Abbreviations
List of abbreviations:
d: number of space dimensions (2D, 3D)
n: number of nodes
ips: number of integration points
R: (d, 1)-Matrix of reference space coordinates
X: (d, 1)-Matrix of real space coordinates
H: (n, 1)-Matrix of shape functions
dH_dR: (n, d)-Jacobian matrix of H with respect to R: \(dh\_dr_{ik} = \partial h_{i} / \partial r_{k}\)
dX_dR: (d, d)-Jacobian matrix of X with respect to R: \(dx\_dr_{ik}= \partial x_{i} / \partial r_{k}\)
dH_dX: (n, d)-Jacobian matrix of H with respect to X: \(dh\_dx_{ik} = \partial h_{i} / \partial x_{k}\)
U: (d,)-Vector of displacements in the real space
U_at_nodes: (n, d)-Matrix of displacements in the real space at the nodes
dU_dX: (d, d)-Jacobian matrix of U with respect to X: \(du\_dx_{ik} = \partial u_{i} / \partial x_{k}\)
S: (d, d)-symmetric Cauchy stress tensor
F: (d, d)-Deformation gradient
P: (d, d)-First Piola-Kirchhoff stress tensor
e: Helmholtz free energy density
CS: (d, d)-Matrix of configurational stresses. Implemented formulations:
mbf: motion based formulation (Eshelby’s formulation)
dbf: displacement based formulation
CF: (d,)-Vector of a configurational force in the real space
CF_at_nodes: (n, d)-Matrix of configurational nodal forces in the real space at the nodes