain concentration matrix which could, in turn, be used to obtain the local subcell strains from the applied, global fields. Following determination of the subcell strains, the subcell stresses are readily calculated using the local constitutive laws, and volume averaging can be used to obtain the homogenized thermomechanical properties of the composite. MOC was later extended to the generalized method of cells by Ref. 3 which accommodated any number of subcells and constituents in two-periodic directions. Ref. 4 adapted the formulation to accommodate triply-periodic materials. Finally, Ref. 5 developed the high-fidelity generalized method of cells which utilized second order displacement field approximations in the subcells, rather than linear. Ref. 6 showed the local elastic field accuracy produced by HFGMC corresponded very well to FEM; whereas, Ref. 7 compared the accuracy of the fields in inelastic phases to composite cylinder assemblage (CCA). Ref. 8 utilized HFGMC to model fiber-matrix debonding in metal matrix composites, and Ref. 9 implemented a multi-axial damage model in HFGMC. Reformulations, of GMC and HFGMC, which reduced the total number of unknowns in the problem were introduced by Refs. 10 and 1, respectively. Herein, the reformulated, doubly-periodic versions of GMC and HFGMC are employed, and the reader is referred to Refs. 10 and 1 for the complete details on these formulations. These semi-analytical methods offer efficient alternatives, compared to fully numerical methods such as FEM, for computing local fields within a composite RUC and are ideal for implementation in a multiscale framework. The generality of the GMC and HFGMC formulations admit any constitutive behavior at the subcell level. However if the response of the subcell material exhibits post-peak strain softening, the tangent stiffness tensor of the subcell loses positive definiteness. This leads to pathologically mesh dependent behavior. This mesh dependency, if not eliminated, will not provide predictive capability, rather, the capability can be only used with confidence for “simulations”. The smeared crack band approach, which introduced a characteristic element length into the post-peak strain softening damage evolution formulation, can be used to alleviate the dependency of failure, and strain localization, on the discretization size within the continuum domain. The tangent slope of the softening stress-strain curve was scaled by the characteristic length to ensure that total strain energy release rate (SERR) upon complete failure (i.e. zero stress) is always equal to the prescribed fracture toughness (or critical SERR), regardless of the size of the discretization. In the original formulation, the band was always oriented perpendicular to the direction of maximum principal stress; thus, the crack band always advanced under pure mode I. Refs. 12 and 13 later reformulated the model to incorporate a fixed crack band that evolved under mixed-mode conditions. Both formulations employ triangular degradation schemes. Later, Ref. 14 incorporated more sophisticated initiation criteria to predict the onset of mixed-mode crack bands. All of these smeared crack formulations assume linear elastic behavior up to the initiation of the crack band, followed by immediate post-peak strain softening. However, Ref. 15 coupled pre-peak plasticity with crack band governed post-peak strain softening to model failure of concrete. Recently, a thermodynamically-based work potential theory was developed which allows for pre-peak progressive damage, as well as, post-peak progressive failure, governed by the smeared crack band approach, in homogenized laminae,, for fiber reinforced laminates. A variation of the crack band theory for concrete structures is implemented here within GMC, HFGMC and FEM to model mesh objective failure of continuous fiber-reinforced PMCs. With this implementation, presented in Section II, two scenarios are considered to determine the mode in which the cracks contained in the crack band grow. If the principal stress that has the highest magnitude is tensile, it is assumed that it is more energetically favorable for the crack band to form perpendicular to the maximum principal stress and for the cracks within the band to advance under mode I conditions. Conversely, if the magnitude of a compressive principal stress is higher than the other principal stresses, the cracks within the crack band evolve under mode II conditions (due to internal, Mohr-Coulomb friction) and are oriented with the plane NASA/TM—2012-217649 2 of maximum shear stress. Previously, similar studies utilized FEM to model the composite microstructure. Plasticity was employed to model the non-linear response of the matrix and failure was introduced by allowing the fiber-matrix interface to debond using cohesive zone elements. The response of a statistical sample of RUCs subjected to a combination of transverse compressive and transverse shear was reported. In the current work, the focus is restricted to the microscale to evaluate the capabilities of the smeared crack band model to predict progressive failure evolution within a composite microstructure, using the semi-analytical methods (GMC and HFGMC), by verifying and validating this implementation, as well as the overall utility of the micromechanics models employed, against available experimental data, and an analogous, fully numerical (FEM) model. The objectivity of failure evolution with respect to the level of discretization used in HFGMC was previously demonstrated in Ref. 20. In Section III, the results for an RUC containing 13, randomly placed fibers subjected to a combination of transverse tension, transverse compression, and transverse shear are presented. The stress-strain response obtained from GMC and HFGMC are compared to experimental data (where available) and FEM results. Furthermore, failure path predictions obtained from the three methods are exhibited. Using GMC to model an RUC containing 13 randomly arranged fibers is beyond the intended use of this ultra-efficient, semi-analytical method, and it was expected that with