Data-Rich, Equation-Free Predictions For Plasticity and Damage of Solids

Data-Rich, Equation-Free Predictions For Plasticity and Damage of Solids

Stefanos Papanikolaou Department of Mechanical and Aerospace Engineering, West Virginia University, Morgantown, WV26506, United States. Department of Physics, West Virginia University, Morgantown, WV26506, United States.
July 1, 2019

Complex systems, far from equilibrium, are characterized by uncommon average behaviors with scenario-specific constitutive laws. In the context of mechanical responses in solids, such as plasticity, damage and crack initiation, constitutive modeling has strong microstructural and loading conditions’ dependence: Mechanical testing is not only characterized by the universality of elasticity, but also by a wealth of inelastic phenomena. In this paper, we propose that inelastic characteristics can be predicted through the in-situ investigation of nominally elastic surfaces. We demonstrate a framework that builds on a loading sequence of spatially resolved strain profiles, to develop equation-free predictions of the mechanical response up to failure. The principal tools are elastic mode fingerprints of the incipient instabilities and a library of pre-existing data sets. In analogy to common fingerprints, we show that these elastic two-dimensional instability-precursor signatures can be used to reconstruct the full mechanical response of unknown sample microstructures up to failure; this feat is achieved by reconstructing appropriate statistical average behaviors with the assistance of a deep convolutional neural network, fine-tuned for image recognition. We illustrate the scalability and robustness of the approach in phase field simulations of single crystalline films with elastoplastic inclusions under mode-I fracture loading.

A major characteristic of human societies’ history has been the intrinsic coupling of material-science developments with technology leaps, an example of which is the widespread use of steel in the Iron Age Waldbaum (1978). Analogously, the Industrial Revolution coincided with the invention of the Bessemer steel Bessemer (1905). However, steelmaking still remains an art Salzbrenner et al. (2017), and finding new alloys with excellent mechanical properties resembles a “Monty-Hall” problem Selvin (1975). A critical bottleneck in the systematic and targeted material discovery for mechanical applications is the lack of widely applicable modeling approaches across material classes beyond elasticity. Such methods require robust and multiscale physical understanding, a scalable ability for predictions with high fidelity, and applicability in technology-relevant extreme conditions, such as high temperatures, pressures and strain-rates.

In the large multidimensional parameter space of possible compositions and loading conditions, a natural requirement for the development of “systematic metallurgy” has been the ability to consistently predict mechanical behaviors of new compounds by testing single-dimensional parameter lines. In this context, a multitude of data science and machine learning Liu et al. (2015, 2017); Ramprasad et al. (2017); Zhang and Ling (2018); Papanikolaou et al. (2019a) approaches have been recently proposed, which focus on consistent fitting and extrapolation. For example, image recognition methods have been efficiently used for the identification of “flaws” in materials and structures Weimer et al. (2016); Ren et al. (2017); Masci et al. (2012); Park et al. (2016), but work remarkably well only when the flaw’s mechanical effects are already well understood Gobel and Pospisil (2009). In this paper, we develop a method that builds upon spatially resolved strain information at the microstructural scale of interest, and at consecutive in-situ mechanical loads, in the nominally elastic, small-deformation and non-invasive regime; such information may be efficiently acquired through microscopy at the scale of interest Schreier et al. (2009). The small-deformation superposition principle Voyiadjis (2007); Asaro and Lubarda (2006) allows for a complete reconstruction of strain profiles by prior existing data, with assistance from a deep convolutional neural network for image recognition. In this way, a far-from-equilibrium statistical average Goldenfeld (1992) mechanical response is formed for an unknown material microstructure. The method is scalable and physically consistent, it remains efficient and minimal in terms of experimental requirements, and can be applied to arbitrary collections of microstructures.

Figure 1: Process diagram of main steps towards equation-free prediction for plasticity and damage. The field typically is a strain invariant field that can be accurately measured through image correlation techniques. The applied strain corresponds to the user-defined loading steps.

In the study of non-linear dynamical systems, the behavior near a stable fixed point may seem ignorant of nearby instabilities. eg.  with appears stable, but the existence of a perturbative subleading term may lead to a bifurcation Strogatz et al. (1994). The subleading term(s) define the instability’s universality class that contains unique spatiotemporal signatures, analogous to fingerprints.

