Numerical laminate analogy models

Harper et al. [42] developed a 2D FE model of randomly oriented 1D fibers embedded in a 2D matrix. A novel meshing method was applied for the definition of the boundary conditions of the interface between fibers and matrix. Planar 2D fiber architectures have been generated by a modified random sequential adsorption scheme, where fiber bundles were deposited over a region of interest (ROI), as shown in Fig. 2.18. Fiber orientations were created using a random number generator.

Fiber intersections were ignored and therefore, no limitation was imposed on the fiber volume fraction. A line clipping algorithm was used to trim the fiber bundles to the RVE

Figure 2.18: Schematic of random sequential adsorption model in the as-deposited state (reprinted from [42]).

the model according to the specified fiber length. Platelets were sequentially deposited until the volume of fiber met the target volume fraction for the inner RVE (shown as a black box in Fig. 2.18). Abaqus [43] was used for simulating the 2D architecture using 1D beam elements with a circular cross-section consisting of fibers and matrix as representation of platelets. The tensile stiffness of each beam was calculated using a rule of mixtures. The matrix material was modeled using a regular array of 2D plane stress continuum elements. Beam elements are fixed to the solid ones using the *EMBEDDED ELEMENT technique, a type of multi-point constraint withinAbaqus [43]. Embedding eliminated the need for a complex meshing algorithm to pair the coincident nodes of fibers to the resin. The novel meshing technique significantly reduced computation time of mesh generation without affecting Von Mises stress results as illustrated in Fig. 2.19.

Figure 2.19: Von Mises tensile stress comparison between a model using an unstruc-tured mesh with tie constraints (left) compared with strucunstruc-tured embedded elements (right) (reprinted from [42]).

In order to further reduce computation times, a Saint-Venant’s principle [44] was adopted, whereby the considered RVE was embedded into a tertiary material and the effective material properties were extracted from the inner region (see Fig. 2.20). Two

Figure 2.20: Saint-Venant’s principle: application of the boundary conditions to a RVE using at homogeneous material by Harper et al. [42] (reprinted from [19]).

different approaches have been investigated: (i) a homogeneous material to model the host material around the boundary and (ii) a heterogeneous material, which was a continuation of the material within the inner RVE. The critical decay length for the heterogeneous approach was found to be two times the fiber length in all cases. This resulted in much larger models, but the effective properties extracted from the inner region after convergence were considered to be more reliable than the homogeneous approach.

Feraboli et al. [1] analyzed CF-SMC in terms of tensile tests combined with a Digital Image Correlation (DIC) system. Full field strain measurements of specimens varying in width revealed highly non-uniform strain distributions. A classical discretization of the specimen to a RVE was not possible due to randomness of strain distributions caused by the heterogeneous structure CF-SMC composites. A novel discretization method was developed by filtering full field strain measurements for critical strain regions, which lead to repetitive strain patterns of high peaks and low valleys. The authors found that the critical strain regions exhibit a repetitive pattern in their randomness. The area of these isolated regions was measured and utilized for discretization of the specimen into Random Representative Volume Elements (RRVEs).

(a) DIC captured image (b) Filtered critical strain regions

Figure 2.21: Results from DIC used for discretization by Feraboli et. al [1]

The properties of the platelets constitute the upper and lower bounds respectively for the discontinuous material. Since measured moduli of CF-SMC were between these bounds, the probability of all platelets in one RRVE having the same direction was con-sidered minimal. This laminate analogy allowed the calculation of mechanical proper-ties using CLT. An algorithm was designed using a stochastic process to model the ran-dom distribution of sizes and orientations for the platelets of the physical plate, where needed input parameters were platelet and specimen dimensions. The randomization algorithm created a random generated platelet fraction with a random orientation angle θassigned. All platelet fractions having the sameθare summed up to the total platelet areaA(θ), dependent on orientation state. This process was continued in a loop until the total volume of fractions reached the volume of the specimen. The thicknesses and orientations of the virtual laminate were used with mechanical properties of platelets for calculation of elastic laminate parameters. Feraboli et al. [1] analyzed tensile tests specimens using FEA in combination with the described algorithm. Therefore, the specimen was partitioned into 48 regions of 0.25×0.25 in according to the measured RRVE size.

Figure 2.22: An example of a tensile specimen divided into several regions, with inde-pendently generated material properties (reprinted from [1]).

Subsequently, the randomization algorithm was applied to each RRVE assigning random thickness values to predefined orientation angles. Once the specimen is fully defined, a FEA was performed usingANSYS to get the macroscopic properties of the virtual tensile test specimens.

2.2.3 3D explicit modeling of platelets

Kravchenko et. al [11] considered the local stress transfer interactions between platelets to define the macroscopic response of the composite. The strength was shown to be dependent on platelet dimensions, overlapping, and orientations. Therefore, researchers characterized CF-SMC material with focus on the mesoscale by conducting tensile tests with specimens varying in thickness in combination with a DIC system to evaluate mechanical performance. Results of Young’s-Modulus and tensile strength exhibited high variations, therefore statistical analysis was performed on the data sets, which indicates an existence of a relationship between mechanical properties and the meso-structure. Digimat FE was used to create virtual CF-SMC specimens by combining geometry and mesh generation in a voxel-based process, where a defined specimen volume is discretized with a pattern of rectangular three-dimensional units (=voxels) of selected size. This pattern (=voxel mesh) is filled sequentially with platelets of defined size and orientation. An example output of this process is shown in Fig. 2.23.