GMC, as with any analytically-based model, that the results obtained from the simulations of this architecture would not be accurate. These GMC simulations were executed to give the authors a measure of how the maximum disorder affected the GMC results. Additional simulations were carried out, utilizing an ordered, square packing architecture to demonstrate that the accuracy lost with the complex could be regained, and GMC is a viable tool for modeling progressive failure in the constituents of a composite. Due to the ultra-efficient nature of GMC, it is far more tractable within a multiscale setting than HFGMC or FEM, and offers fidelity beyond that available with classical mean field theories. Single fiber RUCs were loaded using the same loading scenarios used for the multiple fiber RUCs and the results obtained from GMC, HFGMC, and FEM are compared. The FEM model used to represent the RUC contained a much finer mesh than GMC/HFGMC. This was to ensure that the FEM solutions were as accurate as possible and provided a fully numerical baseline to evaluate the performance of GMC and HFGMC. These results are presented in Section IV. II. Modeling Constituent-Level Post-Peak Strain Softening with the Smeared Crack Band Approach GMC and HFGMC are efficient (relative to fully-numerical methods), semi-analytical tools useful for modeling details of the microstructure of a composite material. Additionally, they are readily amenable for implementation into a multiscale framework. Although, physics-based, discretization objective, constitutive theories for modeling progressive failure must be in place to accurately predict the response of a structure that is failing. For pre-peak loading (i.e. positive-definite tangent stiffness tensor), there are a multitude of non-linear elastic, plastic, continuum damage mechanics, and visoelastic/plastic theories available that can predict the evolution of the appropriate mechanisms in the composite. However when the local fields enter the post-peak regime of the stress-strain laws, most of these theories breakdown in a numerical setting and display pathological mesh dependence. The lack of positive definiteness of the elastic, or inelastic, tangent stiffness tensor leads to imaginary wave speeds in the material. The longitudinal wave speed in an isotropic material is given by cL = √ E(1− ν) ρ(1 + ν)(1− 2ν) (1) where cL is the wave velocity, E is the tangent stiffness of the material, ν is the Poisson’s ratio, and ρ is the material density. A one-dimensional approximation yields v = √ E ρ . The existence of an imaginary wave speed results in a boundary value problem that is ill-posed. Physically, a material must posses a positive-definite tangent stiffness tensor, and in fact, at the micro-scale the material tangent stiffness tensor always remains positive-definite. However for practical purposes, engineers must model structures at scales much larger than the flaws in the material, and the homogenized continuum representation of a material containing the nucleation and propagation of discontinuities, such as cracks or voids, exhibits post-peak strain softening in the macroscopic, homogenized, stress-strain response. This homogenized response is assumed NASA/TM—2012-217649 3 to govern over a suitable volume of the material, appropriate to the microstructure of the material. Loss of positive-definiteness of the tangent stiffness tensor leads to a material instability, which manifests as a localization of damage into the smallest length scale in the continuum problem. In GMC, or HFGMC, this is a single subcell. Thus, the post-peak softening strain energy is dissipated over the volume of the subcell that the damage localizes to. Since a stress-strain relationship prescribes the energy density dissipated during the failure process, the total amount of energy dissipated in the subcell decreases as the size of the subcells is reduced, and in the limit, zero energy is required to fail the structure. An illustration showing this size dependence is given in Figure 1. A simple way to remedy this non-physical behavior in a numerical setting is to judiciously scale the post-peak softening slope of the stress-strain constitutive law. Then, the failure energy density dissipated becomes a function of the characteristic length of discretized continuum. Refs. 25 and 11 first proposed a crack band model in which post-peak softening damage (herein referred to as failure) in the material was assumed to occur within a band. The post-peak slope of the material constitutive law was scaled by the characteristic length of the finite element exhibiting failure; such that, the total SERR in the element, upon reaching a state of zero stress, and the material fracture toughness were coincident. In this reference, equivalence between this smeared crack approach and a line crack approach is presented. Subsequently, Ref. 26 exhibited propagation of a crack band not aligned with the mesh bias. In this work, the crack band theory is implemented within the GMC and HFGMC micromechanics models, in the MAC/GMC suite of micromechanics codes developed at the NASA Glenn Research Center, and used to analyze crack band growth in composite RUCs. The results of the semi-analytical methods are compared to experimental data, where available, and an equivalent FEM models containing the same crack band implementation. The following subsections provide theoretical details on the crack band model. II.A. Physical Behavior of Crack band The smeared crack band model is meant to capture the behavior of a region of a material wherein numerous microcracks have initiated, and they coalesce to form a larger crack. Figure 2 displays a crack band of width wc embedded in a continuum. The domain of the crack band is denoted as Ω′ and the remaining continuum as Ω. The crack band is oriented within the continuum such that, for a given point within the crack band, the unit vector normal to the crack band is n. The total energy dissipated during the failure process is dissipated over Ω′, and the size wc of Ω′ is a material property directly related to the material fracture toughness.