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I am currently working on a presentation about the Cauchy's momentum equation or (also known as?) Cauchy's first law. I need to base my presentation on an equation given by my professor: $ \rho \ddot{u} - \nabla \cdot P = f$. Where $\rho$ is the mass density, $P$ the first Piola-Kirchhoff stress tensor, $f$ an external volume force und $u$ the displacement field so $\ddot{u}$ is the acceleration. Online I found https://en.wikiversity.org/wiki/Continuum_mechanics/Balance_of_linear_momentum this on Wikiversity about the balance of linear momentum. That looks pretty similar to me, except that is uses the body force density, whatever that might be, and the Cauchy stress tensor. Since I study math and not physics I have a pretty hard time, dealing with those equations. My task is to derivate the equation $ \rho \ddot{u} - \nabla \cdot P = f$ and talk about an inverse problem were it can be used. If someone could explain to me the connection and/or difference between the balance of linear momentum and Cauchy's equation of motion, that would be super helpful. Thank you a lot already.

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  • $\begingroup$ What do you mean by the Cauchy equation? Anyway I'll only give a hint in answer here, since you stated that this is a subject of a homework $\endgroup$
    – basics
    Commented Dec 11, 2022 at 14:10
  • $\begingroup$ By Cauchy equation I mean the equation $ \rho \ddot{u} - \nabla \cdot P = f$ But I think that there are many different names for this. Thank you a lot for your help. I think even a hint will help me. $\endgroup$
    – Ole
    Commented Dec 11, 2022 at 14:17
  • $\begingroup$ Ok, I'll leave you an answer below $\endgroup$
    – basics
    Commented Dec 11, 2022 at 14:22

1 Answer 1

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Continuum mechanics can be described with different descriptions, using different sets of coordinates:

  • Lagrangian description, with fields representing physical quantities expressed as a function of the reference space coordinates (that can be interpreted as labels associated to material points: constant reference coordinates, same material point) and time as independent variables, $f^0(\mathbf{r_0},t)$;
  • Eulerian description, with fields representing physical quantities expressed as a function of the physical space coordinates and time as independent variables, $f(\mathbf{r},t)$;
  • arbitrary description.

It looks like you're trying to find the balance equations of a continuous medium using reference coordinates when you talk about Cauchy equation.

Namely, starting from the integral equation of mass and linear momentum for a material volume, expressed in physical space

$\dfrac{d}{dt}\displaystyle \int_{V} \rho = 0$
$\dfrac{d}{dt}\displaystyle \int_{V} \rho \mathbf{u} = \int_V \rho \mathbf{g} + \oint_{\partial V} \mathbf{t_n} = \int_V \rho \mathbf{g} + \oint_{\partial V} \mathbf{\hat{n}} \cdot \mathbb{T} = \int_V \rho \mathbf{g} + \int_{V} \nabla \cdot \mathbb{T} $,

being $\mathbb{T}$ Cauchy stress tensor. Changing coordinates from physical to reference space coordinates, it's possible to recast the integral equations as

$ \displaystyle \int_{V^0} \dfrac{\partial }{\partial t}\bigg|_{\mathbf{r_0}} ( \rho J ) = 0$
$\displaystyle \int_{V^0} (\rho J) \dfrac{\partial }{\partial t}\bigg|_{\mathbf{r_0}} \mathbf{u} = \int_{V^0} \rho J \mathbf{g} + \oint_{\partial V^0} \mathbf{\hat{n}^0} \cdot \mathbb{P} = \int_{V^0} \rho J \mathbf{g} + \int_{V^0} \nabla_0 \cdot \mathbb{P} $,

having used Nanson's formula for the transformation of the surface integral, being $\mathbb{F}$ the nominal stress tensor, $\nabla_0 \cdot$ the divergence in the reference space, and $J$ the determinant of the gradient of the transformation from the reference to the physical coordinates, and exploiting the mass conservation $\frac{\partial}{\partial t} \big|_{\mathbf{r_0}} (\rho J) = 0$ to write $\rho(\mathbf{r_0},t) J(\mathbf{r_0},t) = \overline{\rho}(\mathbf{r_0})$ constant.

Thus, the differential equations using the reference coordinates, remembering that the acceleration fields of material particle is $\mathbf{a} = \big|_{\mathbf{r_0}} \mathbf{u}$ read either

  • using reference coordinates:

    $\rho J = \overline{\rho}$
    $\overline{\rho} \mathbf{a} = \overline{\rho} \mathbf{g} + \nabla_0 \cdot \mathbb{P}$

  • using physical coordinates (in convective form):

    $D_t \rho = - \rho \nabla \cdot \mathbf{u}$
    $\rho D_t \mathbf{u} = \rho \mathbf{g} + \nabla \cdot \mathbb{T}$.

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