A finite difference (FD) scheme has been developed to solve the transport equation numerically for the dislocation density tensor and has been coupled to the commercial finite element software ABAQUS v6.13-1 through a user-defined material subroutine (UMAT).
The UMAT outputs the stress state in the current configuration and the stiffness tangent ma- trix. These are calculated from the material constitutive descriptions which must be specified. At the start of each increment input to the routine includes: the increment time ∆t, element tempera- ture T , element deformation gradient tensor at the start of the increment F
0, element deformation
gradient tensor at the end of the increment F1 and the range of chosen state variables.
In this setup the model requires a state variable for the shear rate ˙γ to be calculated for each increment. This variable can be used within the CP calculations to define the plastic deformation state. A bespoke FORTRAN code was created to handle the FD scheme for dislocation transport; this code was called through a second routine, URDFIL, which can access the state variable values of all elements at the end of each step.
Within this routine the FDM code takes the current increment stress state as an input, allow- ing the dislocation state to evolve for a single timestep and the shear rate to be calculated. The shear rate is saved to the COMMON memory block which is shared between subroutines and is available for the UMAT to call at the start of the next increment. The process is mapped out graphically in Fig. 5.2.1.
As well as the shear rate the COMMON block also carries other variables from the FDM code which are not of direct use in calculations, but which may be of interest when visualised using the ABAQUS viewer (i.e dislocation density, source location etc.). These are saved as state variables on entering the UMAT on the next increment.
Figure 5.2.1: Flow chart for a single time increment of the FE-FDM model.
5.2.1 RVE model and boundary conditions
In the present study RVEs subject to plane strain conditions have been set up with ABAQUS. The domain size and shape was chosen to match the grains of a disc nickel-based superalloy, which typically lie in the range 5-50 µm [13][129], with the average grain shape being equiaxed. A square domain was deemed a reasonable 2D approximation, and side lengths in this work range from 5-30 µm.
The grain was meshed using equally sized tetragonal plane strain elements with reduced inte- gration points (of type CPEG4R). This element type was appropriate for 2D displacement analysis using a UMAT. The enhanced hourglass control and discontinuous analysis options were used to assist in convergence.
With attention to the typical trade-off between element density and processing time, a mesh was selected that would resolve both secondary particles and slip bands but still remain coarse enough
to run simulations to a reasonable strain (∼ 5%) in a cluster wall time of less than fourteen days. A typical slip band width of approximately 100nm has been observed in nickel-based superalloys during the SEM analysis of tensile [130] or compressive [131] tests, and previous CDD simulations have used 100nm as a slip band width for simulations of aluminium crystals [132]. As such a 100nm grid division was employed for the presented simulations, such that a 5 µm grain would contain 2,500 elements.
The following boundary conditions were imposed on the computational domain to reproduce a simple shear mode of deformation: for nodes along y = 0 the displacements u1= u2= 0; for nodes
along y = Y0 the displacement u2 = 0 and the imposed velocity V1app, where X0 and Y0 are the
lateral and vertical side lengths of the domain respectively. The strain rate is given by ˙ = V
app 1
Y0
. This is depicted in Fig. 5.2.2 for Y0 = X0= 10µm.
Figure 5.2.2: Boundary conditions applied within Abaqus.
The counterpart nodes of the left- and right-hand sides, where x = 0 and x = X0, were linked via
a Periodic Boundary Condition (PBC) for displacement. This ensured homogeneous stress fields during elastic deformation by preventing stress concentrations around the pinned corner nodes. In Abaqus this setup is achieved via an intermediary Reference Point node, to which the displacement of both counterpart nodes is linked using an equation constraint.
Figure 5.2.3: Indication of the FDM simulation plane within the standard Crystal reference system
Crystallographically, the planes of both the Abaqus domain and the FDM domain are identical and regard a surface described by the plane normal [112]. This vector is also the tangent vector for the straight edge-type dislocations within the simulations, meaning an element of dislocation density represents a quantity of dislocations viewed end on. The x-axis of the plane is aligned with the slip direction [110] and the y-axis is aligned with the slip plane normal [111]. This is useful in that it confines glide and climb only to the x and y directions respectively (covered in Section 5.3.1). Fig. 5.2.3 illustrates this arrangement more clearly, with the simulation plane depicted as the pink square membrane through which dislocations can penetrate.
Stiffness constants were required to be expressed within the simulation reference frame. To ac- complish this a rotation matrix R was calculated
R = −√1 2 1 √ 3 1 √ 6 1 √ 2 1 √ 3 1 √ 6 0 √1 3 − 2 √ 6 (5.2.1)
This matrix was applied to the elastic stiffness constants C so that the tensor could be expressed within the desired orientation as C∗. This requires the operation
C∗ = R · R · C · RT · RT (5.2.2) The C∗ constants were used for all elastic calculations within the UMAT.