Skip to main content

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


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.

This is a preview of subscription content, access via your institution.


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

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


  1. Annin B, Ostrosablin N (2008) Anisotropy of elastic properties of materials. J Appl Mech Tech Phys 49(6):998–1014

    Article  Google Scholar 

  2. Bathe K (2001) Finite-elemente-methoden, German edn. Springer, New York

    Google Scholar 

  3. Bauer S, Beyer C, Dethlefsen F, Dietrich P, Duttmann R, Ebert M, Feeser V, Görke U, Köber R, Kolditz O et al (2013) Impacts of the use of the geological subsurface for energy storage: an investigation concept. Environ Earth Sci 70(8):3935–3943

    Article  Google Scholar 

  4. Bolcu D, Rizescu S, Ursache M, George N, Bîzdoacă MMS, Rinderu P (2010) Spectral decomposition of the elasticity matrix. Univ Politeh Buchar Sci Bull Ser A - Appl Math Phys 72(4):217–232

    Google Scholar 

  5. Bóna A, Bucataru I, Slawinski MA (2007) Coordinate-free characterization of the symmetry classes of elasticity tensors. J Elast 87(2–3):109–132

    Article  Google Scholar 

  6. Bonet J, Wood RD (1997) Nonlinear continuum mechanics for finite element analysis. Cambridge University Press, Cambridge

    Google Scholar 

  7. de Borst R, Heeres OM (2002) A unified approach to the implicit integration of standard, non-standard and viscous plasticity models. Int J Numer Anal Methods Geomech 26(11):1059–1070

    Article  Google Scholar 

  8. Bucher A, Görke UJ, Kreißig R (2001) Development of a generalized material interface for the simulation of finite elasto-plastic deformations. Int J Solids Struct 38(52):9423–9436

    Article  Google Scholar 

  9. Carcione J, Cavallini F, Helbig K (1998) Anisotropic attenuation and material symmetry. Acta Acust United Acust 84(3):495–502

    Google Scholar 

  10. Cowin S, Doty S (2007) Tissue mechanics. Springer, New York.

  11. Cowin SC (2011) The representation of the linear elastic symmetries by sets of vectors. Math Mech Solids 16(6):615–624

    Article  Google Scholar 

  12. Dellinger J, Vasicek D, Sondergeld C (1998) Kelvin notation for stabilizing elastic-constant inversion. Oil Gas Sci Technol 53(5):709–719

    Google Scholar 

  13. Diner Ç, Kochetov M, Slawinski MA (2011) On choosing effective symmetry classes for elasticity tensors. Quart J Mech Appl Math 64(1):57–74

    Article  Google Scholar 

  14. Dłuzewski P, Rodzik P (1998) Elastic eigenstates in finite element modelling of large anisotropic elasticity. Comput Methods Appl Mech Eng 160(3):325–335

    Article  Google Scholar 

  15. Doghri I (1995) Numerical implementation and analysis of a class of metal plasticity models coupled with ductile damage. Int J Numer Methods Eng 38(20):3403–3431. doi:10.1002/nme.1620382004

    Article  Google Scholar 

  16. Hartmann S, Lührs G, Haupt P (1997) An efficient stress algorithm with applications in viscoplasticity and plasticity. Int J Numer Methods Eng 40(6):991–1013. doi:10.1002/(SICI)1097-0207(19970330)40:6<991:AID-NME98>3.0.CO;2-H

    Article  Google Scholar 

  17. Haupt P (2002) Continuum mechanics and theory of materials. Springer, New York

    Book  Google Scholar 

  18. Helbig K (2013) Review paper: what Kelvin might have written about elasticity. Geophys Prospect 61(1):1–20. doi:10.1111/j.1365-2478.2011.01049.x

    Article  Google Scholar 

  19. Heusermann S, Rolfs O, Schmidt U (2003) Nonlinear finite-element analysis of solution mined storage caverns in rock salt using the LUBBY2 constitutive model. Comput Struct 81(8–11):629–638. doi:10.1016/S0045-7949(02)00415-7, (k.J Bathe 60th Anniversary Issue)

  20. Hughes TJ (2012) The finite element method: linear static and dynamic finite element analysis. Courier Corporation, North Chelmsford

    Google Scholar 

  21. Itskov M (2009) Tensor algebra and tensor analysis for engineers: with applications to continuum mechanics, 2nd edn. Springer, Dordrecht

    Book  Google Scholar 

  22. Itskov M, Aksel N (2002) Elastic constants and their admissible values for incompressible and slightly compressible anisotropic materials. Acta Mech 157(1–4):81–96

    Article  Google Scholar 

  23. Kochetov M, Slawinski MA (2009) On obtaining effective orthotropic elasticity tensors. Quart J Mech Appl Math 62(2):149–166