More specifically, for a dynamical system with evolution equation , the instabilities near a stable fixed point (ie. ) are controlled by the growth rates of generic perturbations which are given by the spectrum of Lyapunov exponents(LE) Lyapunov (1884. Republished in French, University of Toulouse, 1904., 1892, 1992a, 1992b) . LEs are the real parts of the eigenvalues of the Jacobian matrix , which provides the fixed-point dynamical evolution with . The LEs calculation is typically numerical in nature, even though there have been difficulties with limited experimental data Benettin et al. (1980a, b); Brown et al. (1991); Bryant et al. (1990); Bryant (1993); Eckmann and Ruelle (1985); Miller (1964); Schmid (2010a); Shimada and Nagashima (1979)

In the context of elasticity, it is natural to view the loading process of a mechanical system as a non-linear dynamical system with “time” defined by loading. In this definition, the dynamics in-between loading steps is considered fully dissipative, and thus is neglected. Away from the stable elasticity fixed point, a solid may undergo numerous inelastic instabilities, such as necking, buckling, plasticity, crack initiation, damage Asaro and Lubarda (2006). These instabilities of elasticity represent spatially dependent bifurcations, or sometimes more complex fixed point behaviors such as dynamical relaxation oscillations and limit cycles Aifantis (1992); Bigoni (2012); Papanikolaou et al. (2017)

Figure 2: Model description & demonstration: Elastoplastic microstructure with brittle fracture characteristics, through a phase-field approach. (a) Texture, (b) Phase field of damage , (c) Uniaxial stress along the z axis, (d) Uniaxial strain along the z axis, (e) Evolution of stress and damage field vs. axial imposed strain.

Assuming the loading of a sample location through an imposed strain profile , there are various physical quantities that satisfy simple small-strain evolution:


where , is reserved for the (tensor) field of interest, and refer to leading order terms with . Assuming a generic leading-order perturbation, and the identification , one has the normal form:


where is typically an unknown matrix. For example, for the deformation tensor one has,


By subtracting accordingly, this dynamical equation may be written in its normal form with a multi-index tensor that controls the most singular LEs. Analogous considerations may be made for other observables such as damage and stress fields or stress/strain invariants. Without loss of generality, we focus here on two examples, the damage field and the first strain invariant, .

The pursuit in understanding instabilities of elasticity requires the precise identification of (see also Fig. 1). The basic aspects of are understood for exactly solvable cases, such as a dislocation pile-up at a precipitate under shear, an elliptical notch under lateral load, or an Eshelby inclusion in an elastic matrix Asaro and Lubarda (2006). In all these cases, it can be directly shown that captures the long-range stress changes during subtle movements of inelastic defects (dislocations/micro-cracks/damage/inclusions) in the parent elastic system.

The exact solutions of strain induced by individual lattice defects have provided major insights for constructing constitutive equations in models of mechanics Kanouté et al. (2009); Asaro and Lubarda (2006). Here, we develop an automatic framework to capture and develop such solutions towards equation-free predictions: For a particular sample (of unknown microstructure and material), one seeks to predict its mechanical response and failure while only requiring non-invasive mechanical and microscopy testing. One may collect consecutive spatially resolved profiles of an observable which should satisfy the normal form of Eq. 2. These profiles may be used to identify the spatially resolved Jacobian matrix and the corresponding effective growth exponents and eigenmodes . The set of modes of the unknown sample may be considered as the fingerprints of incipient elastic instabilities, that may have either a microstructural or geometric origin. Due to the small-deformation superposition principle, these modes may be directly compared to any superposition of other modes that have been similarly calculated from other microstructures, but with the same loading and geometry conditions as the unknown sample. The modes are stored in the library . The use of a superposition approach implies the existence of a particular, unknown, defect configuration that can generate the mode .

One could use a variety of approaches towards identifying probability weights to satisfy: , eg. eigenmode basis projection or direct fitting Press et al. (1987). In the framework discussed in this manuscript we utilize deep convolutional neural networks to estimate ’s. Moreover, if pre-existing library samples have been tested to failure, with recorded response functions , then at small deformations, it shall formally hold that . If we then promote the statistical averaging hypothesis, then ’s may be extended to estimate average responses at all strains. The identification of ensemble averages in our framework is dependent on the wealth of the modes’ library and their resolution (see Fig. 1).

