Hooke's law

Hooke's experimental arrangement

The Hooke's Law (according to Robert Hooke , it published in 1676 for the first time resolved as anagram and 1678) describes the elastic deformation of solids when their deformation is proportional to the applied load is ( linear-elastic behavior ). This behavior (“Ut tensio sic vis”) is typical for metals , if the load is not too great, and for hard, brittle materials often to the point of breakage (glass, ceramic, silicon).

Hooke's law represents the linear special case of the law of elasticity . The relationship between deformation and stress with a quadratic or higher order cannot be considered here. The non-linear elastic deformation as in the case of rubber , the plastic deformation or the ductile deformation as in the case of metal after the flow limit is exceeded remain outside . However, stress and deformation do not have to be in the same line : deformation in -direction can cause stress in -direction. Hooke's law is therefore generally a tensor relation . ${\ displaystyle x}$${\ displaystyle y}$

In the rheological models , the law is taken into account by the Hooke element.

Hooke's law for spring systems

Spring elongation by weight. (As well as parallel connection of springs)

Hooke's law states that the elongation depends linearly on the acting force and can be expressed as a formula as follows: ${\ displaystyle \ Delta l}$${\ displaystyle F}$

${\ displaystyle F = D \ cdot \ Delta l}$ respectively ${\ displaystyle \ Delta l = {\ frac {F} {D}}}$

The spring constant serves as a proportionality factor and describes the stiffness of the spring. In the case of a helical spring, the linear behavior is shown when loaded with a weight. After doubling the weight, double stretching also occurs. ${\ displaystyle D}$${\ displaystyle \ Delta l}$

This property is decisive, for example, for the use of metal springs as force gauges and in scales . With other materials - such as rubber - the relationship between the force and expansion is not linear.

Hooke's law is not only used in mechanics, but also in other areas of physics. In quantum mechanics, for example, the quantum mechanical harmonic oscillator can be described for sufficiently small ones by applying Hooke's law . Another example is molecular physics . Here, analogous to the spring constant, the linearity can be expressed by a force constant . This force constant then describes the strength of a chemical bond. ${\ displaystyle \ Delta l}$${\ displaystyle \ Delta l}$

The potential energy created in a spring by expansion can be calculated as follows. There is a deflection of the amount that describes the deflection from the rest position ( , equilibrium position). The force is proportional to the deflection, namely . By integrating the force we now get the potential energy: ${\ displaystyle s}$${\ displaystyle s = 0}$${\ displaystyle {\ vec {F}} = - D {\ vec {s}}}$

${\ displaystyle E _ {\ text {pot}} = - \ int \ limits _ {0} ^ {\ vec {s}} {{\ vec {F}} \ cdot d {\ vec {s}} \, ' } = - \ int \ limits _ {0} ^ {\ vec {s}} {\ left ({- D {\ vec {s}} \, '} \ right) \ cdot d {\ vec {s}} \, '} = D \ int \ limits _ {0} ^ {\ vec {s}} {{\ vec {s}} \,' \ cdot d {\ vec {s}} \, '} = {\ frac {1} {2}} Ds ^ {2}}$

This is the harmonic potential (proportional to ) which is important for many model calculations . ${\ displaystyle s ^ {2}}$

One-dimensional case

A tensile or compressive load (force) acts on a rod of length and cross-sectional area along the -axis and causes tension in the rod in the -direction: ${\ displaystyle l_ {0}}$${\ displaystyle A}$${\ displaystyle x}$${\ displaystyle x}$

${\ displaystyle \ sigma _ {x} = {\ frac {F_ {x}} {A}}}$

This results in an elongation of the rod in the direction: ${\ displaystyle \ varepsilon _ {x}}$${\ displaystyle x}$

${\ displaystyle \ varepsilon _ {x} = {\ frac {\ Delta l} {l_ {0}}}}$

The elongation of the rod depends on the force acting, here the tension in the rod. The constant of proportionality represents the modulus of elasticity of the material from which the rod is made. ${\ displaystyle E}$

${\ displaystyle \ sigma _ {x} = E \ cdot \ varepsilon _ {x}}$

By inserting the first two formulas and rearranging them, the following representation results:

${\ displaystyle F_ {x} = E \ cdot A \ cdot {\ frac {\ Delta l} {l_ {0}}}}$

Hooke's law can therefore be used where the acting force depends almost linearly on the deflection or expansion, and is a generalization of Hooke's law for springs.

