Nonlinear Elasticity and Plasticity - TAMU...
Transcript of Nonlinear Elasticity and Plasticity - TAMU...
JN Reddy Nonlin Elast Plastcity: 1
Nonlinear Elasticity and Plasticity
• Nonlinear elasticity• Plasticity• Ideal plasticity and strain hardening plasticity• Stress-strain curves• Finite element models of nonlinear elasticity• Numerical examples• Small deformation theory of plasticity• Finite element formulation• Numerical examples
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Nonlinear Elasticity and Plasticity
Plasticity
Stress
Strain
Loading
Unloading
Plastic strain
Nonlinear Elasticity
Stress
Strain
Linear elastic
Non-linear elastic
Loading and unloading
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Ideal Plasticity and Strain Hardening Plasticity
Ideal (or Perfect) Plasticity
Stress
Strain
Loading
Unloading
Strain Hardening PlasticityStress
Strain
Plastic strain
Loading
Unloading
Nonlin Elast Plastcity: 3
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Ideal Plasticity and Strain Hardening Plasticity
Stress
Strain
dσp
EσY
dεdεp
dεe
Slope ET - Elastic-plastic tangent modulus
EffectiveStress, σ
Plastic Strain, εp
_Ep
σ = σ0 + Ep εp_
Nonlin Elast Plastcity: 4
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Stress-Strain Curves for Boron/Aluminum
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Stress-Strain Curves for Graphite-Epoxy
Nonlin Elast Plastcity: 6
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Stress-Strain Curves for Boron-Epoxy
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Finite Element Models of Nonlinear Elasticity
Virtual Work Statement
Nonlinear Constitutive Equation
¾xx = E F ("xx)
0 =
Z
A
Z xb
xa
¾xx±"xx dxdA ¡Z xb
xa
f±u dx ¡ P e1 ±u(xa) ¡ P e
2 ±u(xb)
=
Z xb
xa
[EA F ("xx)±"xx ¡ f±u] dx ¡ P e1 ±u(xa) ¡ P e
2 ±u(xb)
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Finite Element Models of Nonlinear Elasticity
(continued)
Finite Element Model
Rei =
Z xb
xa
·EA F ("xx)
dÃei
dx¡ fÃi
¸dx ¡ P e
i
Keij =
@Rei
@uej
= EA
Z xb
xa
@F
@"xx
@"xx
@uej
dÃei
dxdx
= EA
Z xb
xa
µ@F
@"xx
¶dÃe
i
dx
dÃej
dxdx
Ramberg-Osgood Model
F ("xx) = ("xx)n
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Numerical Examples
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Numerical Examples
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The theory of plasticity deals with an analytical description of the stress-strain relations of a deformed body after a part or all of the body has yielded.
The stress-strain relations must contain:
1. The elastic stress-strain relations.
2. The stress condition (or yield criterion) which indicates onset of yielding.
3. The stress-strain or stress-strain increment relations after the onset of plastic flow.
Small Deformation Theory of Plasticity
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Small Deformation Theory of Plasticity
Stress
Strain
dσp
EσY
dε
dεp
dεe
Slope ET - Elastic-plastic tangent modulus
¾ij < ¾Y linear elastic behavior
¾ij ¸ ¾Y plastic deformation (not recoverable)
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F (¾ij ; ·) = 0
F (J 02; J
03; ·) = 0; J 0
2 =1
2¾0
ij¾0ij ; J 0
3 =1
3¾0
ij¾0jk¾0
k`
The Tresca yield criterion:
F = 2¹¾ cos µ ¡ Y (·) = 0; ¹¾ =p
J 02
The Huber-von Mises yield criterion:
F =p
3¹¾ ¡ Y (·) = 0
Small Deformation Theory of Plasticity(continued)
General Yield Criterion
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Mathematical Models of the Strain Hardening Behavior
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Small Deformation Theory of Plasticity(continued)
Finite Element Formulation
d" = d"e + d"p; d"e =d¾
E;
d¾
d"= ET
H =d¾
d"p=
d¾d"
1 ¡ d"e
d"
=ET
1 ¡ ET
E
[Ke] =
Z xb
xa
[B]T [De][B]dx
du = hed"xx = he (d"e + d"p)
dF = Ad¾ = AeHd"p
Eep =dF
du=
AeHd"p
he (d"e + d"p)=
EAe
he
·1 ¡ E
(E + H)
¸
[Kep] =
Z xb
xa
[B]T [Dep][B]dxNonlin Elast Plastcity: 16
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Small Deformation Theory of Plasticity(continued)
Nonlin Elast Plastcity: 17
fd"g = fd"eg + fd"pg
fd"eg = [De]¡1fd¾g; fd"pg = d¸
½df
df¾g
¾
fd"eg = fd"g ¡ fd"pg
= fd"g ¡ d¸
½@f
@f¾g
¾
fd¾g = [De]
µfd"g ¡ d¸
½@f
@f¾g
¾¶
0 = f(f¾g; ·) ! 0 = df =
½@f
@f¾g
¾T
fd¾g ¡ Ad¸
0 =
½@f
@f¾g
¾T
[De]
µfd"g ¡ d¸
@f
@f¾g
¶¡ Ad¸
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Small Deformation Theory of Plasticity(continued)
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d¸ =f @f
@f¾ggT[De]fd"g
A + f @f@f¾gg[De]f @f
@f¾gg
A = H =d¹¾
d¹"p=
ET
1 ¡ ET =E
fd¾g = [Dep]fd"g
[Dep] = [De] ¡[De] @f
@f¾gf @f@f¾ggT [De]
H + f @f@f¾ggT [De]f @f
@f¾gg
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Numerical Examples
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Numerical Examples
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Numerical Examples
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Simply Supported Isotropic Plate Under UDL(Combined Material and Geometric Nonlinearity)
E = 106 psi., G = 3.846 E, ν = 0.3, ρ = 0.000259 lb-s2/in4
σi = σ0i + Epi εp (i=1,2,4,5,6); σ01 = σ02 = σ04 = 3 × 104 psiσ04 = σ05 = σ06 = 1.372 × 104 psi, p0 = 300 psi
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Summary
The following topics were discussed:
• Nonlinear elasticity• Plasticity• Ideal plasticity and strain hardening plasticity• Stress-strain curves• Finite element models of nonlinear elasticity• Numerical examples• Small deformation theory of plasticity• Finite element formulation• Numerical examples