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