The progressive failure analysis at the local platelet level was based on the stiff-ness reduction scheme and the smeared crack approach. The developed computational mesoscale approach considered individual platelets having a transversal isotropic mate-rial behavior with the properties of unidirectional prepreg tape. On the platelet level, progressive damage was mathematically described using continuum damage mechanics (CDM). Both initiation and propagation of damage were predicted with CDM without changing the original FE mesh of the system. Kravchenko et. al [11] assumed primary local mesoscale failures of specimens including platelet in-plane and out-of-plane

fail-Figure 2.23: Digimat FE generated CF-SMC specimen using a voxel-based process (reprinted from Kravchenko et. al [11])

model was implemented usingAbaqus in combination with a material model.

Pan et al. [45] assumed uniformly distributed fibers having an elliptical cross-section.

Furthermore, these fibers were set to be either curved or straight, as shown in Fig. 2.24 b). A RVE was generated by stacking three fiber matrix systems as shown in Fig. 2.24, where a single fiber matrix system consisted of 3-matrix and 2-fiber sub-layers. Fiber-rich sub-layers were filled with randomly generated fibers. Curved fibers were generated when an added fiber intersected with other fibers; afterwards, the intersecting part of the new fiber was moved to the next layer (see Fig. 2.24 b)). If no intersection occurred when adding a new fiber, it was set straight.

(a) (b)

Figure 2.24: (a) Layer arrangement of the RVE; (b) Schematic illustration of a side view of intersecting fibers (reprinted from [45]).

Fig. 2.25 illustrates a RVE generated with the described method. A subsequent FEA of the RVE delivered its macroscopic linear elastic properties. Using this model, Pan et al. [45] were able to create RVEs exhibiting realistic fiber volume content levels.

(a) (b)

Figure 2.25: Example of an RVE created by the model proposed by Pan et al. [45]: (a) RVE with curved fibers; (b) RVE fiber orientation distribution.

Sommer et al. [48] described an integrated methodology for analysis of stochastic CF-SMC to develop process-structure-property relationships. Flow-induced fiber ori-entation distributions were predicted using an anisotropic viscous constitutive model implemented in a nonlinear, explicit FE solver. A FEA with an explicitly modeled platelet meso-structure was developed, wherein the platelets were treated as a homo-geneous orthotropic medium, using continuum damage mechanics to model the intra-platelet failure and a cohesive zone model for interlaminar disbonding. A schematic flow chart of the developed model is illustrated in Fig. 2.26.

Figure 2.26: Simulation framework for process-informed progressive failure analysis of CF-SMC (reprinted from [48]).

The flow simulation was performed usingAbaqus Explicit to predict the final platelet orientation distribution function (ODF) from a given partial charge coverage by record-ing of changes from local platelet directions. Digimat FE was used for the generation of the preform. Platelets in the preform were represented by a set of voxels exhibiting the local orientation of the platelet as depicted in Fig. 2.27 (a), where each color repre-sents one platelet. The platelets itself were modeled as an incompressible, anisotropic viscous medium. Subsequently, the voxel mesh was converted to SPH particles (see Fig. 2.27 (b)). A slow flow simulation was carried out neglecting platelet to platelet interface interactions and therefore inhibiting elongation of platelets in the fiber di-rection [49]. An anisotropic viscous constitutive model of an incompressible platelet was used to simulate flow behavior. The model was defined transversely isotropic with three parameters: the suspending fluid viscosity, the anisotropy ratio, and an elastic bulk modulus. Platelets oriented with the flow direction were observed to translate, rather than tending to stretch. Researchers from [48] accounted the large viscosity of the material for this behavior. Platelets were found to stretch rather than translate when oriented transverse to flow direction. These platelets exhibit reduced thickness (see Fig. 2.27).

(a)

(b) (c)

Figure 2.27: (a) Top down view of the flow front progression at different time points, (b) discretization of the voxelized domain with uniformly spaced SPH parti-cles, and (c) evolution of the in-plane fiber orientation distribution at four time points (reprinted from [48]).

A computational damage mechanics approach was utilized in combination with a cohesive zone model to analyze a virtual tensile test in terms of progressive failure behavior. A Monte-Carlo approach was used in order to consider the stochastic distri-butions of effective tensile properties derived from discontinuous platelet orientations.

The platelets were modeled as homogeneous orthotropic material with the properties of the parent UD prepreg tape. The virtual progressive failure analysis accounted for three local meso-structural damage mechanisms: (i) matrix fracture, (ii) fiber fracture, and (iii) interfacial disbonding. The smeared-crack continuum damage model approach, where internal damage variables were used to degrade the stiffness matrix components and for identification of locally failed regions ("smeared cracks"), was adopted from

material.

Salmi [51] discussed a voxel method combined with a stacking algorithm for generat-ing RVEs of CF-SMC composites implemented inDigimat FE. UsingDigimat FE with the orientation tensor for each relevant location (obtained through X-ray computed tomographic scans), RVE was generated, and homogenized properties were computed by means of full field homogenization. Platelets and platelet interfaces were modeled using solid and cohesive elements, respectively. Platelet waviness and resin-rich zones at the platelet ends were accounted [19].

Kilic [52], [53] developed a global–local nonlinear modeling approach for CF-SMC composites at the part level. At the local level, the micro-mechanical unit-cell model generated an effective non-linear response from average responses of two UD layers with axial and transversal fiber orientations. In-plane isotropy assumptions and a weighted average of a matrix-mode layer and fiber-mode layer were used. At the global level, the developed micro-mechanical model was implemented into a 3D FE framework using Abaqus [43]. A 3D Tsai-Wu failure criterion was used for predicting the damage initiation, and the properties of the fiber unit-cells were set to zero while allowing the elastic and non-linear properties of the matrix unit cells to degrade gradually and to represent the matrix determining failure initiation and propagation.