Caveats on formulating finite elasto-plasticity in curvilinear coordinates

Abstract

Tensor analysis provides a frame-invariant foundation for continuum mechanics, yet numerical implementations rely on matrix representations expressed in user-selected bases. When these bases are non-Cartesian and non-orthonormal, additional terms arise that are normally absent or implicit in Cartesian formulations. Using cavity expansion as an initial model problem, this paper clarifies the roles of the deformation gradient, Jacobian, and shifter in finite-strain kinematics under axisymmetry. These quantities, typically straightforward in Cartesian frames, require more careful treatment in curvilinear coordinates, particularly in applications involving large deformations where axisymmetric reductions provide substantial computational savings. The formulation is further complicated when anelastic effects are included: the elastic and anelastic components of the deformation gradient and Jacobian must be distinguished, and the Cauchy stress depends on configuration changes beyond the current elastic state. This increases the complexity of the consistent linearisation required for finite element implementation. The paper provides a clear, step-by-step procedure for handling these contributions. The focus is practical rather than geometric. Instead of adopting a differential-geometric manifold framework, we work within standard Cartesian representations and use explicit changes of basis to obtain the required curvilinear forms. The resulting methodology enables robust finite element analysis of axisymmetric elasto-plastic problems undergoing finite strains.

Figures (8)

• Figure 1: Cross section of the cavity expansion problem and relative quantities of interest. The supports shown permit rotational and translation in one direction.
• Figure 2: Illustration of body $\mathcal{B}$ in the reference and the current configuration. Euclidean space is described via Cartesian common basis vectors ${^0\underline{\boldsymbol{e}}_J} ={^t\underline{\boldsymbol{e}}_j}$ tangent to coordinate lines (grey solid lines). Curvilinear basis vectors ${^0\underline{\boldsymbol{g}}_A}$, ${^t\underline{\boldsymbol{g}}_a}$ are defined as tangent to the coordinate curves at two generic points in the original and current configuration (light blue solid lines).
• Figure 3: Illustration of the initial positions of the point $P$ in the Cartesian ${^0\underline{\boldsymbol{z}}}$ and curvilinear reference ${^0\underline{\boldsymbol{x}}}$ systems. The point $p = \phi \left( P \right)$ is defined via the current positions ${^t\underline{\boldsymbol{z}}}$ and ${^t\underline{\boldsymbol{x}}}$ in the Cartesian and coordinate systems, respectively. The displacement ${^t_0}\underline{\boldsymbol{U}}$ can be defined only as the difference between Cartesian position vectors. Points $P,Q$ belong to the material curve $\mathscr{C}$, which, mapped via $\phi$, result in the points $p,q$ belonging $\mathscr{c}$. The magnification shows a tangential portion of material curves defined by $P$ and $Q$ in the original configuration and how the deformation gradient describes its first-order approximation under the mapping $\phi$ in the current configuration.
• Figure 4: Three-dimensional solid ${^0}\mathcal{B}$, obtained by revolving the in-plane cross section ${^0}\mathcal{B}^{ax}$ around the $Z$-axis. Boundaries ${^0}\tilde{\Gamma}$ and ${^0}\Gamma$ of the in-plane section are also illustrated.
• Figure 5: Three-dimensional setup of the considered hollow cylinder. Displacements varying along $Z$ on the internal wall are applied.