Cauchy's fundamental theorem

Fig. 1: The stress vector T ⁽ⁿ⁾ at a cut surface dA is a linear function of the normal vector n

The Cauchy fundamental theorem (after Augustin-Louis Cauchy ) states that the voltage vector T ⁽ⁿ⁾ , a vector with the dimension force per area, a linear transformation of the unit normal n of the surface on which the force acts, see Fig. 1. According to Cauchy , the linear mapping provides a stress tensor that only depends on the state of stress and not on the orientation n of the interface. Cauchy justified the fundamental theorem with the tetrahedral argument outlined below , which is based on the principle of intersection , Newton's second law and Cauchy's postulate, according to which the stress vector depends only on the normal and not on the curvature of the surface.

With this theorem, which is central to continuum mechanics , Cauchy introduced the most important aid in continuum mechanics - tensor calculus. Four fundamental physical laws , the momentum , angular momentum and energy balance as well as the second law of thermodynamics in the form of the Clausius-Duhem inequality are formulated in continuum mechanics, in their form valid at the material point, with the help of the stress tensor.

introduction

A cylinder is considered on which two opposing, equally large forces F pull on the end faces (1 in Fig. 2), so that mechanical equilibrium prevails.

Fig. 2: Cylinder (gray) under external load (1) with sectional planes (2) and sectional stresses ( 3 / red), which are divided into shear stresses (4 / green) and normal stresses (5 / yellow)

The forces (1) should be the resultant of a stress vector that acts uniformly over the entire surface. The cylinder can now be conceptually divided at cut surfaces (2). Because this is only carried out mentally, each part remains in equilibrium, which is why evenly distributed stress vectors (3) with always the same, resulting cutting force (1) act on each cut surface .

Three things can now be ascertained here:

1. The cut surface is enlarged by oblique cuts, with the result that the length (the amount) of the area-related stress vectors (3) is reduced.
2. The stress vectors (3) do not have to lie in the direction of the normal on the cut surfaces, but are divided into normal stresses (5) in the normal direction and shear stresses (4) in the cut surface perpendicular to the normal.
3. The cutting stresses (4) and (5) depend on the orientation of the normal of the cutting surface.

Fig. 3: Four area-distributed forces (arrows) on the infinitesimally small tetrahedron (gray) must cancel each other out anytime and anywhere.

With suitable cuts, a tetrahedron can be separated out of the cylinder as shown in Fig. 3. Like the entire cylinder, this part of the body is also in equilibrium, which is why the cutting forces (red arrows) cancel each other out on all four surfaces. Purely geometrically it turns out that the normal vectors on the side surfaces also add up to zero when they are multiplied by the content of their triangle side.

Both conditions, the zero sum of the stress vectors and the normal vectors, which are each weighted with the area, can only be met with arbitrarily designed tetrahedra if the stress and normal vectors are linear and unaffected by the selected reference system.

System-invariant linear mappings are performed by tensors .

Cauchy's fundamental theorem summarizes these statements in a formula:

${\displaystyle {\vec {t}}={\boldsymbol {\sigma }}^{\top }\cdot {\hat {n}}={\hat {n}}\cdot {\boldsymbol {\sigma }}}$.

In it is

• ${\displaystyle {\vec {t}}}$the stress vector , T ⁽ⁿ⁾ in Fig. 1 , (3) in Fig. 2 ,
• ${\displaystyle {\boldsymbol {\sigma }}}$the stress tensor and its transposed , ${\displaystyle {\boldsymbol {\sigma }}^{\top }}$
• ${\displaystyle {\hat {n}}}$the normal unit vector , n in Fig. 1, and
• "·" The scalar product .

The tetrahedral argument applies not only in equilibrium with a homogeneous state of tension , but anytime and anywhere in every body, which is explained in more detail below.

