The NavierCauchy , Navier or LaméNavier equations (after Claude Louis Marie Henri Navier , AugustinLouis Cauchy and Gabriel Lamé ) are a mathematical model of the movement  including deformation  of elastic solids . When deriving the model equations, both geometric and physical linearity (linear elasticity) are assumed. The equations are:
 ${\ displaystyle \ rho {\ ddot {\ vec {u}}} = G \ left [\ Delta {\ vec {u}} + {\ frac {1} {12 \ nu}} \ operatorname {grad} {\ bigl (} \ operatorname {div} ({\ vec {u}}) {\ bigr)} \ right] + \ rho {\ vec {k}} \ quad \ Leftrightarrow \ quad \ rho {\ frac {\ partial ^ {2} u_ {i}} {\ partial t ^ {2}}} = G \ sum _ {k = 1} ^ {3} \ left [{\ frac {\ partial ^ {2} u_ {i }} {\ partial x_ {k} ^ {2}}} + {\ frac {1} {12 \ nu}} {\ frac {\ partial ^ {2} u_ {k}} {\ partial x_ { i} \ partial x_ {k}}} \ right] + \ rho k_ {i} \;, \ quad i = 1,2,3 \ ,.}$
The left vector equation is the coordinatefree version that applies in any coordinate system , and the right component equations result in the special case of the Cartesian coordinate system . It is a system of partial differential equations of the second order with three unknown displacements which generally depend on both the position and the time t. Displacements are the paths that the particles of a body cover when moving  including deformation. The material parameters ρ, G and ν are the density , the shear modulus and the Poisson's ratio , grad, div and Δ are the gradient , divergence and Laplace operator and represent a volumedistributed force, such as gravity .
${\ displaystyle {\ vec {u}} ({\ vec {x}}, t) \ ,,}$${\ displaystyle {\ vec {x}}}$${\ displaystyle \ rho {\ vec {k}}}$
Every material in the solid aggregate state has a more or less pronounced linearelastic range, at least in the case of small and slow deformations that occur in many applications, especially in the technical field.
Historical
Claude Louis Marie Henri Navier derived this equation, named after him, in 1821 from a molecular model that is restricted to materials with identical first and second Lamé constants . The more general equation presented here with two different elasticity constants first appeared in a work by Cauchy in 1828.
Derivation
The starting point is the first CauchyEuler law of motion for small displacements
 ${\ displaystyle \ rho {\ ddot {\ vec {u}}} = \ operatorname {div} ({\ boldsymbol {\ sigma}}) + \ rho {\ vec {k}} \ ,,}$
which corresponds to the momentum balance . In addition to the variables described above, the stress tensor, which is symmetrical as a result of the angular momentum balance, occurs here . Its dependence on the displacements results from the linearized strain tensor ${\ displaystyle {\ boldsymbol {\ sigma}}}$
 ${\ displaystyle {\ boldsymbol {\ varepsilon}} = {\ frac {1} {2}} (\ operatorname {grad} ({\ vec {u}}) + \ operatorname {grad} ({\ vec {u} }) ^ {\ top})}$
from Hooke's law :
 ${\ displaystyle {\ begin {aligned} {\ boldsymbol {\ sigma}} = & 2G \ left [{\ boldsymbol {\ varepsilon}} + {\ frac {\ nu} {12 \ nu}} \ mathrm {Sp } ({\ boldsymbol {\ varepsilon}}) \ mathbf {I} \ right] = G \ left [\ operatorname {grad} ({\ vec {u}}) + \ operatorname {grad} ({\ vec {u }}) ^ {\ top} + {\ frac {\ nu} {12 \ nu}} \ mathrm {Sp} \ left (\ operatorname {grad} ({\ vec {u}}) + \ operatorname { grad} ({\ vec {u}}) ^ {\ top} \ right) \ mathbf {I} \ right] \\ = & G \ left [\ operatorname {grad} ({\ vec {u}}) + \ operatorname {grad} ({\ vec {u}}) ^ {\ top} + {\ frac {2 \ nu} {12 \ nu}} \ operatorname {div} ({\ vec {u}}) \ mathbf {I} \ right] \ end {aligned}}}$