The analogy of eigenmodes to fingerprints is insightful: The sought experimentally relevant strain deformation data sets are typically two-dimensional (2D) and only capture small strains of a surface profile. However, in the same way that 3D humans are being recognized by 2D fingerprints, material microstructures may be recognizable by the load-dependent elastic defect signatures in the small strain regime.

Figure 3: Demonstration of Single-Sample Mode (SSM) Identification & Prediction using . (a) Selected (3) EIMs emerging from considering the consecutive damage images until close to failure. (b) real and imaginary parts of for each kept k-mode. (c) Singular value for the j-th SVD-mode. (d) Predicted damage evolution snapshots using only the identified modes.

We label this framework as Stability of Elasticity Analysis (SEA). To explicitly illustrate it, we perform continuum simulations of the mechanical behavior in a model material that resembles a brittle FCC material with material properties of Aluminum that may plastically deform at the microscale Papanikolaou et al. (2019b). Results in these model microstructures can be applicable to virtually all brittle and quasi-brittle material classes Bazant and Planas (1997); Asaro and Lubarda (2006). We use an integrated spectral phase field approach (for crack growth) coupled constitutively to elastoplasticity through the Düsseldorf Advanced MAterial Simulation Kit, DAMASK, open-source software Roters et al. (2019). Typical phenomenological crystal plasticity constitutive laws are utilized, that take into account grain orientation, crystalline structure, and possible slip systems Asaro and Lubarda (2006). This is a multiscale, micromechanical analysis that connects the microstructure to the macroscopic mechanical response.

We consider an exemplary test case by laterally loading a notched thin-film specimen in contact with air in the horizontal direction, with sample dimensions: and a resolution at m in the loading direction, m the horizontal, and m in thickness).The notch facilitates crack growth and it has width mm and height mm, with an elliptical shape. The importance of this example is that its dimensions can be efficiently achieved by current experimentation procedures Schreier et al. (2009). The sample also contains a large variety of needle-like inclusions with distinct material properties than the matrix and there is also stochastically imposed microscale initial damage at the inclusion tips. Naturally, these inclusions behave as severe defects, where micro-cracks may nucleate during mechanical loading.

DAMASK utilizes a phase field model Ambati et al. (2015) in the continuum to solve for material deformation due to damage evolution within the sample. It uses a spectral method to solve for the elastic and plastic deformation, which is very stable in handling microstructural inclusion disorder and damage. For using FFTs, the sample requires a rectangular-gridded mesh and periodic boundary conditions in all three directions. A layer of air is in contact with the notched surface (horizontal direction), in order to satisfy the periodic boundary conditions. The sample is considered infinite in the loading direction, since periodic Papanikolaou et al. (2019b). Here, we focus on the post-processing aspects that are based on the discussed framework (see also Fig. 1). In this way, damage/strain profiles generated in our simulations (Fig. 2 (b-d)) can be easily resembled to crack/void and strain profiles that may be generated by Digital Image Correlation (DIC) techniques Schreier et al. (2009).

For the needs of the framework, and only for demonstration purposes, we consider a particular inclusion realization (see Fig. 2a) as the “unknown” microstructure that requires to be assessed. For the library of pre-existing microstructures , we consider other realizations that include various realizations of the same statistically microstructure, microstructures with larger needle-like inclusions, as well as samples of various materials without any inclusions but different material properties (elastic moduli, strength, critical strain energy release rate). More complex library constructions will be pursued in future works. Loading is monotonically increasing in a displacement-controlled fashion along the lateral direction until fracture. Damage increases across the sample in a highly stochastic manner, both from the notch as well as the damaged inclusions. The overall response is shown in Fig. 2.

In the model, fracture takes place at a loading stress MPa, controlled by both Linear Elastic Fracture Mechanics Sanford and Sanford (2003), as well as quasi-brittle fracture induced by the inclusions. In this complex but realistic scenario, the major question arises in the capacity for data-rich but equation-free predictions of the fracture stress, as well as the overall pathway to failure. Another relevant question involves the prediction of ideal material microstructures with optimal material properties. The latter aspect will be the focus of future works.

