A mixed finite element formulation for elastoplasticity

Nagler, Michaela; Pechstein, Astrid S.; Humer, Alexander

doi:10.1002/nme.7070

Cited by 3 publications

(4 citation statements)

References 46 publications

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“…This process is often called a regularization of the rate independent model, that is, an approximation by a rate dependent one. 62 Due to the choice of the softplus activation function this results all by itself within the training process. Regarding the results for P2, a poor prediction becomes apparent in Figure 19 as soon as the plastic regime is reached.…”

Section: 13mentioning

confidence: 99%

“…Remark It should be noted that an application of the introduced approaches FNN

{}^{\psi +\phi }

and FNN

{}^{\psi +{\phi}^{\ast }}

to data belonging to a rate‐independent material automatically leads to a regularization, 62 that is, an approximation of the data by a rate‐dependent model. This is due to the fact that the chosen activation function cannot represent the non‐smooth dissipation potentials typical for plasticity, compare Table 1.…”

Section: Nn‐based Constitutive Modelsmentioning

confidence: 99%

“…Furthermore, regarding the plots of

𝒟

and

\phi

, one can see that the the original non‐continuous curve shapes at the zero line are approximated by smooth ones. This process is often called a regularization of the rate independent model, that is, an approximation by a rate dependent one 62 . Due to the choice of the softplus activation function this results all by itself within the training process.…”

Section: Applicationsmentioning

confidence: 99%

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A comparative study on different neural network architectures to model inelasticity

Rosenkranz

Kalina

Brummund

et al. 2023

Numerical Meth Engineering

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The mathematical formulation of constitutive models to describe the path‐dependent, that is, inelastic, behavior of materials is a challenging task and has been a focus in mechanics research for several decades. There have been increased efforts to facilitate or automate this task through data‐driven techniques, impelled in particular by the recent revival of neural networks (NNs) in computational mechanics. However, it seems questionable to simply not consider fundamental findings of constitutive modeling originating from the last decades research within NN‐based approaches. Herein, we propose a comparative study on different feedforward and recurrent neural network architectures to model 1D small strain inelasticity. Within this study, we divide the models into three basic classes: black box NNs, NNs enforcing physics in a weak form, and NNs enforcing physics in a strong form. Thereby, the first class of networks can learn constitutive relations from data while the underlying physics are completely ignored, whereas the latter two are constructed such that they can account for fundamental physics, where special attention is paid to the second law of thermodynamics in this work. Conventional linear and nonlinear viscoelastic as well as elastoplastic models are used for training data generation and, later on, as reference. After training with random walk time sequences containing information on stress, strain, and—for some models—internal variables, the NN‐based models are compared to the reference solution, whereby interpolation and extrapolation are considered. Besides the quality of the stress prediction, the related free energy and dissipation rate are analyzed to evaluate the models. Overall, the presented study enables a clear recording of the advantages and disadvantages of different NN architectures to model inelasticity and gives guidance on how to train and apply these models.

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Section: 13mentioning

confidence: 99%

“…Remark It should be noted that an application of the introduced approaches FNN

{}^{\psi +\phi }

and FNN

{}^{\psi +{\phi}^{\ast }}

Section: Nn‐based Constitutive Modelsmentioning

confidence: 99%

“…Furthermore, regarding the plots of

𝒟

and

\phi

Section: Applicationsmentioning

confidence: 99%

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A comparative study on different neural network architectures to model inelasticity

Rosenkranz

Kalina

Brummund

et al. 2023

Numerical Meth Engineering

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“…The remanent polarization is interpolated with discontinous polynomial functions. Displacements and stresses are discretized by elements of mixed continuity that were originally developed for thin elastic structures [16] and later extended to ferroelectric materials [7,8] and elasto-plastic continua [17]. In particular, only the tangential component of displacements and the normal component of the stress vector (TDNNS) are continous accross element faces.…”

Section: Finite-element Formulationmentioning

confidence: 99%

Dissipative Response in Ferroic Materials: Continuum Modeling and Simulation of Electro-Magneto-Mechanical Coupling

Humer,

Pechstein,

Krommer

2023

10th ECCOMAS Thematic Conference on Smart Structures and Materials

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Ferroic materials are characterized by the presence of electric/magnetic dipoles that can be irreversibly re-oriented under sufficiently strong loading. Upon poling, an initially random distribution of dipoles on the microscopic domain scale results in a non-vanishing state of remanent polarization/magnetization on the macroscopic level. The remanent switching of dipoles constitutes the "smartness" in materials and their usage both as sensors and as actuators. Once being poled, material properties are typically assumed to be constant and uni-directional in engineering problems. To extend the operational range of ferroic transducers and open perspectives for novel applications, we seek for an accurate understanding of the evolution of the polarization/magnetization, which requires both physical and geometric non-linearities to be included in the modeling. To describe irreversible changes of the remanent state, we transfer phenomenological concepts and algorithms of associative elasto-plasticity to the field of electro-magneto-mechanics. To describe the dissipative response in a thermodynamically consistent manner, the principle of maximum dissipation is adopted. As in elasto-plasticity, constitutive equations for dissipative internal forces that drive the evolution of the remanent polarization/magnetization then follow as associated flow rules. To simplify the algorithmic treatment, we introduce the notion of dissipation functions, by which the constrained optimization problem is converted into an unconstrained problem, which can be efficiently solved by standard means. The vectors of remanent polarization/magnetization enter the model as additional internal variables. The constitutive response is governed by thermodynamic potentials, i.e., the free energy and dissipation functions. We derive a (mixed) variational formulation, identify appropriate function spaces for the dependent fields involved, and introduce a finite-element discretization. Representative numerical examples demonstrate the efficacy of the framework in electro-magneto-mechanically coupled problems.

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