Bilinear Isotropic Hardening
Behavior
MAE 5700 Final Project
Raghavendar Ranganathan
Bradly Verdant
Ranny Zhao
• Problem Statement
• Illustration of bilinear isotropic hardening plasticity with an example of an
interference fit between a shaft and a bushing assembly
• Plasticity Model
• Yield criterion
• Flow rule
• Hardening rule
• Governing Equations
• Numerical Implementation
• FE Results
Overview
2
Elastic-Plastic Analysis
Elastic Analysis
Quarter model-Plane Stress- interference with an outer rigid body
Elastic Plastic Behavior
3
True Stress vs. True Strain curve
Material Curve
Bilinear: Approximation of the more realistic
multi-linear stress-strain relation
4
• Determines the stress levels at which yield will be initiated
• Given by 𝜎𝑒 =f({𝜎})
• 𝑖𝑓 𝜎𝑒
<𝜎𝑌 ,
=𝜎𝑌 ,
>𝜎𝑌 ,
𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑜𝑛𝑙𝑦
𝑝𝑙𝑎𝑠𝑡𝑖𝑐 𝑑𝑒𝑓𝑜𝑟𝑚𝑎𝑡𝑖𝑜𝑛
𝑛𝑜𝑡 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒
• Written in general as F(𝜎, 𝜖 𝑃𝐿 ) = 0 where F = 𝜎𝑒 -𝜎𝑌
• 𝜎𝑒 =
• 𝐽2 =
3𝐽2 for isotropic hardening (von Mises stress)
1
6
𝜎11 − 𝜎22
2
+ 𝜎22 − 𝜎33
2
+ 𝜎33 − 𝜎11
2
2
2
2
+ 𝛾12
+ 𝛾23
+𝛾31
• 𝜎𝑌 is function of accumulated plastic strain
• For Bilinear: 𝜎𝑌 = 𝜎𝑜 + ℎ ∗ 𝜖𝑃𝐿
Yield Criterion
5
𝐹 𝜎, 𝜖 𝑃𝐿 = 0
𝐹=
3𝐽2 − 𝜎𝑌 = 0
(isotropic hardening)
Yield Surface
6
•
•
𝜕𝑄
𝑑𝜖 = 𝜆{ }
𝜕𝜎
𝜕𝑄
Where { } indicates
𝜕𝜎
𝑃𝐿
the direction
of plastic straining, and 𝜆 is the
magnitude of plastic deformation
• Occurs when 𝜎𝑒 = 𝜎𝑌
• Plastic potential (Q) – a scalar value
function
of
stress
tensor
components and is similar to yield
surface F
• Associative rule: F = Q
Flow Rule (plastic straining)
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• Description of changing of yield surface with progressive yielding
• Allows the yield surface to expand and change shape as the material is
plastically loaded
Plastic
Yield Surface after Loading
Elastic
Initial Yield Surface
Hardening Rule
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Hardening Types
1. Isotropic Hardening
2
Subsequent Yield
Surface
2. Kinematic Hardening
Subsequent Yield Surface
2
Initial Yield
Surface
Initial Yield
Surface
1
1
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• Yield criterion changes with hardening
• Yield surface will expand such that 𝜎𝑒 = 𝜎𝑌
• The 2 values will converge until 𝜎𝑒 can’t be outside of 𝜎𝑌 anymore
• During hardening, stresses should always lie on the yield surface
• 𝑑𝐹 = 0
• 𝑑𝐹 =
𝜕𝐹 𝑇
𝜕𝜎
𝑑𝜎 +
𝜕𝐹 𝑇
𝜕𝜖𝑃𝐿
𝑑𝜖 𝑃𝐿 = 0
Consistency Condition
10
• Strong form
• 𝛻𝑠𝑇 𝜎 + 𝑏 = 0
• Weak form
•
𝑛𝑒𝑙
𝑒=1 Ω
𝛻𝑠 𝑤
𝑇
𝐷𝐸𝑃𝐿 𝛻𝑠 𝑢 𝑑Ω =
𝑛𝑒𝑙
𝑇
𝑒=1 Γ 𝑤
𝑡 𝑑Γ +
𝜎 = 𝐷 𝐸𝑃𝐿 𝜖
• 𝜖 = 𝛻𝑠 𝑢 = [B]d
𝑛𝑒𝑙
𝑇
𝑒=1 Ω 𝑤
𝑏 𝑑Ω
• Matrix form
• 𝑤𝑇
𝑛𝑒𝑙 𝑇
𝑒=1 𝐿 { Ω
• 𝐾 =
• 𝑓 =
𝑇
𝐵
𝑇
𝐷𝐸𝑃𝐿 𝐵 𝑑Ω 𝐿𝑑 −
𝐷 𝐸𝑃𝐿 𝐵 𝑑Ω
Ω
𝐵
Γ
𝑁 𝑇 𝑡 𝑑Γ +
Ω
• Where 𝐷𝐸𝑃𝐿 = 𝐷 −
Γ
𝑁 𝑇 𝑡 𝑑Γ −
Ω
𝑁 𝑇 𝑏 𝑑Ω} = 0
𝑁 𝑇 𝑏 𝑑Ω
𝜕𝑓 𝜕𝑓 𝑇
𝐷
𝐷
𝜕𝜎 𝜕𝜎
𝜕𝑓 𝑇
𝜕𝑓
𝐷
𝜕𝜎
𝜕𝜎
Governing Equations
11
Stress and strain states at load step ‘n’ at disposal
The material yield from previous step is used as
basis
Load step ‘n+1’ with Δ𝐹 load increment
Compute 𝜖𝑡𝑟 from Δ𝑢 and 𝜎𝑡𝑟 from 𝜖𝑡𝑟
Trail Displacement Δ𝑢 || Updated Displacement Δ𝑢new
If 𝜎𝑒 < 𝜎𝑌 : 𝑁𝑜 𝑃𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
Compute 𝜎𝑒
If 𝜎𝑒 ≥ 𝜎𝑌 ∶ 𝑃𝑙𝑎𝑠𝑡𝑖𝑐𝑖𝑡𝑦
Compute 𝜆 using NRI such that dF = 0
Compute restoring forces and Residual
𝜕F
Δ𝜖 𝑃𝐿 = 𝜆{ }
𝜕𝜎
Perform Newton Rapshon iterations for equilibrium by updating Δ𝑢
𝑃𝐿
𝜖𝑛𝑃𝐿 = {𝜖𝑛−1
} + 𝑑𝜖 𝑃𝐿
Update stresses and strains
𝜖 𝑒𝑙 = {𝜖 𝑡𝑟 } + 𝑑𝜖 𝑃𝐿
Proceed to next load step
𝜎 = 𝐷 𝜖 𝑒𝑙
Implementation
𝑃𝐿
𝜎𝑌 = 𝑓(𝜖 )
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Elastic-Plastic Analysis
Elastic Analysis
Geometry: Quarter model- OD = 10in; ID = 6in; Boundary-Rigid- OD=9.9in
Material: E=30e6psi; 𝜈=0.3; 𝜎𝑌 = 36300psi; 𝐸 𝑇 = 75000psi (tangent modulus)
ANSYS RESULTS- Von Mises Stress
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Elastic-Plastic Analysis
Elastic Analysis
ANSYS Results- Radial Stress (X-Plot)
14
Elastic-Plastic Analysis
Elastic Analysis
ANSYS Results- Hoop Stress (Y-Plot) 15
Elastic-Plastic
Elastic-Plastic Analysis
Analysis
Elastic
Elastic Analysis
Analysis
ANSYS Results- Deformation
16
• Question?
Thank You
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