In continuum mechanics , the most commonly used measure of stress is the Cauchy stress tensor , often called simply the stress tensor or "true stress". However, several alternative measures of stress can be defined:[ 1] [ 2] [ 3]
The Kirchhoff stress ( τ {\displaystyle {\boldsymbol {\tau }}} ). The nominal stress ( N {\displaystyle {\boldsymbol {N}}} ). The Piola–Kirchhoff stress tensors The first Piola–Kirchhoff stress ( P {\displaystyle {\boldsymbol {P}}} ). This stress tensor is the transpose of the nominal stress ( P = N T {\displaystyle {\boldsymbol {P}}={\boldsymbol {N}}^{T}} ). The second Piola–Kirchhoff stress or PK2 stress ( S {\displaystyle {\boldsymbol {S}}} ). The Biot stress ( T {\displaystyle {\boldsymbol {T}}} ) Consider the situation shown in the following figure. The following definitions use the notations shown in the figure.
Quantities used in the definition of stress measures
In the reference configuration Ω 0 {\displaystyle \Omega _{0}} , the outward normal to a surface element d Γ 0 {\displaystyle d\Gamma _{0}} is N ≡ n 0 {\displaystyle \mathbf {N} \equiv \mathbf {n} _{0}} and the traction acting on that surface (assuming it deforms like a generic vector belonging to the deformation) is t 0 {\displaystyle \mathbf {t} _{0}} leading to a force vector d f 0 {\displaystyle d\mathbf {f} _{0}} . In the deformed configuration Ω {\displaystyle \Omega } , the surface element changes to d Γ {\displaystyle d\Gamma } with outward normal n {\displaystyle \mathbf {n} } and traction vector t {\displaystyle \mathbf {t} } leading to a force d f {\displaystyle d\mathbf {f} } . Note that this surface can either be a hypothetical cut inside the body or an actual surface. The quantity F {\displaystyle {\boldsymbol {F}}} is the deformation gradient tensor , J {\displaystyle J} is its determinant.
The Cauchy stress (or true stress) is a measure of the force acting on an element of area in the deformed configuration. This tensor is symmetric and is defined via
d f = t d Γ = σ T ⋅ n d Γ {\displaystyle d\mathbf {f} =\mathbf {t} ~d\Gamma ={\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} ~d\Gamma } or
t = σ T ⋅ n {\displaystyle \mathbf {t} ={\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} } where t {\displaystyle \mathbf {t} } is the traction and n {\displaystyle \mathbf {n} } is the normal to the surface on which the traction acts.
The quantity,
τ = J σ {\displaystyle {\boldsymbol {\tau }}=J~{\boldsymbol {\sigma }}} is called the Kirchhoff stress tensor , with J {\displaystyle J} the determinant of F {\displaystyle {\boldsymbol {F}}} . It is used widely in numerical algorithms in metal plasticity (where there is no change in volume during plastic deformation). It can be called weighted Cauchy stress tensor as well.
Piola–Kirchhoff stress[ edit ] Nominal stress/First Piola–Kirchhoff stress[ edit ] The nominal stress N = P T {\displaystyle {\boldsymbol {N}}={\boldsymbol {P}}^{T}} is the transpose of the first Piola–Kirchhoff stress (PK1 stress, also called engineering stress) P {\displaystyle {\boldsymbol {P}}} and is defined via
d f = t d Γ = N T ⋅ n 0 d Γ 0 = P ⋅ n 0 d Γ 0 {\displaystyle d\mathbf {f} =\mathbf {t} ~d\Gamma ={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {P}}\cdot \mathbf {n} _{0}~d\Gamma _{0}} or
t 0 = t d Γ d Γ 0 = N T ⋅ n 0 = P ⋅ n 0 {\displaystyle \mathbf {t} _{0}=\mathbf {t} {\dfrac {d{\Gamma }}{d\Gamma _{0}}}={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {P}}\cdot \mathbf {n} _{0}} This stress is unsymmetric and is a two-point tensor like the deformation gradient. The asymmetry derives from the fact that, as a tensor, it has one index attached to the reference configuration and one to the deformed configuration.[ 4]
