∂ρ
∂t
þ ρ
∂e v x
∂x
þ
∂e v y
∂y
þ
∂e v z
∂z
!
¼ 0
2.16 The Incompressible Materials
When the material is incompressible,
dρ
dt
¼ 0
Therefore,
∂e v x
∂x
þ
∂e v x
∂y
þ
∂e v z
∂z
¼ 0
must be satisfied. It is important to point out that in the theory of plasticity during
plastic deformation, materials, like metals, are considered to be incompressible.
2.17 Conservation of Momentum Principle
“The expression for equilibrium of forces is obtained by using the conservation of
momentum principle of a collection of particles. This principle applies to a collection
of particles as well as the continuous medium. This principle states that the vector
sum of all the external forces acting on the free body is equal to the rate of change of
the total momentum.”
Normal stresses, σ ii , are positive when the vector is in the positive axis direction.
Shear stresses σ ij are positive when the vector component is in the positive direction
on the positive face of the block and in the negative direction on the negative face of
the block. Shear stresses σ ij are negative when the vector component is in the
negative direction on the positive face of the block and in the positive direction on
the negative face of the block.
Tensile stress is defined to be positive and compression is defined as negative.
However, we realize that this definition is not universal. For example, in
geo-mechanics, tension is negative and compression is positive.
While tension and compression lead to completely different results when applied
on any object. Negative shear stress and positive shear stress are considered to be the
same kind of loading in the opposite directions from a physical point of view. There
is no compression shear and tension shear.
2.17 Conservation of Momentum Principle
65
∂t
þ ρ
∂e v x
∂x
þ
∂e v y
∂y
þ
∂e v z
∂z
!
¼ 0
2.16 The Incompressible Materials
When the material is incompressible,
dρ
dt
¼ 0
Therefore,
∂e v x
∂x
þ
∂e v x
∂y
þ
∂e v z
∂z
¼ 0
must be satisfied. It is important to point out that in the theory of plasticity during
plastic deformation, materials, like metals, are considered to be incompressible.
2.17 Conservation of Momentum Principle
“The expression for equilibrium of forces is obtained by using the conservation of
momentum principle of a collection of particles. This principle applies to a collection
of particles as well as the continuous medium. This principle states that the vector
sum of all the external forces acting on the free body is equal to the rate of change of
the total momentum.”
Normal stresses, σ ii , are positive when the vector is in the positive axis direction.
Shear stresses σ ij are positive when the vector component is in the positive direction
on the positive face of the block and in the negative direction on the negative face of
the block. Shear stresses σ ij are negative when the vector component is in the
negative direction on the positive face of the block and in the positive direction on
the negative face of the block.
Tensile stress is defined to be positive and compression is defined as negative.
However, we realize that this definition is not universal. For example, in
geo-mechanics, tension is negative and compression is positive.
While tension and compression lead to completely different results when applied
on any object. Negative shear stress and positive shear stress are considered to be the
same kind of loading in the opposite directions from a physical point of view. There
is no compression shear and tension shear.
2.17 Conservation of Momentum Principle
65