Numerical and Experimental Investigation of Hygrothermal Aging in Laminated Composites

Although being a crucial step in structural design of laminated composites, prediction of their long-term mechanical performance remains a challenging task for which no comprehensive and reliable solution is currently available. Nevertheless, structures such as wind turbine blades, of which laminated composites constitute the main load bearing parts, must be designed to withstand 20 years of service while being subjected to a combination of fatigue loads and interaction with often extreme environmental conditions. In the end, a compromise is reached by compensating the lack of knowledge on the complex material degradation and failure mechanisms spanning multiple spatial and time scales that determine mechanical performance by adopting higher safety factors. This in turn leads to heavier, less efficient and more expensive designs. A better understanding of these mechanisms through discerning experiments and the development of fast and accurate numerical prediction tools are therefore necessary.This work focuses on the phenomenon of hygrothermal aging (a combination of high temperatures and moisture ingression) on unidirectional laminated composites. The complexities of the aging problem, a combination of physical and chemical degradation mechanisms that affect fibers, resin and interface differently, are investigated through a combination of experiments, microscopic observation techniques and state-of-the-art numerical modeling. The result is an efficient multiscale and multiphysics framework for the prediction of failure and hygrothermal degradation in composites.First, an experimental campaign is conducted on unidirectional glass/epoxy composite samples and on pure epoxy specimens immersed in water at 50C and tested quasi-statically and in fatigue. By comparing results of unaged, partially saturated, saturated and redried samples, the contributions of reversible and irreversible hygrothermal aging mechanisms are measured. The results indicate a strong correlation of degradation with the water concentration field inside the specimens. Furthermore, significant differences in strength reduction between composites and pure resin specimens point to damage in the fiber-matrix interfaces.In order to realistically model the diffusion process that drives degradation, an experimental/numerical study is conducted on the anisotropic diffusion behavior of laminated composites. Thin material slices extracted from a thick composite panel are immersed until saturation and the obtained anisotropic diffusivity parameters are numerically reproduced through a microscopic diffusion model with periodic concentration field. The existence of an interphase transition region around the fibers is confirmed through microscopic experiments and included in the model through a level set field.Since both the diffusion process and the resultant material degradation are highly influenced by the microstructure of the material, a multiphysics and multiscale analysis approach becomes necessary. A numerical framework for modeling of the aging process is proposed combining a macroscopic Fickian diffusion analysis with a multiscale stress equilibrium analysis based on the FE2 method. Since the multiscale approach does not rely on any constitutive hypotheses at the macroscale, complex failure behavior combined with plasticization and differential swelling can be accurately captured. In order to expand the framework to allow for modeling of cyclic loading and cyclic environmental exposure, a number of additional model ingredients are developed. Firstly, a new constitutive model for epoxy combining viscoelasticity, viscoplasticity and a damage formulation with rate-dependent fracture onset is presented. The model is calibrated through a series of quasi-static and fatigue experiments on pure resin specimens at multiple strain rates and both before and after hygrothermal aging. The calibrated model is able to accurately capture the observed strain rate dependency and stiffness and strength degradations after aging, as well as correctly capturing damage activation in low-cycle fatigue. Secondly, the significant computational cost associated with the use of a cyclic multiphysics/multiscale analysis with nested micromodels is alleviated through a number of acceleration techniques. Time homogenization is used to explicitly divide the loading into a nonlinear macrochronological part and a linear computationally inexpensive microchronological one. Furthermore, the size of the microscopic boundary value problem is reduced through a combination of Proper Orthogonal Decomposition (POD) and the Empirical Cubature Method (ECM), resulting in a hyper-reduced model. The resultant reduced and time homogenized micromodel allows for speed-ups higher than 1000, dramatically accelerating the solution of the problem. The modified version of the framework is used to numerically reproduce the experimentally obtained interlaminar shear behavior of composite samples aged for different durations. Use of the multiphysics/multiscale approach allows for accurately describing the stress state in specimens with non-uniform water concentration fields. The viscoelsatic/viscoplastic resin model is capable of capturing differences in stress response between the very slow conditioning phase and the much faster mechanical test. The model is completed by a cohesive-zone model for fiber-matrix interface debonding including friction calibrated with a set of Single Fiber Fragmentation tests performed on dry and saturated samples.Applied Mechanic