The tetrahedron

Intersectional stress vectors (red) act on a tetrahedron (gray)

Cauchy's tetrahedron argument for the introduction of the stress tensor can be seen on a tetrahedron cut out of a body, see figure on the right. Let three of the surfaces of the tetrahedron be aligned parallel to any orthonormal basis and the origin (o in the picture) of a Cartesian coordinate system with x, y and z directions lies at the intersection of these three surfaces . Seen from the origin, the edges of the tetrahedron have the lengths a, b and c in the negative x, y and z directions. The surfaces have the surface elements (surface area times normal):

${\displaystyle {\vec {a}}_{x}={\frac {bc}{2}}{\vec {e}}_{x}\;,\quad {\vec {a}}_{y}={\frac {ac}{2}}{\vec {e}}_{y}\;,\quad {\vec {a}}_{z}={\frac {ab}{2}}{\vec {e}}_{z}}$

The surface element of the fourth, inclined surface is calculated using the cross product "×":

${\displaystyle {\vec {a}}_{g}=|{\vec {a}}_{g}|{\vec {n}}_{g}={\frac {1}{2}}{\begin{pmatrix}-a\\b\\0\end{pmatrix}}\times {\begin{pmatrix}0\\b\\-c\end{pmatrix}}={\frac {1}{2}}{\begin{pmatrix}-bc\\-ac\\-ab\end{pmatrix}}=-{\vec {a}}_{x}-{\vec {a}}_{y}-{\vec {a}}_{z}}$

This is the Frobenius norm of a vector and the normal vector of the surface. The surface elements add up to the zero vector . ${\displaystyle |(\cdot )|}$ ${\displaystyle {\vec {n}}_{g}}$

The Euler-Cauchy tension principle

According to the stress principle of Euler-Cauchy , a field of stress vectors exists on the cut surfaces, which replaces the effect of the part of the body that has been cut away there, see cutting principle . The voltage vectors have the shape

${\displaystyle {\vec {t}}_{i}=\sum _{j=1}^{3}({\vec {t}}_{i}\cdot {\hat {e}}_{j}){\hat {e}}_{j}=:\sum _{j=1}^{3}\sigma _{ij}{\hat {e}}_{j}\;,\quad i=1,2,3}$

if - as usual - the x, y and z coordinates are numbered 1, 2 and 3. The first index i therefore relates to the surface normal and the second j to the component of the stress vector. If the tetrahedron is sufficiently (infinitesimally) small, these stress vectors can be assumed to be constant over the areas.

According to Cauchy's postulate, these stress vectors depend exclusively on the normal vectors in the relevant point of the surface and not e.g. B. of their curvature . ${\displaystyle {\hat {n}}}$

A consequence of this is Cauchy's fundamental lemma, according to which stress vectors acting on different sides of the same surface are oppositely equal:

${\displaystyle {\vec {t}}(-{\hat {n}})=-{\vec {t}}({\hat {n}})}$

This lemma corresponds to Newton ’s principle of Actio and Reactio .

Newton's second law

Newton's second law says that the forces acting on the tetrahedron from the outside - in addition to volume-distributed forces multiplying them by the area, stress vectors - accelerate the tetrahedron. If the edge lengths of the tetrahedron are of the order L, then the areas and the stress vectors multiplied by the area are of the order of L ² and the volume of the tetrahedron is of the order of L ³ . In the case of infinitesimally small edge lengths (with L → 0), the quantities proportional to the volume and thus to the mass ( weight force , momentum change or inertia force ) can be neglected compared to the surface forces . So on the infinitesimally small tetrahedron the sum of the surface forces must vanish in all spatial directions. In other words, the surface forces on the tetrahedron must be in balance anytime and anywhere .

Cauchy's stress tensor

The summation of the surface forces acting on the tetrahedron in the x, y and z directions yields:

${\displaystyle {\begin{aligned}{\frac {bc}{2}}{\vec {t}}_{x}+{\frac {ac}{2}}{\vec {t}}_{y}+{\frac {ab}{2}}{\vec {t}}_{z}=&{\frac {bc}{2}}{\begin{pmatrix}\sigma _{xx}\\\sigma _{xy}\\\sigma _{xz}\end{pmatrix}}+{\frac {ac}{2}}{\begin{pmatrix}\sigma _{yx}\\\sigma _{yy}\\\sigma _{yz}\end{pmatrix}}+{\frac {ab}{2}}{\begin{pmatrix}\sigma _{zx}\\\sigma _{zy}\\\sigma _{zz}\end{pmatrix}}\\=&\underbrace {\begin{pmatrix}\sigma _{xx}&\sigma _{yx}&\sigma _{zx}\\\sigma _{xy}&\sigma _{yy}&\sigma _{zy}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}\end{pmatrix}} _{=:{\boldsymbol {\sigma }}^{\top }}\underbrace {{\frac {1}{2}}{\begin{pmatrix}bc\\ac\\ab\end{pmatrix}}} _{=-{\vec {a}}_{g}}\\=&-{\boldsymbol {\sigma }}^{\top }\cdot {\vec {a}}_{g}=-|{\vec {a}}_{g}|{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}_{g}\end{aligned}}}$

