MECHANICS  THEORY



One Dimensional Hooke's Law

1D Hooke's Law 

Recall, Hooke's Law in one dimension (uniaxial loading), relates the normal stress and normal strain as
The constant E is Young's modulus and represents the stiffness of the the material. 





Poisson's Ratio

Poisson's Effect
Contraction in y and zdirection
when Stressed in xdirection 

When analyzing more than one dimension, interaction between all directions
needs to be considered. This is done through Poisson's ratio. Basically, Poisson's
ratio is the amount of transverse contraction, or negative strain, when strained
in a given direction. For a basic object pulled or strained in the xdirection,
the Poisson's Ratio is defined as
Thus, when a member is pulled in the xdirection, there is a contraction strain
in the ydirection (and zdirection). If it is pulled in the ydirection, then
the contraction strain will be in the xdirection (and zdirection).
For a three dimensional object, Poisson's ratio will occur in equally in both
perpendicular directions. If the load is in the xdirection, then strain in the
y and zdirection will be
ε_{y} = ε_{z} = νε_{x} 





Two Dimensional StressStrain

Stress in Two Directions


If a material is isotropic (homogenous in all directions, such as a solid
metal) and is pulled in two directions, then due to Poisson's ratio, the overall
normal strain will be the total of the two strains. For example, if there are normal stresses
in both the x and ydirections, then the total normal strain in the xdirection is
ε_{x total }
= ε_{x due to σx} + ε_{x
due to σy} = σ_{x} /E
 νσ_{y}/E
ε_{x} = (σ_{x}  νσ_{y})
/ E
Similarly, the normal strain in the ydirection would be
ε_{y} = (σ_{y}  νσ_{x})
/ E 



Pure Shear Stress in a 2D plane
Shear Angle due to Shear Stress 

However, Hooke's Law also relates shear strain and shear stress. If the shear
stress and strain occurs in a plane then the stress and strain are related as
or
γ = τ/G
= [2(1 + ν)/E] τ
where G is the shear modulus (a material property) and γ
is the shear strain. The shear strain is defined as the angle (radians) caused
by the shear stress as shown in the diagram at the left.
The shear modulus is
related to Young modulus and Poisson's ratio,
Two dimensional stressstrain relationships are summarized in the table below. 



2D Hooke's Law (StressStrain Relationship)

Compliance Format 

Stiffness Format 



or in matrix form


or in matrix form










Three Dimensional StressStrain

Stress Directions in 3D
(τ_{xy} = τ_{yx}, τ_{yz} = τ_{zy}, τ_{xz} = τ_{zx}) 

Just like 1D or 2D, Hooke's Law can also be applied to material undergoing
three dimensional stress (triaxial loading). The development of 3D equations
is similar to 2D, sum the total normal strain in one direction due to loads in all three
directions. For the xdirection, this gives,
ε_{x total } = ε_{x
due to σx} + ε_{x
due to σy} + ε_{x
due to σz}
= σ_{x} /E  νσ_{y}/E  νσ_{z}/E
ε_{x} = (σ_{x}  νσ_{y}  νσ_{y}) / E
Similarly, the other directions can also be determined. The final equations are summarized in the table below. 



3D Hooke's Law (StressStrain Relationship) 
Compliance Format 

Stiffness Format 









3D Elasticity

3D Stress and Deflection
using FEA Analysis Tool


In addition to the Hooke's Law, complex stresses can be determined using the theory of elasticity. This topic is beyond this text, but through the use of compatibility and equilibrium equations, complex 3D stresses can be determined by numerical methods. The numerical method most commonly used is finite element analysis (FEA) and is widely used in industry. Just a few of the many commercial FEA codes include ANSYS, Cosmos, NISA, Abaqus, and AutoDesk, SolidWorks, many more.






