Piola–Kirchhoff stress tensors

E825430

Piola–Kirchhoff stress tensor continuum mechanics concept stress measure tensor two-point tensor

Piola–Kirchhoff stress tensors are alternative measures of stress in continuum mechanics that describe forces with respect to a material’s reference configuration rather than its current deformed state.

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Observed surface forms (3)

Surface form	Occurrences
Piola–Kirchhoff stress tensor	0
first Piola–Kirchhoff stress tensor	0
second Piola–Kirchhoff stress tensor	0

Statements (49)

Predicate	Object
instanceOf	Piola–Kirchhoff stress tensor ⓘ Piola–Kirchhoff stress tensor ⓘ continuum mechanics concept ⓘ stress measure ⓘ tensor ⓘ two-point tensor ⓘ
advantage	convenient for formulations in reference configuration ⓘ simplifies constitutive laws for hyperelastic materials ⓘ
alsoDefinedBy	S = F^{-1} · σ · F^{-T} · J ⓘ
appearsIn	balance of linear momentum in reference configuration ⓘ finite element formulations for large deformations ⓘ
contrastsWith	Cauchy stress tensor ⓘ
coordinateSystem	material (Lagrangian) coordinates ⓘ
definedBy	P = J · σ · F^{-T} ⓘ S = F^{-1} · P ⓘ
definedOn	reference configuration ⓘ reference configuration and current configuration ⓘ
describes	stress with respect to reference configuration ⓘ
field	continuum mechanics ⓘ nonlinear elasticity ⓘ solid mechanics ⓘ
hasType	first Piola–Kirchhoff stress tensor ⓘ second Piola–Kirchhoff stress tensor ⓘ
isGenerally	non-symmetric ⓘ symmetric for non-polar materials ⓘ
namedAfter	Gabrio Piola NERFINISHED ⓘ
order	second-order tensor ⓘ second-order tensor ⓘ
relatedConcept	Cauchy stress tensor NERFINISHED ⓘ Green–Lagrange strain tensor NERFINISHED ⓘ Kirchhoff stress tensor NERFINISHED ⓘ deformation gradient ⓘ
relatedTo	Cauchy stress tensor ⓘ Cauchy stress tensor ⓘ first Piola–Kirchhoff stress tensor ⓘ
relates	forces and areas both in reference configuration ⓘ forces in current configuration to areas in reference configuration ⓘ
symbol	P ⓘ S ⓘ
transformsVia	deformation gradient ⓘ
usedIn	Lagrangian formulations of elasticity ⓘ finite deformation theory ⓘ hyperelastic material models ⓘ large strain analysis ⓘ
where	F is deformation gradient ⓘ J is determinant of deformation gradient ⓘ σ is Cauchy stress tensor ⓘ
workConjugateTo	Green–Lagrange strain tensor ⓘ deformation gradient ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cauchy stress tensor → relatedTo → Piola–Kirchhoff stress tensors ⓘ