bilinear isotropic hardening behavior mae 5700 final project raghavendar ranganathan bradly verdant...
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Bilinear Isotropic Hardening Behavior
MAE 5700 Final Project
Raghavendar Ranganathan
Bradly Verdant
Ranny Zhao
2Overview
• 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
3
Elastic-Plastic Analysis Elastic Analysis
Quarter model-Plane Stress- interference with an outer rigid body
Elastic Plastic Behavior
4Material Curve
Bilinear: Approximation of the more realistic multi-linear stress-strain relationTrue Stress vs. True Strain curve
5Yield Criterion
• Determines the stress levels at which yield will be initiated• Given by f({
• Written in general as F() = 0 where F = -• for isotropic hardening (von Mises stress)• +
• is function of accumulated plastic strain• For Bilinear:
6Yield Surface
(isotropic hardening)
7
Flow Rule (plastic straining)
• Where indicates the direction of plastic straining, and is the magnitude of plastic deformation
• Occurs when • Plastic potential (Q) – a scalar
value function of stress tensor components and is similar to yield surface F
• Associative rule: F = Q
8Hardening Rule
• Description of changing of yield surface with progressive yielding• Allows the yield surface to expand and change shape as the material is
plastically loaded
Elastic
Plastic
Elastic
Plastic
Yield Surface after Loading
Initial Yield Surface
9
Hardening Types1. Isotropic Hardening 2. Kinematic Hardening
2
Initial Yield Surface
1
Subsequent Yield Surface2
Initial Yield Surface
1
Subsequent Yield Surface
10Consistency Condition
11Governing Equations
• Strong form
• Weak form• • = [B]d
• Matrix form
•
• Where
12
Stress and strain states at load step ‘n’ at disposal
Load step ‘n+1’ with load increment
Trail Displacement Updated Displacement
Compute restoring forces and Residual
Perform Newton Rapshon iterations for equilibrium by updating
Update stresses and strains
Proceed to next load step
Implementation
The material yield from previous step is used as basis
Compute from and from
Compute
If <
If
𝜎 𝑌= 𝑓 (𝜖𝑃𝐿)
Compute using NRI such that dF = 0
Δ𝜖𝑃𝐿=𝜆 {𝜕F𝜕𝜎
}
{𝜎 }=[𝐷 ] {𝜖𝑒𝑙 }
13
ANSYS RESULTS- Von Mises Stress
Elastic-Plastic Analysis Elastic Analysis
Geometry: Quarter model- OD = 10in; ID = 6in; Boundary-Rigid- OD=9.9inMaterial: E=30e6psi; =0.3; = 36300psi; = 75000psi (tangent modulus)
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ANSYS Results- Radial Stress (X-Plot)
Elastic-Plastic Analysis Elastic Analysis
15
ANSYS Results- Hoop Stress (Y-Plot)
Elastic-Plastic Analysis Elastic Analysis
16
ANSYS Results- Deformation
Elastic-Plastic Analysis Elastic Analysis Elastic-Plastic Analysis Elastic Analysis
17Thank You
•Question?