Generalized Hooke's Law

In the general case, Hooke's law is expressed by a linear tensor equation (4th level):

${\ displaystyle {\ tilde {\ sigma}} = {\ tilde {\ tilde {C}}} {\ tilde {\ varepsilon}}}$,

with the elasticity tensor , which characterizes the elastic properties of the deformed matter. Since the tensor 81 has components , it is difficult to handle. Due to the symmetry of the strain and stress tensor , the number of independent components is reduced to 36 after conversion into constants based on the scheme 11 → 1, 22 → 2, 33 → 3, 23 → 4, 31 → 5, 12 → 6 Hooke's law can be converted into an easier-to-use matrix equation, where the elastic constants are represented in a matrix, as well as the distortion and the stress as six-component vectors: ${\ displaystyle {\ tilde {\ tilde {C}}}}$${\ displaystyle {\ tilde {\ tilde {C}}}}$${\ displaystyle C_ {ijkl}, \; i, j, k, l = 1, \ dotsc, 3}$${\ displaystyle C_ {ijkl}}$${\ displaystyle C_ {IJ}}$${\ displaystyle 6 \ times 6}$

${\ displaystyle {\ begin {bmatrix} \ sigma _ {1} \\\ sigma _ {2} \\\ sigma _ {3} \\\ sigma _ {4} \\\ sigma _ {5} \\\ sigma _ {6} \ end {bmatrix}} = {\ begin {bmatrix} C_ {11} & C_ {12} & C_ {13} & C_ {14} & C_ {15} & C_ {16} \\ C_ {21} & C_ { 22} & C_ {23} & C_ {24} & C_ {25} & C_ {26} \\ C_ {31} & C_ {32} & C_ {33} & C_ {34} & C_ {35} & C_ {36} \\ C_ {41} & C_ {42} & C_ {43} & C_ {44} & C_ {45} & C_ {46} \\ C_ {51} & C_ {52} & C_ {53} & C_ {54} & C_ {55} & C_ {56} \\ C_ { 61} & C_ {62} & C_ {63} & C_ {64} & C_ {65} & C_ {66} \ end {bmatrix}} {\ begin {bmatrix} \ varepsilon _ {1} \\\ varepsilon _ {2} \\ \ varepsilon _ {3} \\\ varepsilon _ {4} \\\ varepsilon _ {5} \\\ varepsilon _ {6} \ end {bmatrix}}}$

From energetic considerations it follows that this matrix is ​​also symmetrical . The number of independent ( elastic constants ) is thus further reduced to a maximum of 21. ${\ displaystyle 6 \ times 6}$${\ displaystyle C_ {ij}, \; i, j = 1, \ dotsc, 6}$

The maximum of six independents of the two symmetrical tensors for strain and tension are thus distributed over two six-component vectors ( Voigt notation ). In and you must, watch out because here an additional factor 2 comes in and not only the indices are adjusted. ${\ displaystyle \ varepsilon _ {4} = 2 \ varepsilon _ {23}, \ varepsilon _ {5}}$${\ displaystyle \ varepsilon _ {6}}$

Isotropic media

In the special case of isotropic media, the number of independent elastic constants is reduced from 21 to 2. Essential properties of the deformation can then be characterized by the Poisson's ratio. Hooke's law can then be represented in the form

