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Longitudinal elastic deformation
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When a force is applied to the surface of a body, it creates a zone where the material either compresses or expands, resulting in a motion parallel to the normal vector of the surface. This is what is referred to as longitudinal deformation.
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Longitudinal elastic deformation
Storyboard
When a force is applied to the surface of a body, it creates a zone where the material either compresses or expands, resulting in a motion parallel to the normal vector of the surface. This is what is referred to as longitudinal deformation.
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If we use the equation to calculate the work ($W$) as the integral of the elastic Force ($F_k$) along the path during deformation:
$W=\displaystyle\int_0^u \vec{F}\cdot d\vec{s}$
And we employ the equation for the elastic Force ($F_k$) with the modulus of Elasticity ($E$), the body Section ($S$), the elongation ($u$), the body length ($L$), and the elongation ($u$)
where we sum along the path traveled. In the case of elastic deformation, the relationship is linear and becomes:
$W=\displaystyle\frac{ES}{L}\displaystyle\int_0^u d\vec{s}\cdot\vec{s}$
This leads to:
$W=\displaystyle\frac{ES}{2L}u^2$
By using the equation for the deformation ($\epsilon$)
and the equation for the volume ($V$)
we obtain:
With Hooke's Law for the elastic Force ($F_k$), the hooke Constant ($k$), and the elongation ($u$) as follows:
and the expression for the hooke Constant ($k$) in terms of the body length ($L$), the body Section ($S$), the microscopic length of spring ($l$), the microscopic section of spring ($s$), and the microscopic Hook constant ($k_m$):
combined with the expression for the modulus of Elasticity ($E$):
the result is:
Since the deformation energy ($W$) is related to the volume ($V$), the modulus of Elasticity ($E$), and the deformation ($\epsilon$) as follows:
If we replace the deformation ($\epsilon$) with the strain ($\sigma$) in the equation:
We obtain:
The elastic Force ($F_k$) is a function that depends on the modulus of Elasticity ($E$), the body Section ($S$), the elongation ($u$), and the body length ($L$).
This function can be expressed using the definition of the strain ($\sigma$)
and the definition of the deformation ($\epsilon$)
resulting in
The deformation energy ($W$) is expressed as a function of the volume ($V$), the modulus of Elasticity ($E$), and the deformation ($\epsilon$) as follows:
And with the deformation energy density ($w$) defined as:
We obtain:
ERROR:8844 with ERROR:8843, ERROR:8838, the deformation in the direction perpendicular to the force ($\epsilon_{\perp}$), and the poisson coefficient ($\nu$) is expressed as:
If we replace the deformation in the direction perpendicular to the force ($\epsilon_{\perp}$) using the equation
We obtain the initial expression:
Examples
As Hooke's Law relates the elastic Force ($F_k$) through the hooke Constant ($k$) and the elongation ($u$) in the following manner:
you can replace the hooke Constant ($k$) with the microscopic expression and using the definition of the modulus of Elasticity ($E$), you obtain with the body length ($L$) and the body Section ($S$) that:
The elastic Force ($F_k$) is a function of the modulus of Elasticity ($E$), the body Section ($S$), the elongation ($u$), and the body length ($L$).
In this case, the ratio between the elongation ($u$) and the body length ($L$) is represented by the deformation ($\epsilon$), which can be defined as follows:
In general, the deformation ($\epsilon$) is defined as the variation of the elongation ($u$) in proportion to the body length ($L$):
This concept can be generalized in the microscopic limit, where the deformation in the coordinate $i$ ($\epsilon_i$) is introduced as the variation of displacement in i ($\partial u_i$) over the length of an element in i ($\partial x_i$) in the direction $i$, and it would be expressed as:
The reason for using a different symbol to denote the differential
$d \rightarrow \partial$
is that there are several differentials that affect different variables in the model. The use of the symbol $\partial$ indicates that one should perform one variation at a time, meaning that when considering one variable, the remaining variables are assumed to have their initial values.
The elastic Force ($F_k$) is a function that depends on the modulus of Elasticity ($E$), the body Section ($S$), the elongation ($u$), and the body length ($L$).
Similarly, just as the deformation ($\epsilon$) is introduced to avoid using the dimension the body length ($L$), we can construct a factor that expresses the elastic Force ($F_k$) in terms of the body Section ($S$) as the strain ($\sigma$).
The elastic Force ($F_k$) is a function that depends on the modulus of Elasticity ($E$), the body Section ($S$), the elongation ($u$), and the body length ($L$).
This function can be rewritten using the definitions of the strain ($\sigma$) and the deformation ($\epsilon$), resulting in the continuous version of Hooke's Law:
The Hooke's Law for ERROR:8845, ERROR:8843, and ERROR:8838 is expressed as:
This law can be generalized for the stress on axis $i$ ($\sigma_i$) and the deformation in the coordinate $i$ ($\epsilon_i$) as:
The total mass the volume ($V$) of the body is calculated using the body Section ($S$) and the body length ($L$):
Similar to a spring, deforming a material requires energy. The energy the work ($W$) required to compress or expand the material is calculated as the integral of the elastic Force ($F_k$) along the path $ds$ during deformation:
$W=\displaystyle\int_0^u \vec{F}\cdot d\vec{s}$
In the case of the continuous Hooke's Law, this reduces to:
As the deformation energy ($W$) is related to the volume ($V$), the modulus of Elasticity ($E$), and the deformation ($\epsilon$), it can be expressed as:
Using Hooke's Law, we can replace the deformation ($\epsilon$) in terms of the strain ($\sigma$), resulting in:
For the deformation energy ($W$) contained within a volume ($V$), we can define the deformation energy density ($w$) as:
The deformation energy ($W$) as a function of the volume ($V$), the modulus of Elasticity ($E$), and the deformation ($\epsilon$) is equal to
So, if we divide by the volume ($V$), we obtain the deformation energy density ($w$), which is defined as
Lateral deformation is directly proportional to the deformation it causes. The proportionality coefficient is denoted as the poisson coefficient ($\nu$) [1] and typically falls within the range of 0.15 to 0.4.
If the original deformation is the deformation ($\epsilon$) and the generated one is the deformation in the direction perpendicular to the force ($\epsilon_{\perp}$), the following relationship is established:
In the linear approximation, the Poisson's coefficient represents the relationship between lateral and longitudinal deformations.
where the sign indicates that the deformation is in the opposite direction to the cause.
[1] This concept was introduced by Sim on Denis Poisson in a statistical analysis work, in which he mentioned, among other unrelated topics to mechanics, what was later referred to as the Poisson's coefficient in an elasticity example. The work is titled "Recherches sur la Probabilit des Jugements en Mati re Criminelle et en Mati re Civile" (Research on the Probability of Judgments in Criminal and Civil Matters), authored by Sim on Denis Poisson (1837).
ERROR:8844 as a function of ERROR:8843 and ERROR:8838 is equal to
This equation expresses ERROR:8844 without considering the deformation in the direction perpendicular to the force ($\epsilon_{\perp}$), which is associated with ERROR:8838 through the Poisson's coefficient. ERROR:8844 can be expressed as a function of ERROR:8838 and the deformation in the direction perpendicular to the force ($\epsilon_{\perp}$) using the following equation:
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