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विभिन्न हाइपरलास्टिक भौतिकी मॉडल के लिए तनाव घटता है।
हाइपरलास्टिक या अतिप्रत्यास्थ भौतिकी [1] प्रत्यास्थ भौतिकी के लिए एक प्रकार का प्रलक्षित समीकरण है जिसमे किरचॉफ तनाव और लग्रांगियन विरूपण तनाव मे संबंध तनाव ऊर्जा घनत्व से प्राप्त होता है। हाइपरलास्टिक भौतिकी कॉची प्रत्यास्थ भौतिकी की एक विशेष स्थिति है।
कई भौतिकी मॉडल रैखिक प्रत्यास्थ मॉडल के लिए भौतिक क्रियाविधि का वर्णन नहीं करते हैं। इस प्रकार की भौतिकी का सबसे सामान्य उदाहरण घर्षण है जिसमे किरचॉफ तनाव और लग्रांगियन विरूपण तनाव संबंध को गैर-रैखिक रूप से प्रत्यास्थ, समदैशिक और असंपीड्यता के रूप में परिभाषित किया जा सकता है। हाइपरलास्टिक ऐसी भौतिकी तनाव को मॉडलिंग करने का एक साधन प्रदान करता है।[2] वल्केनाइज्ड प्रत्यास्थता की क्रियाविधि प्रायः हाइपरलास्टिक भौतिकी के अनुरूप होती है। प्रत्यास्थता भौतिकी और जैविक ऊतक [3] [4]
रोनाल्ड रिवलिन और मेल्विन मूनी ने नियो-हुकेन और मूनी-रिवलिन ठोस यांत्रिकी मॉडल के पहले हाइपरलास्टिक मॉडल को विकसित किया था इसके बाद से कई अन्य हाइपरलास्टिक मॉडल विकसित किए गए हैं। अन्य व्यापक रूप से उपयोग किए जाने वाले हाइपरलास्टिक भौतिकी मॉडल में ओग्डेन मॉडल और अरुडा-बॉयस मॉडल सम्मिलित हैं।
हाइपरलास्टिक भौतिकी मॉडल सेंट वेनेंट-किरचॉफ मॉडल सबसे सामान्य हाइपरलास्टिक भौतिकी मॉडल सेंट वेनेंट-किरचॉफ मॉडल है जो ज्यामितीय रूप से गैर-रैखिक मॉडल के लिए ज्यामितीय रूप से रैखिक प्रत्यास्थ भौतिकी मॉडल का विस्तार है। इस मॉडल का क्रमशः सामान्य और समदैशिक रूप है:
S = C : E S = λ tr ( E ) I + 2 μ E . {\displaystyle {\begin{aligned}{\boldsymbol {S}}&={\boldsymbol {C}}:{\boldsymbol {E}}\\{\boldsymbol {S}}&=\lambda ~{\text{tr}}({\boldsymbol {E}}){\boldsymbol {\mathit {I}}}+2\mu {\boldsymbol {E}}{\text{.}}\end{aligned}}} जहाँ
: {\displaystyle \mathbin {:} } टेंसर संकुचन
S {\displaystyle {\boldsymbol {S}}} है दूसरा पिओला-किरचॉफ तनाव
C : R 3 × 3 → R 3 × 3 {\displaystyle {\boldsymbol {C}}:\mathbb {R} ^{3\times 3}\to \mathbb {R} ^{3\times 3}} है :और चौथा क्रम कठोरता टेन्सर
E {\displaystyle {\boldsymbol {E}}} है, जिसे लग्रांगियन ग्रीन स्ट्रेन द्वारा दिया गया है:
E = 1 2 [ ( ∇ X u ) T + ∇ X u + ( ∇ X u ) T ⋅ ∇ X u ] {\displaystyle \mathbf {E} ={\frac {1}{2}}\left[(\nabla _{\mathbf {X} }\mathbf {u} )^{\textsf {T}}+\nabla _{\mathbf {X} }\mathbf {u} +(\nabla _{\mathbf {X} }\mathbf {u} )^{\textsf {T}}\cdot \nabla _{\mathbf {X} }\mathbf {u} \right]\,\!} λ {\displaystyle \lambda } μ {\displaystyle \mu } I {\displaystyle {\boldsymbol {\mathit {I}}}}
W ( E ) = λ 2 [ tr ( E ) ] 2 + μ tr ( E 2 ) {\displaystyle W({\boldsymbol {E}})={\frac {\lambda }{2}}[{\text{tr}}({\boldsymbol {E}})]^{2}+\mu {\text{tr}}{\mathord {\left({\boldsymbol {E}}^{2}\right)}}} और दूसरा पिओला-किरचॉफ तनाव संबंध से प्राप्त किया जा सकता है:
S = ∂ W ∂ E . {\displaystyle {\boldsymbol {S}}={\frac {\partial W}{\partial {\boldsymbol {E}}}}~.} हाइपरलास्टिक भौतिकी मॉडल का वर्गीकरण हाइपरलास्टिक भौतिकी मॉडल को इस प्रकार वर्गीकृत किया जा सकता है:
हाइपरलास्टिक भौतिकी गतिविधि का घटनात्मक विवरण
फंग
मूनी-रिवलिन
ओग्डेन (हाइपरलास्टिक मॉडल)
बहुपद (हाइपरलास्टिक मॉडल) सेंट वेनेंट-किरचॉफ
योह (हाइपरलेस्टिक मॉडल)
मार्लो (हाइपरलास्टिक मॉडल)
भौतिकी की अंतर्निहित संरचना के विषय में तर्कों से प्राप्त यांत्रिकीय मॉडल
अरुडा-बॉयस मॉडल[5]
नियो-हुकेन मॉडल [1]
बुके-सिल्बरस्टीन मॉडल[6]
यांत्रिकीय और परिघटनात्मक मॉडल के हाइब्रिड
जेंट (हाइपरलास्टिक मॉडल)
वैन डेर वाल्स (हाइपरेलेटिक मॉडल) सामान्यतः एक हाइपरलास्टिक मॉडल को ड्रकर स्थिरता मानदंड को पूर्ण करने की आवश्यकता होती है। क्योकि कुछ हाइपरलास्टिक मॉडल वैलेनिस-लैंडल परिकल्पना को सिद्ध करते हैं जो प्रदर्शित करते है कि तनाव ऊर्जा कार्य को प्रमुख भागों ( λ 1 , λ 2 , λ 3 ) {\displaystyle (\lambda _{1},\lambda _{2},\lambda _{3})}
W = f ( λ 1 ) + f ( λ 2 ) + f ( λ 3 ) . {\displaystyle W=f(\lambda _{1})+f(\lambda _{2})+f(\lambda _{3})\,.} किरचॉफ तनाव और लग्रांगियन विरूपण मे संबंध संकुचित हाइपरलास्टिक भौतिकी पहला पिओला-किरचॉफ तनाव यदि W ( F ) {\displaystyle W({\boldsymbol {F}})}
P = ∂ W ∂ F or P i K = ∂ W ∂ F i K {\displaystyle {\boldsymbol {P}}={\frac {\partial W}{\partial {\boldsymbol {F}}}}\qquad {\text{or}}\qquad P_{iK}={\frac {\partial W}{\partial F_{iK}}}} जहाँ F {\displaystyle {\boldsymbol {F}}} विरूपण प्रवणता है। लग्रांगियन तनाव E {\displaystyle {\boldsymbol {E}}}