${\ displaystyle {\ bar {\ varepsilon}} = L ^ {- 1} {\ bar {\ sigma}}}$, With
${\ displaystyle L ^ {- 1} = {\ frac {1} {E}} {\ begin {bmatrix} 1 & - \ nu & - \ nu & 0 & 0 & 0 \\\ cdot & 1 & - \ nu & 0 & 0 & 0 \\\ cdot & \ cdot & 1 & 0 & 0 & 0 \\\ cdot & \ cdot & \ cdot & 2 (1+ \ nu) & 0 & 0 \\\ cdot & \ cdot & \ cdot & \ cdot & 2 (1+ \ nu) & 0 \\\ cdot & \ cdot & \ cdot & \ cdot & \ cdot & 2 (1+ \ nu) \ end {bmatrix}}}$, or.
${\ displaystyle L = {\ frac {E} {1+ \ nu}} {\ begin {bmatrix} {\ frac {1- \ nu} {1-2 \ nu}} & {\ frac {\ nu} { 1-2 \ nu}} & {\ frac {\ nu} {1-2 \ nu}} & 0 & 0 & 0 \\\ cdot & {\ frac {1- \ nu} {1-2 \ nu}} & {\ frac {\ nu} {1-2 \ nu}} & 0 & 0 & 0 \\\ cdot & \ cdot & {\ frac {1- \ nu} {1-2 \ nu}} & 0 & 0 & 0 \\\ cdot & \ cdot & \ cdot & {\ frac {1} {2}} & 0 & 0 \\\ cdot & \ cdot & \ cdot & 0 & {\ frac {1} {2}} & 0 \\\ cdot & \ cdot & \ cdot & 0 & 0 & {\ frac {1} {2}} \\\ end {bmatrix}}}$,

where the modulus of elasticity (also Young's modulus) and the Poisson's ratio are. Both are determined by the material. For one-dimensional deformations, the relationship is simplified to ${\ displaystyle E}$${\ displaystyle \ nu}$

${\ displaystyle \ varepsilon = {\ frac {1} {E}} \ sigma}$.

Notation with Lamé constants

Often there is also a notation for the generalized Hooke's law for isotropic media with the help of the Lamé constants :

${\ displaystyle \ sigma = 2 \ mu \ varepsilon + \ lambda \; \ operatorname {track} (\ varepsilon) I}$

or written out:

${\ displaystyle \ left ({\ begin {array} {ccc} \ sigma _ {xx} & \ sigma _ {xy} & \ sigma _ {xz} \\\ sigma _ {xy} & \ sigma _ {yy} & \ sigma _ {yz} \\\ sigma _ {xz} & \ sigma _ {yz} & \ sigma _ {zz} \ end {array}} \ right) = 2 \ mu \ left ({\ begin {array } {ccc} \ varepsilon _ {xx} & \ varepsilon _ {xy} & \ varepsilon _ {xz} \\\ varepsilon _ {xy} & \ varepsilon _ {yy} & \ varepsilon _ {yz} \\\ varepsilon _ {xz} & \ varepsilon _ {yz} & \ varepsilon _ {zz} \ end {array}} \ right) + \ lambda (\ varepsilon _ {xx} + \ varepsilon _ {yy} + \ varepsilon _ {zz }) \ left ({\ begin {array} {ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \ end {array}} \ right)}$.

The equation is to be understood component-wise, e.g. B. applies . The reverse relationship is ${\ displaystyle \ sigma _ {xx} = 2 \ mu \ varepsilon _ {xx} + \ lambda (\ varepsilon _ {xx} + \ varepsilon _ {yy} + \ varepsilon _ {zz}) {\ cdot} 1}$

${\ displaystyle \ left ({\ begin {array} {ccc} \ varepsilon _ {xx} & \ varepsilon _ {xy} & \ varepsilon _ {xz} \\\ varepsilon _ {xy} & \ varepsilon _ {yy} & \ varepsilon _ {yz} \\\ varepsilon _ {xz} & \ varepsilon _ {yz} & \ varepsilon _ {zz} \ end {array}} \ right) = {\ frac {1} {2 \ mu} } \ left ({\ begin {array} {ccc} \ sigma _ {xx} & \ sigma _ {xy} & \ sigma _ {xz} \\\ sigma _ {xy} & \ sigma _ {yy} & \ sigma _ {yz} \\\ sigma _ {xz} & \ sigma _ {yz} & \ sigma _ {zz} \ end {array}} \ right) - {\ frac {\ nu} {E}} (\ sigma _ {xx} + \ sigma _ {yy} + \ sigma _ {zz}) \ left ({\ begin {array} {ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \ end {array}} \ right)}$.

