A scale transition plasticity theory - from crystal to continuum
Citation for published version (APA):Poh, L. H., Peerlings, R. H. J., & Geers, M. G. D. (2010). A scale transition plasticity theory - from crystal to continuum. Poster session presented at Mate Poster Award 2010 : 15th Annual Poster Contest.
Document status and date: Published: 01/01/2010 Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne
Take down policy
If you believe that this document breaches copyright please contact us at:
openaccess@tue.nl
providing details and we will investigate your claim.
/w
Mechanics of Materials
1.
Introduction
Gradient crystal plasticity models take into account the ad-ditional work done by the geometrically necessary disloca-tions (GNDs). However, these models are not efficient when solving for large problems. We developed an 1D continuum gradient plasticity model by homogenizing the (microscopic) work done in a single-slip crystal.
2.
Homogenization theory
Polycrystalline metals compose of grains separated by grain boundaries, with the interior of grains having similar struc-ture as that of a single crystal. In the continuum model, each macroscopic point represents the averaged response of a single crystal centered atx with micro-coordinates y, as depicted in Fig1.
Figure 1: Continuum model with unit cells of lengthL
The plastic slip within the crystal (γ) is decomposed into a slow-varying component (˜εp) and its gradient (∂ ˜ε∂xp) as well as a fluctuation functionf(y) and its amplitude g(x), shown schematically in Fig2.
γ(x + y) = ˜εp(x) +∂ ˜εp(x)
∂x y + g(x)f(y)
Figure 2: Decomposition of microscopic plastic slipγ
This expression forγ is substituted into the single crystal plasticity model by [1]. The continuum power expenditure is derived by homogenizing the microscopic free energy and dissipation. The homogenized micro-force balance is given by:
(Ga + 2μ)˜εp− (Gl2+μL2
2 )∇2˜εp= Gaεp
whereG is the shear modulus, a = kLl22 (k is a fluctuation
parameter),μ is the resistance modulus at grain boundaries andl is a length scale parameter. This expression has the same form as the "implicit" gradient formulation which is widely used in softening models [2].
3.
Influence of grain boundaries
The grain boundary resistance plays a major role in the ma-terial response. This is easily observed in an uniform defor-mation where the gradient term vanishes. The stress-strain graphs of a rigid plastic material for two extreme cases, the micro-free (μ → 0) and the micro-hard (μ → ∞) boundary conditions are shown in Fig3.
Figure 3: Stress-strain graphs (σ0= yield stress,H = hardening modulus)
References
[1] Cermelli, P., Gurtin, M. E., 2002. Geometrically nec-essary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations. Int J Solid Struct 39, 6281-6309.
[2] Peerlings, R. H. J., De Borst, R., Brekelmans, W. A. M., De Vree, J. H. P., 1996. Gradient enhanced damage for quasi-brittle materials. Int J Numer Methods Eng 39, 3391-3403.
