Progress Report: Syed Taha Khursheed Date: 17th November 2023 Summary: This report highlights the hardening models available and the issues that are existing with the hardening model in the user material subroutine. It also talks about the tests performed on high stresses as well as changing the orientation and loading direction according to the instructions provided. Hardening Issues A hardening law is a crucial part of the constitutive model of the material. It is used to control the plastic flow from becoming abnormally large. The hardening laws however, are not well defined when it comes to crystal plasticity. There are several models available which give an approximation of the hardening behaviour. The literature on crystal plasticity finite element model states that as slip gets activated, the critically resolved shear stress value increases due to hardening. Hardening Matrix: This matrix has the two components, self hardening (diagonal components) and Latent Hardening (non-diagonal components) Self Hardening: This is the hardening ratio of the slip system due to it and its identical slip systems getting activated, without effects of any other slip systems Latent Hardening: This is the hardening ratio of the slip system due to other non-identical slip systems getting activated. Since this code is being tested on the basal slip system at the moment, the hardening law for that is rather linear [1]. The Values for q are 1 for self hardening, 1.4 for latent [1]. One Element Test – High Stresses (45,45,45) The main objective of this test was to increase the stresses to 400 MPa. This was being done gradually. From 5 MPa, to 10 MPa, so on and so forth. The user material subroutine has shear strain on slip system equations which have exponents of 40. This is the main area of focus as it controls the plastic flow. A simple hardening law for the basal slip according to the literature is mentioned above. Since there is only the basal slip system, only the self hardening will be included in this test. A 100 MPa Stress was applied in 5000 load steps. The code unconverged in between. It only gave values till the Mises stress levels reached 14 MPa. After that the values became extremely large. This again points towards the hardening laws adapted. Stress vs Strain 1.60E+01 1.40E+01 1.20E+01 1.00E+01 8.00E+00 6.00E+00 4.00E+00 2.00E+00 0.00E+00 0.00E+00 2.00E-04 4.00E-04 6.00E-04 8.00E-04 1.00E-03 1.20E-03 Figure 1. Stress Strain Graph (45,45,45) Euler Angles Next steps: Discuss another hardening model in detail with Dr. Zhu. Another hardening model by Bassani and Wu is available, which is a three-stage hardening model. It follows the graph as follows. This is suitable for high stresses and strains however, upon thorough research, it was discovered that the parameters for the Basal Slip system of Magnesium needed to model this hardening behaviour are not available yet. It is mainly used by FCC and BCC models [2]. Figure 2. Bassani and Wu 3-Stage Hardening Model [2] One Element Test – (90,90,0)[1,-1,0,0] Loading Direction: Euler Angle: 90,90,0 Loading Direction: [1,-1,0,0] Loading Direction in Cartesian: [1.5,-0.866,0] (Obtained from Stabix) Figure 3. One Element Test Ambiguities: The ANSYS APDL lets one apply the stresses on planes. Since this is a one element test, the planes were only the x,y,z. Hence, this loading direction was not possible just with the stresses. For this, instead of stress, forces were applied on the top 4 nodes of the element. The equation for the force that would give loading in this direction is presented below. P is the increment of force applied. 𝐹 = 1.5𝑃 𝑥 − 0.866𝑃 𝑦 P = 0 – 3 mega N Element size = 1x1x1 Corresponding Applied Stress = 0 – 3 MPa As per the instructions of Dr. Zhu, the tests were performed on the element in different directions to see the functionality of the code. The boundary conditions of this test were kept the same as all the previous ones. Results: The results of this test were unsatisfactory. They did not show any plasticity or yielding. Potential causes of failure: The loading conditions might not have been well defined to initiate a plastic response. Eventhough the stresses were exceeding the CRSS value of the basal slip, there were no signs of plasticity. When the loading was increased to force a plastic response, the code stopped responding. Stress vs Strain 3.00E+00 2.50E+00 2.00E+00 1.50E+00 1.00E+00 5.00E-01 0.00E+00 0.00E+005.00E-061.00E-051.50E-052.00E-052.50E-053.00E-053.50E-054.00E-054.50E-05 Figure 4. Stress Strain Graph (90,90,0) [1,-1,0,0] LD References [1] Xi, BL., Fang, G. Crystal plasticity behavior of single-crystal pure magnesium under planestrain compression. Rare Met. 36, 541–549 (2017). https://doi.org/10.1007/s12598-016-0856-7 [2] Bassani, John L., and Tien-Yue Wu. “Latent Hardening in Single Crystals II. Analytical Characterization and Predictions.” Proceedings: Mathematical and Physical Sciences, vol. 435, no. 1893, 1991, pp. 21–41. JSTOR, http://www.jstor.org/stable/52000. Accessed 18 Nov. 2023.