Therein is the modulus of elasticity. The material constant is called the shear module in German-speaking countries and has the symbol here . ${\ displaystyle E = 2 \ mu (1+ \ nu)}$${\ displaystyle \ mu}$${\ displaystyle G}$

Flat state of stress and strain

Slices are flat surface supports which, by definition, are only loaded in their plane. Rods and beams are slender girders with two dimensions small compared to the third axial dimension. If there are no loads perpendicular to the plane or longitudinal axis of these girders, they are in a plane stress state (ESZ) in which all stress components perpendicular to the plane under consideration can be neglected.

Surface beams that are also loaded perpendicular to their plane are called panels. If this plate is so thick that it is not noticeably compressed by the load acting on it perpendicularly, a plane state of distortion (EVZ) prevails in its plane, in which all distortion components perpendicular to the plane under consideration can be neglected.

Bars, beams, discs and plates are widespread construction elements in mechanical engineering and construction. It is therefore worthwhile to write down the elasticity relationship for the ESZ and EVZ.

Level state of tension

The ESZ corresponds to the condition above . This simplifies the elasticity relationship to ${\ displaystyle \ sigma _ {xz} = \ sigma _ {yz} = \ sigma _ {zz} = 0}$

${\ displaystyle \ left ({\ begin {array} {c} \ varepsilon _ {xx} \\\ varepsilon _ {yy} \\ 2 \ varepsilon _ {xy} \ end {array}} \ right) = {\ frac {1} {E}} \ left ({\ begin {array} {ccc} 1 & - \ nu & 0 \\ - \ nu & 1 & 0 \\ 0 & 0 & 2 (1+ \ nu) \ end {array}} \ right) \ left ({\ begin {array} {c} \ sigma _ {xx} \\\ sigma _ {yy} \\\ sigma _ {xy} \ end {array}} \ right)}$

or.

${\ displaystyle \ left ({\ begin {array} {c} \ sigma _ {xx} \\\ sigma _ {yy} \\\ sigma _ {xy} \ end {array}} \ right) = {\ frac {E} {1- \ nu ^ {2}}} \ left ({\ begin {array} {ccc} 1 & \ nu & 0 \\\ nu & 1 & 0 \\ 0 & 0 & {\ frac {1- \ nu} {2} } \ end {array}} \ right) \ left ({\ begin {array} {c} \ varepsilon _ {xx} \\\ varepsilon _ {yy} \\ 2 \ varepsilon _ {xy} \ end {array} } \ right)}$

and . ${\ displaystyle \ varepsilon _ {zz} = - {\ frac {\ nu} {1- \ nu}} (\ varepsilon _ {xx} + \ varepsilon _ {yy}) = - {\ frac {\ nu} { E}} (\ sigma _ {xx} + \ sigma _ {yy})}$

Flat state of distortion

The following applies in the EVZ . The following relationships can then be derived from this: ${\ displaystyle \ varepsilon _ {xz} = \ varepsilon _ {yz} = \ varepsilon _ {zz} = 0}$

${\ displaystyle \ left ({\ begin {array} {c} \ varepsilon _ {xx} \\\ varepsilon _ {yy} \\\ varepsilon _ {xy} \ end {array}} \ right) = {\ frac {1} {2 \ mu}} \ left ({\ begin {array} {ccc} 1- \ nu & - \ nu & 0 \\ - \ nu & 1- \ nu & 0 \\ 0 & 0 & 1 \ end {array}} \ right) \ left ({\ begin {array} {c} \ sigma _ {xx} \\\ sigma _ {yy} \\\ sigma _ {xy} \ end {array}} \ right)}$.

or.

${\ displaystyle \ left ({\ begin {array} {c} \ sigma _ {xx} \\\ sigma _ {yy} \\\ sigma _ {xy} \ end {array}} \ right) = {\ frac {2 \ mu} {1-2 \ nu}} \ left ({\ begin {array} {ccc} 1- \ nu & \ nu & 0 \\\ nu & 1- \ nu & 0 \\ 0 & 0 & 1-2 \ nu \ end {array}} \ right) \ left ({\ begin {array} {c} \ varepsilon _ {xx} \\\ varepsilon _ {yy} \\\ varepsilon _ {xy} \ end {array}} \ right )}$

with . ${\ displaystyle \ sigma _ {zz} = \ nu (\ sigma _ {xx} + \ sigma _ {yy}) = \ lambda (\ varepsilon _ {xx} + \ varepsilon _ {yy})}$