56 research outputs found
Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model
The purpose of this paper is to provide analytical and numerical solutions of
the formation and evolution of the localized plastic zone in a uniaxially
loaded bar with variable cross-sectional area. An energy-based variational
approach is employed and the governing equations with appropriate physical
boundary conditions, jump conditions, and regularity conditions at evolving
elasto-plastic interface are derived for a fourth-order explicit gradient
plasticity model with linear isotropic softening. Four examples that differ by
regularity of the yield stress and stress distributions are presented. Results
for the load level, size of the plastic zone, distribution of plastic strain
and its spatial derivatives, plastic elongation, and energy balance are
constructed and compared to another, previously discussed non-variational
gradient formulation.Comment: 41 pages, 24 figures; moderate revision after the first round of
review, Appendix A re-written completel
eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation
Lattice networks with dissipative interactions are often employed to analyze
materials with discrete micro- or meso-structures, or for a description of
heterogeneous materials which can be modelled discretely. They are, however,
computationally prohibitive for engineering-scale applications. The
(variational) QuasiContinuum (QC) method is a concurrent multiscale approach
that reduces their computational cost by fully resolving the (dissipative)
lattice network in small regions of interest while coarsening elsewhere. When
applied to damageable lattices, moving crack tips can be captured by adaptive
mesh refinement schemes, whereas fully-resolved trails in crack wakes can be
removed by mesh coarsening. In order to address crack propagation efficiently
and accurately, we develop in this contribution the necessary generalizations
of the variational QC methodology. First, a suitable definition of crack paths
in discrete systems is introduced, which allows for their geometrical
representation in terms of the signed distance function. Second, special
function enrichments based on the partition of unity concept are adopted, in
order to capture kinematics in the wakes of crack tips. Third, a summation rule
that reflects the adopted enrichment functions with sufficient degree of
accuracy is developed. Finally, as our standpoint is variational, we discuss
implications of the mesh refinement and coarsening from an energy-consistency
point of view. All theoretical considerations are demonstrated using two
numerical examples for which the resulting reaction forces, energy evolutions,
and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after
review, paper title change
The Peierls--Nabarro FE model in two-phase microstructures -- a comparison with atomistics
This paper evaluates qualitatively as well as quantitatively the accuracy of
a recently proposed Peierls--Nabarro Finite Element (PN-FE) model for
dislocations by a direct comparison with an equivalent molecular statics
simulation. To this end, a two-dimensional microstructural specimen subjected
to simple shear is considered, consisting of a central soft phase flanked by
two hard-phase regions. A hexagonal atomic structure with equal lattice spacing
is adopted, the interactions of which are described by the Lennard--Jones
potential with phase specific depths of its energy well. During loading, edge
dislocation dipoles centred in the soft phase are introduced, which progress
towards the phase boundaries, where they pile up. Under a sufficiently high
external shear load, the leading dislocation is eventually transmitted into the
harder phase. The homogenized PN-FE model is calibrated to an atomistic model
in terms of effective elasticity constants and glide plane properties as
obtained from simple uniform deformations. To study the influence of different
formulations of the glide plane potential, multiple approaches are employed,
ranging from a simple sinusoidal function of the tangential disregistry to a
complex model that couples the influence of the tangential and the normal
disregistries. The obtained results show that, qualitatively, the dislocation
structure, displacement, strain fields, and the dislocation evolution are
captured adequately. The simplifications of the PN-FE model lead, however, to
some discrepancies within the dislocation core. Such discrepancies play a
dominant role in the dislocation transmission process, which thus cannot
quantitatively be captured properly. Despite its simplicity, the PN-FE model
proves to be an elegant tool for a qualitative study of edge dislocation
behaviour in two-phase microstructures, although it may not be quantitatively
predictive.Comment: 29 pages, 11 figures, 5 tables, abstract shortened to fulfill 1920
character limit, small changes after revie
Extended Quasicontinuum Methodology for Highly Heterogeneous Discrete Systems
Lattice networks are indispensable to study heterogeneous materials such as
concrete or rock as well as textiles and woven fabrics. Due to the discrete
character of lattices, they quickly become computationally intensive. The
QuasiContinuum (QC) Method resolves this challenge by interpolating the
displacement of the underlying lattice with a coarser finite element mesh and
sampling strategies to accelerate the assembly of the resulting system of
governing equations. In lattices with complex heterogeneous microstructures
with a high number of randomly shaped inclusions the QC leads to an almost
fully-resolved system due to the many interfaces. In the present study the QC
Method is expanded with enrichment strategies from the eXtended Finite Element
Method (XFEM) to resolve material interfaces using nonconforming meshes. The
goal of this contribution is to bridge this gap and improve the computational
efficiency of the method. To this end, four different enrichment strategies are
compared in terms of their accuracy and convergence behavior. These include the
Heaviside, absolute value, modified absolute value and the corrected XFEM
enrichment. It is shown that the Heaviside enrichment is the most accurate and
