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On advantages of the Kelvin mapping in finite element implementations of deformation processes

Abstract

Classical continuum mechanical theories operate on three-dimensional Euclidian space using scalar, vector, and tensor-valued quantities usually up to the order of four. For their numerical treatment, it is common practice to transform the relations into a matrix–vector format. This transformation is usually performed using the so-called Voigt mapping. This mapping does not preserve tensor character leaving significant room for error as stress and strain quantities follow from different mappings and thus have to be treated differently in certain mathematical operations. Despite its conceptual and notational difficulties having been pointed out, the Voigt mapping remains the foundation of most current finite element programmes. An alternative is the so-called Kelvin mapping which has recently gained recognition in studies of theoretical mechanics. This article is concerned with benefits of the Kelvin mapping in numerical modelling tools such as finite element software. The decisive difference to the Voigt mapping is that Kelvin’s method preserves tensor character, and thus the numerical matrix notation directly corresponds to the original tensor notation. Further benefits in numerical implementations are that tensor norms are calculated identically without distinguishing stress- or strain-type quantities, and tensor equations can be directly transformed into matrix equations without additional considerations. The only implementational changes are related to a scalar factor in certain finite element matrices, and hence, harvesting the mentioned benefits comes at very little cost.

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Abbreviations

\({\varvec{{\upepsilon }}},\ \varvec{\epsilon },\ \underline{{\epsilon }}\) :

Small strain tensor/its Kelvin mapping/its Voigt mapping

\({\mathrm {d}}\varGamma\) :

Area element

\(\lambda\) :

First Lamé constant

\(\mu\) :

Second Lamé constant, shear modulus of linear elasticity

\({\mathrm {d}}\varOmega\) :

Volume element

\(\varPsi ^i\) :

Residual in iteration i

\(\rho\) :

Mass density

\({\varvec{\upsigma }},\ \varvec{\sigma }, \ \underline{{\sigma }}\) :

Cauchy stress tensor/its Kelvin mapping/its Voigt mapping

\({\mathbf {a}} \cdot {\mathbf {b}}\) :

Dot product of \({\mathbf {a}}\) and \({\mathbf {b}}\)

\({\mathbf {A}} {\mathbf {\,:\,}}{\mathbf {B}}\) :

Double contraction of \({\mathbf {A}}\) and \({\mathbf {B}}\)

\({\mathbf {a}} \otimes {\mathbf {b}}\) :

Dyadic product of \({\mathbf {a}}\) and \({\mathbf {b}}\)

\({\mathbf {A}} \underline{\odot } {\mathbf {B}}\) :

\(\frac{1}{2}[ ({\mathbf {A}} \otimes {\mathbf {B}})^{\mathop {{\mathrm {T}}}\limits ^{23}} + ({\mathbf {A}} \otimes {\mathbf {B}}^{\mathrm {T}})^{\mathop {{\mathrm {T}}}\limits ^{24}}]\)

\({\mathbf {A}} \odot {\mathbf {B}}\) :

\(\frac{1}{2}[ ({\mathbf {A}} \otimes {\mathbf {B}}^\mathrm {T})^{\mathop {{\mathrm {T}}}\limits ^{23}} + ({\mathbf {A}} \otimes {\mathbf {B}})^{\mathop {{\mathrm {T}}}\limits ^{24}}]\)

\(\,\hbox {div}\,\) :

Divergence operator

\(\,\hbox {grad}\,\) :

Gradient operator

\((\bullet )^{\mathrm {D}}\) :

Deviatoric part of a tensor

\((\bullet )^{\mathrm {T}}\), \((\bullet )^{\mathop {{\mathrm {T}}}\limits ^{\mathrm {ab}}}\) :

Transpose operator; transposition of ath and bth base vector

\({\mathbf {b}}\) :

External body force

\(\varvec{{\mathscr {C}}}\), \({\varvec{\mathsf{{ C}}}}\), \(\underline{\underline{{C}}}\) :

Tangent moduli (fourth-order tensor)/its Kelvin mapping/its Voigt mapping

\({\mathbf {E}}\) :

Green Lagrange strain tensor

e :

Linear volume strain

\({\mathbf {F}}\) :

Deformation gradient

F :

Yield function

G :

Plastic potential

J :

Volume ratio of material volume elements in the current and the reference configuration

K :

Bulk modulus of linear elasticity

\({\mathbf {n}}\) :

Outward unit normal vector

\(N_a\), \({\mathbf {N}}\) :

Nodal shape function, element matrix of nodal shape functions

\(\varvec{{\mathscr {{P}}}}^{\mathrm {S}},\varvec{{\mathscr {{P}}}}^{\mathrm {D}}\) :

Spherical and deviatoric projection tensors

p :

Hydrostatic pressure

\({\mathbf {S}}\) :

Second Piola–Kirchhoff stress tensor

\({\mathbf {t}}\) :

Surface traction

\({\mathbf {u}}\) :

Displacement vector

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Acknowledgments

The authors would gratefully like to acknowledge the funding provided by the German Ministry of Education and Research (BMBF) for the ANGUS+ project, Grant Number 03EK3022, as well as the support of the Project Management Jülich (PTJ). Additional funding was provided by the Helmholtz Initiating and Networking Fund through the NUMTHECHSTORE project.