P = F ⋅ ∂ W ∂ E or P i K = F i L ∂ W ∂ E L K {\displaystyle {\boldsymbol {P}}={\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}\qquad {\text{or}}\qquad P_{iK}=F_{iL}~{\frac {\partial W}{\partial E_{LK}}}~} परिमित कॉची-ग्रीन विरूपण टेन्सर C {\displaystyle {\boldsymbol {C}}}
P = 2 F ⋅ ∂ W ∂ C or P i K = 2 F i L ∂ W ∂ C L K . {\displaystyle {\boldsymbol {P}}=2~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\qquad {\text{or}}\qquad P_{iK}=2~F_{iL}~{\frac {\partial W}{\partial C_{LK}}}~.} दूसरा पियोला-किरचॉफ तनाव यदि S {\displaystyle {\boldsymbol {S}}}
S = F − 1 ⋅ ∂ W ∂ F or S I J = F I k − 1 ∂ W ∂ F k J . {\displaystyle {\boldsymbol {S}}={\boldsymbol {F}}^{-1}\cdot {\frac {\partial W}{\partial {\boldsymbol {F}}}}\qquad {\text{or}}\qquad S_{IJ}=F_{Ik}^{-1}{\frac {\partial W}{\partial F_{kJ}}}~.} लग्रांगियन तनाव के संदर्भ में
S = ∂ W ∂ E or S I J = ∂ W ∂ E I J . {\displaystyle {\boldsymbol {S}}={\frac {\partial W}{\partial {\boldsymbol {E}}}}\qquad {\text{or}}\qquad S_{IJ}={\frac {\partial W}{\partial E_{IJ}}}~.}
परिमित कॉची-ग्रीन विरूपण टेंसर के संदर्भ में
S = 2 ∂ W ∂ C or S I J = 2 ∂ W ∂ C I J . {\displaystyle {\boldsymbol {S}}=2~{\frac {\partial W}{\partial {\boldsymbol {C}}}}\qquad {\text{or}}\qquad S_{IJ}=2~{\frac {\partial W}{\partial C_{IJ}}}~.}
उपरोक्त संबंध को भौतिक विरूपण में "डॉयल-एरिक्सन सूत्र" के रूप में भी जाना जाता है।
कॉची तनाव इसी प्रकार, यह तनाव (भौतिकी) द्वारा दिया जाता है:
σ = 1 J ∂ W ∂ F ⋅ F T ; J := det F or σ i j = 1 J ∂ W ∂ F i K F j K . {\displaystyle {\boldsymbol {\sigma }}={\frac {1}{J}}~{\frac {\partial W}{\partial {\boldsymbol {F}}}}\cdot {\boldsymbol {F}}^{\textsf {T}}~;~~J:=\det {\boldsymbol {F}}\qquad {\text{or}}\qquad \sigma _{ij}={\frac {1}{J}}~{\frac {\partial W}{\partial F_{iK}}}~F_{jK}~.}
लग्रांगियन ग्रीन तनाव के संदर्भ में
σ = 1 J F ⋅ ∂ W ∂ E ⋅ F T or σ i j = 1 J F i K ∂ W ∂ E K L F j L . {\displaystyle {\boldsymbol {\sigma }}={\frac {1}{J}}~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}\cdot {\boldsymbol {F}}^{\textsf {T}}\qquad {\text{or}}\qquad \sigma _{ij}={\frac {1}{J}}~F_{iK}~{\frac {\partial W}{\partial E_{KL}}}~F_{jL}~.}
परिमित सही कॉची-ग्रीन विरूपण टेंसर के संदर्भ में
σ = 2 J F ⋅ ∂ W ∂ C ⋅ F T or σ i j = 2 J F i K ∂ W ∂ C K L F j L . {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{J}}~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{\textsf {T}}\qquad {\text{or}}\qquad \sigma _{ij}={\frac {2}{J}}~F_{iK}~{\frac {\partial W}{\partial C_{KL}}}~F_{jL}~.}
उपरोक्त भाव विषमदैशिक मीडिया के लिए भी मान्य हैं जिस स्थिति में, संभावित कार्य को प्रारंभिक फाइबर अभिविन्यास जैसे संदर्भ दिशात्मक राशियों पर निहित रूप से निर्भर करने के लिए समझा जाता है। समदैशिक की विशेष स्थिति में, कॉची तनाव को बाएं कॉची-ग्रीन विरूपण टेंसर के संदर्भ में निम्नानुसार व्यक्त किया जा सकता है:
[7] σ = 2 J ∂ W ∂ B ⋅ B or σ i j = 2 J B i k ∂ W ∂ B k j . {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{J}}{\frac {\partial W}{\partial {\boldsymbol {B}}}}\cdot ~{\boldsymbol {B}}\qquad {\text{or}}\qquad \sigma _{ij}={\frac {2}{J}}~B_{ik}~{\frac {\partial W}{\partial B_{kj}}}~.} असंपीड्य हाइपरलास्टिक भौतिकी एक असंपीड्य भौतिकी J := det F = 1 {\displaystyle J:=\det {\boldsymbol {F}}=1} J − 1 = 0 {\displaystyle J-1=0}
W = W ( F ) − p ( J − 1 ) {\displaystyle W=W({\boldsymbol {F}})-p~(J-1)} जहां स्थैतिक दाब
p {\displaystyle p} असंपीड्यता अवरोध को प्रयुक्त करने के लिए
लैग्रेंज गुणक के रूप में कार्य करता है। अब पिओला-किरचॉफ तनाव पहला तनाव बन गया है:
P = − p J F − T + ∂ W ∂ F = − p F − T + F ⋅ ∂ W ∂ E = − p F − T + 2 F ⋅ ∂ W ∂ C . {\displaystyle {\boldsymbol {P}}=-p~J{\boldsymbol {F}}^{-{\textsf {T}}}+{\frac {\partial W}{\partial {\boldsymbol {F}}}}=-p~{\boldsymbol {F}}^{-{\textsf {T}}}+{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}=-p~{\boldsymbol {F}}^{-{\textsf {T}}}+2~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}~.}