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

Concurrent multiscale finite element analysis (FE2) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models for FE2 based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macroscopic integration points. The Bayesian formalism inherent to GP models provides a natural tool for online uncertainty estimation through which new observations or inclusion of new anchor micromodels are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity and the framework is further demonstrated with the elastoplastic analysis of a plate with multiple cutouts and with a crack growth example for mixed-mode bending. Although not able to handle non-monotonic strain paths in its current form, the framework is found to be a promising approach in reducing the computational cost of FE2, with significant efficiency gains being obtained without resorting to offline training.</p

Machine learning of evolving physics-based material models for multiscale solid mechanics

In this work we present a hybrid physics-based and data-driven learning approach to construct surrogate models for concurrent multiscale simulations of complex material behavior. We start from robust but inflexible physics-based constitutive models and increase their expressivity by allowing a subset of their material parameters to change in time according to an evolution operator learned from data. This leads to a flexible hybrid model combining a data-driven encoder and a physics-based decoder. Apart from introducing physics-motivated bias to the resulting surrogate, the internal variables of the decoder act as a memory mechanism that allows path dependency to arise naturally. We demonstrate the capabilities of the approach by combining an FNN encoder with several plasticity decoders and training the model to reproduce the macroscopic behavior of fiber-reinforced composites. The hybrid models are able to provide reasonable predictions of unloading/reloading behavior while being trained exclusively on monotonic data. Furthermore, in contrast to traditional surrogates mapping strains to stresses, the specific architecture of the hybrid model allows for lossless dimensionality reduction and straightforward enforcement of frame invariance by using strain invariants as the feature space of the encoder.</p

Neural networks meet physics-based material models: Accelerating concurrent multiscale simulations of path-dependent composite materials

In a concurrent (FE2) multiscale modeling is an increasingly popular approach for modeling complex materials. As such, it is especially suited for modeling composites, as their complex microstructure can be explicitly modeled and nested to each integration point of the macroscale. However, this generality is often associated with exceedingly high computational costs for real-scale applications. One way to tackle the issue is to employ a cheaper-to-evaluate surrogate model for the microstructure based on few observations of the high-fidelity solution. On this note, Neural Networks (NN) are by far the most popular technique in building constitutive surrogates. However, conventional NNs assume a unique mapping between strains and stresses, limiting their ability to reproduce path-dependent behavior. Moreover, their data-driven nature severely limits their ability to extrapolate away from their training spaces. To circumvent these drawbacks, the alternative explored in this work is to reintroduce some of the physics-based knowledge of the problem into the NN. This is done by employing actual material models used in the full-order micromodel as the activation function of one of the layers of the network. Thus, path-dependency arises naturally since every material model in the layer has its own internal variables. To assess its capabilities, the network is employed as the surrogate model for a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material. First, for a single micromodel, the performance of the network is compared to that of a state-of-the-art Recurrent Neural Network (RNN) in a number of challenging scenarios for data-driven models. Then, the proposed framework is applied to an FE2 example and the results are compared to the full-order solution in terms of accuracy and computational cost. An important outcome of the physics-infused network is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with highly popular but data-hungry models such as RNN.Applied Mechanic

Interpreting the single fiber fragmentation test with numerical simulations

Characterization of the mechanical properties of the fiber/matrix interface is a challenge that needs to be addressed to enable accurate micromechanical modeling of failure in composite materials. In this paper a numerical investigation is presented into one of the tests that has been proposed for measuring these interfacial properties. A new cohesive zone model with friction is presented, as well as an original numerical framework for modeling of embedded fibers. The research generates new insight into the meaning of the single fiber fragmentation test, confirming the applicability of shear lag theory also in presence of multiple cracks, and emphasizing the relevance of matrix plasticity for the development of friction in the test. Although the frictional stress that can be obtained from the test should not be confused with the cohesive strength of the fiber/matrix interface, measurements of fracture process zone length can give indirect information on this cohesive strength.Accepted Author ManuscriptApplied Mechanic