Figure 4: Equation-Free Predictions with : The SSM prediction extends only to the behavior of the damage field. EM-CNN uses the library of prior samples and modes to reconstruct a predicted behavior that is very accurate for the currently modeled stochastic microstructures.

The primary aspect of SEA involves the identification of elastic instability modes (EIM) through a sequence of spatially resolved strain profiles, which should solve the corresponding eigenproblem of Eq. 2. Then, EIMs may promote equation-free predictions of overall mechanical behaviors in unknown samples. For this purpose, any observable field may be used. In this work, we consider the phase-field damage field (which takes locally the value when material remains locally undamaged and if fully damaged) and the first strain invariant . The results for are shown in Figs. 3 and 4, while the results for are shown in Figs. 5 and 6.

Figure 5: Demonstration of Single-Sample Mode (SSM) Identification & Prediction using . (a) Selected (3) EIMs emerging from considering the consecutive damage images until close to failure. (b) real and imaginary parts of for each kept k-mode. (c) Singular value for the j-th SVD-mode. (d) Predicted damage evolution snapshots using only the identified modes.
Figure 6: Equation-Free Predictions with : The SSM prediction extends only to the behavior of the damage field. EM-CNN uses the library of prior samples and modes to reconstruct a predicted behavior that is very accurate for the currently modeled stochastic microstructures.

Given the selection of the field , EIMs may be identified by using Singular Value Decomposition to attempt an approximate but formal reconstruction of  Rowley et al. (2009); Schmid (2010b); Guckenheimer and Holmes (1983) The principal idea is to assume that the evolution operator can be self-consistently diagonalized, after defining it in terms of consecutive time-step snapshot vectors and :


where is the Moore-Penrose pseudo-inverse of . The matrix is the attempted solution of . is the best-fit linear system evolution operator that takes to . The eigenvectors and eigenvalues of can be estimated by taking the reduced SVD of :


where , and with the rank of . In this work, we maintain (cf. Figs. 3c, 5c). Clearly, the singular value amplitudes of the SVD modes capture the variability in the strain evolution for both studied cases (see Figs. 3c, 5c). The eigendecomposition of can be directly calculated, giving a set of eigenvectors and eigenvalues . The operator has eigenvalues and eigenvectors (cf. Figs. 3a, 5a for some example modes). This low-rank approximation of eigenvalues and eigenvectors of allows for the approximate reconstruction of the time evolution as:


with the coefficients characterize the initial condition and . The quantity has a real part, which if larger than 0 signifies a finite instability growth rate, dominated by mode . An imaginary part signifies additional oscillatory response (see Figs. 3b, 5b). The Single Sample Mode (SSM) prediction is promoted by considering the formal mode expansion into modes and then extrapolating the modes into the future as in any non-linear dynamical system Strogatz et al. (1994); Guckenheimer and Holmes (1983). The SSM is able to capture the incipient instability, even though equation-free predictions of average quantities typically miss the true response (see Figs.4 and 6). This is naturally expected for investigations of precursors in bifurcation dynamics Papanikolaou et al. (2017).

The resolution scale shall be at the characteristic scale of the elastic fluctuations (eg. at the scale of a Representative Volume Element (RVE)) and thus, the dimensions of are necessarily finite. In a formal sense, represents the projection of the operator onto a Krylov space given by the collected snapshots.

The efficient but approximate Elastic Instability Eigenmode Analysis (EIEA) promotes well resolved eigenmodes that should form orthogonal and linearly independent basis functions; however, their approximate character makes them neither exactly linearly independent neither necessarily orthogonal. The eigenvalues control the instabilities, for which the real part determines the growth rate whilst its imaginary part identifies the frequency. If the snapshots are being gathered in a regime that , then future solutions shall be linear superpositions of the resolved eigenmodes. Moreover, due to the existence of elastic superposition principle Asaro and Lubarda (2006), any linear EIM superposition corresponds to an unknown but realistic microstructure with identical loading and boundary conditions. The latter feature of the discussed method is key to its ultimate usefulness.

The eigenmodes are ranked by their amplitudes (see Fig. 4). Highly damped modes make negligibly small contribution regardless their amplitude in a long term. A way to estimate the error made by the method is to estimate relative mode contributions in large enough strain intervals (up to strain ) .