straightforward to implement. A first-order interaction based summation rule is
applied and adapted for the extended QC for elements intersected by a material
interface to complement the Heaviside enrichment. The developed methodology is
demonstrated by three numerical examples in comparison with the standard QC and
the full solution. The extended QC is also able to predict the results with 5
percent error compared to the full solution, while employing almost one order
of magnitude fewer degrees of freedom than the standard QC and even more
compared to the fully-resolved system.Comment: 36 pages, 20 figure
Learning constitutive models from microstructural simulations via a non-intrusive reduced basis method
In order to optimally design materials, it is crucial to understand the
structure-property relations in the material by analyzing the effect of
microstructure parameters on the macroscopic properties. In computational
homogenization, the microstructure is thus explicitly modeled inside the
macrostructure, leading to a coupled two-scale formulation. Unfortunately, the
high computational costs of such multiscale simulations often render the
solution of design, optimization, or inverse problems infeasible. To address
this issue, we propose in this work a non-intrusive reduced basis method to
construct inexpensive surrogates for parametrized microscale problems; the
method is specifically well-suited for multiscale simulations since the coupled
simulation is decoupled into two independent problems: (1) solving the
microscopic problem for different (loading or material) parameters and learning
a surrogate model from the data; and (2) solving the macroscopic problem with
the learned material model. The proposed method has three key features. First,
the microscopic stress field can be fully recovered. Second, the method is able
to accurately predict the stress field for a wide range of material parameters;
furthermore, the derivatives of the effective stress with respect to the
material parameters are available and can be readily utilized in solving
optimization problems. Finally, it is more data efficient, i.e. requiring less
training data, as compared to directly performing a regression on the effective
stress. For the microstructures in the two test problems considered, the mean
approximation error of the effective stress is as low as 0.1% despite using a
relatively small training dataset. Embedded into the macroscopic problem, the
reduced order model leads to an online speed up of approximately three orders
of magnitude while maintaining a high accuracy as compared to the FE
solver
A Variational Formulation of Dissipative Quasicontinuum Methods
Lattice systems and discrete networks with dissipative interactions are
successfully employed as meso-scale models of heterogeneous solids. As the
application scale generally is much larger than that of the discrete links,
physically relevant simulations are computationally expensive. The
QuasiContinuum (QC) method is a multiscale approach that reduces the
computational cost of direct numerical simulations by fully resolving complex
phenomena only in regions of interest while coarsening elsewhere. In previous
work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally
conservative QC methodology was generalized to a virtual-power-based QC
approach that includes local dissipative mechanisms. In this contribution, the
virtual-power-based QC method is reformulated from a variational point of view,
by employing the energy-based variational framework for rate-independent
processes (Mielke and Roub\'i\v{c}ek, Rate-Independent Systems: Theory and
Application, Springer-Verlag, 2015). By construction it is shown that the QC
method with dissipative interactions can be expressed as a minimization problem
of a properly built energy potential, providing solutions equivalent to those
of the virtual-power-based QC formulation. The theoretical considerations are
demonstrated on three simple examples. For them we verify energy consistency,
quantify relative errors in energies, and discuss errors in internal variables
obtained for different meshes and two summation rules.Comment: 38 pages, 21 figures, 4 tables; moderate revision after review, one
example in Section 5.3 adde
A reduced order model for geometrically parameterized two-scale simulations of elasto-plastic microstructures under large deformations
In recent years, there has been a growing interest in understanding complex
microstructures and their effect on macroscopic properties. In general, it is
difficult to derive an effective constitutive law for such microstructures with
reasonable accuracy and meaningful parameters. One numerical approach to bridge
the scales is computational homogenization, in which a microscopic problem is
solved at every macroscopic point, essentially replacing the effective
constitutive model. Such approaches are, however, computationally expensive and
typically infeasible in multi-query contexts such as optimization and material
design. To render these analyses tractable, surrogate models that can
accurately approximate and accelerate the microscopic problem over a large
design space of shapes, material and loading parameters are required. In
previous works, such models were constructed in a data-driven manner using
methods such as Neural Networks (NN) or Gaussian Process Regression (GPR).
However, these approaches currently suffer from issues, such as need for large
amounts of training data, lack of physics, and considerable extrapolation
errors. In this work, we develop a reduced order model based on Proper
Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a
geometrical transformation method with the following key features: (i) large
shape variations of the microstructure are captured, (ii) only relatively small
amounts of training data are necessary, and (iii) highly non-linear
history-dependent behaviors are treated. The proposed framework is tested and
examined in two numerical examples, involving two scales and large geometrical
variations. In both cases, high speed-ups and accuracies are achieved while
observing good extrapolation behavior
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