Second Piola–Kirchhoff stress[ edit ] If we pull back d f {\displaystyle d\mathbf {f} } to the reference configuration we obtain the traction acting on that surface before the deformation d f 0 {\displaystyle d\mathbf {f} _{0}} assuming it behaves like a generic vector belonging to the deformation. In particular we have
d f 0 = F − 1 ⋅ d f {\displaystyle d\mathbf {f} _{0}={\boldsymbol {F}}^{-1}\cdot d\mathbf {f} } or,
d f 0 = F − 1 ⋅ N T ⋅ n 0 d Γ 0 = F − 1 ⋅ t 0 d Γ 0 {\displaystyle d\mathbf {f} _{0}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}~d\Gamma _{0}} The PK2 stress ( S {\displaystyle {\boldsymbol {S}}} ) is symmetric and is defined via the relation
d f 0 = S T ⋅ n 0 d Γ 0 = F − 1 ⋅ t 0 d Γ 0 {\displaystyle d\mathbf {f} _{0}={\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}~d\Gamma _{0}} Therefore,
S T ⋅ n 0 = F − 1 ⋅ t 0 {\displaystyle {\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {F}}^{-1}\cdot \mathbf {t} _{0}} The Biot stress is useful because it is energy conjugate to the right stretch tensor U {\displaystyle {\boldsymbol {U}}} . The Biot stress is defined as the symmetric part of the tensor P T ⋅ R {\displaystyle {\boldsymbol {P}}^{T}\cdot {\boldsymbol {R}}} where R {\displaystyle {\boldsymbol {R}}} is the rotation tensor obtained from a polar decomposition of the deformation gradient. Therefore, the Biot stress tensor is defined as
T = 1 2 ( R T ⋅ P + P T ⋅ R ) . {\displaystyle {\boldsymbol {T}}={\tfrac {1}{2}}({\boldsymbol {R}}^{T}\cdot {\boldsymbol {P}}+{\boldsymbol {P}}^{T}\cdot {\boldsymbol {R}})~.} The Biot stress is also called the Jaumann stress.
The quantity T {\displaystyle {\boldsymbol {T}}} does not have any physical interpretation. However, the unsymmetrized Biot stress has the interpretation
R T d f = ( P T ⋅ R ) T ⋅ n 0 d Γ 0 {\displaystyle {\boldsymbol {R}}^{T}~d\mathbf {f} =({\boldsymbol {P}}^{T}\cdot {\boldsymbol {R}})^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}} Relations between Cauchy stress and nominal stress [ edit ] From Nanson's formula relating areas in the reference and deformed configurations:
n d Γ = J F − T ⋅ n 0 d Γ 0 {\displaystyle \mathbf {n} ~d\Gamma =J~{\boldsymbol {F}}^{-T}\cdot \mathbf {n} _{0}~d\Gamma _{0}} Now,
σ T ⋅ n d Γ = d f = N T ⋅ n 0 d Γ 0 {\displaystyle {\boldsymbol {\sigma }}^{T}\cdot \mathbf {n} ~d\Gamma =d\mathbf {f} ={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}} Hence,
σ T ⋅ ( J F − T ⋅ n 0 d Γ 0 ) = N T ⋅ n 0 d Γ 0 {\displaystyle {\boldsymbol {\sigma }}^{T}\cdot (J~{\boldsymbol {F}}^{-T}\cdot \mathbf {n} _{0}~d\Gamma _{0})={\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}} or,
N T = J ( F − 1 ⋅ σ ) T = J σ T ⋅ F − T {\displaystyle {\boldsymbol {N}}^{T}=J~({\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }})^{T}=J~{\boldsymbol {\sigma }}^{T}\cdot {\boldsymbol {F}}^{-T}} or,
N = J F − 1 ⋅ σ and N T = P = J σ T ⋅ F − T {\displaystyle {\boldsymbol {N}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\qquad {\text{and}}\qquad {\boldsymbol {N}}^{T}={\boldsymbol {P}}=J~{\boldsymbol {\sigma }}^{T}\cdot {\boldsymbol {F}}^{-T}} In index notation,
N I j = J F I k − 1 σ k j and P i J = J σ k i F J k − 1 {\displaystyle N_{Ij}=J~F_{Ik}^{-1}~\sigma _{kj}\qquad {\text{and}}\qquad P_{iJ}=J~\sigma _{ki}~F_{Jk}^{-1}} Therefore,
J σ = F ⋅ N = F ⋅ P T . {\displaystyle J~{\boldsymbol {\sigma }}={\boldsymbol {F}}\cdot {\boldsymbol {N}}={\boldsymbol {F}}\cdot {\boldsymbol {P}}^{T}~.} Note that N {\displaystyle {\boldsymbol {N}}} and P {\displaystyle {\boldsymbol {P}}} are (generally) not symmetric because F {\displaystyle {\boldsymbol {F}}} is (generally) not symmetric.