इस तनाव टेन्सर को बाद में किसी भी अन्य भौतिकी तनाव टेंसर में परिवर्तित किया जा सकता है, जैसे
कॉची तनाव टेन्सर जो इसके द्वारा दिया जाता है
σ = P ⋅ F T = − p 1 + ∂ W ∂ F ⋅ F T = − p 1 + F ⋅ ∂ W ∂ E ⋅ F T = − p 1 + 2 F ⋅ ∂ W ∂ C ⋅ F T . {\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {P}}\cdot {\boldsymbol {F}}^{\textsf {T}}=-p~{\boldsymbol {\mathit {1}}}+{\frac {\partial W}{\partial {\boldsymbol {F}}}}\cdot {\boldsymbol {F}}^{\textsf {T}}=-p~{\boldsymbol {\mathit {1}}}+{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}\cdot {\boldsymbol {F}}^{\textsf {T}}=-p~{\boldsymbol {\mathit {1}}}+2~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{\textsf {T}}~.} कॉची तनाव के लिए अभिव्यक्तियाँ संपीड्य समदैशिक हाइपरलास्टिक भौतिकी संपीड्य समदैशिक हाइपरलास्टिक भौतिकी के लिए कॉची तनाव को बाएं कॉची-ग्रीन विरूपण टेंसर या दाएं कॉची-ग्रीन विरूपण टेंसर के संपीड्यता के सिद्धांत के रूप में व्यक्त किया जा सकता है। यदि तनाव ऊर्जा घनत्व फलन है:
W ( F ) = W ^ ( I 1 , I 2 , I 3 ) = W ¯ ( I ¯ 1 , I ¯ 2 , J ) = W ~ ( λ 1 , λ 2 , λ 3 ) , {\displaystyle W({\boldsymbol {F}})={\hat {W}}(I_{1},I_{2},I_{3})={\bar {W}}({\bar {I}}_{1},{\bar {I}}_{2},J)={\tilde {W}}(\lambda _{1},\lambda _{2},\lambda _{3}),} तब
σ = 2 I 3 [ ( ∂ W ^ ∂ I 1 + I 1 ∂ W ^ ∂ I 2 ) B − ∂ W ^ ∂ I 2 B ⋅ B ] + 2 I 3 ∂ W ^ ∂ I 3 1 = 2 J [ 1 J 2 / 3 ( ∂ W ¯ ∂ I ¯ 1 + I ¯ 1 ∂ W ¯ ∂ I ¯ 2 ) B − 1 J 4 / 3 ∂ W ¯ ∂ I ¯ 2 B ⋅ B ] + [ ∂ W ¯ ∂ J − 2 3 J ( I ¯ 1 ∂ W ¯ ∂ I ¯ 1 + 2 I ¯ 2 ∂ W ¯ ∂ I ¯ 2 ) ] 1 = 2 J [ ( ∂ W ¯ ∂ I ¯ 1 + I ¯ 1 ∂ W ¯ ∂
इन प्रतीकों की परिभाषाओं के लिए बाएँ कॉची-ग्रीन विरूपण टेन्सर सिद्धान्त को देखें।
Proof 1
The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by
S = 2 ∂ W ∂ C {\displaystyle {\boldsymbol {S}}=2~{\frac {\partial W}{\partial {\boldsymbol {C}}}}}
where
C = F T ⋅ F {\displaystyle {\boldsymbol {C}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}} is the
right Cauchy–Green deformation tensor and
F {\displaystyle {\boldsymbol {F}}} is the
deformation gradient . The
Cauchy stress is given by
σ = 1 J F ⋅ S ⋅ F T = 2 J F ⋅ ∂ W ∂ C ⋅ F T {\displaystyle {\boldsymbol {\sigma }}={\frac {1}{J}}~{\boldsymbol {F}}\cdot {\boldsymbol {S}}\cdot {\boldsymbol {F}}^{T}={\frac {2}{J}}~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}}
where
J = det F {\displaystyle J=\det {\boldsymbol {F}}} . Let
I 1 , I 2 , I 3 {\displaystyle I_{1},I_{2},I_{3}} be the three principal invariants of
C {\displaystyle {\boldsymbol {C}}} . Then
∂ W ∂ C = ∂ W ∂ I 1 ∂ I 1 ∂ C + ∂ W ∂ I 2 ∂ I 2 ∂ C + ∂ W ∂ I 3 ∂ I 3 ∂ C . {\displaystyle {\frac {\partial W}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial I_{1}}}~{\frac {\partial I_{1}}{\partial {\boldsymbol {C}}}}+{\frac {\partial W}{\partial I_{2}}}~{\frac {\partial I_{2}}{\partial {\boldsymbol {C}}}}+{\frac {\partial W}{\partial I_{3}}}~{\frac {\partial I_{3}}{\partial {\boldsymbol {C}}}}~.}
The
derivatives of the invariants of the symmetric tensor
C {\displaystyle {\boldsymbol {C}}} are
∂ I 1 ∂ C = 1 ; ∂ I 2 ∂ C = I 1 1 − C ; ∂ I 3 ∂ C = det ( C ) C − 1 {\displaystyle {\frac {\partial I_{1}}{\partial {\boldsymbol {C}}}}={\boldsymbol {\mathit {1}}}~;~~{\frac {\partial I_{2}}{\partial {\boldsymbol {C}}}}=I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {C}}~;~~{\frac {\partial I_{3}}{\partial {\boldsymbol {C}}}}=\det({\boldsymbol {C}})~{\boldsymbol {C}}^{-1}}
Therefore, we can write
∂ W ∂ C = ∂ W ∂ I 1 1 + ∂ W ∂ I 2 ( I 1 1 − F T ⋅ F ) + ∂ W ∂ I 3 I 3 F − 1 ⋅ F − T . {\displaystyle {\frac {\partial W}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial I_{1}}}~{\boldsymbol {\mathit {1}}}+{\frac {\partial W}{\partial I_{2}}}~(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})+{\frac {\partial W}{\partial I_{3}}}~I_{3}~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {F}}^{-T}~.}
Plugging into the expression for the Cauchy stress gives
