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A finite strain elastoplastic model based on Flory’s decomposition and 3D FEM applications


The Flory’s decomposition is an important mathematical tool used to write hyperelastic constitutive models. As far as the author’s knowledge goes, it has not been used to write plastic flow directions in elastoplastic models and this study is an opportunity to introduce this simple strategy in so important subject. Adopting this decomposition it is possible to write an alternative total Lagrangian elastoplastic framework for finite strains with simple implementation and good response. Using Flory’s decomposition, strains are split into one volumetric and two isochoric parts. The volumetric part is considered elastic along all strain range and isochoric parts are treated as elastoplastic, i.e., the isochoric plastic flow direction is directly defined by the Flory’s decomposition. Assuming this plastic flow direction it is not necessary to employ the classical Kröner-Lee multiplicative decomposition to consider elastic and plastic parts of finite strains. The proposed model is implemented in a 3D geometrical nonlinear positional FEM code and results are compared with literature experimental and numerical data for validation purposes and applications.

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This research has been supported by the São Paulo Research Foundation, Brazil—Grant #2020/05393–4.

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Correspondence to Humberto Breves Coda.

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Appendix A

Second Piola–Kirchhoff stress components

In order to be complete, this appendix shows that the second Piola–Kirchhoff stress components \(\left( {{\varvec{S}}^{J} ,{\varvec{S}}^{1} ,{\varvec{S}}^{2} } \right)\) defined by Eq. (12) are related, respectively, to the hydrostatic and deviatoric Cauchy stress components \(\left( {{\varvec{\sigma}}_{{}}^{vol} ,{\varvec{\sigma}}^{iso1} ,{\varvec{\sigma}}^{iso2} } \right)\). This is done applying the well known relation among the second Piola-Kirchhof stress and the Cauchy stress, as follows:

$$ {\varvec{\sigma}} = \frac{1}{J}{\varvec{A}} \cdot {\varvec{S}} \cdot {\varvec{A}}^{t} = \frac{1}{J}{\varvec{A}} \cdot \left( {{\varvec{S}}^{J} + {\varvec{S}}^{1} + {\varvec{S}}^{2} } \right) \cdot {\varvec{A}}^{t} = \frac{1}{J}{\varvec{A}} \cdot {\varvec{S}}^{J} \cdot {\varvec{A}}^{t} + \frac{1}{J}{\varvec{A}} \cdot {\varvec{S}}^{1} \cdot {\varvec{A}}^{t} + \frac{1}{J}{\varvec{A}} \cdot {\varvec{S}}^{2} \cdot {\varvec{A}}^{t} $$

To make operation simple one starts rewriting Eq. (11) in a general form:

$$ \psi^{J} = \psi^{J} (J),\psi_{{}}^{1} = \psi_{{}}^{1} (\overline{I}_{1} ),\psi_{{}}^{2} = \psi_{{}}^{2} (\overline{I}_{2} ) $$

So, for each component one writes:

  • Volumetric

The candidate to volumetric component of the second Piola Kirchhoff stress is written as:

$$ {\varvec{S}}_{{}}^{J} = \frac{{\partial \psi_{{}}^{J} }}{\partial J}\frac{\partial J}{{\partial {\varvec{E}}}} = \alpha \,{\mathfrak{E}}^{J} $$

In which \(\alpha\) is a scalar and

$$ {\mathfrak{E}}^{J} = \frac{\partial J}{{\partial {\varvec{E}}}} = J{\varvec{C}}^{ - 1} $$

Applying Eq. (55) over the second Piola–Kirchhoff stress of Eq. (57) results:

$$ {\varvec{\sigma}}_{{}}^{vol} = \frac{1}{J}{\varvec{A}} \cdot {\varvec{S}}_{{}}^{J} \cdot {\varvec{A}}^{t} = \frac{1}{J}{\varvec{A}} \cdot \alpha \,J{\varvec{C}}^{ - 1} \cdot {\varvec{A}}^{t} = \alpha \,{\varvec{I}} = \sigma^{hid} $$

which means that \({\mathfrak{E}}^{J}\) is the Lagrangian strain direction corresponding to the hydrostatic stress in the Cauchy space and that \({\varvec{A}}^{J}\) is hydrostatic in Lagrangian sense.