The superscript stands for the transposition , I for the unit tensor, and the operator Sp extracts the trace unaffected by the transposition , which is equal to the divergence of the vector field with a gradient of a vector field. The divergence occurring in CauchyEuler's first law of motion is provided:
${\ displaystyle \ top}$
 ${\ displaystyle {\ begin {aligned} \ operatorname {div} ({\ boldsymbol {\ sigma}}) = & G \ left [\ operatorname {div (grad} ({\ vec {u}})) + \ operatorname { div (grad} ({\ vec {u}}) ^ {\ top}) + {\ frac {2 \ nu} {12 \ nu}} \ operatorname {div} \ left (\ operatorname {div} ( {\ vec {u}}) \ mathbf {I} \ right) \ right] \\ = & G \ left [\ Delta {\ vec {u}} + {\ frac {1} {12 \ nu}} \ operatorname {grad (div} ({\ vec {u}})) \ right] \ end {aligned}}}$
In combination with the above law of motion , this leads to the NavierCauchy equations:
${\ displaystyle (\ rho {\ ddot {\ vec {u}}} = \ operatorname {div} ({\ boldsymbol {\ sigma}}) + \ rho {\ vec {k}})}$
 ${\ displaystyle \ rho {\ ddot {\ vec {u}}} = G \ left [\ Delta {\ vec {u}} + {\ frac {1} {12 \ nu}} \ operatorname {grad} {\ bigl (} \ operatorname {div} ({\ vec {u}}) {\ bigr)} \ right] + \ rho {\ vec {k}} = \ mu \ Delta {\ vec {u}} + (\ lambda + \ mu) \ operatorname {grad} {\ bigl (} \ operatorname {div} ({\ vec {u}}) {\ bigr)} + \ rho {\ vec {k}} \ ,.}$
In the equation on the right, the first and second Lamé constants λ and μ were used alternatively . Occasionally it is convenient to use the identity :
${\ displaystyle \ Delta {\ vec {u}} = \ operatorname {grad (div} ({\ vec {u}}))  \ operatorname {red (red} ({\ vec {u}}))}$
 ${\ displaystyle \ rho {\ ddot {\ vec {u}}} = (\ lambda +2 \ mu) \ operatorname {grad (div} ({\ vec {u}})))  \ mu \ operatorname {red ( red} ({\ vec {u}})) + \ rho {\ vec {k}} \ ,.}$
The operator rot forms the rotation of a vector field .
boundary conditions
In the specific case of calculation of the NavierCauchy equations, boundary conditions must be defined. The displacement is specified in the supports as geometric or Dirichlet boundary conditions , and often completely suppressed. The dynamic or Neumann boundary conditions correspond to areadistributed forces (vectors with the dimension force per area) that act on surfaces of the body.
${\ displaystyle {\ vec {t}}}$
Solution methods
For simple cases, see the example below, straight bars and flat disks, analytical solutions can be given. In the case of irregularly shaped bodies, a numerical tool is the displacement method in the finite element method .
Special cases
Harmonic gravity
The NavierCauchy equations are written in equilibrium
 ${\ displaystyle {\ vec {0}} = (\ lambda +2 \ mu) \ operatorname {grad (div} ({\ vec {u}}))  \ mu \ operatorname {red (red} ({\ vec {u}})) + \ rho {\ vec {k}} \ ,.}$
The divergence and rotation of this equation give:
 ${\ displaystyle {\ begin {aligned} (\ lambda +2 \ mu) \ Delta \ operatorname {div} ({\ vec {u}}) = (\ lambda +2 \ mu) \ operatorname {div} (\ Delta {\ vec {u}}) = &  \ operatorname {div} (\ rho {\ vec {k}}) \\\ mu \ operatorname {red} (\ Delta {\ vec {u}}) = &  \ operatorname {red} (\ rho {\ vec {k}}) \ end {aligned}}}$
If gravity is both divergence and rotation free, then it results
${\ displaystyle \ rho {\ vec {k}}}$
 ${\ displaystyle \ operatorname {div} (\ Delta {\ vec {u}}) = 0 \ quad {\ text {and}} \ quad \ operatorname {red} (\ Delta {\ vec {u}}) = { \ vec {0}} \ ,.}$
A vector field whose divergence and rotation vanish is harmonious, so that in equilibrium from and on
${\ displaystyle \ operatorname {div} (\ rho {\ vec {k}}) = 0}$${\ displaystyle \ operatorname {red} (\ rho {\ vec {k}}) = {\ vec {0}}}$
 ${\ displaystyle \ Delta \ Delta {\ vec {u}} = {\ vec {0}}}$
