Thermodynamics of Phase Change from Solid to Liquid P M V Subbarao Professor Mechanical Engineering Department I I T Delhi A shift from cutting to casting…..

Mathematical Definitions of Strain Strain is a second-order tensor related to the displacement u of two points. Euler’s Tensor  for small displacements of points(<10%) Green-Lagrange tensor  for large displacements of points: Define T as the tangent linear transformation of point Q in initial configuration into a point Q’ in deformed configuration.

Mathematical Definitions of Stress Stress, a second order tensor dual of the strain tensor. Its contracted product by the strain rate tensor is the power involved in the manufacturing process. Cauchy stress tensor  for small displacements of points:

Equation of State in Tensor form for Solids A solid work piece acts as house of generation of Strain field. Two independent variable the strain field are stress and temperature.

EoS for LINEAR ELASTIC SOLIDS

Matrix form of models of linear elastic materials

Hypoelastic Solids Also called as incremental elastic solids. Their EoS always written in an incremental form of irreversible material equations:

Plastic Solids Many important engineering materials (metals, polymers, soils) will have quite different mechanical behavior when their physical state comes on its load bearing capacity. The material mostly lost its elastic characteristics. This is called as plastic phenomena. The size of this plastic range depends on the internal microstructure of the material.

Melting Melting is a vibrational instability released when the root mean square amplitude of vibration reaches a critical fraction (10%) of the interatomic distance. Several investigators attempted to calculate the critical temperature for different planes when this instability will appear. The onset of melting of these particles is found to take place significantly below the bulk melting point. This is called as Debye temperature .