  • First isochoric

The candidate to first isochoric component of the second Piola–Kirchhoff stress is written as:

$$ {\varvec{S}}_{{}}^{1} = \frac{{\partial \psi_{{}}^{1} }}{{\partial \overline{I}_{1} }}\frac{{\partial \overline{I}_{1} }}{{\partial {\varvec{E}}}} = \beta \,{\mathfrak{E}}^{1} $$

where \(\beta\) is a scalar and \(\mathfrak{E}^{1}\) is a symmetric tensor of the same order of the Green strain. It is not usual to see the following expressions, but it is well known and straightforward to show that:

$$ {\mathfrak{E}}^{1} = \frac{{\partial \overline{I}_{1} }}{{\partial {\varvec{E}}}} = \frac{{\partial Trac\left( {\overline{{C}}} \right)}}{{\partial {\varvec{E}}}} = \frac{{\partial \left( {J^{ - 2/3} Trac\left( {\varvec{C}} \right)} \right)}}{{\partial {\varvec{E}}}} $$

from which one finds:

$$ {\mathfrak{E}}^{1} = - \frac{2}{3}J^{ - 2/3} Tr({\varvec{C}}){\varvec{C}}^{ - 1} + 2J^{ - 2/3} {\varvec{I}} $$

Considering Eq. (62) and applying the transformation (55) over Eq. (60), results:

$$ {\varvec{\sigma}}^{1} = \beta \left\{ {2J^{ - 5/3} \left( {{\varvec{A}} \cdot {\varvec{A}}^{t} - \frac{{tr\left( {{\varvec{A}}^{t} \cdot {\varvec{A}}} \right)}}{3}{\varvec{I}}} \right)} \right\} $$

And, as

$$ tr\left( {{\varvec{A}}^{t} \cdot {\varvec{A}}} \right) = {\varvec{A}}:{\varvec{A}}^{t} = {\varvec{A}}^{t} :{\varvec{A}} = tr({\varvec{A}} \cdot {\varvec{A}}^{t} ) $$

one achieves:

$$ {\varvec{\sigma}}^{iso1} = \beta \left\{ {2J^{ - 5/3} \left( {{\varvec{A}} \cdot {\varvec{A}}^{t} - \frac{{tr\left( {{\varvec{A}} \cdot {\varvec{A}}^{t} } \right)}}{3}{\varvec{I}}} \right)} \right\} = {\varvec{\sigma}}_{desv} $$

which means that \({\mathfrak{E}}^{1}\) is the first isochoric Lagrangian strain direction and \({\varvec{S}}^{1}\) is the first isochoric second Piola–Kirchhoff stress.

  • Second isochoric

The candidate to second isochoric component of the second Piola–Kirchhoff stress is written as:

$$ {\varvec{S}}^{2} = \frac{{\partial \psi^{2} }}{{\partial \overline{I}_{2} }}\frac{{\partial \overline{I}_{2} }}{{\partial {\varvec{E}}}} = \gamma \,{\mathfrak{E}}^{2} $$

with \(\gamma\) and \({\mathfrak{E}}^{2}\) being respectively a scalar and a symmetric tensor of the same order as the Green–Lagrange strain tensor. It is interesting to write:

$$ \begin{aligned}\overline{I}_{2} &= J^{ - 4/3} I_{2} = J^{ - 4/3} \{ ( c_{11} c_{22} - c_{12} c_{21}) + ( c_{11} c_{33} - c_{13} c_{31} ) \\ &\quad + ( c_{22} c_{33} - c_{23} c_{32} )\}\end{aligned} $$

in which \(I_{2}\) is the second invariant of the Cauchy stretch. With some algebraic effort, over Eq. (68), results:

$$ {\mathfrak{E}}^{2} = \frac{{\partial \overline{I}_{2} }}{{\partial {\varvec{E}}}} = 2J^{ - 4/3} \left( { - \frac{2}{3}{\varvec{C}}^{ - 1} I_{2} + \left\{ {Tr({\varvec{C}}){\varvec{I}} - {\varvec{C}}^{t} } \right\}} \right) $$