can be closed. The latter is the socalled biharmonic differential equation .
Incompressibility
In the case of incompressibility, the trace of the strain tensor disappears because it indicates the volume expansion :
 ${\ displaystyle \ operatorname {Sp} ({\ boldsymbol {\ varepsilon}}) = {\ frac {1} {2}} \ operatorname {Sp} (\ operatorname {grad} ({\ vec {u}}) + \ operatorname {grad} ({\ vec {u}}) ^ {\ top}) = \ operatorname {Sp} (\ operatorname {grad} ({\ vec {u}})) = \ operatorname {div} ({ \ vec {u}}) = 0 \ ,.}$
In the case of incompressibility, the spherical component of the stress tensor is indefinite and is summarized as the pressure tensor :
 ${\ displaystyle {\ boldsymbol {\ sigma}} =  p \ mathbf {I} + 2G {\ boldsymbol {\ varepsilon}} =  p \ mathbf {I} + G \ operatorname {grad} ({\ vec {u }}) + G \ operatorname {grad} ({\ vec {u}}) ^ {\ top} \ ,.}$
The scalar p is the pressure that only results from the boundary conditions and natural laws in the specific calculation case. This has the consequence for the divergence of the stress tensor (see the comments above):
 ${\ displaystyle {\ begin {aligned} \ rightarrow \ operatorname {div} ({\ boldsymbol {\ sigma}}) = &  \ operatorname {div} (p \ mathbf {I}) + G \ operatorname {div (grad } ({\ vec {u}})) + G \ operatorname {div (grad} ({\ vec {u}}) ^ {\ top}) \\ = &  \ operatorname {grad} (p) + G \ operatorname {grad (div} ({\ vec {u}})) + G \ Delta {\ vec {u}} \\ = &  \ operatorname {grad} (p) + G \ Delta {\ vec {u }} \ end {aligned}}}$
The first CauchyEuler law of motion is then written
 ${\ displaystyle \ rho {\ ddot {\ vec {u}}} = \ operatorname {div} ({\ boldsymbol {\ sigma}}) + \ rho {\ vec {k}} =  \ operatorname {grad} ( p) + G \ Delta {\ vec {u}} + \ rho {\ vec {k}} \ ,.}$
These three equations in the four unknowns are still needed in conclusion.
${\ displaystyle \ {p, {\ vec {u}} \}}$${\ displaystyle \ operatorname {div} ({\ vec {u}}) = 0}$
Wave equations
According to the Helmholtz theorem , every vector field that declines sufficiently quickly at infinity can be clearly broken down into a divergencefree and a rotationfree part:
 ${\ displaystyle {\ vec {u}} = {\ vec {u}} _ {l} + {\ vec {u}} _ {t} \ quad {\ text {with}} \ quad \ operatorname {red} ({\ vec {u}} _ {l}) = {\ vec {0}} \ quad {\ text {and}} \ quad \ operatorname {div} ({\ vec {u}} _ {t}) = 0 \ ,.}$
Inserting this into the NavierCauchy equation and dividing by the density yields, with negligible gravity:
 ${\ displaystyle {\ ddot {\ vec {u}}} = {\ frac {\ lambda +2 \ mu} {\ rho}} \ operatorname {grad (div} ({\ vec {u}}))  { \ frac {\ mu} {\ rho}} \ operatorname {rot (red} ({\ vec {u}})) =: c_ {l} ^ {2} \ operatorname {grad (div} ({\ vec { u}} _ {l}))  c_ {t} ^ {2} \ operatorname {red (red} ({\ vec {u}} _ {t})) \ ,,}$
The factors c _{l} and c _{t} have the dimension of a speed. For the rotationfree part there is a scalar potential , whose gradient field it is, and for the divergencefree part there is a vector field whose rotation it is:
 ${\ displaystyle {\ vec {u}} _ {l} = \ operatorname {grad} (\ varphi) \ ,, \ quad {\ vec {u}} _ {t} = \ operatorname {red} ({\ vec {a}}) \ ,.}$
With shows like this:
${\ displaystyle \ operatorname {rot \ circ rot} = \ operatorname {grad \ circ div}  \ Delta}$
 ${\ displaystyle {\ begin {aligned} {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} \ operatorname {grad} (\ varphi) + {\ frac {\ partial ^ {2} } {\ partial t ^ {2}}} \ operatorname {rot} ({\ vec {a}}) = & c_ {l} ^ {2} \ operatorname {grad (div (grad} (\ varphi)))  c_ {t} ^ {2} \ operatorname {rot (rot (rot} ({\ vec {a}}))) \\ = & c_ {l} ^ {2} \ operatorname {grad} (\ Delta \ varphi) c_ {t} ^ {2} \ operatorname {grad (div (red} ({\ vec {a}}))) + c_ {t} ^ {2} \ Delta \ operatorname {red} ({\ vec { a}}) \\ = & c_ {l} ^ {2} \ operatorname {grad} (\ Delta \ varphi) + c_ {t} ^ {2} \ operatorname {red} (\ Delta {\ vec {a}} ) \ end {aligned}}}$
