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1.033/1.57 Mechanics of Material Systems (Mechanics and Durability of Solids I) Franz-Josef Ulm Lecture: MWF1 // Recitation: F 3:00-4:30 1.033/1.57 Mechanics.

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Presentation on theme: "1.033/1.57 Mechanics of Material Systems (Mechanics and Durability of Solids I) Franz-Josef Ulm Lecture: MWF1 // Recitation: F 3:00-4:30 1.033/1.57 Mechanics."— Presentation transcript:

1 1.033/1.57 Mechanics of Material Systems (Mechanics and Durability of Solids I) Franz-Josef Ulm Lecture: MWF1 // Recitation: F 3:00-4: /1.57 Mechanics of Material Systems

2 Part III: Elasticity and Elasticity Bounds 6. The Theorem of Virtual Work and Variational Methods in Elasticity

3 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 1D-Plasticity – An Energy Approac 8 Plasticity Models 9 Limit Analysis and Yield Design

4 1.033/1.57 Mechanics of Material Systems Hill-Mandel Lemma Theorem of Virtual Work applied to a Heterogeneous Material System

5 1.033/1.57 Mechanics of Material Systems Convexity of a function Secant Tangent

6 1.033/1.57 Mechanics of Material Systems Convexity: Applied to Free Energy State E q u a t i o n

7 1.033/1.57 Mechanics of Material Systems Theorem of Minimum Potential Energy 1-Parameter System Upper Energy Bound

8 1.033/1.57 Mechanics of Material Systems Ex: Heterogeneous Material System I Macroscale Displacement Field (KA) Stored Energy External Work Upper Energy Bound

9 1.033/1.57 Mechanics of Material Systems Theorem of Minimum Complementary Energy 1-Parameter System Upper Energy Bound

10 1.033/1.57 Mechanics of Material Systems Ex: Heterogeneous Material System II Macroscale Complementary Energy External Work Lower Energy Bound Stress Field (SA)

11 1.033/1.57 Mechanics of Material Systems Elements of Elastic Energy Bounds Statically Admissible Stress Field Solution Elastic Material Law Kinematically Admissible Displacement Field

12 1.033/1.57 Mechanics of Material Systems Elastic Energy Bounds (Contd) Statically Admissible Stress Field Solution Elastic Material Law Kinematically Admissible Displacement Field

13 1.033/1.57 Mechanics of Material Systems Training Set: Effective Modulus Heterogeneous Microstructure Tension Sample

14 1.033/1.57 Mechanics of Material Systems Voigt-Reuss Bounds 2-Phase Material System (ν=const) VoigtReuss Voigt Reuss

15 1.033/1.57 Mechanics of Material Systems Problem Set Recitation


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