4. Expression of the tangent matrix in the global base#

The paragraph § 3 allows you to build the consistent tangent matrix in the main base, noted \(T\). It is now appropriate to bring this matrix back into the global (Cartesian) base, which will be noted \(\stackrel{̄}{T}\).

4.1. Introductory remark#

It should be noted that the construction of this consistent tangent matrix is a crucial step both for the robustness and the performance of the algorithm:

  • First, it is well known that such a matrix allows for a quadratic convergence rate for the Newton process;

Second, this matrix reports the rotation of the main directions during an increment. Without it, the formulation of Rankine’s law in terms of the main constraints described in paragraph**§*1 would not be complete, since the main constraints, kept fixed during the local integration of the law (§ 2), could not operate at the global level of the structure.

In this section, we describe in detail the method for building \(\stackrel{ˉ}{T}\) from \(T\). In the appendix (§ 5), we will find the elements of theory necessary for the transformation of tensor quantities from one base to another.

4.2. Application to the case of Rankine#

The transposition of the formulas in Appendix § 5 to digital implementation requires some clarification. First of all, we have the following correspondence:

  • \(X={\widehat{\mathrm{\epsilon }}}^{\mathit{pred}}\) and \({x}_{\alpha }={\epsilon }_{\alpha }^{\mathit{pred}}\);

  • \(Y={\widehat{\mathrm{\sigma }}}^{\text{+}}\) and \({y}_{\alpha }={\sigma }_{\alpha }^{\text{+}}\);

  • \({E}_{\mathrm{\alpha }}={\widehat{d}}_{\mathrm{\alpha }}^{\mathit{pred}}\otimes {\widehat{d}}_{\mathrm{\alpha }}^{\mathit{pred}}\);

:math:`{left(Tright)}_{alpha beta }=frac{partial {y}_{alpha }}{partial {x}_{beta }}` is the consistent tangent matrix in the main base calculated in paragraph**§* 3;

The notation \({}^{\mathit{pred}}\) indicates that we are working with « predicted » quantities given as input by the Newton process, the notation \({}^{\text{+}}\) with quantities resulting from the local resolution of the law of behavior, and the notation \(\widehat{\text{}}\) with the Voigt base. Note that the main directions predicted \({\widehat{d}}_{\mathrm{\alpha }}^{\mathit{pred}}\) are fixed during local resolution, which is consistent with the isotropy hypothesis adopted (see the explanations in paragraph § 5).

Having all this information at the end of the local resolution of the law of behavior, we deduce the consistent tangent matrix \(\stackrel{ˉ}{T}\mathrm{=}\stackrel{ˉ}{D}\) expressed in the projection base \(\stackrel{ˉ}{b}\) defined in paragraph § 6 :

  • The equation () in the two-dimensional case in plane constraints (C_ PLAN);

  • The equation () in the two-dimensional case in plane deformation (D_ PLAN) or axisymmetric (AXIS);

  • The equation () in the three-dimensional (3D) case;

The second important piece of information concerns the writing convention of the various tensors. Indeed, for the sake of generality, the notation used for tensors throughout paragraph § 4 is the classical notation, causing tensors up to order four to appear. This writing is unsuitable for numerical resolution, where we prefer to use condensed notations made possible by the fact that we work with symmetric tensors of order two (constraints and deformations always are). There are two forms of condensed notations corresponding to two projection bases (see § 6):

  • The orthonormal basis \(\stackrel{ˉ}{b}\) for symmetric tensors of order two. It is in this database that the constraints and deformations at the input and output of the local resolution of behavior are given;

  • The so-called Voigt \(\widehat{b}\) database, much more convenient to use when solving the local numerical behavior because it avoids having to manipulate coefficients in \(\sqrt{2}\) during matrix operations;

4.3. Scheme for resolving the law of behavior#

Entries:

  • Constraints expressed in the global database \({\overline{\mathrm{\sigma }}}^{-}\);

  • Deformation increment expressed in the global base \(\mathrm{\Delta }\overline{\mathrm{\epsilon }}\);

Calculations:

  • We evaluate the elastic stresses \({\widehat{\mathrm{\sigma }}}^{e}\) and the elastic deformations \({\widehat{\mathrm{\epsilon }}}^{e}\) in the Voigt base;

  • We transform them in the main base, we get \({\mathrm{\sigma }}^{e}\) and \({\mathrm{\epsilon }}^{e}\);

  • We integrate the law of behavior and we obtain the plastic deformation increment \(\mathrm{\Delta }{\mathrm{\epsilon }}^{p}\) and the stresses in the Voigt base \({\widehat{\mathrm{\sigma }}}^{+}\);

  • Calculation of the tangent matrix in the main base:

    • \(\left({\widehat{\mathrm{\sigma }}}^{+},{\widehat{\mathrm{\epsilon }}}^{e}\right)\to {\widehat{E}}_{\mathrm{\alpha }}\) then \({T}_{\mathrm{\alpha }\mathrm{\beta }}=\frac{{\partial \mathrm{\sigma }}_{\mathrm{\alpha }}}{{\partial \mathrm{\epsilon }}_{\mathrm{\beta }}}\)

  • Transfer of the tangent matrix into the Voigt base:

    • \(\left({\widehat{\mathrm{\sigma }}}^{+},{\widehat{\mathrm{\epsilon }}}^{e},{\widehat{E}}_{\mathrm{\alpha }},T\right)\to {\widehat{T}}_{\mathit{ij}}=\frac{{\partial \mathrm{\sigma }}_{i}}{{\partial \mathrm{\epsilon }}_{j}}\)

  • Transfer of the tangent matrix into the global base:

    • \(\widehat{T}\to \overline{T}\)