or
 ${\ displaystyle \ operatorname {grad} \ left ({\ frac {\ partial ^ {2} \ varphi} {\ partial t ^ {2}}}  c_ {l} ^ {2} \ Delta \ varphi \ right) + \ operatorname {rot} \ left ({\ frac {\ partial ^ {2} {\ vec {a}}} {\ partial t ^ {2}}}  c_ {t} ^ {2} \ Delta {\ vec {a}} \ right) = {\ vec {0}} \ ,.}$
This equation is certainly satisfied when the terms in the brackets disappear, which represent the wave equations :
 ${\ displaystyle {\ begin {aligned} {\ frac {\ partial ^ {2} \ varphi} {\ partial t ^ {2}}}  c_ {l} ^ {2} \ Delta \ varphi = & 0 \\ { \ frac {\ partial ^ {2} {\ vec {a}}} {\ partial t ^ {2}}}  c_ {t} ^ {2} \ Delta {\ vec {a}} = & {\ vec {0}} \,. \ End {aligned}}}$
The equation above describes longitudinal waves that move with speed
 ${\ displaystyle c_ {l} = {\ sqrt {\ frac {\ lambda +2 \ mu} {\ rho}}}}$
propagate and the lower transverse waves that move with speed
 ${\ displaystyle c_ {t} = {\ sqrt {\ frac {\ mu} {\ rho}}}}$
spread. Because of this , longitudinal waves are called P waves (primary waves) and the transverse waves are called S waves (secondary waves), because they arrive later.
${\ displaystyle c_ {l}> c_ {t}}$
example
Longitudinal wave of an elastic rod in the second mode
With the longitudinal wave of the straight rod, which lies in the 1direction (perpendicular in the picture), all crosssectional areas move parallel to the 1direction and shear distortions do not occur. The field of displacement lies in the form
 ${\ displaystyle {\ vec {u}} = {\ begin {pmatrix} u_ {1} (x_ {1}, t) \\ u_ {2} (x_ {2}, t) \\ u_ {3} ( x_ {3}, t) \ end {pmatrix}}}$
in front. If the acceleration due to gravity is neglected, the NavierCauchy equations are:
 ${\ displaystyle {\ begin {aligned} \ rho {\ frac {\ partial ^ {2} u_ {1}} {\ partial t ^ {2}}} & = & G \ left [{\ frac {\ partial ^ { 2} u_ {1}} {\ partial x_ {1} ^ {2}}} + {\ frac {\ partial ^ {2} u_ {1}} {\ partial x_ {2} ^ {2}}} + {\ frac {\ partial ^ {2} u_ {1}} {\ partial x_ {3} ^ {2}}} + {\ frac {1} {12 \ nu}} \ left ({\ frac { \ partial ^ {2} u_ {1}} {\ partial x_ {1} \ partial x_ {1}}} + {\ frac {\ partial ^ {2} u_ {2}} {\ partial x_ {1} \ partial x_ {2}}} + {\ frac {\ partial ^ {2} u_ {3}} {\ partial x_ {1} \ partial x_ {3}}} \ right) \ right] \\\ rho {\ frac {\ partial ^ {2} u_ {2}} {\ partial t ^ {2}}} & = & G \ left [{\ frac {\ partial ^ {2} u_ {2}} {\ partial x_ {1 } ^ {2}}} + {\ frac {\ partial ^ {2} u_ {2}} {\ partial x_ {2} ^ {2}}} + {\ frac {\ partial ^ {2} u_ {2 }} {\ partial x_ {3} ^ {2}}} + {\ frac {1} {12 \ nu}} \ left ({\ frac {\ partial ^ {2} u_ {1}} {\ partial x_ {2} \ partial x_ {1}}} + {\ frac {\ partial ^ {2} u_ {2}} {\ partial x_ {2} \ partial x_ {2}}} + {\ frac {\ partial ^ {2} u_ {3}} {\ partial x_ {2} \ partial x_ {3}}} \ right) \ right] \\\ rho {\ frac {\ partial ^ {2} u_ {3}} {\ partial t ^ {2}}} & = & G \ left [{\ frac {\ partial ^ {2} u_ {3}} {\ partial x_ {1} ^ {2}}} + {\ frac {\ part ial ^ {2} u_ {3}} {\ partial x_ {2} ^ {2}}} + {\ frac {\ partial ^ {2} u_ {3}} {\ partial x_ {3} ^ {2} }} + {\ frac {1} {12 \ nu}} \ left ({\ frac {\ partial ^ {2} u_ {1}} {\ partial x_ {3} \ partial x_ {1}}} + {\ frac {\ partial ^ {2} u_ {2}} {\ partial x_ {3} \ partial x_ {2}}} + {\ frac {\ partial ^ {2} u_ {3}} {\ partial x_ {3} \ partial x_ {3}}} \ right) \ right] \,. \ end {aligned}}}$
With the above displacement approach, an equation of the form is derived in all three spatial directions
