| 1. |
- Carlsson, Kristoffer, 1989, et al.
(author)
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A Comparison of Computational Formats of Gradient-Extended Crystal Viscoplasticity in the Context of Selective Homogenization
- 2016
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In: Eccomas, 5-10 juni, Kreta, 2016 (1 page abstract).
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Conference paper (peer-reviewed)abstract
- Crystal (visco)plasticity is the accepted model framework for incorporating microstructural information incontinuum theory with application to crystalline metals, where dislocations constitute the physicalmechanism behind inelastic deformation. In order to account for the size effects due to the existence ofgrain boundaries in a polycrystal, it is convenient to include some sort of gradient-extension of the flowproperties along the slip directions, either in the dragstress or backstress (from GND’s). Various explicitmodels based on this conceptual background have been proposed, e.g. Gurtin et al.[1], Gottschalk et al.[2];however, several modeling issues still await its resolution. A comprehensive unifying account of gradienttheory for a variety of application models was presented by Miehe[3]. When applied to a polycrystal, it isdesirable that the homogenization strategy will result in a standard continuum format on the macroscale,whereas micro-stresses are confined to the mesoscale and and automatically "suppressed" during theprocedure of (selective) homogenization. This can be achieved within a fairly general setting of variationallyconsistent homogenization. In this contribution we focus on issues related to the computational format ofgradient-extended crystal viscoplasticity that constitutes the RVE-problem. A few different variationalformats are thereby investigated. The so-called “primal” format exploits the slip on each slip system togetherwith the displacement field as the unknown global fields. An alternative format is coined the “semi-dualformat”, in which the slip variables are replaced by the microstresses as global fields, thereby defining amixed variational problem. For both the primal and semi-dual formulations, we establish variationalprinciples for the time incremental FE-problems which ensure symmetry of the corresponding tangentproblems. We note that a mixed method that bears strong resemblance with the semi-dual format has beenused extensively in our research group in recent years, e.g. Bargmann et al.[4]; however, without possessing awell-defined variational structure. We compare the primal and semi-dual variational formats in terms of prosand cons from various aspects. We also discuss the pertinent FE-spaces that appear as the natural/possiblechoices and assess the computational efficiency of the FE-approximations with the aid of numericalexamples pertaining to a single crystal as well as to a polycrystal in the homogenization context.References:[1] M. E. Gurtin, L. Anand, S. P. Lele, Journal of the Mechanics and Physics of Solids, 55, 1853, (2007)[2] D. Gottschalk, A. McBride, B.D. Reddy, A. Javili, P. Wriggers, C.B. Hirschberger, ComputationalMaterial Science, 111, 443, (2016)[3] C. Miehe, J. Mech. Phys. Solids, 59 898, (2011)[4] S. Bargmann, M. Ekh, K. Runesson, B. Svendsen, Philosophical Magazine, 90, 1263, (2010)
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| 2. |
- Carlsson, Kristoffer, 1989, et al.
(author)
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A comparison of the primal and semi-dual variational formats of gradient-extended crystal inelasticity
- 2017
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In: Computational Mechanics. - : Springer Science and Business Media LLC. - 1432-0924 .- 0178-7675. ; 60:4, s. 531-548
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Journal article (peer-reviewed)abstract
- In this paper we discuss issues related to the theoretical as well as the computational format of gradient-extended crystal viscoplasticity. The so-called primal format uses the displacements, the slip of each slip system and the dissipative stresses as the primary unknown fields. An alternative format is coined the semi-dual format, which in addition includes energetic microstresses among the primary unknown fields. We compare the primal and semi-dual variational formats in terms of advantages and disadvantages from modeling as well as numerical viewpoints. Finally, we perform a series of representative numerical tests to investigate the rate of convergence with finite element mesh refinement. In particular, it is shown that the commonly adopted microhard boundary condition poses a challenge in the special case that the slip direction is parallel to a grain boundary.
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| 3. |
- Carlsson, Kristoffer, 1989, et al.