This is where Cauchy's stress tensor manifests itself

${\displaystyle {\boldsymbol {\sigma }}={\begin{pmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{yx}&\sigma _{yy}&\sigma _{yz}\\\sigma _{zx}&\sigma _{zy}&\sigma _{zz}\end{pmatrix}}={\begin{pmatrix}{\vec {t}}_{x}&{\vec {t}}_{y}&{\vec {t}}_{z}\end{pmatrix}}^{\top }=\sum _{i=1}^{3}{\vec {e}}_{i}\otimes {\vec {t}}_{i}}$,

in which the voltage vectors in the x, y and z directions are entered line by line. The arithmetic symbol is the dyadic product of two vectors. ${\displaystyle \otimes }$

Cauchy's stress tensor is symmetrical , but that only results from the principle of twist , which is not taken into account here.

The tetrahedral argument

According to Newton's second law, as explained above, the sum of all forces must vanish in every spatial direction:

${\displaystyle {\begin{aligned}{\frac {bc}{2}}{\vec {t}}_{x}+{\frac {ac}{2}}{\vec {t}}_{y}+{\frac {ab}{2}}{\vec {t}}_{z}+|{\vec {a}}_{g}|{\vec {t}}_{g}=|{\vec {a}}_{g}|(-{\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}_{g}+{\vec {t}}_{g})={\vec {0}}\;\;\\\rightarrow {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}_{g}={\vec {t}}_{g}\end{aligned}}}$

If so , then the forces are in equilibrium. ${\displaystyle {\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}_{g}={\vec {t}}_{g}}$

Now it is also the other way around

${\displaystyle {\begin{aligned}{\vec {0}}=&{\boldsymbol {\sigma }}^{\top }\cdot {\vec {0}}={\boldsymbol {\sigma }}^{\top }\cdot ({\vec {a}}_{x}+{\vec {a}}_{y}+{\vec {a}}_{z}+{\vec {a}}_{g})\\=&{\boldsymbol {\sigma }}^{\top }\cdot {\vec {a}}_{x}+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {a}}_{y}+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {a}}_{z}+{\boldsymbol {\sigma }}^{\top }\cdot {\vec {a}}_{g}\\=&{\frac {bc}{2}}{\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{x}+{\frac {ac}{2}}{\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{y}+{\frac {ab}{2}}{\boldsymbol {\sigma }}^{\top }\cdot {\hat {e}}_{z}+|{\vec {a}}_{g}|{\boldsymbol {\sigma }}^{\top }\cdot {\hat {n}}_{g}\end{aligned}}}$

The reverse is also true: If the stress vectors are the normal vectors transformed by the stress tensor, then equilibrium prevails at the infinitesimally small tetrahedron. Because the orientation of the tetrahedron and the slope of its fourth face can be chosen arbitrarily, Cauchy's fundamental theorem follows

${\displaystyle {\vec {t}}={\boldsymbol {\sigma }}^{\top }\cdot {\vec {n}}}$   or equivalent    . ${\displaystyle {\vec {t}}={\vec {n}}\cdot {\boldsymbol {\sigma }}}$

See also

• Formula collection tensor algebra

Footnotes

1. ↑ ^{a b} Haupt (2000) calls the postulate Cauchy's tension principle
2. ↑ ^{a b} Sometimes, as in Haupt (2000), the transposed relationship is used. Then the indices of the components of the stress vectors at this point do not match those of the stress components in Cauchy's stress tensor. Because Cauchy's stress tensor is symmetrical due to the angular momentum balance , this discrepancy has no effect.
   ${\displaystyle {\vec {t}}={\boldsymbol {\sigma }}^{*}\cdot {\hat {n}}\quad \rightarrow \quad {\boldsymbol {\sigma }}^{*}={\boldsymbol {\sigma }}^{\top }}$
   

literature

• H. Altenbach: Continuum Mechanics . Springer, 2012, ISBN 978-3-642-24118-5 .
• P. Haupt: Continuum Mechanics and Theory of Materials . Springer, 2000, ISBN 3-540-66114-X .
• WH Müller: Forays through the continuum theory . Springer, 2011, ISBN 978-3-642-19869-4 .