Piola–Kirchhoff stress tensors
E825430
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Piola–Kirchhoff stress tensors canonical | 1 |
| first Piola–Kirchhoff stress tensor | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9843816 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Piola–Kirchhoff stress tensors Context triple: [Cauchy stress tensor, relatedTo, Piola–Kirchhoff stress tensors]
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A.
Cauchy stress tensor
The Cauchy stress tensor is a fundamental concept in continuum mechanics that mathematically represents the internal distribution of forces (stresses) within a deformable material at a point.
-
B.
Les tenseurs en mécanique et en élasticité
"Les tenseurs en mécanique et en élasticité" is a technical work by Léon Brillouin that presents the mathematical theory and applications of tensors in continuum mechanics and elasticity.
-
C.
Mooney-Rivlin theory
Mooney-Rivlin theory is a constitutive model in continuum mechanics that describes the nonlinear elastic behavior of rubber-like materials using a specific strain energy function.
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D.
The Non-Linear Field Theories of Mechanics
The Non-Linear Field Theories of Mechanics is a foundational monograph by Clifford Truesdell that rigorously develops the mathematical theory of nonlinear continuum mechanics and its applications to elastic and fluid media.
-
E.
Finite Elements of Nonlinear Continua
"Finite Elements of Nonlinear Continua" is a foundational textbook by J. Tinsley Oden that develops the theory and application of finite element methods for analyzing nonlinear solid and structural mechanics problems.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Piola–Kirchhoff stress tensors Target entity description: 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.
-
A.
Cauchy stress tensor
The Cauchy stress tensor is a fundamental concept in continuum mechanics that mathematically represents the internal distribution of forces (stresses) within a deformable material at a point.
-
B.
Les tenseurs en mécanique et en élasticité
"Les tenseurs en mécanique et en élasticité" is a technical work by Léon Brillouin that presents the mathematical theory and applications of tensors in continuum mechanics and elasticity.
-
C.
Mooney-Rivlin theory
Mooney-Rivlin theory is a constitutive model in continuum mechanics that describes the nonlinear elastic behavior of rubber-like materials using a specific strain energy function.
-
D.
The Non-Linear Field Theories of Mechanics
The Non-Linear Field Theories of Mechanics is a foundational monograph by Clifford Truesdell that rigorously develops the mathematical theory of nonlinear continuum mechanics and its applications to elastic and fluid media.
-
E.
Finite Elements of Nonlinear Continua
"Finite Elements of Nonlinear Continua" is a foundational textbook by J. Tinsley Oden that develops the theory and application of finite element methods for analyzing nonlinear solid and structural mechanics problems.
- F. None of above. chosen
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 ⓘ |
| order |
second-order tensor
ⓘ
second-order tensor ⓘ |
| relatedConcept |
Cauchy stress tensor
ⓘ
Green–Lagrange strain tensor ⓘ Kirchhoff stress tensor ⓘ 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 ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Piola–Kirchhoff stress tensors Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Piola–Kirchhoff stress tensors