The aforementioned approach leads to the formation of EIMs that can lead to a large-strain solution for a single sample, which approximately predicts incipient elastic instabilities. There is a natural way to develop a dramatic scalability of this approach by utilizing a fundamental aspect of small mechanical deformations: the superposition principle Asaro and Lubarda (2006). Namely, various different microstructures, tested at total strain with identical boundary conditions may exactly model the mechanical response of an unknown microstructure, using appropriately weighted sums. For making accurate but equation-free predictions of damage and plasticity we utilize a library of dynamical modes and complete responses up to failure. Major understanding of elastic instabilities originates in the fundamental works by Eshelby Eshelby (1957). Here, we propose a systematic capture of a scalable set of defects in a consistently tracked manner that extend those studies and understanding. Thus, it is assumed that modes for are precisely known for sample that were previously tested up to complete failure for desired loading conditions, each providing a functional form where denotes the loading response of interest (, , etc.) while provides the loading probe of interest (eg. loading strain in a particular direction). Analogous data sets, for different tested microstructures, including mode-sets and responses are stored in the library . The data sets are produced for different microstructures but same loading and boundary conditions for precise comparison purposes. Small-deformation superposition and eigen-decomposition principles dictate that there must be weights s so that at small deformations,


The identification of probability weights requires a projection of the library EIMs on the new EIM . Given s so that is effectively reconstructed by pre-existing library eigenmodes, then far-from-equilibrium statistical mechanics dictates that it is critical to perform averages only of systems that share analogous defect content Goldenfeld (1992). For elasticity, the key step consists in identifying samples that contain similar dynamical features, despite the possibility of location (or/and orientation) differences in the sample. It is expected that in homogeneously distributed samples, such differences will not influence significantly the material response. Thus, a natural statistical averaging hypothesis for the equation-free prediction of the unknown sample’s mechanical response should be,


For estimating the probability weights , we utilize a deep convolutional neural network (d-CNN). The implementation is straight-forward in that mode identification is treated as a face-recognition problem. While there are multiple approaches towards identifying appropriate mode projections on the emerging basis Cariolaro (2015), we find that d-CNNs are efficient and robust. CNNs were originally suggested for handling two dimensional inputs (e.g. image), in which features learning were achieved by stacking convolutional layers and pooling layers.LeCun et al. (1998) dCNNs (dCNN) were introduced Krizhevsky A and GE. (2012), for improving performance in image recognition. CNNs and dCNNs are well fit for automated defect identification in surface integration inspection, and their optimization is based on backpropagation and stochastic gradient descent algorithms. Weimer et al. (2016); Ren et al. (2017); Masci et al. (2012); Park et al. (2016) We apply a standard deep convolutional neural net for image recognition of similar resolution to our images, by using the TensorFlow software tf (). The predictions of this Elastic Mode Convolutional Neural Network (EM-CNN) approach for the average field (either damage field or first strain invariant ) and loading stress field average are shown in Figs. 4 and 6, whereas true mechanical response (cf. Fig. 2e) is overlayed. The EM-CNN results are efficient and robust with impressive agreement with the true mechanical response. The success of the method is clearly connected to the fact that the library of pre-existing data contained similar microstructures to the ones tested.

The interpretation of the weights should be made in terms of the identification of an appropriate combined set of defects that provide similar modes to the sample of interest. While the method is not expected to be numerically exact (even for model simulations), it is a fundamentally robust approach. The scalability of the implemented approach is guaranteed by the fact that given the wealth of data in the library , equation-free predictions will contain enough parameters to fully model unknown sample microstructures. In this way, it becomes transparent that the testing of a large variety of microstructures in advance, may allow, in principle, for the identification of EIMs for a completely unknown microstructure that may include completely different compounds through cooperation of different types of microstructures. The combination of the small deformation superposition and the eigen-decomposition principles allows for a complete and scalable –in principle exact– identification of the precise spatiotemporal evolution of an elasticity-driven sample, by using only local test strain information at small, non-invasive deformation strains. The ability of this method to produce accurate predictions depends primarily on the library of available information and the required resolution but not on any fundamentally uncontrolled constraint.