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Correspondence to Thomas Nagel.

Additional information

This article is part of a Topical Collection in Environmental Earth Sciences on “Subsurface Energy Storage”, guest edited by Sebastian Bauer, Andreas Dahmke, and Olaf Kolditz.

Throughout the article bold face symbols denote tensors and vectors. Normal face letters represent scalar quantities.

Appendices

Appendix 1: Spherical and deviatoric projections

Any second-order tensor \({\mathbf {A}}\) can be additively decomposed into a spherical and a deviatoric part:

$${\mathbf {A}} = {\mathbf {A}}^\mathrm {S} + {\mathbf {A}}^\mathrm {D} \quad \mathrm {with} \quad {\mathbf {A}}^\mathrm {S} = \frac{1}{3} ({\mathbf {A}} {\mathbf {\,:\,}}{\mathbf {I}}){\mathbf{I}}$$
(60)

where \({\mathbf {I}} = {\mathbf {g}}^k \otimes {\mathbf {g}}_k\) is the metric tensor formed by the contra- and covariant basis vectors, respectively. The mapping can also be written in terms of fourth-order tensors:

$${\mathbf {A}}^\mathrm {D} = \varvec{{\mathscr {P}}}^\mathrm {D} {\mathbf {\,:\,}}{\mathbf {A}} \quad \mathrm {and} \quad {\mathbf {A}}^\mathrm {S} = \varvec{{\mathscr {P}}}^\mathrm {S} {\mathbf {\,:\,}}{\mathbf {A}}$$
(61)

with the fourth-order tensors

$$\varvec{{\mathscr {{P}}}}^\mathrm {S} = \frac{1}{3} {\mathbf {I}} \otimes {\mathbf {I}}$$
(62)
$$\varvec{{\mathscr {{P}}}}^\mathrm {D} = ({\mathbf {I}} \otimes {\mathbf {I}})^{\mathop {{\mathrm {T}}}\limits ^{23}} - \frac{1}{3} {\mathbf {I}} \otimes {\mathbf {I}} = \varvec{{\mathscr {I}}} - \varvec{{\mathscr {{P}}}}^\mathrm {S}$$
(63)

Note in passing that the deviatoric representation of a quantity \(\underline{{a}}\) can be calculated with the projection

$$\underline{{a}}^\mathrm {D} = \underline{\underline{{P}}}^\mathrm {D} \underline{{a}} \quad \mathrm {with} \quad \underline{\underline{{P}}}^\mathrm {D} = \left( \begin{array}{lllccc} \frac{2}{3} &{} -\frac{1}{3} &{} -\frac{1}{3} &{} 0 &{} 0 &{} 0 \\ -\frac{1}{3} &{} \frac{2}{3} &{} -\frac{1}{3} &{} 0 &{} 0 &{} 0 \\ -\frac{1}{3} &{} -\frac{1}{3} &{} \frac{2}{3} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{array} \right)$$
(64)

which is independent of the mapping used, i.e., \(\underline{\underline{{P}}}^\mathrm {D} = {\varvec{\mathsf{{ P}}}}^\mathrm {D}\) and thus \({\varvec{\mathsf{{ a}}}}^\mathrm {D} = {\varvec{\mathsf{{ P}}}}^\mathrm {D}{\varvec{\mathsf{{ a}}}}\).

Appendix 2: Notes on the pullback of the weak form

The left-hand side of Eq. (35) is pulled back into the reference configuration by using \(\mathrm {d}\varOmega = J \mathrm {d}\varOmega ^0\) where \(J = \det {\mathbf {F}}\) is the volume ratio. The pullback proceeds as follows:

$$\begin{aligned}\int \limits _\varOmega {{\varvec{\upsigma }} {\mathbf {\,:\,}}\,\hbox {grad}\,{\mathbf {v}}}\, \mathrm {d}\varOmega&= \int \limits _{\varOmega ^0} {J {\varvec{\upsigma }} {\mathbf {\,:\,}}\,\hbox {sym}\,\,\hbox {grad}\,{\mathbf {v}}}\, \mathrm {d}\varOmega ^0 \\&= \int \limits _{\varOmega ^0} {J {\varvec{\upsigma }} {\mathbf {\,:\,}}{\mathbf {F}}^{-\mathrm {T}} \underbrace{\frac{1}{2} \left[ (\,\hbox {Grad}\,{\mathbf {v}})^\mathrm {T} {\mathbf {F}} + {\mathbf {F}}^\mathrm {T} \,\hbox {Grad}\,{\mathbf {v}} \right] }_{ := {\mathbf {E}}({\mathbf {v}};{\mathbf {u}})} {\mathbf {F}}^{-1}}\, \mathrm {d}\varOmega ^0\end{aligned}$$
(65)
$$= \int \limits _{\varOmega ^0} {J {\mathbf {F}}^{-1} {\varvec{\upsigma }} {\mathbf {F}}^{-\mathrm {T}} {\mathbf {\,:\,}}{\mathbf {E}}({\mathbf {u}};{\mathbf {v}})}\, \mathrm {d}\varOmega ^0 = \int \limits _{\varOmega ^0} {{\mathbf {S}} {\mathbf {\,:\,}}{\mathbf {E}}({\mathbf {u}};{\mathbf {v}})}\, \mathrm {d}\varOmega ^0$$
(66)

where the second Piola–Kirchhoff stress \({\mathbf {S}}\) appears. The volume integral on the right-hand side of Eq. (35) is transformed similarly noticing that \(\rho _0 = J \rho\) and that a vector can be expressed in both the reference and the current configuration by the use of shifters: \({\mathbf {a}} = a^i {\mathbf {g}}_i = a^i g_i^K {\mathbf {G}}_K = a^K {\mathbf {G}}_K\), where the coordinates of the shifter are \(g^K_i = {\mathbf {g}}_i \cdot {\mathbf {G}}^K\). The surface traction \({\mathbf {t}} = {\mathbf {\sigma n}}\) acting on the current (deforming) Neumann boundary with (deformation dependent) area elements \(\mathrm {d}\varGamma\) can be transformed to the reference configuration with (constant) area elements \(\mathrm {d}\varGamma ^0\) with the help of Nanson’s formula:

$$\varvec{\sigma } {\mathbf {n}} \mathrm {d}\varGamma ^{t+\Delta t} = \varvec{\sigma } J {\mathbf {F}}^{-T} {\mathbf {N}} \mathrm {d}\varGamma ^0 = {\mathbf {P}} {\mathbf {N}} \mathrm {d}\varGamma ^0 = \bar{{\mathbf {T}}} \mathrm {d}\varGamma ^0$$
(67)

Therefore, the pulled-back version of the linear momentum balance reads

$$\int \limits _{\varOmega ^0} {{\mathbf {S}} {\mathbf {\,:\,}}{\mathbf {E}}({\mathbf {U}};{\mathbf {V}})}\, \mathrm {d}\varOmega ^0 = \int \limits _{\varOmega ^0} {\rho _0 {\mathbf {F}} \cdot {\mathbf {V}}}\, \mathrm {d}\varOmega ^0 + \int \limits _{\partial \varOmega _{{\mathbf {T}}}^0} {\bar{{\mathbf {T}}} \cdot {\mathbf {V}}} \, \mathrm {d}\varGamma ^0$$
(68)

where the vectors \({\mathbf {v}}\), \({\mathbf {u}},\) and \({\mathbf {f}}\) have been capitalised due to the fact that the basis of the reference configuration is chosen throughout.

Appendix 3: Additional definitions

The stress matrix is defined as

$$\bar{{\varvec{\mathsf{{ S}}}}} = \left( \begin{array}{ccccccccc} S_{11} &{} S_{12} &{} S_{13}\\ S_{12} &{} S_{22} &{} S_{23} &{} &{} {\varvec{\mathsf{{ 0}}}} &{} &{} &{} {\varvec{\mathsf{{ 0}}}}\\ S_{13} &{} S_{23} &{} S_{33}\\ &{} &{} &{} S_{11} &{} S_{12} &{} S_{13}\\ &{} {\varvec{\mathsf{{ 0}}}} &{} &{} S_{12} &{} S_{22} &{} S_{23} &{} &{} {\varvec{\mathsf{{ 0}}}}\\ &{} &{} &{} S_{13} &{} S_{23} &{} S_{33}\\ &{} &{} &{} &{} &{} &{} S_{11} &{} S_{12} &{} S_{13}\\ &{} {\varvec{\mathsf{{ 0}}}}&{} &{} &{} {\varvec{\mathsf{{ 0}}}} &{} &{} S_{12} &{} S_{22} &{} S_{23}\\ &{} &{} &{} &{} &{} &{} S_{13} &{} S_{23} &{} S_{33} \end{array} \right)$$
(69)

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Nagel, T., Görke, UJ., Moerman, K.M. et al. On advantages of the Kelvin mapping in finite element implementations of deformation processes. Environ Earth Sci 75, 937 (2016). https://doi.org/10.1007/s12665-016-5429-4

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Keywords

  • Kelvin mapping
  • Voigt mapping
  • Finite elements
  • Numerical algorithms
  • OpenGeoSys