σ = 2 J [ ∂ W ∂ I 1 F ⋅ F T + ∂ W ∂ I 2 ( I 1 F ⋅ F T − F ⋅ F T ⋅ F ⋅ F T ) + ∂ W ∂ I 3 I 3 1 ] {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{J}}~\left[{\frac {\partial W}{\partial I_{1}}}~{\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}+{\frac {\partial W}{\partial I_{2}}}~(I_{1}~{\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}-{\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T})+{\frac {\partial W}{\partial I_{3}}}~I_{3}~{\boldsymbol {\mathit {1}}}\right]}
Using the
left Cauchy–Green deformation tensor B = F ⋅ F T {\displaystyle {\boldsymbol {B}}={\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}} and noting that
I 3 = J 2 {\displaystyle I_{3}=J^{2}} , we can write
σ = 2 I 3 [ ( ∂ W ∂ I 1 + I 1 ∂ W ∂ I 2 ) B − ∂ W ∂ I 2 B ⋅ B ] + 2 I 3 ∂ W ∂ I 3 1 . {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{\sqrt {I_{3}}}}~\left[\left({\frac {\partial W}{\partial I_{1}}}+I_{1}~{\frac {\partial W}{\partial I_{2}}}\right)~{\boldsymbol {B}}-{\frac {\partial W}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+2~{\sqrt {I_{3}}}~{\frac {\partial W}{\partial I_{3}}}~{\boldsymbol {\mathit {1}}}~.}
For an
incompressible material
I 3 = 1 {\displaystyle I_{3}=1} and hence
W = W ( I 1 , I 2 ) {\displaystyle W=W(I_{1},I_{2})} .Then
∂ W ∂ C = ∂ W ∂ I 1 ∂ I 1 ∂ C + ∂ W ∂ I 2 ∂ I 2 ∂ C = ∂ W ∂ I 1 1 + ∂ W ∂ I 2 ( I 1 1 − F T ⋅ F ) {\displaystyle {\frac {\partial W}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial I_{1}}}~{\frac {\partial I_{1}}{\partial {\boldsymbol {C}}}}+{\frac {\partial W}{\partial I_{2}}}~{\frac {\partial I_{2}}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial I_{1}}}~{\boldsymbol {\mathit {1}}}+{\frac {\partial W}{\partial I_{2}}}~(I_{1}~{\boldsymbol {\mathit {1}}}-{\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}})}
Therefore, the Cauchy stress is given by
σ = 2 [ ( ∂ W ∂ I 1 + I 1 ∂ W ∂ I 2 ) B − ∂ W ∂ I 2 B ⋅ B ] − p 1 . {\displaystyle {\boldsymbol {\sigma }}=2\left[\left({\frac {\partial W}{\partial I_{1}}}+I_{1}~{\frac {\partial W}{\partial I_{2}}}\right)~{\boldsymbol {B}}-{\frac {\partial W}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]-p~{\boldsymbol {\mathit {1}}}~.}
where
p {\displaystyle p} is an undetermined pressure which acts as a
Lagrange multiplier to enforce the incompressibility constraint.
If, in addition, I 1 = I 2 {\displaystyle I_{1}=I_{2}} W = W ( I 1 ) {\displaystyle W=W(I_{1})}
∂ W ∂ C = ∂ W ∂ I 1 ∂ I 1 ∂ C = ∂ W ∂ I 1 1 {\displaystyle {\frac {\partial W}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial I_{1}}}~{\frac {\partial I_{1}}{\partial {\boldsymbol {C}}}}={\frac {\partial W}{\partial I_{1}}}~{\boldsymbol {\mathit {1}}}}
In that case the Cauchy stress can be expressed as
σ = 2 ∂ W ∂ I 1 B − p 1 . {\displaystyle {\boldsymbol {\sigma }}=2{\frac {\partial W}{\partial I_{1}}}~{\boldsymbol {B}}-p~{\boldsymbol {\mathit {1}}}~.} Proof 2
The isochoric deformation gradient is defined as F ¯ := J − 1 / 3 F {\displaystyle {\bar {\boldsymbol {F}}}:=J^{-1/3}{\boldsymbol {F}}} B ¯ := F ¯ ⋅ F ¯ T = J − 2 / 3 B {\displaystyle {\bar {\boldsymbol {B}}}:={\bar {\boldsymbol {F}}}\cdot {\bar {\boldsymbol {F}}}^{T}=J^{-2/3}{\boldsymbol {B}}} B ¯ {\displaystyle {\bar {\boldsymbol {B}}}}
I ¯ 1 = tr ( B ¯ ) = J − 2 / 3 tr ( B ) = J − 2 / 3 I 1 I ¯ 2 = 1 2 ( tr ( B ¯ ) 2 − tr ( B ¯ 2 ) ) = 1 2 ( ( J − 2 / 3 tr ( B ) ) 2 − tr ( J − 4 / 3 B 2 ) ) = J − 4 / 3 I 2 I ¯ 3 = det ( B ¯ ) = J − 6 / 3 det ( B ) = J − 2 I 3 = J − 2 J 2 = 1 {\displaystyle {\begin{aligned}{\bar {I}}_{1}&={\text{tr}}({\bar {\boldsymbol {B}}})=J^{-2/3}{\text{tr}}({\boldsymbol {B}})=J^{-2/3}I_{1}\\{\bar {I}}_{2}&={\frac {1}{2}}\left({\text{tr}}({\bar {\boldsymbol {B}}})^{2}-{\text{tr}}({\bar {\boldsymbol {B}}}^{2})\right)={\frac {1}{2}}\left(\left(J^{-2/3}{\text{tr}}({\boldsymbol {B}})\right)^{2}-{\text{tr}}(J^{-4/3}{\boldsymbol {B}}^{2})\right)=J^{-4/3}I_{2}\\{\bar {I}}_{3}&=\det({\bar {\boldsymbol {B}}})=J^{-6/3}\det({\boldsymbol {B}})=J^{-2}I_{3}=J^{-2}J^{2}=1\end{aligned}}}
The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add
J {\displaystyle J} into the fray to describe the volumetric behaviour.