    Article  Google Scholar 

  24. Kolditz O, Bauer S, Bilke L, Böttcher N, Delfs J, Fischer T, Görke U, Kalbacher T, Kosakowski G, McDermott C et al (2012) OpenGeoSys: an open-source initiative for numerical simulation of thermo-hydro-mechanical/chemical (THM/C) processes in porous media. Environ Earth Sci 67(2):589–599

    Article  Google Scholar 

  25. Kowalczyk-Gajewska K, Ostrowska-Maciejewska J (2014) Review on spectral decomposition of hooke’s tensor for all symmetry groups of linear elastic material. Engineering Transactions 57(3-4),

  26. Li M, Zhang H, Xing W, Hou Z, Were P (2015) Study of the relationship between surface subsidence and internal pressure in salt caverns. Environ Earth Sci 73(11):6899–6910

    Article  Google Scholar 

  27. Ma H, Yang C, Li Y, Shi X, Liu J, Wang T (2015) Stability evaluation of the underground gas storage in rock salts based on new partitions of the surrounding rock. Environ Earth Sci 73(11):6911–6925

    Article  Google Scholar 

  28. Mehrabadi MM, Cowin SC (1990) Eigentensors of linear anisotropic elastic materials. Quart J Mech Appl Mathe 43(1):15–41. doi:10.1093/qjmam/43.1.15,,

  29. Minkley W, Mühlbauer J (2007) Constitutive models to describe the mechanical behavior of salt rocks and the imbedded weakness planes. In: Wallner M, Lux K, Minkley W, Hardy H (eds) The mechanical behaviour of salt—understanding of THMC processes in salt: 6th conference (SaltMech6). Hannover, Germany, pp 119–127

    Google Scholar 

  30. Moakher M (2008) Fourth-order cartesian tensors: old and new facts, notions and applications. Quart J Mech Appl Math 61(2):181–203. doi:10.1093/qjmam/hbm027,,

  31. Moakher M, Norris AN (2006) The closest elastic tensor of arbitrary symmetry to an elasticity tensor of lower symmetry. J Elast 85(3):215–263

    Article  Google Scholar 

  32. Moesen M, Cardoso L, Cowin SC (2012) A symmetry invariant formulation of the relationship between the elasticity tensor and the fabric tensor. Mech Mater 54:70–83

    Article  Google Scholar 

  33. Norris A (2006) The isotropic material closest to a given anisotropic material. J Mech Mater Struct 1(2):223–238

    Article  Google Scholar 

  34. Planas J, Romero I, Sancho J (2012) B free. Comput Methods Appl Mech Eng 217–220:226 – 235. doi:10.1016/j.cma.2012.01.019,

  35. Rychlewski J (1984) On Hooke’s law. J Appl Math Mech 48(3):303–314

    Article  Google Scholar 

  36. Safaei M, Lee MG, Waele WD (2015) Evaluation of stress integration algorithms for elastic–plastic constitutive models based on associated and non-associated flow rules. Comput Methods Appl Mech Eng 295:414–445. doi:10.1016/j.cma.2015.07.014,

  37. Simo JC, Hughes TJR (1998) Objective integration algorithms for rate formulations of elastoplasticity. In: Computational inelasticity, interdisciplinary applied mathematics, vol 7, Springer, New York, pp 276–299. doi:10.1007/0-387-22763-6_8

  38. Theocaris P (2000) Sorting out the elastic anisotropy of transversely isotropic materials. Acta Mech 143(3–4):129–140

    Article  Google Scholar 

  39. Theocaris P, Philippidis T (1991) Spectral decomposition of compliance and stiffness fourth-rank tensors suitable for orthotropic materials. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik 71(3):161–171

    Google Scholar 

  40. Thomson W (1856) Elements of a mathematical theory of elasticity. part i. on stresses and strains. Philosophical Transactions of the Royal Society of London 146: 481–498.

  41. Voigt W (1966) Lehrbuch der Kristallphysik Teubner, Leipzig 1910; reprinted (1928) with an additional appendix. Leipzig, Teubner; New York, Johnson Reprint

  42. Wriggers P (2008) Nonlinear finite element methods. Springer, Berlin, Heidelberg

    Google Scholar 

  43. Zienkiewicz O, Cormeau I (1974) Visco-plasticity–plasticity and creep in elastic solids—a unified numerical solution approach. Int J Numer Methods Eng 8(4):821–845. doi:10.1002/nme.1620080411/abstract

    Article  Google Scholar 

  44. Zienkiewicz OC, Taylor RL, Zhu JZ (2006) The finite element method set, 6th edn. Elsevier Butterworth-Heinemann, Oxford.

Download references


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.

Author information



Corresponding author

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.


Appendix 1: Spherical and deviatoric projections

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

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

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:

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

with the fourth-order tensors

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

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

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

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:

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

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:

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

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

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

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

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

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation


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