Plasticity theoremsstaff.bath.ac.uk/abscjkw/LectureNotes/OtherNotes...Microsoft Word - Plasticity...
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1 Plasticity theorems Proof of virtual work theorem Stress σ Body load per unit volume p Boundary load per unit area F Displacement u Strain increment ε σ = σ T ∇ • σ + p = 0 within body F = n • σ on boundary ε = 1 2 ∇u + ∇u ( ) T Using divergence theorem p • udV V ∫ = − ∇ • σ ( ) • udV V ∫ = − ∇ • σ • u ( ) dV V ∫ + σ ••∇udV V ∫ = − n • σ • udA ∂V ∫ + σ •• εdV V ∫ Thus p • udV V ∫ + F • udA ∂V ∫ = σ •• εdV V ∫ . This is the virtual work theorem. Upper bound theorem Assume distributions of loads are known, but collapse load factor, λ collapse , is unknown. Thus p collapse = λ collapse p known F collapse = λ collapse F known and assume a collapse mode, u , ε :
Transcript of Plasticity theoremsstaff.bath.ac.uk/abscjkw/LectureNotes/OtherNotes...Microsoft Word - Plasticity...
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Plasticity theorems
Proof of virtual work theorem
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Stress σBody load per unit volume pBoundary load per unit area FDisplacement uStrain increment ε
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σ = σT
∇ •σ + p = 0 within bodyF = n•σ on boundary
ε =12∇u + ∇u( )
T
Using divergence theorem
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p•udVV∫ = − ∇ •σ( ) •udV
V∫
= − ∇ • σ •u( )dVV∫ + σ ••∇udV
V∫
= − n•σ •udA∂V∫ + σ ••εdV
V∫
Thus
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p•udVV∫ + F •udA
∂V∫ = σ ••εdV
V∫ .
This is the virtual work theorem.
Upper bound theorem
Assume distributions of loads are known, but collapse load factor,
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λcollapse, is unknown.
Thus
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pcollapse = λcollapsepknownFcollapse = λcollapseFknown
and assume a collapse mode,
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u ,
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ε :
2
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λcollapse pknown •udVV∫ + Fknown •udA
∂V∫
= σ ••εdV
V∫
Note that this will give us the CORRECT value of
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λcollapse even if we use the WRONG (virtual)
collapse mode shape PROVIDED that we use the CORRECT
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σ .
However
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σ ••εdVV∫ ≤ σYield Surface ••εdV
V∫
where
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σYield Surface is the stress corresponding
to the plastic strain increment
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ε . The
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≤ is
there if we assume the normality condition
and that the yield surface is convex.
The normality condition states that the plastic
strain increment is perpendicular to the yield
surface.
Thus if we use
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σYield Surface ••εdVV∫ on the right hand side we must obtain an upper bound for
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λcollapse.
Lower bound theorem
Postulate state of stress which satisfies
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∇ •σ + λcollapsepknown = 0 within bodyλcollapseFknown = n•σ on boundary
everywhere and does not violate the yield condition.
Then
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λcollapse pknown •udVV∫ + Fknown •udA
∂V∫
= σ ••εdV
V∫
for any (virtual) mode of deformation,
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u,
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ε .
If we have the true collapse state of stress then the right hand side is equal to
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σYield Surface ••εdVV∫ . Again using
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σ ••εdVV∫ ≤ σYield Surface ••εdV
V∫ we must now have a
lower bound on the collapse load.