Relations between nominal stress and second P–K stress[ edit ] Recall that
N T ⋅ n 0 d Γ 0 = d f {\displaystyle {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0}=d\mathbf {f} } and
d f = F ⋅ d f 0 = F ⋅ ( S T ⋅ n 0 d Γ 0 ) {\displaystyle d\mathbf {f} ={\boldsymbol {F}}\cdot d\mathbf {f} _{0}={\boldsymbol {F}}\cdot ({\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}~d\Gamma _{0})} Therefore,
N T ⋅ n 0 = F ⋅ S T ⋅ n 0 {\displaystyle {\boldsymbol {N}}^{T}\cdot \mathbf {n} _{0}={\boldsymbol {F}}\cdot {\boldsymbol {S}}^{T}\cdot \mathbf {n} _{0}} or (using the symmetry of S {\displaystyle {\boldsymbol {S}}} ),
N = S ⋅ F T and P = F ⋅ S {\displaystyle {\boldsymbol {N}}={\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}\qquad {\text{and}}\qquad {\boldsymbol {P}}={\boldsymbol {F}}\cdot {\boldsymbol {S}}} In index notation,
N I j = S I K F j K T and P i J = F i K S K J {\displaystyle N_{Ij}=S_{IK}~F_{jK}^{T}\qquad {\text{and}}\qquad P_{iJ}=F_{iK}~S_{KJ}} Alternatively, we can write
S = N ⋅ F − T and S = F − 1 ⋅ P {\displaystyle {\boldsymbol {S}}={\boldsymbol {N}}\cdot {\boldsymbol {F}}^{-T}\qquad {\text{and}}\qquad {\boldsymbol {S}}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {P}}} Relations between Cauchy stress and second P–K stress[ edit ] Recall that
N = J F − 1 ⋅ σ {\displaystyle {\boldsymbol {N}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}} In terms of the 2nd PK stress, we have
S ⋅ F T = J F − 1 ⋅ σ {\displaystyle {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}} Therefore,
S = J F − 1 ⋅ σ ⋅ F − T = F − 1 ⋅ τ ⋅ F − T {\displaystyle {\boldsymbol {S}}=J~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}\cdot {\boldsymbol {F}}^{-T}={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}} In index notation,
S I J = F I k − 1 τ k l F J l − 1 {\displaystyle S_{IJ}=F_{Ik}^{-1}~\tau _{kl}~F_{Jl}^{-1}} Since the Cauchy stress (and hence the Kirchhoff stress) is symmetric, the 2nd PK stress is also symmetric.
Alternatively, we can write
σ = J − 1 F ⋅ S ⋅ F T {\displaystyle {\boldsymbol {\sigma }}=J^{-1}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}} or,
τ = F ⋅ S ⋅ F T . {\displaystyle {\boldsymbol {\tau }}={\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}~.} Clearly, from definition of the push-forward and pull-back operations, we have
S = φ ∗ [ τ ] = F − 1 ⋅ τ ⋅ F − T {\displaystyle {\boldsymbol {S}}=\varphi ^{*}[{\boldsymbol {\tau }}]={\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\tau }}\cdot {\boldsymbol {F}}^{-T}} and
τ = φ ∗ [ S ] = F ⋅ S ⋅ F T . {\displaystyle {\boldsymbol {\tau }}=\varphi _{*}[{\boldsymbol {S}}]={\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}~.} Therefore, S {\displaystyle {\boldsymbol {S}}} is the pull back of τ {\displaystyle {\boldsymbol {\tau }}} by F {\displaystyle {\boldsymbol {F}}} and τ {\displaystyle {\boldsymbol {\tau }}} is the push forward of S {\displaystyle {\boldsymbol {S}}} .