In conclusion, we presented and illustrated a data-rich but equation-free, fundamentally sound approach towards predictions of the complete mechanical responses in generic solids. Stability of Elasticity Analysis is a framework that combines two principal features: i) the identification of instability fingerprints by utilizing spatial strain profiles at consecutive mechanical loads, ii) the fingerprint comparison and probability weight identification for pre-existing library data, defining appropriate far-from-equilibrium ensemble averages. For the former step, standard SVD methods are used, while the latter step required state-of-art image recognition software by using a dCNN. While we illustrated how this method explicitly works for FCC thin film modelmetals, it is expected that SEA should have generic material class and loading condition applicability.


We would like to cordially thank E. Barbero and E. Van der Giessen for inspiring conversations. This work is supported by NSF, DMR-MPS, Award No #1709568 and DOE-BES #DE-SC0014109.


  • Waldbaum (1978) J. C. Waldbaum, From bronze to iron. The transition from the Bronze Age to the Iron Age in the Eastern Mediterranean (Coronet Books Inc, 1978), vol. 54, pp. 106–106.
  • Bessemer (1905) H. Bessemer, Sir Henry Bessemer, FRS: An Autobiography, with a Concluding Chapter, vol. 451 (Maney Pub, 1905), ISBN 0901462497.
  • Salzbrenner et al. (2017) B. C. Salzbrenner, J. M. Rodelas, J. D. Madison, B. H. Jared, L. P. Swiler, Y.-L. Shen, and B. L. Boyce, Journal of Materials Processing Technology 241, 1 (2017).
  • Selvin (1975) S. Selvin (1975).
  • Liu et al. (2015) R. Liu, Y. C. Yabansu, A. Agrawal, S. R. Kalidindi, and A. N. Choudhary, Integrating Materials and Manufacturing Innovation 4, 13 (2015).
  • Liu et al. (2017) Y. Liu, T. Zhao, W. Ju, and S. Shi, Journal of Materiomics 3, 159 (2017).
  • Ramprasad et al. (2017) R. Ramprasad, R. Batra, G. Pilania, A. Mannodi-Kanakkithodi, and C. Kim, npj Computational Materials 3, 54 (2017).
  • Zhang and Ling (2018) Y. Zhang and C. Ling, Npj Computational Materials 4, 25 (2018).
  • Papanikolaou et al. (2019a) S. Papanikolaou, M. Tzimas, A. Reid, and S. Langer, Phys. Rev. E xx, xxxx (2019a).
  • Weimer et al. (2016) D. Weimer, B. Scholz-Reiter, and M. Shpitalni, CIRP Ann Manuf Technol 65, 417 (2016).
  • Ren et al. (2017) R. Ren, T. Hung, and K. Tan, IEEE Trans Cybern 99, 1 (2017).
  • Masci et al. (2012) J. Masci, U. Meier, D. Ciresan, J. Schmidhuber, G. Fricout, and A. Mittal, IEEE international joint conference on neural networks (IJCNN) 20 (2012).
  • Park et al. (2016) J. K. Park, B. K. Kwon, J. H. Park, and D. J. Kang, Int J Precision Eng Manuf Green Technol 3, 303 (2016).
  • Gobel and Pospisil (2009) M. Gobel and O. Pospisil, MacTech Magazine p. 18 (2009).
  • Schreier et al. (2009) H. Schreier, J.-J. Orteu, and M. A. Sutton, Image correlation for shape, motion and deformation measurements: Basic concepts, theory and applications, vol. 1 (Springer, 2009).
  • Voyiadjis (2007) G. Z. Voyiadjis, Review of fundamentals of micromechanics of solids by j. qu and m. cherkaoui: John wiley, new york; 2006; 386 pp. (2007).
  • Asaro and Lubarda (2006) R. Asaro and V. Lubarda, Mechanics of solids and materials (Cambridge University Press, 2006), ISBN 1139448994.
  • Goldenfeld (1992) N. Goldenfeld (1992).
  • Strogatz et al. (1994) S. Strogatz, M. Friedman, A. J. Mallinckrodt, and S. McKay, Computers in Physics 8, 532 (1994).
  • Lyapunov (1884. Republished in French, University of Toulouse, 1904.) A. M. Lyapunov, Master’s thesis, University of St. Petersburg (1884. Republished in French, University of Toulouse, 1904.).
