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This paper describes a scientific workflow for ab initio calculations of elastic coefficients (Cij) of crystalline materials implemented in the VLab Cyberinfrastructure [da Silveira et al., 2008]. This workflow has recently been upgraded to treat crystals of all symmetries and integrated in the XSEDE. First we review the underlying Cij calculations and list explicitly different requirements for each Bravais lattice. We also describe the workflow management and its general method for handling actions. We illustrate the Cij application with a calculation of diamond's elastic coefficients at high pressures. We conclude with an outlook of future implementation plans.