Motion and Stress Analysis by Vector Mechanics Edward C. Ting Professor Emeritus of Applied Mechanics Purdue University, West Lafayette, IN National Central University, ChungLi, Taiwan

a computer framework for the study of a multi-component structural system with component motion component interactions: connection, contact, collision, penetration geometrical changes: deformation, displacement, fragmentation, collapse stress distribution behavior and material property changes

a physics approach of mechanics motion analysis and VFIFE * vector mechanics ---- particle mechanics * discrete description * intrinsic finite element ---- physical structural element

example: a rod in plane motion Newton’s law 1. displacement is a motion

analytical mechanics: For motion analysis, assume 1. rigid body, 2. functional description

pendulum problem: hinged at end 1 motion analysis 1. general formulation: 2. complete formulation :

stress analysis assume: 1. deformable body, 2. Hooke’s law

1. An approximation → separate motion analysis and stress analysis ► continuous bodies: motion--rigid body; stress--deformable body ► variables: motion--displacement; stress--deformation ► governing equations: motion--translation and rotation; stress--equilibrium 2. Described by continuous functions → discretization computation based on analytical mechanics

1. Newton’s law 2. behavior model 3. kinematics 4. Hooke’s law 5. pendulum: constraint conditions hinged end: straight rod: vector mechanics

properties: 1. structure: a set of particles 2. always a dynamic process 3. always deformable advantages: 1. suitable for computation 2. a general and systematic formulation 3. explicit constraint conditions

development needs: 1. describe structural geometry: intrinsic finite element 2. kinematics: fictitious reversed motion 3. continuity requirements 4. mechanics requirements 5. material model: standard tests elements: plane rod, plane frame, plane solid, space rod, space frame, 3d membrane, 3d solid, 3d plate shell V-5 research group: e. c. ting, c. y. wang, t. y. wu, r. z. wang, c. j. chuang

motion analysis procedure: a simple rod structure

discrete model: mass particles and structural elements

vector form equation of motion

path element 1. element geometry remains unchanged 2. small deformation discrete path:

kinematics and force calculation 1 material frame: configuration at 2. variable: nodal deformation 3. fictitious reversed motion to define deformation 4. infinitesimal strain and engineering stress 5. nodal forces: use finite element 6. internal forces are in equilibrium

reversed motion for nodal deformation

governing equations

difference equation (symmetrical case)

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4 node plane solid element

estimate the rigid body motion

fictitious reversed motion

nodal deformation

deformation coordinates to define independent variables

shape functions:

nodal forces

stress

Thank You