To express the Cauchy stress in terms of the invariants I ¯ 1 , I ¯ 2 , J {\displaystyle {\bar {I}}_{1},{\bar {I}}_{2},J}
I ¯ 1 = J − 2 / 3 I 1 = I 3 − 1 / 3 I 1 ; I ¯ 2 = J − 4 / 3 I 2 = I 3 − 2 / 3 I 2 ; J = I 3 1 / 2 . {\displaystyle {\bar {I}}_{1}=J^{-2/3}~I_{1}=I_{3}^{-1/3}~I_{1}~;~~{\bar {I}}_{2}=J^{-4/3}~I_{2}=I_{3}^{-2/3}~I_{2}~;~~J=I_{3}^{1/2}~.}
The chain rule of differentiation gives us
∂ W ∂ I 1 = ∂ W ∂ I ¯ 1 ∂ I ¯ 1 ∂ I 1 + ∂ W ∂ I ¯ 2 ∂ I ¯ 2 ∂ I 1 + ∂ W ∂ J ∂ J ∂ I 1 = I 3 − 1 / 3 ∂ W ∂ I ¯ 1 = J − 2 / 3 ∂ W ∂ I ¯ 1 ∂ W ∂ I 2 = ∂ W ∂ I ¯ 1 ∂ I ¯ 1 ∂ I 2 + ∂ W ∂ I ¯ 2 ∂ I ¯ 2 ∂ I 2 + ∂ W ∂ J ∂ J ∂ I 2 = I 3 − 2 / 3 ∂ W ∂ I ¯ 2 = J − 4 / 3 ∂ W ∂ I ¯ 2 ∂ W ∂ I 3 = ∂ W ∂ I ¯ 1 ∂
Recall that the Cauchy stress is given by
σ = 2 I 3 [ ( ∂ W ∂ I 1 + I 1 ∂ W ∂ I 2 ) B − ∂ W ∂ I 2 B ⋅ B ] + 2 I 3 ∂ W ∂ I 3 1 . {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{\sqrt {I_{3}}}}~\left[\left({\frac {\partial W}{\partial I_{1}}}+I_{1}~{\frac {\partial W}{\partial I_{2}}}\right)~{\boldsymbol {B}}-{\frac {\partial W}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+2~{\sqrt {I_{3}}}~{\frac {\partial W}{\partial I_{3}}}~{\boldsymbol {\mathit {1}}}~.}
In terms of the invariants
I ¯ 1 , I ¯ 2 , J {\displaystyle {\bar {I}}_{1},{\bar {I}}_{2},J} we have
σ = 2 J [ ( ∂ W ∂ I 1 + J 2 / 3 I ¯ 1 ∂ W ∂ I 2 ) B − ∂ W ∂ I 2 B ⋅ B ] + 2 J ∂ W ∂ I 3 1 . {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{J}}~\left[\left({\frac {\partial W}{\partial I_{1}}}+J^{2/3}~{\bar {I}}_{1}~{\frac {\partial W}{\partial I_{2}}}\right)~{\boldsymbol {B}}-{\frac {\partial W}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+2~J~{\frac {\partial W}{\partial I_{3}}}~{\boldsymbol {\mathit {1}}}~.}
Plugging in the expressions for the derivatives of
W {\displaystyle W} in terms of
I ¯ 1 , I ¯ 2 , J {\displaystyle {\bar {I}}_{1},{\bar {I}}_{2},J} , we have
σ = 2 J [ ( J − 2 / 3 ∂ W ∂ I ¯ 1 + J − 2 / 3 I ¯ 1 ∂ W ∂ I ¯ 2 ) B − J − 4 / 3 ∂ W ∂ I ¯ 2 B ⋅ B ] + 2 J [ − 1 3 J − 2 ( I ¯ 1 ∂ W ∂ I ¯ 1 + 2 I ¯ 2 ∂ W ∂ I ¯ 2 ) + 1 2 J − 1 ∂ W ∂ J ] 1 {\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&={\frac {2}{J}}~\left[\left(J^{-2/3}~{\frac {\partial W}{\partial {\bar {I}}_{1}}}+J^{-2/3}~{\bar {I}}_{1}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)~{\boldsymbol {B}}-J^{-4/3}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]+\\&\qquad 2~J~\left[-{\frac {1}{3}}~J^{-2}~\left({\bar {I}}_{1}~{\frac {\partial W}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)+{\frac {1}{2}}~J^{-1}~{\frac {\partial W}{\partial J}}\right]~{\boldsymbol {\mathit {1}}}\end{aligned}}}
or,
σ = 2 J [ 1 J 2 / 3 ( ∂ W ∂ I ¯ 1 + I ¯ 1 ∂ W ∂ I ¯ 2 ) B − 1 J 4 / 3 ∂ W ∂ I ¯ 2 B ⋅ B ] + [ ∂ W ∂ J − 2 3 J ( I ¯ 1 ∂ W ∂ I ¯ 1 + 2 I ¯ 2 ∂ W ∂ I ¯ 2 ) ] 1 {\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&={\frac {2}{J}}~\left[{\frac {1}{J^{2/3}}}~\left({\frac {\partial W}{\partial {\bar {I}}_{1}}}+{\bar {I}}_{1}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)~{\boldsymbol {B}}-{\frac {1}{J^{4/3}}}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]\\&\qquad +\left[{\frac {\partial W}{\partial J}}-{\frac {2}{3J}}\left({\bar {I}}_{1}~{\frac {\partial W}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)\right]{\boldsymbol {\mathit {1}}}\end{aligned}}}
In terms of the deviatoric part of
B {\displaystyle {\boldsymbol {B}}} , we can write
σ = 2 J [ ( ∂ W ∂ I ¯ 1 + I ¯ 1 ∂ W ∂ I ¯ 2 ) B ¯ − ∂ W ∂ I ¯ 2 B ¯ ⋅ B ¯ ] + [ ∂ W ∂ J − 2 3 J ( I ¯ 1 ∂ W ∂ I ¯ 1 + 2 I ¯ 2 ∂ W ∂ I ¯ 2 ) ] 1 {\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&={\frac {2}{J}}~\left[\left({\frac {\partial W}{\partial {\bar {I}}_{1}}}+{\bar {I}}_{1}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)~{\bar {\boldsymbol {B}}}-{\frac {\partial W}{\partial {\bar {I}}_{2}}}~{\bar {\boldsymbol {B}}}\cdot {\bar {\boldsymbol {B}}}\right]\\&\qquad +\left[{\frac {\partial W}{\partial J}}-{\frac {2}{3J}}\left({\bar {I}}_{1}~{\frac {\partial W}{\partial {\bar {I}}_{1}}}+2~{\bar {I}}_{2}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)\right]{\boldsymbol {\mathit {1}}}\end{aligned}}}
For an
incompressible material
J = 1 {\displaystyle J=1} and hence
W = W ( I ¯ 1 , I ¯ 2 ) {\displaystyle W=W({\bar {I}}_{1},{\bar {I}}_{2})} .Then
the Cauchy stress is given by
σ = 2 [ ( ∂ W ∂ I ¯ 1 + I 1 ∂ W ∂ I ¯ 2 ) B ¯ − ∂ W ∂ I ¯ 2 B ¯ ⋅ B ¯ ] − p 1 . {\displaystyle {\boldsymbol {\sigma }}=2\left[\left({\frac {\partial W}{\partial {\bar {I}}_{1}}}+I_{1}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)~{\bar {\boldsymbol {B}}}-{\frac {\partial W}{\partial {\bar {I}}_{2}}}~{\bar {\boldsymbol {B}}}\cdot {\bar {\boldsymbol {B}}}\right]-p~{\boldsymbol {\mathit {1}}}~.}