Using Eq. (55) over Eq. (67) and considering Eq. (69), one writes:

$$ {\varvec{\sigma}}^{iso2} = \gamma \,2J^{ - 7/3} \left[ {\left( {tr({\varvec{C}})\left( {{\varvec{A}} \cdot {\varvec{A}}^{t} } \right) - \left( {{\varvec{A}} \cdot {\varvec{A}}^{t} } \right) \cdot \left( {{\varvec{A}} \cdot {\varvec{A}}^{t} } \right)} \right) - \left( {\frac{2}{3}I_{2} } \right){\varvec{I}}} \right] $$

Using equation (70) results:

$$ tr({\varvec{A}} \cdot {\varvec{A}}^{t} )tr({\varvec{A}} \cdot {\varvec{A}}^{t} ) - tr(({\varvec{A}} \cdot {\varvec{A}}^{t} ) \cdot ({\varvec{A}} \cdot {\varvec{A}}^{t} )) = 2I_{2} $$

From Eq. (71) one calculates \(Tr\left( {\sigma^{iso2} } \right)\) as

$$ Tr\left( {{\varvec{\sigma}}^{iso2} } \right) = 2\gamma J^{ - 7/3} \left( {2I_{2} - 2I_{2} } \right) = 0 $$


$$ {\varvec{\sigma}}^{iso2} = {\varvec{\sigma}}_{des} $$

meaning that \({\mathfrak{E}}^{2}\) is the second isochoric Lagrangian strain direction and \({\varvec{S}}^{2}\) is the second isochoric component of the second Piola–Kirchhoff stress.

Appendix B

Axial test and stress limits

In a simple axial test (following direction \(x_{1}\)) for a quasi isochoric material one writes:

$$ J = \lambda_{1} \lambda_{2} \lambda_{3} \cong 1 $$

In which \(\lambda_{i}\) are stretches in direction \(i\). Considering isotropy, for this test one writes \(\lambda_{2} = \lambda_{3} = \lambda\) and, from (73) one assumes:

$$ \lambda^{2} = \frac{1}{{\lambda_{1} }} $$

Making some algebraic simple steps one achieves from Eqs. (13) and (14):

$$ {\mathfrak{E}}^{1} = \frac{2}{3}\left[ {\begin{array}{*{20}c} {2(1 - \lambda_{1}^{ - 3} )} & 0 & 0 \\ 0 & {1 - \lambda_{1}^{3} } & 0 \\ 0 & 0 & {1 - \lambda_{1}^{3} } \\ \end{array} } \right] $$


$$ {\mathfrak{E}}^{2} = \frac{2}{{3\lambda_{1}^{{}} }}\left[ {\begin{array}{*{20}c} {2(1 - \lambda_{1}^{ - 3} )} & 0 & 0 \\ 0 & {1 - \lambda_{1}^{3} } & 0 \\ 0 & 0 & {1 - \lambda_{1}^{3} } \\ \end{array} } \right] $$

Thus, using Eq. (12), the following particular relation is achieved for very small volume change in elastic situation and simple axial stretch:

$$ {\varvec{S}}^{1} = \lambda_{1} {\varvec{S}}^{2} $$

For metallic materials it is expected that the yielding starts in the interval \(0.001 < \lambda_{1} - 1 < 0.003\). From this reasoning it is fair to split the von-Mises criterion given by Eq. (18) into two expressions given in Eq. (19) with \(\overline{\tau }^{1} = \overline{\tau }^{2} = \overline{\tau }\). If finite elastic strain is present and \(G^{1}\) is different from \(G^{2}\), than \(\overline{\tau }_{i}\) can be calibrated accordingly.

Appendix C

In this appendix two tables referred to example 1 are presented in order to simplify graphical reproduction (Tables 2 and 3).

Table 2 Cauchy and First Piola–Kirchhoff stresses for Fig. 5
Table 3 Cauchy and Piola–Kirchhoff stresses for Fig. 7

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Coda, H.B. A finite strain elastoplastic model based on Flory’s decomposition and 3D FEM applications. Comput Mech (2021).

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  • Finite strain plasticity
  • Flory’s decomposition
  • Finite elements
  • Plastic flow