Physically recurrent neural networks for path-dependent heterogeneous materials: Embedding constitutive models in a data-driven surrogate

Driven by the need to accelerate numerical simulations, the use of machine learning techniques is rapidly growing in the field of computational solid mechanics. Their application is especially advantageous in concurrent multiscale finite element analysis (FE2) due to the exceedingly high computational costs often associated with it and the high number of similar micromechanical analyses involved. To tackle the issue, using surrogate models to approximate the microscopic behavior and accelerate the simulations is a promising and increasingly popular strategy. However, several challenges related to their data-driven nature compromise the reliability of surrogate models in material modeling. The alternative explored in this work is to reintroduce some of the physics-based knowledge of classical constitutive modeling into a neural network by employing the actual material models used in the full-order micromodel to introduce non-linearity. Thus, path-dependency arises naturally since every material model in the layer keeps track of its own internal variables. For the numerical examples, a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material is used as the microscopic model. The network is tested in a series of challenging scenarios and its performance is compared to that of a state-of-the-art Recurrent Neural Network (RNN). A remarkable outcome of the novel framework is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with popular but data-hungry models such as RNNs. Finally, the proposed network is applied to FE2 examples to assess its robustness for application in nonlinear finite element analysis.</p

Micromechanics-based surrogate models for the response of composites: A critical comparison between a classical mesoscale constitutive model, hyper-reduction and neural networks

Although being a popular approach for the modeling of laminated composites, mesoscale constitutive models often struggle to represent material response for arbitrary load cases. A better alternative in terms of accuracy is to use the FE2 technique to upscale microscopic material behavior without loss of generality, but the associated computational effort can be extreme. It is therefore interesting to explore alternative surrogate modeling strategies that maintain as much of the fidelity of FE2 as possible while still being computationally efficient. In this work, three surrogate modeling approaches are compared in terms of accuracy, efficiency and calibration effort: the state-of-the-art mesoscopic plasticity model by Vogler et al. (Vogler et al., 2013), regularized feed-forward neural networks and hyper-reduced-order models obtained by combining the Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) techniques. Training datasets are obtained from a Representative Volume Element (RVE) model of the composite microstructure with a number of randomly-distributed linear-elastic fibers surrounded by a matrix with pressure-dependent plasticity. The approaches are evaluated with a comprehensive set of numerical tests comprising pure stress cases and three different stress combinations relevant in the design of laminated composites. The models are assessed on their ability to accurately reproduce the training cases as well as on how well they are able to predict unseen stress combinations. Gains in execution time are compared by using the trained surrogates in the FE2 model of an interlaminar shear test.Applied Mechanic

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

Concurrent multiscale finite element analysis (FE2) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models for FE2 based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macroscopic integration points. The Bayesian formalism inherent to GP models provides a natural tool for online uncertainty estimation through which new observations or inclusion of new anchor micromodels are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity and the framework is further demonstrated with the elastoplastic analysis of a plate with multiple cutouts and with a crack growth example for mixed-mode bending. Although not able to handle non-monotonic strain paths in its current form, the framework is found to be a promising approach in reducing the computational cost of FE2, with significant efficiency gains being obtained without resorting to offline training.Applied Mechanic

An adaptive domain-based POD/ECM hyper-reduced modeling framework without offline training

This work presents a reduced-order modeling framework that precludes the need for offline training and adaptively adjusts its lower-order solution space as the analysis progresses. The analysis starts with a fully-solved step and elements are clustered based on their strain response. Elements with the highest strains are solved with a local/global approach in which degrees of freedom from elements undergoing the highest amount of nonlinearity are fully-solved and the rest is approximated by a Proper Orthogonal Decomposition (POD) reduced model with full integration. Elements belonging to the remaining clusters are subjected to a hyper-reduction step using the Empirical Cubature Method (ECM). Online error estimators are used to trigger a retraining process once the reduced solution space becomes inadequate. The performance of the framework is assessed through a series of numerical examples featuring a material model with pressure-dependent plasticity.Applied MechanicsMaterials- Mechanics- Management & Desig