  • Lyapunov (1892) A. M. Lyapunov, Ph.D. thesis, (in Russian). Kharkov Mathematical Society (250 pp.), Collected Works II, 7. Republished by the University of Toulouse 1908 and Princeton University Press 1949 (in French), republished in English by Int. J. Control 1992 (1892).
  • Lyapunov (1992a) A. M. Lyapunov, Int. J. Control 55, 531 (1992a).
  • Lyapunov (1992b) A. Lyapunov, The General Problem of the Stability of Motion (Taylor and Francis, London, 1992b).
  • Benettin et al. (1980a) G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Meccanica 15, 9 (1980a).
  • Benettin et al. (1980b) G. Benettin, L. Galgani, A. Giorgilli, and J.-M. Strelcyn, Meccanica 15, 9 (1980b).
  • Brown et al. (1991) R. Brown, P. Bryant, and H. D. Abarbanel, Physical Review A 43, 2787 (1991).
  • Bryant et al. (1990) P. Bryant, R. Brown, and H. D. Abarbanel, Physical Review Letters 65, 1523 (1990).
  • Bryant (1993) P. H. Bryant, Physics Letters A 179, 186 (1993).
  • Eckmann and Ruelle (1985) J.-P. Eckmann and D. Ruelle, Ergodic theory of chaos and strange attractors (Springer, 1985), pp. 273–312.
  • Miller (1964) R. Miller, The Astrophysical Journal 140, 250 (1964).
  • Schmid (2010a) P. J. Schmid, Journal of fluid mechanics 656, 5 (2010a).
  • Shimada and Nagashima (1979) I. Shimada and T. Nagashima, Progress of theoretical physics 61, 1605 (1979).
  • Aifantis (1992) E. C. Aifantis, International Journal of Engineering Science 30, 1279 (1992).
  • Bigoni (2012) D. Bigoni, Nonlinear solid mechanics: bifurcation theory and material instability (Cambridge University Press, 2012), ISBN 1139536982.
  • Papanikolaou et al. (2017) S. Papanikolaou, Y. Cui, and N. Ghoniem, arXiv preprint arXiv:1705.06843 (2017).
  • Kanouté et al. (2009) P. Kanouté, D. Boso, J. Chaboche, and B. Schrefler, Archives of Computational Methods in Engineering 16, 31 (2009).
  • Press et al. (1987) W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, and P. B. Kramer, Numerical recipes: the art of scientific computing (AIP, 1987), ISBN 0031-9228.
  • Papanikolaou et al. (2019b) S. Papanikolaou, P. Shanthraj, J. Thibault, and F. Roters, Materials Theory (2019b).
  • Bazant and Planas (1997) Z. P. Bazant and J. Planas, Fracture and size effect in concrete and other quasibrittle materials, vol. 16 (CRC press, 1997), ISBN 084938284X.
  • Roters et al. (2019) F. Roters, M. Diehl, P. Shanthraj, P. Eisenlohr, C. Reuber, S. L. Wong, T. Maiti, A. Ebrahimi, T. Hochrainer, and H.-O. Fabritius, Computational Materials Science 158, 420 (2019).
  • Ambati et al. (2015) M. Ambati, T. Gerasimov, and L. De Lorenzis, Computational Mechanics 55, 383 (2015).
  • Sanford and Sanford (2003) R. J. Sanford and R. Sanford, Principles of fracture mechanics (Prentice Hall Upper Saddle River, NJ, 2003), ISBN 0130929921.
  • Rowley et al. (2009) C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, Journal of fluid mechanics 641, 115 (2009).
  • Schmid (2010b) P. J. Schmid, Journal of fluid mechanics 656, 5 (2010b).
  • Guckenheimer and Holmes (1983) J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector elds, Applied Mathematical Sciences (Book 42) (Springer-Verlag, New York, NY, 1983).
  • Eshelby (1957) J. D. Eshelby, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 241, 376 (1957).
  • Cariolaro (2015) G. Cariolaro, Vector and Hilbert Spaces (Springer, 2015), pp. 21–75.
  • LeCun et al. (1998) Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, Proceedings of the IEEE 86, 2278 (1998).
  • Krizhevsky A and GE. (2012) S. I. Krizhevsky A and H. GE., in International conference on neural information processing systems (2012), vol. 25, pp. 1097–105.
  • (50) Google tensorflow. 2019.
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description