Traction fields and stress matrices representing them

Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be a spatial set and let $I\subseteq \mathbb{R}$ be a time interval. No openness of $\mathrm{\Omega }$ or regularity of a traction map is assumed by this definition; consumers impose any needed additional hypotheses. Let $S^2:=\{n\in {\mathbb{R}}^3:\mathrm{\parallel }n{\mathrm{\parallel }}_2=1\}$ be the unit sphere of Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$.

A traction field is a map

$t:S^2⟶{\mathbb{R}}^3,$

read with the following convention: $t(n)$ is the force per unit area exerted by the material on the side towards which $n$ points upon the material on the other side, at the point of a surface whose unit normal is $n$. The dependence of $t$ on position and time is suppressed in this notation; a traction field on a set $\mathrm{\Omega }\times I$ is a map $t:\mathrm{\Omega }\times I\times S^2\to {\mathbb{R}}^3$, and all statements below apply to each fixed $(y,s)$.

For $v\in {\mathbb{R}}^3$ use one-based component labels $v_i:=v(i-1)$, and for $\sigma \in M_3(\mathbb{R})$ use ${\sigma }_{ij}:=\sigma (i-1,j-1)$, where $1\le i,j\le 3$; $e_i$ is the vector with value $1$ at $i-1$ and $0$ at the other two indices. Define the matrix-vector action $\sigma v\in {\mathbb{R}}^3$ by $(\sigma v)(k):={\sum }_{j<3}\sigma (k,j)v(j)$ for $0\le k<3$, equivalently $(\sigma v{)}_i={\sum }_{j=1}^3{\sigma }_{ij}{v}_j$. It is the rectangular product of Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes after identifying $v$ with the column matrix $C_v(j,0):=v(j)$ and the result with its sole column.

A matrix $\sigma \in M_3(\mathbb{R})$ (The vector space $M_{m\times n}(F):=F^{m\times n}$ of $m$ by $n$ matrices over a field, with entrywise operations) represents the traction field $t$ when

$t(n)=\sigma n\text{for every }n\in S^2,$

the product being the matrix-vector action defined above. A matrix representing a traction field is called a stress matrix for it.

The reversed normal convention is the identity $t(-n)=-t(n)$; it is a property some traction fields have and is not part of the definition. A traction field is normal when $t(n)$ is a real multiple of $n$ for every $n$, and purely tangential when $⟨t(n),n{⟩}_2=0$ for every $n$ (The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$).

• Continuity and dependence on the normal alone do not imply existence of a representing matrix: the continuous map $t(n)=(n_1^3,0,0)$ has $t(e_1)=e_1$ and $t(e_2)=t(e_3)=0$, forcing any representing matrix to be $\mathrm{diag}(1,0,0)$, but this matrix disagrees with $t$ at $n=(1\mathrm{/}\sqrt{2},1\mathrm{/}\sqrt{2},0)$. Conditional Cauchy tetrahedron arguments also require balance and limiting hypotheses, stated by their consuming items. If a representing matrix exists, it is unique because its columns are $t(e_1),t(e_2),t(e_3)$.

• Stress positivity is not asserted: for $\lambda >0$, the matrix $-\lambda I_3$ represents the normal-dependent map $t(n)=-\lambda n$ under the sign convention above. This is a mathematical example and imposes no constitutive or force-balance law.

• This definition does not identify a traction with an acceleration or assert any momentum balance; those are separate model data and results with their own hypotheses.

• The notation $t(n)$ suppresses the point of application. A traction field depending only on $(y,s)$ cannot distinguish two surfaces through the same point with the same normal, which is exactly the restriction Cauchy's hypothesis imposes; the failure of that hypothesis for general contact interactions is not a defect of this definition.