 ${\ displaystyle \ rho {\ frac {\ partial ^ {2} u (x, t)} {\ partial t ^ {2}}} = G \ left (1 + {\ frac {1} {12 \ nu}} \ right) {\ frac {\ partial ^ {2} u (x, t)} {\ partial x ^ {2}}} = (\ lambda +2 \ mu) {\ frac {\ partial ^ { 2} u (x, t)} {\ partial x ^ {2}}}}$
from. With the wave propagation speed
 ${\ displaystyle c = {\ sqrt {\ frac {\ lambda +2 \ mu} {\ rho}}}}$
the oscillation equation for the straight rod results:
 ${\ displaystyle {\ frac {\ partial ^ {2}} {\ partial t ^ {2}}} u (x, t) = c ^ {2} {\ frac {\ partial ^ {2}} {\ partial x ^ {2}}} u (x, t) \ ,.}$
The product approach with free parameters a and C , which result in the equation above and serve to adapt to boundary conditions, as well as two functions T and U still to be determined result:
${\ displaystyle u (x, t) = aT (t) U (x) + C}$
 ${\ displaystyle {\ ddot {T}} U = c ^ {2} TU '' \ quad \ rightarrow \ quad {\ frac {\ ddot {T}} {T}} = c ^ {2} {\ frac { U ''} {U}}}$
As usual, the dash () 'represents the derivation according to the xcoordinate. Because the functions on the left of the last equation only depend on time and those on the right only on the xcoordinate, the fractions are constants:
${\ displaystyle {\ begin {aligned} {\ frac {\ ddot {T}} {T}} = &  \ omega ^ {2} & \ rightarrow T (t) = & \ sin (\ omega t + \ alpha) \\ {\ frac {U ''} {U}} = &  \ lambda ^ {2} & \ rightarrow U (x) = & u \ sin (\ lambda x + \ beta) \\  \ omega ^ {2} = &  c ^ {2} \ lambda ^ {2} & \ rightarrow \ omega = & c \ lambda \,. \ end {aligned}}}$
The amplitude of the function T and the factor a is the amplitude and the function U slammed shut. Values for the angular frequency ω with the opposite sign are possible, but lead to equivalent solutions. The amplitude u , the displacement C , the angular frequencies ω and λ as well as the phase angles α and β must be adapted to the initial and boundary conditions.
With firm clamping is
 ${\ displaystyle U (x_ {0}) = u \ sin (\ lambda x_ {0} + \ beta) = 0 \ quad \ rightarrow \ quad \ beta =  \ lambda x_ {0} \ ,.}$
Other values for β are possible, but lead to equivalent solutions and translations are implemented with the parameter C. At a free end at x = x _{0} the normal force is given, where the factor E is the modulus of elasticity and A is the crosssectional area of the rod. The derivative of the function U at the free end is determined with the force :
${\ displaystyle N = EAU '(x_ {0})}$
 ${\ displaystyle U '(x_ {0}) = u \ lambda \ cos (\ lambda x_ {0} + \ beta) = {\ frac {N} {EA}}}$
In the specific case here, the maximum deflection is initially with
 ${\ displaystyle T (0) = \ sin (\ alpha) = 1 \ quad \ rightarrow \ quad \ alpha = {\ frac {\ pi} {2}} \ quad \ rightarrow \ quad T (t) = \ cos ( \ omega t) \ ,,}$
fixed restraint at C = x = 0, an unloaded free end at x = L and initial deflection at the free end around R assumed:
 ${\ displaystyle {\ begin {aligned} U (0) = & 0 \ quad \ rightarrow \ quad \ beta = 0 \\ U '(L) = & u \ lambda \ cos (\ lambda L) \ equiv 0 \ quad \ rightarrow \ quad \ lambda = {\ frac {(2n1) \ pi} {2L}} = {\ frac {\ omega} {c}} \\ U (L) = & R = u \ sin \ left ({\ frac {(2n1) \ pi} {2}} \ right) = ( 1) ^ {n + 1} u \,. \ end {aligned}}}$
The counter indicates the oscillation mode. The final form of the motion function is thus:
${\ displaystyle n = 1,2, \ ldots}$
 ${\ displaystyle u (x, t) = ( 1) ^ {n + 1} R \ cos \ left ({\ frac {(2n1) \ pi} {2}} {\ frac {c \, t } {L}} \ right) \ sin \ left ({\ frac {(2n1) \ pi} {2}} {\ frac {x} {L}} \ right) \ ,.}$
The picture shows the solution calculated with the parameters from the table.
parameter

