Coefficient of elasticity

The coefficient of elasticity (sometimes called the Hooke coefficient, the coefficient of stiffness or the stiffness of the spring ) is the coefficient connecting the extension of the elastic body in the Hooke law and the elastic force resulting from this extension. It is used in solid mechanics in the section of elasticity . It is denoted by the letter k ^[1] , sometimes D ^[2] or c ^[3] . It has a unit of measurement N / m or kg / s ² (in SI ), dyn / cm or g / s ² (in GHS ).

The coefficient of elasticity is numerically equal to the force that must be applied to the spring so that its length changes by a unit of distance .

Content

Definition and properties

The elastic coefficient by definition is equal to the elastic force divided by the change in spring length: $k=F_{\mathrm {e} }/\Delta l.$ ${\ displaystyle k = F _ {\ mathrm {e}} / \ Delta l.}$ ^[4] The coefficient of elasticity depends both on the properties of the material and on the size of the elastic body. So, for an elastic rod, we can distinguish the dependence on the size of the rod (cross-sectional $S$ ${\ displaystyle S}$ and lengths $L$ ${\ displaystyle L}$ ), writing the coefficient of elasticity as $k=E\cdot S/L.$ ${\ displaystyle k = E \ cdot S / L.}$ Value $E$ ${\ displaystyle E}$ is called the Young's modulus and, in contrast to the coefficient of elasticity, depends only on the properties of the rod material ^[5] .

The rigidity of deformable bodies when they are joined

Parallel connection of springs.

Series connection of springs.

When several elastically deformable bodies are connected (hereinafter, for brevity, springs ), the total rigidity of the system will change. With a parallel connection, the stiffness increases, with a serial connection it decreases.

Parallel Connection

With parallel connection $n$ ${\ displaystyle n}$ springs with stiffnesses equal to $k_{1},k_{2},k_{3},...,k_{n},$ ${\ displaystyle k_ {1}, k_ {2}, k_ {3}, ..., k_ {n},}$ the rigidity of the system is equal to the sum of the stiffnesses, i.e. $k=k_{1}+k_{2}+k_{3}+\ldots +k_{n}.$ ${\ displaystyle k = k_ {1} + k_ {2} + k_ {3} + \ ldots + k_ {n}.}$

Evidence

In parallel connection there is $n$ ${\ displaystyle n}$ stiff springs $k_{1},k_{2},...,k_{n}.$ ${\ displaystyle k_ {1}, k_ {2}, ..., k_ {n}.}$ From III law of Newton, $F=F_{1}+F_{2}+\ldots +F_{n}.$ ${\ displaystyle F = F_ {1} + F_ {2} + \ ldots + F_ {n}.}$ ( Force is applied to them $F$ ${\ displaystyle F}$ . In this case, a force is applied to the spring 1 $F_{1},$ ${\ displaystyle F_ {1},}$ to spring 2 force $F_{2},$ ${\ displaystyle F_ {2},}$ ... to the spring $n$ ${\ displaystyle n}$ strength $F_{n}.$ ${\ displaystyle F_ {n}.}$ )

Now from Hooke’s Law ( $F=-kx$ ${\ displaystyle F = -kx}$ , where x is the elongation) we derive: $F=kx;F_{1}=k_{1}x;F_{2}=k_{2}x;...;F_{n}=k_{n}x.$ ${\ displaystyle F = kx; F_ {1} = k_ {1} x; F_ {2} = k_ {2} x; ...; F_ {n} = k_ {n} x.}$ We substitute these expressions into equality (1): $kx=k_{1}x+k_{2}x+\ldots +k_{n}x;$ ${\ displaystyle kx = k_ {1} x + k_ {2} x + \ ldots + k_ {n} x;}$ reducing by $x,$ ${\ displaystyle x,}$ we get: $k=k_{1}+k_{2}+\ldots +k_{n},$ ${\ displaystyle k = k_ {1} + k_ {2} + \ ldots + k_ {n},}$ Q.E.D.

