For elastic constants you need to strain your crystal and calculate the total-energy response to the strain. The nature of the strain dictates the specific elastic constant or a linear combination of elastic constants. For VASP the structure is defined in the POSCAR file. For the specific strains you can use what has been published. For example for cubic: Phys. Rev. B 48, 5844 (1993); hexagonal: Phys. Rev. B 51, 17431 (1995); orthorhombic: Phys. Rev. B 66, 085113 (2002); monoclinic: Phys. Rev. B 79, 104110 (2009). Hope this helps.

thanks for you suggestion, i will follow it,

You can also check the ElaStic code (http://exciting-code.org/elastic), which can generate suitable strains and analyze the results for any structure. There is no VASP interface in the official version, but you can look in their paper about it for more information Computer Physics Communications 184 (2013) 1861–1873, and you can also find out more about their method for generating strains in the original reference: Comput. Phys. Comm. 181 (3) (2010) 671–675.

Edit: Also beware that you need tighter than normal convergence parameters for accurate elastic constants. With VASP, I would go for

PREC = Accurate

ADDGRID = .True.

EDIFF = 1e-6

EDIFFG = -0.002

Remember to check your k-point convergence carefully.

this is my INCAR file

where is the problema?

system = Li2TiO3-15 relaxation calculation

PREC = High

ISTART = 0 # job : 0-new 1-cont 2-samecut

ICHARG = 0  ## charge from orbitals

ISMEAR = -5 #( ISMEAR=-5 ). This method also gives a good account for the electronic density of states (DOS)

ENCUT = 550 # energy cut-off for the calculation

IBRION = 6  # calculate th, 2 use CG algorithm

EDIFF = 5E-7 # stopping-criterion for ELM

EDIFFG = 5E-6 #threshold for forces and stress tensor,stopping-criterion for IOM

SIGMA = 0.05 # broadening in eV -4-tet -1-fermi 0-gaus

ISPIN  = 1    #spin polarized calculation (2-yes 1-no)

NSW    = 3   #number of steps for ionic upd

ISIF = 3

ADDGRID = .TRUE.

NGX = 45

NGY = 80

NGZ = 48

RWIGS = 1.376 0.82 1.323

LREAL = .FALSE.

LWAVE = .FALSE. # Do not write WAVECAR

LCHARG= .FALSE. # Do not write CHGCAR, CHG

LVPOT = .FALSE. # Do not write LOCPOT

LVHAR = .FALSE. # Do not write LOCPOT

NELMIN = 3

NFREE = 2

POTIM = 0.02

Your INCAR file seems good.

If you don't obtain in OUTCAR file the elastic constants,

add in INCAR file next command

SYMPREC=1e-7 # positions in the POSCAR file can be less accurate, easier find symmetry

which was useful in my case.

What are the relevant or good INCAR file for PHONON Despersion and combined elastic properties and density of states?

For PHONON calculations, a high converged structure has to be used as input. EDIFF=1e-8; EDIFFG=-1e-6