Lecture - Week 5
3.4 Thermal Effect
Shear strain
In general, increase in temperature causes the atoms to vibrate more and makes average atomic spacing greater. Macroscopically, this will lead to dimension increase in solid: T + T +V T – T – V
In an isotropic material, temperature change only introduces variation of direct (normal) strain. In other words, thermal deformation does not cause shear strain. where T0 is a reference temperature and = coefficient of thermal expansion (C)-1
Hooke’s law under thermal condition
Thus the Hooke’s law is changed as
Example 3.5
A prismatic bar restrained in the x-direction and free to expand in y and z diections as shown. Show that the only non-vanishing stress and strain components due to the temperature change are: and
Soln
Step 1: Known:
Bar is restrained in x. No deformation in this direction (but )
On the other hand, the surfaces of bar normal to y- and z-directions are stress free: (but because of the Poisson’s effect)
No shear stress due to thermal expansion:
Thus:
Step 2: Apply Hooke’s law with thermal expansion: (negative indicates in compression) .
Chapter 4 Modelling and Solution
4.1 Difficulties
Unknowns and equations
15 Unknowns:
6 stresses:
6 strains:
3 displacements:
Basic equations:
3 equilibrium equations:
6 strain-displacement eqns:
6 strain-stress eqns:
Plus compatibility equations
These equations are called “basic equations”.
Difficulties:
(1) how to translate an engineering problem to its mechanics model – namely mechanics modelling;
(2) difficult to solve for 15 equations simultaneously
4.2 Mechanics Modelling
Displacement and Stress boundary condition (B.C.)
Let’s look at an example in manufacturing: Cutting tool in a lathe must be sufficiently stiff to avoid vibration and too-tip deflection. When we want to know whether the tool is good enough for a turning operation, we need to analyse its
Shear strain
In general, increase in temperature causes the atoms to vibrate more and makes average atomic spacing greater. Macroscopically, this will lead to dimension increase in solid: T + T +V T – T – V
In an isotropic material, temperature change only introduces variation of direct (normal) strain. In other words, thermal deformation does not cause shear strain. where T0 is a reference temperature and = coefficient of thermal expansion (C)-1
Hooke’s law under thermal condition
Thus the Hooke’s law is changed as
Example 3.5
A prismatic bar restrained in the x-direction and free to expand in y and z diections as shown. Show that the only non-vanishing stress and strain components due to the temperature change are: and
Soln
Step 1: Known:
Bar is restrained in x. No deformation in this direction (but )
On the other hand, the surfaces of bar normal to y- and z-directions are stress free: (but because of the Poisson’s effect)
No shear stress due to thermal expansion:
Thus:
Step 2: Apply Hooke’s law with thermal expansion: (negative indicates in compression) .
Chapter 4 Modelling and Solution
4.1 Difficulties
Unknowns and equations
15 Unknowns:
6 stresses:
6 strains:
3 displacements:
Basic equations:
3 equilibrium equations:
6 strain-displacement eqns:
6 strain-stress eqns:
Plus compatibility equations
These equations are called “basic equations”.
Difficulties:
(1) how to translate an engineering problem to its mechanics model – namely mechanics modelling;
(2) difficult to solve for 15 equations simultaneously
4.2 Mechanics Modelling
Displacement and Stress boundary condition (B.C.)
Let’s look at an example in manufacturing: Cutting tool in a lathe must be sufficiently stiff to avoid vibration and too-tip deflection. When we want to know whether the tool is good enough for a turning operation, we need to analyse its