Serial Connection

With serial connection $n$ ${\ displaystyle n}$ springs with stiffnesses equal to $k_{1},k_{2},k_{3},...,k_{n},$ ${\ displaystyle k_ {1}, k_ {2}, k_ {3}, ..., k_ {n},}$ total stiffness is determined from the equation: $1/k=(1/k_{1}+1/k_{2}+1/k_{3}+\ldots +1/k_{n}).$ ${\ displaystyle 1 / k = (1 / k_ {1} + 1 / k_ {2} + 1 / k_ {3} + \ ldots + 1 / k_ {n}).}$

Evidence

The serial connection has $n$ ${\ displaystyle n}$ stiff springs $k_{1},k_{2},...,k_{n}.$ ${\ displaystyle k_ {1}, k_ {2}, ..., k_ {n}.}$ From Hooke's Law ( $F=-kl$ ${\ displaystyle F = -kl}$ , where l is the elongation) it follows that $F=k\cdot l.$ ${\ displaystyle F = k \ cdot l.}$ The sum of the elongations of each spring is equal to the total elongation of the entire joint $l_{1}+l_{2}+\ldots +l_{n}=l.$ ${\ displaystyle l_ {1} + l_ {2} + \ ldots + l_ {n} = l.}$

The same force acts on each spring $F .$ ${\ displaystyle F.}$ According to Hooke’s Law, $F=l_{1}\cdot k_{1}=l_{2}\cdot k_{2}=\ldots =l_{n}\cdot k_{n}.$ ${\ displaystyle F = l_ {1} \ cdot k_ {1} = l_ {2} \ cdot k_ {2} = \ ldots = l_ {n} \ cdot k_ {n}.}$ From the previous expressions we deduce: $l=F/k,\quad l_{1}=F/k_{1},\quad l_{2}=F/k_{2},\quad ...,\quad l_{n}=F/k_{n}.$ ${\ displaystyle l = F / k, \ quad l_ {1} = F / k_ {1}, \ quad l_ {2} = F / k_ {2}, \ quad ..., \ quad l_ {n} = F / k_ {n}.}$ Substituting these expressions in (2) and dividing by $F,$ ${\ displaystyle F,}$ we get $1/k=1/k_{1}+1/k_{2}+\ldots +1/k_{n},$ ${\ displaystyle 1 / k = 1 / k_ {1} + 1 / k_ {2} + \ ldots + 1 / k_ {n},}$ Q.E.D.

The rigidity of some deformable bodies

Constant Section Rod

A uniform rod of constant cross section, elastically deformable along the axis, has a stiffness coefficient

k={\frac {E\,S}{L_{0}}},

{\ displaystyle k = {\ frac {E \, S} {L_ {0}}},}

Where

E - Young's modulus , depending only on the material from which the rod is made;

S is the cross-sectional area;

L ₀ is the length of the rod.

Cylindrical coil spring

Twisted coil compression spring.

A coiled compression or tensile coil spring wound from a cylindrical wire and elastically deformable along the axis has a stiffness coefficient

k={\frac {G\cdot d_{\mathrm {D} }^{4}}{8\cdot d_{\mathrm {F} }^{3}\cdot n}},

{\ displaystyle k = {\ frac {G \ cdot d _ {\ mathrm {D}} ^ {4}} {8 \ cdot d _ {\ mathrm {F}} ^ {3} \ cdot n}},}

Where

d _D is the diameter of the wire;

d _F - winding diameter (measured from the axis of the wire);

n is the number of turns;

G is the shear modulus (for ordinary steel G ≈ 80 GPa , for spring steel G ≈ 78.5 GPa, for copper ~ 45 GPa ).

Sources and notes

↑ Elastic deformation (Russian) . Archived June 30, 2012.
↑ Dieter Meschede, Christian Gerthsen. Physik. - Springer, 2004 .-- P. 181 ..
↑ Bruno Assmann. Technische Mechanik: Kinematik und Kinetik. - Oldenbourg, 2004 .-- P. 11 ..
↑ Dynamics, Strength of elasticity (Russian) . Archived June 30, 2012.
↑ Mechanical properties of bodies (Russian) . Archived June 30, 2012.

[1] Elastic deformation (Russian) . Archived June 30, 2012.

[2] Dieter Meschede, Christian Gerthsen. Physik. - Springer, 2004 .-- P. 181 ..

[3] Bruno Assmann. Technische Mechanik: Kinematik und Kinetik. - Oldenbourg, 2004 .-- P. 11 ..

[4] Dynamics, Strength of elasticity (Russian) . Archived June 30, 2012.

[5] Mechanical properties of bodies (Russian) . Archived June 30, 2012.