Key: J = det ( F ) , C = F T F = U 2 , F = R U , R T = R − 1 , {\displaystyle J=\det \left({\boldsymbol {F}}\right),\quad {\boldsymbol {C}}={\boldsymbol {F}}^{T}{\boldsymbol {F}}={\boldsymbol {U}}^{2},\quad {\boldsymbol {F}}={\boldsymbol {R}}{\boldsymbol {U}},\quad {\boldsymbol {R}}^{T}={\boldsymbol {R}}^{-1},} P = J σ F − T , τ = J σ , S = J F − 1 σ F − T , T = R T P , M = C S {\displaystyle {\boldsymbol {P}}=J{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T},\quad {\boldsymbol {\tau }}=J{\boldsymbol {\sigma }},\quad {\boldsymbol {S}}=J{\boldsymbol {F}}^{-1}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T},\quad {\boldsymbol {T}}={\boldsymbol {R}}^{T}{\boldsymbol {P}},\quad {\boldsymbol {M}}={\boldsymbol {C}}{\boldsymbol {S}}}
Conversion formulae Equation for σ {\displaystyle {\boldsymbol {\sigma }}} τ {\displaystyle {\boldsymbol {\tau }}} P {\displaystyle {\boldsymbol {P}}} S {\displaystyle {\boldsymbol {S}}} T {\displaystyle {\boldsymbol {T}}} M {\displaystyle {\boldsymbol {M}}} σ = {\displaystyle {\boldsymbol {\sigma }}=\,} σ {\displaystyle {\boldsymbol {\sigma }}} J − 1 τ {\displaystyle J^{-1}{\boldsymbol {\tau }}} J − 1 P F T {\displaystyle J^{-1}{\boldsymbol {P}}{\boldsymbol {F}}^{T}} J − 1 F S F T {\displaystyle J^{-1}{\boldsymbol {F}}{\boldsymbol {S}}{\boldsymbol {F}}^{T}} J − 1 R T F T {\displaystyle J^{-1}{\boldsymbol {R}}{\boldsymbol {T}}{\boldsymbol {F}}^{T}} J − 1 F − T M F T {\displaystyle J^{-1}{\boldsymbol {F}}^{-T}{\boldsymbol {M}}{\boldsymbol {F}}^{T}} (non isotropy) τ = {\displaystyle {\boldsymbol {\tau }}=\,} J σ {\displaystyle J{\boldsymbol {\sigma }}} τ {\displaystyle {\boldsymbol {\tau }}} P F T {\displaystyle {\boldsymbol {P}}{\boldsymbol {F}}^{T}} F S F T {\displaystyle {\boldsymbol {F}}{\boldsymbol {S}}{\boldsymbol {F}}^{T}} R T F T {\displaystyle {\boldsymbol {R}}{\boldsymbol {T}}{\boldsymbol {F}}^{T}} F − T M F T {\displaystyle {\boldsymbol {F}}^{-T}{\boldsymbol {M}}{\boldsymbol {F}}^{T}} (non isotropy) P = {\displaystyle {\boldsymbol {P}}=\,} J σ F − T {\displaystyle J{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}} τ F − T {\displaystyle {\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}} P {\displaystyle {\boldsymbol {P}}} F S {\displaystyle {\boldsymbol {F}}{\boldsymbol {S}}} R T {\displaystyle {\boldsymbol {R}}{\boldsymbol {T}}} F − T M {\displaystyle {\boldsymbol {F}}^{-T}{\boldsymbol {M}}} S = {\displaystyle {\boldsymbol {S}}=\,} J F − 1 σ F − T {\displaystyle J{\boldsymbol {F}}^{-1}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}} F − 1 τ F − T {\displaystyle {\boldsymbol {F}}^{-1}{\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}} F − 1 P {\displaystyle {\boldsymbol {F}}^{-1}{\boldsymbol {P}}} S {\displaystyle {\boldsymbol {S}}} U − 1 T {\displaystyle {\boldsymbol {U}}^{-1}{\boldsymbol {T}}} C − 1 M {\displaystyle {\boldsymbol {C}}^{-1}{\boldsymbol {M}}} T = {\displaystyle {\boldsymbol {T}}=\,} J R T σ F − T {\displaystyle J{\boldsymbol {R}}^{T}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}} R T τ F − T {\displaystyle {\boldsymbol {R}}^{T}{\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}} R T P {\displaystyle {\boldsymbol {R}}^{T}{\boldsymbol {P}}} U S {\displaystyle {\boldsymbol {U}}{\boldsymbol {S}}} T {\displaystyle {\boldsymbol {T}}} U − 1 M {\displaystyle {\boldsymbol {U}}^{-1}{\boldsymbol {M}}} M = {\displaystyle {\boldsymbol {M}}=\,} J F T σ F − T {\displaystyle J{\boldsymbol {F}}^{T}{\boldsymbol {\sigma }}{\boldsymbol {F}}^{-T}} (non isotropy) F T τ F − T {\displaystyle {\boldsymbol {F}}^{T}{\boldsymbol {\tau }}{\boldsymbol {F}}^{-T}} (non isotropy) F T P {\displaystyle {\boldsymbol {F}}^{T}{\boldsymbol {P}}} C S {\displaystyle {\boldsymbol {C}}{\boldsymbol {S}}} U T {\displaystyle {\boldsymbol {U}}{\boldsymbol {T}}} M {\displaystyle {\boldsymbol {M}}}
^ J. Bonet and R. W. Wood, Nonlinear Continuum Mechanics for Finite Element Analysis , Cambridge University Press. ^ R. W. Ogden, 1984, Non-linear Elastic Deformations , Dover. ^ L. D. Landau, E. M. Lifshitz, Theory of Elasticity , third edition ^ Three-Dimensional Elasticity . Elsevier. 1 April 1988. ISBN 978-0-08-087541-5 .