where
p {\displaystyle p} is an undetermined pressure-like Lagrange multiplier term. In addition, if
I ¯ 1 = I ¯ 2 {\displaystyle {\bar {I}}_{1}={\bar {I}}_{2}} , we have
W = W ( I ¯ 1 ) {\displaystyle W=W({\bar {I}}_{1})} and hence
the Cauchy stress can be expressed as
σ = 2 ∂ W ∂ I ¯ 1 B ¯ − p 1 . {\displaystyle {\boldsymbol {\sigma }}=2{\frac {\partial W}{\partial {\bar {I}}_{1}}}~{\bar {\boldsymbol {B}}}-p~{\boldsymbol {\mathit {1}}}~.} Proof 3
To express the Cauchy stress in terms of the stretches λ 1 , λ 2 , λ 3 {\displaystyle \lambda _{1},\lambda _{2},\lambda _{3}}
∂ λ i ∂ C = 1 2 λ i R T ⋅ ( n i ⊗ n i ) ⋅ R ; i = 1 , 2 , 3 . {\displaystyle {\frac {\partial \lambda _{i}}{\partial {\boldsymbol {C}}}}={\frac {1}{2\lambda _{i}}}~{\boldsymbol {R}}^{T}\cdot (\mathbf {n} _{i}\otimes \mathbf {n} _{i})\cdot {\boldsymbol {R}}~;~~i=1,2,3~.}
The chain rule gives
∂ W ∂ C = ∂ W ∂ λ 1 ∂ λ 1 ∂ C + ∂ W ∂ λ 2 ∂ λ 2 ∂ C + ∂ W ∂ λ 3 ∂ λ 3 ∂ C = R T ⋅ [ 1 2 λ 1 ∂ W ∂ λ 1 n 1 ⊗ n 1 + 1 2 λ 2 ∂ W ∂ λ 2 n 2 ⊗ n 2 + 1 2 λ 3 ∂ W ∂ λ 3 n 3 ⊗ n 3 ] ⋅ R {\displaystyle {\begin{aligned}{\frac {\partial W}{\partial {\boldsymbol {C}}}}&={\frac {\partial W}{\partial \lambda _{1}}}~{\frac {\partial \lambda _{1}}{\partial {\boldsymbol {C}}}}+{\frac {\partial W}{\partial \lambda _{2}}}~{\frac {\partial \lambda _{2}}{\partial {\boldsymbol {C}}}}+{\frac {\partial W}{\partial \lambda _{3}}}~{\frac {\partial \lambda _{3}}{\partial {\boldsymbol {C}}}}\\&={\boldsymbol {R}}^{T}\cdot \left[{\frac {1}{2\lambda _{1}}}~{\frac {\partial W}{\partial \lambda _{1}}}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\frac {1}{2\lambda _{2}}}~{\frac {\partial W}{\partial \lambda _{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\frac {1}{2\lambda _{3}}}~{\frac {\partial W}{\partial \lambda _{3}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}\right]\cdot {\boldsymbol {R}}\end{aligned}}}
The Cauchy stress is given by
σ = 2 J F ⋅ ∂ W ∂ C ⋅ F T = 2 J ( V ⋅ R ) ⋅ ∂ W ∂ C ⋅ ( R T ⋅ V ) {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{J}}~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}={\frac {2}{J}}~({\boldsymbol {V}}\cdot {\boldsymbol {R}})\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\cdot ({\boldsymbol {R}}^{T}\cdot {\boldsymbol {V}})}
Plugging in the expression for the derivative of
W {\displaystyle W} leads to
σ = 2 J V ⋅ [ 1 2 λ 1 ∂ W ∂ λ 1 n 1 ⊗ n 1 + 1 2 λ 2 ∂ W ∂ λ 2 n 2 ⊗ n 2 + 1 2 λ 3 ∂ W ∂ λ 3 n 3 ⊗ n 3 ] ⋅ V {\displaystyle {\boldsymbol {\sigma }}={\frac {2}{J}}~{\boldsymbol {V}}\cdot \left[{\frac {1}{2\lambda _{1}}}~{\frac {\partial W}{\partial \lambda _{1}}}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\frac {1}{2\lambda _{2}}}~{\frac {\partial W}{\partial \lambda _{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\frac {1}{2\lambda _{3}}}~{\frac {\partial W}{\partial \lambda _{3}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}\right]\cdot {\boldsymbol {V}}}
Using the
spectral decomposition of
V {\displaystyle {\boldsymbol {V}}} we have
V ⋅ ( n i ⊗ n i ) ⋅ V = λ i 2 n i ⊗ n i ; i = 1 , 2 , 3. {\displaystyle {\boldsymbol {V}}\cdot (\mathbf {n} _{i}\otimes \mathbf {n} _{i})\cdot {\boldsymbol {V}}=\lambda _{i}^{2}~\mathbf {n} _{i}\otimes \mathbf {n} _{i}~;~~i=1,2,3.}
Also note that
J = det ( F ) = det ( V ) det ( R ) = det ( V ) = λ 1 λ 2 λ 3 . {\displaystyle J=\det({\boldsymbol {F}})=\det({\boldsymbol {V}})\det({\boldsymbol {R}})=\det({\boldsymbol {V}})=\lambda _{1}\lambda _{2}\lambda _{3}~.}
Therefore, the expression for the Cauchy stress can be written as
σ = 1 λ 1 λ 2 λ 3 [ λ 1 ∂ W ∂ λ 1 n 1 ⊗ n 1 + λ 2 ∂ W ∂ λ 2 n 2 ⊗ n 2 + λ 3 ∂ W ∂ λ 3 n 3 ⊗ n 3 ] {\displaystyle {\boldsymbol {\sigma }}={\frac {1}{\lambda _{1}\lambda _{2}\lambda _{3}}}~\left[\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda _{2}~{\frac {\partial W}{\partial \lambda _{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}\right]}
For an
incompressible material
λ 1 λ 2 λ 3 = 1 {\displaystyle \lambda _{1}\lambda _{2}\lambda _{3}=1} and hence
W = W ( λ 1 , λ 2 ) {\displaystyle W=W(\lambda _{1},\lambda _{2})} . Following Ogden
[1] p. 485, we may write
σ = λ 1 ∂ W ∂ λ 1 n 1 ⊗ n 1 + λ 2 ∂ W ∂ λ 2 n 2 ⊗ n 2 + λ 3 ∂ W ∂ λ 3 n 3 ⊗ n 3 − p 1 {\displaystyle {\boldsymbol {\sigma }}=\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda _{2}~{\frac {\partial W}{\partial \lambda _{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}-p~{\boldsymbol {\mathit {1}}}~}
Some care is required at this stage because, when an eigenvalue is repeated, it is in general only
Gateaux differentiable , but not
Fréchet differentiable .
[8] [9] A rigorous
tensor derivative can only be found by solving another eigenvalue problem.
