Bilinear Isotropic Hardening Behavior

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## Bilinear Isotropic Hardening Behavior

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**Bilinear Isotropic Hardening Behavior**MAE 5700 Final Project RaghavendarRanganathan 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**Elastic Analysis**Elastic-Plastic Analysis Elastic Plastic Behavior Quarter model-Plane Stress- interference with an outer rigid body**Bilinear: Approximation of the more realistic multi-linear**stress-strain relation • True Stress vs. True Strain curve Material Curve**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: Yield Criterion**(isotropic hardening)**Yield Surface**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 Flow Rule (plastic straining)**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 Plastic Yield Surface after Loading Elastic Elastic Initial Yield Surface Hardening Rule**Subsequent Yield Surface**2 Initial Yield Surface 1 Subsequent Yield Surface 2 Initial Yield Surface 1 1. Isotropic Hardening 2. Kinematic Hardening Hardening Types**Strong form**• Weak form • = [B]d • Matrix form • Where Governing Equations**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 load increment Compute from and from Trail Displacement Updated Displacement If < Compute If Compute using NRI such that dF = 0 Compute restoring forces and Residual Perform Newton Rapshon iterations for equilibrium by updating Update stresses and strains Proceed to next load step Implementation**Elastic-Plastic Analysis**Elastic Analysis ANSYS RESULTS- Von Mises Stress Geometry: Quarter model- OD = 10in; ID = 6in; Boundary-Rigid- OD=9.9in Material: E=30e6psi; =0.3; = 36300psi; = 75000psi (tangent modulus)**Elastic-Plastic Analysis**Elastic Analysis ANSYS Results- Radial Stress (X-Plot)**Elastic-Plastic Analysis**Elastic Analysis ANSYS Results- Hoop Stress (Y-Plot)**Elastic-Plastic Analysis**Elastic Analysis Elastic Analysis Elastic-Plastic Analysis ANSYS Results- Deformation**Question?**Thank You