Bilinear Isotropic Hardening Behavior

MAE 5700 Final ProjectRaghavendar RanganathanBradly VerdantRanny 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

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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

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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 }

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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?