1 0331 57 Mechanics of Material Systems Mechanics

Part IV: Plasticity and Yield Design 8. Plasticity Models 1. 033/1. 57: Mechanics of

Content 1. 033/1. 57 Part I. Deformation and Strain 1 Description of Finite Deformation

1 D → Stress Strain Plastic Strain Hardening Variable Hardening Force etc 1. 033/1.

1 D → 3 D Extension Stress Strain Plastic Strain Stress Tensor Strain Plastic

Convexity of Elasticity Domain 1. 033/1. 57: Mechanics of Material Systems

Example: Crystal Structure of Steel Impurity = defects in the crystal structure / network

Sliding in a Monocrystal (“Kinematics”) Plastic Elastic When ? 1. 033/1. 57: Mechanics of

Stress Vector on Deviator Plane Orientation of hydrostatic axis 1. 033/1. 57: Mechanics of

Von-Mises Plasticity: Yield Criterion Principal Stress Space Deviator Plane 1. 033/1. 57: Mechanics of

Von-Mises Plasticity: “Kinematics” Shear Test During Plastic Loading 1. 033/1. 57: Mechanics of Material

Drucker-Prager Plasticity: Yield Criterion 1. 033/1. 57: Mechanics of Material Systems

Drucker-Prager Plasticity: “Kinematics” During Plastic Loading After Unloading 1. 033/1. 57: Mechanics of Material

Drucker-Prager Plasticity: Thermodynamic Restrictions Thermodynamically Admissible Associated Plasticity Non-Associated Plasticity 1. 033/1. 57: Mechanics

Plastic Hardening Models Isotropic Hardening Kinematical Hardening 1. 033/1. 57: Mechanics of Material Systems

Cam-Clay Model: Yield Criterion 1. 033/1. 57: Mechanics of Material Systems

Cam-Clay Model: “Kinematics” 1. 033/1. 57: Mechanics of Material Systems

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1. 033/1. 57 Mechanics of Material Systems (Mechanics and Durability of Solids I) Franz-Josef Ulm Lecture: MWF 1 // Recitation: F 3: 00 -4: 30 1. 033/1. 57: Mechanics of Material Systems

Part IV: Plasticity and Yield Design 8. Plasticity Models 1. 033/1. 57: Mechanics of Material Systems

Content 1. 033/1. 57 Part I. Deformation and Strain 1 Description of Finite Deformation 2 Infinitesimal Deformation Part II. Momentum Balance and Stresses 3 Momentum Balance 4 Stress States / Failure Criterion Part III. Elasticity and Elasticity Bounds 5 Thermoelasticity, 6 Variational Methods Part IV. Plasticity and Yield Design 7 1 D-Plasticity – An Energy Approach 8 Plasticity Models 9 Limit Analysis and Yield Design 1. 033/1. 57: Mechanics of Material Systems

1 D → Stress Strain Plastic Strain Hardening Variable Hardening Force etc 1. 033/1. 57: Mechanics of Material Systems

1 D → 3 D Extension Stress Strain Plastic Strain Stress Tensor Strain Plastic Strain Tensor Hardening Variable Hardening Force 1. 033/1. 57: Mechanics of Material Systems

Convexity of Elasticity Domain 1. 033/1. 57: Mechanics of Material Systems

Example: Crystal Structure of Steel Impurity = defects in the crystal structure / network Overcome an Obstacle “when? ” Slippage planes = Direction of permanent deformation “how? ” 1. 033/1. 57: Mechanics of Material Systems

Sliding in a Monocrystal (“Kinematics”) Plastic Elastic When ? 1. 033/1. 57: Mechanics of Material Systems How ?

Stress Vector on Deviator Plane Orientation of hydrostatic axis 1. 033/1. 57: Mechanics of Material Systems

Von-Mises Plasticity: Yield Criterion Principal Stress Space Deviator Plane 1. 033/1. 57: Mechanics of Material Systems

Von-Mises Plasticity: “Kinematics” Shear Test During Plastic Loading 1. 033/1. 57: Mechanics of Material Systems After Unloading

Drucker-Prager Plasticity: Yield Criterion 1. 033/1. 57: Mechanics of Material Systems

Drucker-Prager Plasticity: “Kinematics” During Plastic Loading After Unloading 1. 033/1. 57: Mechanics of Material Systems Plastic Dilatation

Drucker-Prager Plasticity: Thermodynamic Restrictions Thermodynamically Admissible Associated Plasticity Non-Associated Plasticity 1. 033/1. 57: Mechanics of Material Systems

Plastic Hardening Models Isotropic Hardening Kinematical Hardening 1. 033/1. 57: Mechanics of Material Systems

Cam-Clay Model: Yield Criterion 1. 033/1. 57: Mechanics of Material Systems

Cam-Clay Model: “Kinematics” 1. 033/1. 57: Mechanics of Material Systems