Navier-Cauchy equations

The Navier-Cauchy , Navier or Lamé-Navier equations (after Claude Louis Marie Henri Navier , Augustin-Louis 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:

${\displaystyle \rho {\ddot {\vec {u}}}=G\left[\Delta {\vec {u}}+{\frac {1}{1-2\nu }}\operatorname {grad} {\bigl (}\operatorname {div} ({\vec {u}}){\bigr )}\right]+\rho {\vec {k}}\quad \Leftrightarrow \quad \rho {\frac {\partial ^{2}u_{i}}{\partial t^{2}}}=G\sum _{k=1}^{3}\left[{\frac {\partial ^{2}u_{i}}{\partial x_{k}^{2}}}+{\frac {1}{1-2\nu }}{\frac {\partial ^{2}u_{k}}{\partial x_{i}\partial x_{k}}}\right]+\rho k_{i}\;,\quad i=1,2,3\,.}$

The left vector equation is the coordinate-free 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 volume-distributed 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 linear-elastic 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 Cauchy-Euler 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 :

${\displaystyle {\begin{aligned}{\boldsymbol {\sigma }}=&2G\left[{\boldsymbol {\varepsilon }}+{\frac {\nu }{1-2\nu }}\mathrm {Sp} ({\boldsymbol {\varepsilon }})\mathbf {I} \right]=G\left[\operatorname {grad} ({\vec {u}})+\operatorname {grad} ({\vec {u}})^{\top }+{\frac {\nu }{1-2\nu }}\mathrm {Sp} \left(\operatorname {grad} ({\vec {u}})+\operatorname {grad} ({\vec {u}})^{\top }\right)\mathbf {I} \right]\\=&G\left[\operatorname {grad} ({\vec {u}})+\operatorname {grad} ({\vec {u}})^{\top }+{\frac {2\nu }{1-2\nu }}\operatorname {div} ({\vec {u}})\mathbf {I} \right]\end{aligned}}}$

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 Cauchy-Euler's first law of motion is provided: ${\displaystyle \top }$

${\displaystyle {\begin{aligned}\operatorname {div} ({\boldsymbol {\sigma }})=&G\left[\operatorname {div(grad} ({\vec {u}}))+\operatorname {div(grad} ({\vec {u}})^{\top })+{\frac {2\nu }{1-2\nu }}\operatorname {div} \left(\operatorname {div} ({\vec {u}})\mathbf {I} \right)\right]\\=&G\left[\Delta {\vec {u}}+{\frac {1}{1-2\nu }}\operatorname {grad(div} ({\vec {u}}))\right]\end{aligned}}}$

In combination with the above law of motion , this leads to the Navier-Cauchy equations: ${\displaystyle (\rho {\ddot {\vec {u}}}=\operatorname {div} ({\boldsymbol {\sigma }})+\rho {\vec {k}})}$

${\displaystyle \rho {\ddot {\vec {u}}}=G\left[\Delta {\vec {u}}+{\frac {1}{1-2\nu }}\operatorname {grad} {\bigl (}\operatorname {div} ({\vec {u}}){\bigr )}\right]+\rho {\vec {k}}=\mu \Delta {\vec {u}}+(\lambda +\mu )\operatorname {grad} {\bigl (}\operatorname {div} ({\vec {u}}){\bigr )}+\rho {\vec {k}}\,.}$

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 : ${\displaystyle \Delta {\vec {u}}=\operatorname {grad(div} ({\vec {u}}))-\operatorname {rot(rot} ({\vec {u}}))}$

${\displaystyle \rho {\ddot {\vec {u}}}=(\lambda +2\mu )\operatorname {grad(div} ({\vec {u}}))-\mu \operatorname {rot(rot} ({\vec {u}}))+\rho {\vec {k}}\,.}$

The operator rot forms the rotation of a vector field .

boundary conditions

→ Main article : boundary condition

In the specific case of calculation of the Navier-Cauchy 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 area-distributed 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 Navier-Cauchy equations are written in equilibrium

${\displaystyle {\vec {0}}=(\lambda +2\mu )\operatorname {grad(div} ({\vec {u}}))-\mu \operatorname {rot(rot} ({\vec {u}}))+\rho {\vec {k}}\,.}$

The divergence and rotation of this equation give:

${\displaystyle {\begin{aligned}(\lambda +2\mu )\Delta \operatorname {div} ({\vec {u}})=(\lambda +2\mu )\operatorname {div} (\Delta {\vec {u}})=&-\operatorname {div} (\rho {\vec {k}})\\\mu \operatorname {rot} (\Delta {\vec {u}})=&-\operatorname {rot} (\rho {\vec {k}})\end{aligned}}}$