If we express the stress in terms of differences between components,
σ 11 − σ 33 = λ 1 ∂ W ∂ λ 1 − λ 3 ∂ W ∂ λ 3 ; σ 22 − σ 33 = λ 2 ∂ W ∂ λ 2 − λ 3 ∂ W ∂ λ 3 {\displaystyle \sigma _{11}-\sigma _{33}=\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}-\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}~;~~\sigma _{22}-\sigma _{33}=\lambda _{2}~{\frac {\partial W}{\partial \lambda _{2}}}-\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}}
If in addition to incompressibility we have
λ 1 = λ 2 {\displaystyle \lambda _{1}=\lambda _{2}} then a possible solution to the problem
requires
σ 11 = σ 22 {\displaystyle \sigma _{11}=\sigma _{22}} and we can write the stress differences as
σ 11 − σ 33 = σ 22 − σ 33 = λ 1 ∂ W ∂ λ 1 − λ 3 ∂ W ∂ λ 3 {\displaystyle \sigma _{11}-\sigma _{33}=\sigma _{22}-\sigma _{33}=\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}-\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}} असंपीड्य समदैशिक हाइपरलास्टिक भौतिकी असंपीड्य समदैशिक हाइपरलास्टिक भौतिकी के लिए, तनाव ऊर्जा घनत्व कार्य W ( F ) = W ^ ( I 1 , I 2 ) {\displaystyle W({\boldsymbol {F}})={\hat {W}}(I_{1},I_{2})}
σ = − p 1 + 2 [ ( ∂ W ^ ∂ I 1 + I 1 ∂ W ^ ∂ I 2 ) B − ∂ W ^ ∂ I 2 B ⋅ B ] = − p 1 + 2 [ ( ∂ W ∂ I ¯ 1 + I 1 ∂ W ∂ I ¯ 2 ) B ¯ − ∂ W ∂ I ¯ 2 B ¯ ⋅ B ¯ ] = − p 1 + λ 1 ∂ W ∂ λ 1 n 1 ⊗ n 1 + λ 2 ∂ W ∂ λ 2 n 2 ⊗ n 2 + λ 3 ∂ W ∂ λ 3 n 3 ⊗ n 3 {\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}&=-p~{\boldsymbol {\mathit {1}}}+2\left[\left({\frac {\partial {\hat {W}}}{\partial I_{1}}}+I_{1}~{\frac {\partial {\hat {W}}}{\partial I_{2}}}\right){\boldsymbol {B}}-{\frac {\partial {\hat {W}}}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]\\&=-p~{\boldsymbol {\mathit {1}}}+2\left[\left({\frac {\partial W}{\partial {\bar {I}}_{1}}}+I_{1}~{\frac {\partial W}{\partial {\bar {I}}_{2}}}\right)~{\bar {\boldsymbol {B}}}-{\frac {\partial W}{\partial {\bar {I}}_{2}}}~{\bar {\boldsymbol {B}}}\cdot {\bar {\boldsymbol {B}}}\right]\\&=-p~{\boldsymbol {\mathit {1}}}+\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda _{2}~{\frac {\partial W}{\partial \lambda _{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}\end{aligned}}}
जहाँ
p {\displaystyle p} एक अनिश्चित दाब है। तनाव के संदर्भ में
σ 11 − σ 33 = λ 1 ∂ W ∂ λ 1 − λ 3 ∂ W ∂ λ 3 ; σ 22 − σ 33 = λ 2 ∂ W ∂ λ 2 − λ 3 ∂ W ∂ λ 3 {\displaystyle \sigma _{11}-\sigma _{33}=\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}-\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}~;~~\sigma _{22}-\sigma _{33}=\lambda _{2}~{\frac {\partial W}{\partial \lambda _{2}}}-\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}}
यदि इसके अतिरिक्त
I 1 = I 2 {\displaystyle I_{1}=I_{2}} तब,
σ = 2 ∂ W ∂ I 1 B − p 1 . {\displaystyle {\boldsymbol {\sigma }}=2{\frac {\partial W}{\partial I_{1}}}~{\boldsymbol {B}}-p~{\boldsymbol {\mathit {1}}}~.}
यदि
λ 1 = λ 2 {\displaystyle \lambda _{1}=\lambda _{2}} , तब
σ 11 − σ 33 = σ 22 − σ 33 = λ 1 ∂ W ∂ λ 1 − λ 3 ∂ W ∂ λ 3 {\displaystyle \sigma _{11}-\sigma _{33}=\sigma _{22}-\sigma _{33}=\lambda _{1}~{\frac {\partial W}{\partial \lambda _{1}}}-\lambda _{3}~{\frac {\partial W}{\partial \lambda _{3}}}} रैखिक प्रत्यास्थता के साथ संगतता रैखिक प्रत्यास्थता के साथ संगतता का उपयोग प्रायः हाइपरलास्टिक भौतिकी मॉडल के कुछ मापदंडों को निर्धारित करने के लिए किया जाता है। इन संगतता स्थितियों को हुक के सिद्धान्त की तुलना छोटे तनाव पर रैखिककृत अतिप्रत्यास्थता के साथ प्रयोग करके प्राप्त किया जा सकता है।
संपीड्य प्रत्यास्थ भौतिकी मॉडल के लिए संगतता की स्थिति संपीड्य प्रत्यास्थ भौतिकी मॉडल के लिए संपीड्य रैखिक प्रत्यास्थता के अनुरूप होने के लिए, किरचॉफ तनाव और लग्रांगियन विरूपण के संबंध में अतिसूक्ष्म तनाव सिद्धांत सीमा में निम्न रूप होना चाहिए:
σ = λ t r ( ε ) 1 + 2 μ ε {\displaystyle {\boldsymbol {\sigma }}=\lambda ~\mathrm {tr} ({\boldsymbol {\varepsilon }})~{\boldsymbol {\mathit {1}}}+2\mu {\boldsymbol {\varepsilon }}} जहाँ
λ , μ {\displaystyle \lambda ,\mu } "लमे" स्थिरांक हैं। उपरोक्त संबंध के अनुरूप तनाव ऊर्जा घनत्व कार्य है:
[1] W = 1 2 λ [ t r ( ε ) ] 2 + μ t r ( ε 2 ) . {\displaystyle W={\tfrac {1}{2}}\lambda ~[\mathrm {tr} ({\boldsymbol {\varepsilon }})]^{2}+\mu ~\mathrm {tr} {\mathord {\left({\boldsymbol {\varepsilon }}^{2}\right)}}.} एक असंपीड्य भौतिकी के लिए
t r ( ε ) = 0 {\displaystyle \mathrm {tr} ({\boldsymbol {\varepsilon }})=0} और
W = μ t r ( ε 2 ) . {\displaystyle W=\mu ~\mathrm {tr} {\mathord {\left({\boldsymbol {\varepsilon }}^{2}\right)}}.}