Length L 
End shift R

Fashion n 
Wave speed c

unit

mm 
mm 
 
mm / s

value

100 
10 
2 
1

See also
Footnotes

↑ ^{a } ^{b} The identities are used and thus there is also a divergence operator in the literature for which and is, which therefore first transposes a tensor argument. With the divergence operator it is also true that the end result is the same.${\ displaystyle \ operatorname {div (grad} ({\ vec {u}})) = \ operatorname {grad (div} ({\ vec {u}})) \ ,,}$${\ displaystyle \ operatorname {div (grad} ({\ vec {f}}) ^ {\ top}) = \ Delta {\ vec {f}}}$${\ displaystyle \ operatorname {div} (f \ mathbf {I}) = \ operatorname {grad} f \ ,,}$${\ displaystyle \ operatorname {div (div} ({\ vec {u}}) \ mathbf {I}) = \ operatorname {grad (div} ({\ vec {u}})) \ ,.}$${\ displaystyle \ operatorname {{\ tilde {div}} (grad} ({\ vec {f}}) ^ {\ top}) = \ operatorname {grad ({\ tilde {div}}} ({\ vec { f}}))}$${\ displaystyle \ operatorname {{\ tilde {div}} (grad} ({\ vec {f}})) = \ Delta {\ vec {f}}}$${\ displaystyle \ operatorname {{\ tilde {div}} (div} ({\ vec {u}}) \ mathbf {I}) = \ operatorname {grad ({\ tilde {div}}} ({\ vec { u}})) \ ,,}$

↑ The fact that a rotation field is always divergencefree, a gradient field is always rotationfree and with is used . With the identity already used above follows${\ displaystyle \ Delta f = \ operatorname {div} (\ operatorname {grad} (f))}$${\ displaystyle f = \ operatorname {div} {\ vec {u}}}$${\ displaystyle \ Delta {\ vec {u}} = \ operatorname {grad (div} ({\ vec {u}}))  \ operatorname {red (red} ({\ vec {u}}))}$${\ displaystyle \ operatorname {red} (\ Delta {\ vec {u}}) =  \ operatorname {red (red (red} ({\ vec {u}}))) \ ,.}$
Individual evidence

↑ ME Gurtin (1972), p. 90
literature
 Holm Altenbach: Continuum Mechanics. Introduction to the materialindependent and materialdependent equations . 2nd Edition. Springer Vieweg, Berlin et al. 2012, ISBN 9783642241185 .
 P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 354066114X .
 Ralf Greve: Continuum Mechanics . Springer, 2003, ISBN 3540007601 .
 ME Gurtin: The Linear Theory of Elasticity . In: S. Flügge (Ed.): Handbuch der Physik . Volume VI2 / a, Volume Editor C. Truesdell. Springer, 1972, ISBN 3540055355 .
Web links