If gravity is both divergence and rotation free, then it results ${\displaystyle \rho {\vec {k}}}$

${\displaystyle \operatorname {div} (\Delta {\vec {u}})=0\quad {\text{und}}\quad \operatorname {rot} (\Delta {\vec {u}})={\vec {0}}\,.}$

A vector field whose divergence and rotation vanish is harmonious, so that in equilibrium from and on ${\displaystyle \operatorname {div} (\rho {\vec {k}})=0}$ ${\displaystyle \operatorname {rot} (\rho {\vec {k}})={\vec {0}}}$

${\displaystyle \Delta \Delta {\vec {u}}={\vec {0}}}$

can be closed. The latter is the so-called 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 Cauchy-Euler 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 divergence-free and a rotation-free part:

${\displaystyle {\vec {u}}={\vec {u}}_{l}+{\vec {u}}_{t}\quad {\text{mit}}\quad \operatorname {rot} ({\vec {u}}_{l})={\vec {0}}\quad {\text{und}}\quad \operatorname {div} ({\vec {u}}_{t})=0\,.}$

Inserting this into the Navier-Cauchy equation and dividing by the density yields, with negligible gravity:

${\displaystyle {\ddot {\vec {u}}}={\frac {\lambda +2\mu }{\rho }}\operatorname {grad(div} ({\vec {u}}))-{\frac {\mu }{\rho }}\operatorname {rot(rot} ({\vec {u}}))=:c_{l}^{2}\operatorname {grad(div} ({\vec {u}}_{l}))-c_{t}^{2}\operatorname {rot(rot} ({\vec {u}}_{t}))\,,}$

The factors c _l and c _t have the dimension of a speed. For the rotation-free part there is a scalar potential , whose gradient field it is, and for the divergence-free part there is a vector field whose rotation it is:

${\displaystyle {\vec {u}}_{l}=\operatorname {grad} (\varphi )\,,\quad {\vec {u}}_{t}=\operatorname {rot} ({\vec {a}})\,.}$

With shows like this: ${\displaystyle \operatorname {rot\circ rot} =\operatorname {grad\circ div} -\Delta }$

${\displaystyle {\begin{aligned}{\frac {\partial ^{2}}{\partial t^{2}}}\operatorname {grad} (\varphi )+{\frac {\partial ^{2}}{\partial t^{2}}}\operatorname {rot} ({\vec {a}})=&c_{l}^{2}\operatorname {grad(div(grad} (\varphi )))-c_{t}^{2}\operatorname {rot(rot(rot} ({\vec {a}})))\\=&c_{l}^{2}\operatorname {grad} (\Delta \varphi )-c_{t}^{2}\operatorname {grad(div(rot} ({\vec {a}})))+c_{t}^{2}\Delta \operatorname {rot} ({\vec {a}})\\=&c_{l}^{2}\operatorname {grad} (\Delta \varphi )+c_{t}^{2}\operatorname {rot} (\Delta {\vec {a}})\end{aligned}}}$

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 1-direction (perpendicular in the picture), all cross-sectional areas move parallel to the 1-direction 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 Navier-Cauchy equations are:

${\displaystyle {\begin{aligned}\rho {\frac {\partial ^{2}u_{1}}{\partial t^{2}}}&=&G\left[{\frac {\partial ^{2}u_{1}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}u_{1}}{\partial x_{2}^{2}}}+{\frac {\partial ^{2}u_{1}}{\partial x_{3}^{2}}}+{\frac {1}{1-2\nu }}\left({\frac {\partial ^{2}u_{1}}{\partial x_{1}\partial x_{1}}}+{\frac {\partial ^{2}u_{2}}{\partial x_{1}\partial x_{2}}}+{\frac {\partial ^{2}u_{3}}{\partial x_{1}\partial x_{3}}}\right)\right]\\\rho {\frac {\partial ^{2}u_{2}}{\partial t^{2}}}&=&G\left[{\frac {\partial ^{2}u_{2}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}u_{2}}{\partial x_{2}^{2}}}+{\frac {\partial ^{2}u_{2}}{\partial x_{3}^{2}}}+{\frac {1}{1-2\nu }}\left({\frac {\partial ^{2}u_{1}}{\partial x_{2}\partial x_{1}}}+{\frac {\partial ^{2}u_{2}}{\partial x_{2}\partial x_{2}}}+{\frac {\partial ^{2}u_{3}}{\partial x_{2}\partial x_{3}}}\right)\right]\\\rho {\frac {\partial ^{2}u_{3}}{\partial t^{2}}}&=&G\left[{\frac {\partial ^{2}u_{3}}{\partial x_{1}^{2}}}+{\frac {\partial ^{2}u_{3}}{\partial x_{2}^{2}}}+{\frac {\partial ^{2}u_{3}}{\partial x_{3}^{2}}}+{\frac {1}{1-2\nu }}\left({\frac {\partial ^{2}u_{1}}{\partial x_{3}\partial x_{1}}}+{\frac {\partial ^{2}u_{2}}{\partial x_{3}\partial x_{2}}}+{\frac {\partial ^{2}u_{3}}{\partial x_{3}\partial x_{3}}}\right)\right]\,.\end{aligned}}}$