(author)
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Bounds on the effective response for gradient crystal inelasticity based on homogenization and virtual testing
- 2019
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In: International Journal for Numerical Methods in Engineering. - : Wiley. - 0029-5981 .- 1097-0207. ; 119:4, s. 281-304
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Journal article (peer-reviewed)abstract
- This paper presents the application of variationally consistent selective homogenization applied to a polycrystal with a subscale model of gradient-enhanced crystal inelasticity. Although the full gradient problem is solved on Statistical Volume Elements (SVEs), the resulting macroscale problem has the formal character of a standard local continuum. A semi-dual format of gradient inelasticity is exploited, whereby the unknown global variables are the displacements and the energetic micro-stresses on each slip-system. The corresponding time-discrete variational formulation of the SVE-problem defines a saddle point of an associated incremental potential. Focus is placed on the computation of statistical bounds on the effective energy, based on virtual testing on SVEs and an argument of ergodicity. As it turns out, suitable combinations of Dirichlet and Neumann conditions pertinent to the standard equilibrium and the micro-force balance, respectively, will have to be imposed. Whereas arguments leading to the upper bound are quite straightforward, those leading to the lower bound are significantly more involved; hence, a viable approximation of the lower bound is computed in this paper. Numerical evaluations of the effective strain energy confirm the theoretical predictions. Furthermore, heuristic arguments for the resulting macroscale stress-strain relations are numerically confirmed.
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| 4. |
- Carlsson, Kristoffer, 1989, et al.
(author)
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COMPUTATIONAL ISSUES OF GRADIENT-EXTENDED CRYSTAL INELASTICITY
- 2016
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In: NSCM, 26 - 28 October, Göteborg, 2016 4 page extended abstract.
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Conference paper (peer-reviewed)abstract
- . In this paper we discuss issues related to the theoretical as well as the computationalformat of gradient-extended crystal viscoplasticity. The so-called “primal”format uses the internal variables and the displacements as the primary unknown fields.An alternative format is coined the “semi-dual format”, which in addition includes microstresses,thereby defining a mixed variational problem. We compare the primal andsemi-dual variational formats in terms of pros and cons from a modeling as well as anumerical viewpoint. We perform a set of numerical benchmarks to investigate the rateof convergence for errors in different norms.
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| 5. |
- Carlsson, Kristoffer, 1989, et al.
(author)
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Computational modeling issues of gradient-extended viscoplasticity
- 2015
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In: Svenska Mekanikdagar, 10-12 juni, Linköping, 2015 (1 page abstract).
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Conference paper (peer-reviewed)abstract
- Crystal (visco)plasticity is the accepted model framework for incorporating microstructural in-formation in continuum theory with application to crystalline metals where dislocations constitute thephysical mechanism behind inelastic deformation. In order to account for the size effects due to the ex-istence of grain boundaries in a polycrystal, it is convenient to include some sort of gradient-extensionof the flow properties along the slip directions, either in the dragstress or backstress (from GND, whichare generally of two types: edge and screw dislocations). Various explicit models based on this con-ceptual background have been proposed, not the least by Gurtin and coworkers 1 ; however, severalmodeling issues still await its resolution. An elegant way of unifying gradient theory for differentapplication models was presented by Miehe 2 .In this contribution we focus on issues related to the theoretical as well as the computationalformat, while (for the sake of clarity) restricting to gradient-extended viscoplasticity for a standardcontinuum. Thereby, we avoid the additional complications associated with the proper version ofcrystal (visco) plasticity, such as higher order boundary conditions. The so-called “primal” formatexploits the internal variables as the primary unknown field together with the displacement field. Analternative format is coined the “semi-dual format”, which exploits (in addition) the microstresses,thereby defining a mixed variational problem. We note that a mixed method that bears resemblancewith the semi-dual format has been used extensively in our research group in recent years 3 ; however,without possessing a well-defined variational structure.We compare the primal and semi dual variational formats in terms of pros and cons from variousaspects. We also discuss the pertinent FE-spaces that appear as the natural/possible choices. In partic-ular, for the semi-dual format we investigate the possibility to use a minimal degree of regularity thathas so far not been discussed in the literature.
