Hello everyone, I want to calculate elastic constants (C11, C22, C12, and C66) for a single layer graphene sheet by MD simulation in LAMMPS. I am using the following equation:
E = (C11*C22-C12*C21)/C22 and Poisson's ratio = C12/C22 from this paper:
http://www.sciencedirect.com/science/article/pii/S0261306909003082
My strain in y and z direction is zero. I calculated sigma11, and sigma22 and tau12 in x, y and xy direction from LAMMPS by compute stress/atom command. The equations to solve from the stress-strain relationship would be:
sigma11 = C11*epsilon11 + C22*epsilon22
sigma22 = C12*epsilon11 + C22*epsilon22
tau12 = C66*gamma12
But as, epsilon22 = 0, sigma reduces to sigma11 = C11*epsilon11 and from this equation we can calculate C11.
My question is, what is the actual value of sigma11 and epsilon11 to be used here. If they are fracture stress and strain, then my C11 = 170 Gpa. If, I take very low values of strain, for example 0.04, then C11 = 700 Gpa. But it should be 977 Gpa as the paper says.
Am I understanding the calculation correctly? I can share my input file here if it is required.
Dear, Baig Abdullah Al Muhit,
stiffnesses are linear characteristics. They are defined for infinitesimal (very small) strains. Practically small mean that the stress-strain diagram should be linear. I would recommend epsilon ~ 0.01-0.001.
In order to measure elastic constants you could, for example, specify strain (epsiln11) and measure stress (sigma11). Then substitute the results into the Hooke's law (your system of equations). You may find this paper useful - https://www.researchgate.net/publication/298395624_Lattice_with_vacancies_elastic_fields_and_effective_properties_in_frameworks_of_discrete_and_continuum_models
Section 4 contains description of the similar problem.
Best regards,
Vitaly
Article Lattice with vacancies: Elastic fields and effective propert...
Hello Kuzkin,
Thank you for sending me your paper. I have approximately constant value of E, mu and lame's constant at 0.01-0.02 strain. Below 0.01, poisson ratio fluctuates quite a lot. And I found out the calculation of elastic constants after I had sent the question. Thanks a lot for replying. I found out that until 0.04 strain, my graphene sheet have linear behavior.
I think your equations are wrong.
They should be:
nu = c12/(c11+c12)
E = (c11^2 + c11 * c12 - 2c12^2)/(c11 + c12)
from Schultz, Jerold M., ed. Properties of Solid Polymeric Materials: Treatise on Materials Science and Technology. Vol. 10. Elsevier, 2017 for example.
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