With the above displacement approach, an equation of the form is derived in all three spatial directions

${\displaystyle \rho {\frac {\partial ^{2}u(x,t)}{\partial t^{2}}}=G\left(1+{\frac {1}{1-2\nu }}\right){\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}=(\lambda +2\mu ){\frac {\partial ^{2}u(x,t)}{\partial x^{2}}}}$

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 x-coordinate. Because the functions on the left of the last equation only depend on time and those on the right only on the x-coordinate, 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 cross-sectional 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:

${\displaystyle {\begin{aligned}U(0)=&0\quad \rightarrow \quad \beta =0\\U'(L)=&u\lambda \cos(\lambda L)\equiv 0\quad \rightarrow \quad \lambda ={\frac {(2n-1)\pi }{2L}}={\frac {\omega }{c}}\\U(L)=&R=u\sin \left({\frac {(2n-1)\pi }{2}}\right)=(-1)^{n+1}u\,.\end{aligned}}}$

The counter indicates the oscillation mode. The final form of the motion function is thus: ${\displaystyle n=1,2,\ldots }$

${\displaystyle u(x,t)=(-1)^{n+1}R\cos \left({\frac {(2n-1)\pi }{2}}{\frac {c\,t}{L}}\right)\sin \left({\frac {(2n-1)\pi }{2}}{\frac {x}{L}}\right)\,.}$

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

• Euler's equations (fluid mechanics)
• Navier-Stokes equations
• Tensoranalysis formula collection
• Formula collection tensor algebra

Footnotes

1. ↑ ^{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. ${\displaystyle \operatorname {div(grad} ({\vec {u}}))=\operatorname {grad(div} ({\vec {u}}))\,,}$ ${\displaystyle \operatorname {div(grad} ({\vec {f}})^{\top })=\Delta {\vec {f}}}$ ${\displaystyle \operatorname {div} (f\mathbf {I} )=\operatorname {grad} f\,,}$ ${\displaystyle \operatorname {div(div} ({\vec {u}})\mathbf {I} )=\operatorname {grad(div} ({\vec {u}}))\,.}$ ${\displaystyle \operatorname {{\tilde {div}}(grad} ({\vec {f}})^{\top })=\operatorname {grad({\tilde {div}}} ({\vec {f}}))}$ ${\displaystyle \operatorname {{\tilde {div}}(grad} ({\vec {f}}))=\Delta {\vec {f}}}$ ${\displaystyle \operatorname {{\tilde {div}}(div} ({\vec {u}})\mathbf {I} )=\operatorname {grad({\tilde {div}}} ({\vec {u}}))\,,}$
2. ↑ The fact that a rotation field is always divergence-free, a gradient field is always rotation-free and with is used . With the identity already used above follows ${\displaystyle \Delta f=\operatorname {div} (\operatorname {grad} (f))}$ ${\displaystyle f=\operatorname {div} {\vec {u}}}$ ${\displaystyle \Delta {\vec {u}}=\operatorname {grad(div} ({\vec {u}}))-\operatorname {rot(rot} ({\vec {u}}))}$ ${\displaystyle \operatorname {rot} (\Delta {\vec {u}})=-\operatorname {rot(rot(rot} ({\vec {u}})))\,.}$

Individual evidence

1. ↑ ME Gurtin (1972), p. 90

literature

• Holm Altenbach: Continuum Mechanics. Introduction to the material-independent and material-dependent equations . 2nd Edition. Springer Vieweg, Berlin et al. 2012, ISBN 978-3-642-24118-5 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X .
• Ralf Greve: Continuum Mechanics . Springer, 2003, ISBN 3-540-00760-1 .
• ME Gurtin: The Linear Theory of Elasticity . In: S. Flügge (Ed.): Handbuch der Physik . Volume VI2 / a, Volume Editor C. Truesdell. Springer, 1972, ISBN 3-540-05535-5 .

Web links

• Lamé-Navier equations. University of Stuttgart, accessed on October 30, 2015 .