Although Abaqus finite element analysis suite is not a open source program, there are a lot user subroutines through which you can enhance the function of Abaqus. Similarly to FEAPpv, you can write a Fortran subroutine to add a new constitutive model to the main Abaqus program. To get started, you shall study the example user subroutines provided in the Abaqus documentation. For example, the Neo-Hookean material model has been provided as an example in Abaqus documentation. You can download a copy of the source code for Neo-Hookean material model online.
In Abaqus, user subroutines are provided for specific purposes. For example, if an isotropic hyperelastic material model is to be implemented, then the user subroutine interface
UHYPER should be used. The user subroutine interface
UMAT should be used only if you are not able to use all other material subroutine interfaces, see the Abaqus user’s manual on UMAT for more information.