## Simple Stress Strain & Elastic Constants [Strength Of Material]

### Stress:

Stress is defined as the
magnitude of internal resisting force developed or induced at a point against
the deformation caused due to load acting on the member.

F=Total internal resistance force developed on the plane of cross-section of the member.

When deformation or strain
occur freely in a particular direction then stress developed in that direction
will be zero. Because there is no resisting force against the deformation.

#### Strength:

Strength is defined as the
maximum value of stress that a material can withstand without any failure.

From above two equation,
the following conclusion can be made:

1)Stress is a second order
tensor.

2)Normal stress and Shear
stress on any oblique plane passing through a point under uniaxial state of
stress can be determined when normal stress on the plane of cross section is
known.

3)To determine normal and
shear stress on any oblique plane passing through a point under biaxial state
of stress, we should know stress developed on two mutual perpendicular plane
passing through that point.

#### Principal Planes:

Principal Planes are
complementary oblique plane on which shear stress is zero, but normal stress is
either maximum or minimum. Hence these planes are also known as planes of zero
shear stress or planes of maximum and minimum normal stress.

#### Stress Tensor:

Stress Tensor is used to
define the state of stress at a point.

Stress Tensor is a square
matrix. That is-

Under triaxial Stress
Tensor is 3x3 Matrix

Under biaxial Stress Tensor
is 2x2 Matrix

Under uniaxial Stress
Tensor is 1x1 Matrix.

Stress Tensor is
symmetrical about the diagonal.

Symmetry of Stress Tensor
is obtained by using moment equilibrium equation.

The number of stress
components in stress tensor at a point under triaxial state of stress are 9 ( 3
normal stresses and 6 shear stresses).

Number of stress components
in a stress tensor to define triaxial state of stress at a point are 6 (3
normal stresses and 3 shear stresses).

The number of stress
components in stress tensor at a point under biaxial state of stress are 4. ( 2
normal stresses and 2 shear stresses).

Number of stress components
in a stress tensor to define biaxial state of stress at a point are 3 (2 normal
stresses and 1 shear stress).

Every Shear stress should
be associated with a complimentary shear stress of equal magnitude but in
opposite direction.

Strain Tensor:

Strain Tensor is used to
define the state of strain at a point.

Strain analysis at a point
is similar to stress analysis at that point that is to get strain analysis
equations normal stress should be replaced by corresponding normal strain

**Total Number of Pages= 9**

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