किसी भी नाव ऊर्जा घनत्व फलन
W ( λ 1 , λ 2 , λ 3 ) {\displaystyle W(\lambda _{1},\lambda _{2},\lambda _{3})} के लिए छोटे लग्रांगियन तनाव विरूपण के लिए उपरोक्त रूपों को कम करने के लिए निम्नलिखित शर्तों को पूर्ण करना आवश्यक होता है।
[1] W ( 1 , 1 , 1 ) = 0 ; ∂ W ∂ λ i ( 1 , 1 , 1 ) = 0 ∂ 2 W ∂ λ i ∂ λ j ( 1 , 1 , 1 ) = λ + 2 μ δ i j {\displaystyle {\begin{aligned}&W(1,1,1)=0~;~~{\frac {\partial W}{\partial \lambda _{i}}}(1,1,1)=0\\&{\frac {\partial ^{2}W}{\partial \lambda _{i}\partial \lambda _{j}}}(1,1,1)=\lambda +2\mu \delta _{ij}\end{aligned}}}
यदि भौतिकी असंपीड्य है, तो उपरोक्त शर्तों को निम्नलिखित रूप में व्यक्त किया जा सकता है:
W ( 1 , 1 , 1 ) = 0 ∂ W ∂ λ i ( 1 , 1 , 1 ) = ∂ W ∂ λ j ( 1 , 1 , 1 ) ; ∂ 2 W ∂ λ i 2 ( 1 , 1 , 1 ) = ∂ 2 W ∂ λ j 2 ( 1 , 1 , 1 ) ∂ 2 W ∂ λ i ∂ λ j ( 1 , 1 , 1 ) = i n d e p e n d e n t o f i , j ≠ i ∂ 2 W ∂ λ i 2 ( 1 , 1 , 1 ) − ∂ 2 W ∂ λ i ∂ λ j ( 1 , 1 , 1 ) + ∂ W ∂ λ i ( 1 , 1 , 1 ) = 2 μ ( i ≠ j ) {\displaystyle {\begin{aligned}&W(1,1,1)=0\\&{\frac {\partial W}{\partial \lambda _{i}}}(1,1,1)={\frac {\partial W}{\partial \lambda _{j}}}(1,1,1)~;~~{\frac {\partial ^{2}W}{\partial \lambda _{i}^{2}}}(1,1,1)={\frac {\partial ^{2}W}{\partial \lambda _{j}^{2}}}(1,1,1)\\&{\frac {\partial ^{2}W}{\partial \lambda _{i}\partial \lambda _{j}}}(1,1,1)=\mathrm {independentof} ~i,j\neq i\\&{\frac {\partial ^{2}W}{\partial \lambda _{i}^{2}}}(1,1,1)-{\frac {\partial ^{2}W}{\partial \lambda _{i}\partial \lambda _{j}}}(1,1,1)+{\frac {\partial W}{\partial \lambda _{i}}}(1,1,1)=2\mu ~~(i\neq j)\end{aligned}}}
इन स्थितियों का उपयोग किसी दिए गए अतिप्रत्यास्थ मॉडल, कर्तनी मॉडल और स्थूल मोडुली के पैरामीटर के बीच संबंधों को खोजने के लिए किया जा सकता है।
असंपीड्य I 1 कई इलास्टोमर्स को तनाव ऊर्जा घनत्व फलन द्वारा पर्याप्त रूप से तैयार किया जाता है जो केवल I 1 {\displaystyle I_{1}} W = W ( I 1 ) {\displaystyle W=W(I_{1})} I 1 = 3 , λ i = λ j = 1 {\displaystyle I_{1}=3,\lambda _{i}=\lambda _{j}=1}
W ( I 1 ) | I 1 = 3 = 0 and ∂ W ∂ I 1 | I 1 = 3 = μ 2 . {\displaystyle \left.W(I_{1})\right|_{I_{1}=3}=0\quad {\text{and}}\quad \left.{\frac {\partial W}{\partial I_{1}}}\right|_{I_{1}=3}={\frac {\mu }{2}}\,.} ऊपर दी गई दूसरी संगतता की स्थिति को ध्यान में रखते हुए प्राप्त किया जा सकता है कि
∂ W ∂ λ i = ∂ W ∂ I 1 ∂ I 1 ∂ λ i = 2 λ i ∂ W ∂ I 1 and ∂ 2 W ∂ λ i ∂ λ j = 2 δ i j ∂ W ∂ I 1 + 4 λ i λ j ∂ 2 W ∂ I 1 2 . {\displaystyle {\frac {\partial W}{\partial \lambda _{i}}}={\frac {\partial W}{\partial I_{1}}}{\frac {\partial I_{1}}{\partial \lambda _{i}}}=2\lambda _{i}{\frac {\partial W}{\partial I_{1}}}\quad {\text{and}}\quad {\frac {\partial ^{2}W}{\partial \lambda _{i}\partial \lambda _{j}}}=2\delta _{ij}{\frac {\partial W}{\partial I_{1}}}+4\lambda _{i}\lambda _{j}{\frac {\partial ^{2}W}{\partial I_{1}^{2}}}\,.}
इन संबंधों को तब समदैशिक असंपीड्य हाइपरलास्टिक भौतिकी के लिए संगतता की स्थिति में प्रतिस्थापित किया जा सकता है।
संदर्भ
↑ 1.0 1.1 1.2 1.3 1.4 R.W. Ogden, 1984, Non-Linear Elastic Deformations , ISBN 0-486-69648-0 , Dover.
↑ Muhr, A. H. (2005). "Modeling the stress–strain behavior of rubber". Rubber Chemistry and Technology . 78 (3): 391–425. doi :10.5254/1.3547890 . ↑ Gao, H; Ma, X; Qi, N; Berry, C; Griffith, BE; Luo, X (2014). "द्रव-संरचना अंतःक्रिया के साथ एक परिमित तनाव अरैखिक मानव माइट्रल वाल्व मॉडल" . Int J Numer Method Biomed Eng . 30 (12): 1597–613. doi :10.1002/cnm.2691 . PMC 4278556 PMID 25319496 . ↑ Jia, F; Ben Amar, M; Billoud, B; Charrier, B (2017). "Morphoelasticity in the development of brown alga Ectocarpus siliculosus : from cell rounding to branching" . J R Soc Interface . 14 (127): 20160596. doi :10.1098/rsif.2016.0596 . PMC 5332559 PMID 28228537 . ↑ Arruda, E.M.; Boyce, M.C. (1993). "रबर लोचदार सामग्री के बड़े खिंचाव व्यवहार के लिए एक त्रि-आयामी मॉडल" (PDF) . J. Mech. Phys. Solids . 41 : 389–412. doi :10.1016/0022-5096(93)90013-6 . S2CID 136924401 . ↑ Buche, M.R.; Silberstein, M.N. (2020). "Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble". Phys. Rev. E . 102 (1): 012501. arXiv :2004.07874 doi :10.1103/PhysRevE.102.012501 . PMID 32794915 . S2CID 215814600 . ↑ Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.
↑ Fox & Kapoor, Rates of change of eigenvalues and eigenvectors , AIAA Journal , 6 (12) 2426–2429 (1968)
↑ Friswell MI. The derivatives of repeated eigenvalues and their associated eigenvectors. Journal of Vibration and Acoustics (ASME) 1996; 118:390–397.
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