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| 6. |
- Carlsson, Kristoffer, 1989
(author)
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On computational homogenization of gradient crystal inelasticity
- 2017
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Licentiate thesis (other academic/artistic)abstract
- Crystal inelasticity is the main modeling technique employed to study the mechanical behavior of polycrystalline metallic materials. This class of models has the capability to represent micromechanical phenomena such as plastic slip, grain boundary interactions, and dislocation pile-up. An extended class of models, known as gradient crystal inelasticity, can also be used to predict size dependence of the grains, which is a property that has been observed experimentally. An increased understanding, in terms of the challenges in modeling polycrystalline materials, could aid in reducing costs and resources needed to determine the properties of these type of materials experimentally.The length scales that are characteristic for typical engineering applications and the length scale of the underlying microstructure often differ by many orders of magnitude. As a consequence, it is not computationally feasible to fully resolve the model at a fine enough scale to capture the microstructural characteristics. Instead, computational homogenization is a suitable framework for modeling structures exhibiting these scale separations. Homogenization allows for bridging the microstructural to the effective properties that pertain to the (structural) scale of engineering interest.In this work, different modeling aspects of gradient crystal inelasticity and their modeling capabilities, in a computational homogenization setting, are investigated. In particular, two variational formats are compared, specifically in terms of convergence rate with respect to mesh refinements, and the effect of applying certain boundary conditions. Furthermore, it is shown that certain effective properties (properties for sufficiently large microscopic models, called Representative Volume Elements) can be bounded from above and below based on simulations performed on finite size models (Statistical Volum Elements), that are amenable to simulation. The bounding property can be used towards estimating how large microscopic models that are needed to produce accurate results in the computational homogenization analysis. Several numerical examples, applied to both two and three-dimensional models, are given, demonstrating the validity of the theoretically made predictions.
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| 7. |
- Carlsson, Kristoffer, 1989, et al.
(author)
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ON THE PRIMAL AND MIXED DUAL FORMATS IN VARIATIONALLY CONSISTENT COMPUTATIONAL HOMOGENIZATION WITH EMPHASIS ON FLUX BOUNDARY CONDITIONS
- 2020
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In: International Journal for Multiscale Computational Engineering. - 1543-1649. ; 18:6, s. 651-675
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Journal article (peer-reviewed)abstract
- In this paper, we view homogenization within the framework of variational multiscale methods. The standard (primal) variational format lends itself naturally to the choice of Dirichlet boundary conditions on the Representative Volume Element (RVE). However, how to impose flux boundary conditions, treated as Neumann conditions in the standard variational format, is less obvious. Therefore, in this paper we propose and investigate a novel mixed variational setting, where the fluxes are treated as additional primary fields, in order to provide the natural variational environment for such flux boundary conditions. This mixed dual formulation allows for a conforming implementation of (lower bound) flux boundary conditions in the framework of discretization-based homogenization. To focus on essential features, a very simple problem is studied: the classical stationary linear heat equation. Furthermore, we consider the standard context of model-based homogenization (without loss of generality), since we are only concerned with the RVE problem and merely assume that the relevant macroscale fields are properly prolonged. Numerical results from the primary and the mixed dual variational formats are compared and their convergence properties for mesh finite element (FE) refinement and RVE size are assessed.
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| 8. |
- Carlsson, Kristoffer, 1989, et al.
(author)
-
Tensors.jl — Tensor Computations in Julia
- 2019
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In: Journal of Open Research Software. - : Ubiquity Press, Ltd.. - 2049-9647. ; 7:1
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Journal article (peer-reviewed)abstract
- Tensors.jl is a Julia package that provides efficient computations with symmetric and non-symmetric tensors. The focus is on the kind of tensors commonly used in e.g. continuum mechanics and fluid dynamics. Exploiting Julia’s ability to overload Unicode infix operators and using Unicode in identifiers, implemented tensor expressions commonly look very similar to their mathematical writing. This possibly reduces the number of bugs in implementations. Operations on tensors are often compiled into the minimum assembly instructions required, and, when beneficial, SIMD-instructions are used. Computations involving symmetric tensors take symmetry into account to reduce computational cost. Automatic differentiation is supported, which means that most functions written in pure Julia can be efficiently differentiated without having to implement the derivative by hand. The package is useful in applications where efficient tensor operations are required, e.g. in the Finite Element Method.
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