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Existence, uniqueness and symmetry of the stress matrix from the adopted balance principles

Physical scope

Conditional on the adopted force model and balance principles: on every bounded spatial region inside the current configuration the continuous traction field is represented by a unique continuous stress matrix, the reversed-normal identity t(-n) = -t(n) holds, and the couple-free torque balance forces the stress matrix to be symmetric. The result is a conditional deduction from the adopted balance principles and the cited mathematical Cauchy-tetrahedron and angular-balance theorems; it asserts no positivity, no constitutive relation and no measured stress.

Statement

Assume the Axiom of Countable Choice ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )), as used by the surface and divergence machinery of the cited theorems. Let the model and its fields be as in Adopted balance principles of the continuum model: force and couple-free torque balance on every admissible region: a motion $χ$ with current regions $Ω_{s}$ , a continuous density field $ρ$ , spatial velocity $u$ and material acceleration $a = D_{t} u$ , a continuous body-force field $b$ and a continuous traction field $t$ , with the balance principles (F) and (T) of that postulate holding for every admissible region at every time $s$ .

Then for every time $s$ and every bounded open set $W \subseteq Ω_{s}$ with $\overline{W}$ contained in the open domain of the fields at time $s$ there is a unique continuous matrix field

$σ (s, \cdot) : W ⟶ M_{3} (R)$

such that

$t (s, y, n) = σ (s, y) n for every (y, n) \in W \times S^{2},$

and necessarily

$t (s, y, - n) = - t (s, y, n) ((y, n) \in W \times S^{2}), σ (s, y)^{T} = σ (s, y) (y \in W) .$

The stress matrix is continuous and unique per time; positivity of its isotropic part, a constitutive relation and any measured stress are not asserted.

Facts & Assumptions

Given: ${A C}_{ω}$ ; the model fields of Adopted balance principles of the continuum model: force and couple-free torque balance on every admissible region; the balance principles (F) and (T).

[F1]

Field regularity and force description (Adopted balance principles of the continuum model: force and couple-free torque balance on every admissible region, Adopted force model of the continuum branch: continuous body forces and Cauchy contact tractions, The density field of the continuum model and the mass of a material subregion, Traction fields and stress matrices of the continuum model, with units and sign conventions, Euclidean spatial calculus: gradient, divergence and material derivative): $ρ$ , $u$ and $a = D_{t} u$ are continuous and $b$ is continuous on the swept region domain; the traction field $t$ is of class $C^{1}$ , so each $y \mapsto t (s, y, e_{j})$ is $C^{1}$ ; and $t (s, y, n)$ is force per unit area depending on the surface at $(s, y)$ only through $n$ .

[F2]

Balance principles (Adopted balance principles of the continuum model: force and couple-free torque balance on every admissible region clauses (iii)-(iv)): for every admissible region $D$ at time $s$ , $\int_{\partial D} t (s, y, ν_{D}) d S + \int_{D} (ρ b - ρ a) (s, y) d y = 0$ and the same with integrands crossed with $y - x_{0}$ for every fixed origin $x_{0}$ .

[F3]

Cauchy tetrahedron (A continuous traction field balanced on every piecewise $C^{1}$ region is given by a continuous stress matrix, and $t (- n) = - t (n)$ ): if $O$ is open, $t : O \times S^{2} \to R^{3}$ is continuous, $g : O \to R^{3}$ is continuous and bounded with $C_{g} < \infty$ , and $\int_{\partial D} t (y, ν_{D}) d S + \int_{D} g d y = 0$ for every bounded piecewise $C^{1}$ Euclidean domain with a specified finite presentation whose closure and pieces lie in $O$ (Specified finite piecewise C1 boundary presentations), then there is a unique continuous $σ : O \to M_{3} (R)$ with $t (y, n) = σ (y) n$ , and $t (y, - n) = - t (y, n)$ .

[F4]

Couple-free symmetry (Angular-momentum balance without couples forces the stress matrix to be symmetric): if in addition $σ$ is $C^{1}$ , $b_{0} : O \to R^{3}$ is continuous and for every bounded $C^{1}$ domain $D$ with $\overline{D} \subseteq O$ (Bounded C1 domains and their outward normals) one has $\int_{\partial D} σ ν d S + \int_{D} b_{0} d y = 0$ and $\int_{\partial D} y \times (σ ν) d S + \int_{D} y \times b_{0} d y = 0$ , then $σ (y)^{T} = σ (y)$ for every $y \in O$ .

Proof

technique · direct; fix a time and a bounded spatial region, apply the Cauchy-tetrahedron theorem to the force balance with the inertial term moved into the volume force, then apply the angular-momentum symmetry theorem to the torque balance

1.1givenF1F2construct

Fix a time $s$ in the window and a bounded open $W \subseteq Ω_{s}$ with $\overline{W}$ in the field domain. Define $g : W \to R^{3}$ by $g (y) : = ρ (s, y) b (s, y) - ρ (s, y) a (s, y)$ ; by [F1] the fields $ρ, b, a$ are continuous, hence $g$ is continuous, and since $W$ is bounded with closure in the field domain, $g$ is bounded on $W$ . For every bounded piecewise $C^{1}$ domain $D$ with a specified finite presentation, closure in $W$ and pieces in $W$ , the region $D$ is an admissible region at time $s$ , so the force balance (F) of [F2] reads $\int_{\partial D} t (s, y, ν_{D}) d S (y) + \int_{D} g (y) d y = 0$ .

2.1step 1.1F3

Existence, uniqueness and the reversed-normal identity: step 1.1 provides exactly the hypotheses of [F3] with $O : = W$ and the continuous traction field $y \mapsto t (s, y, \cdot)$ ; hence there is a unique continuous $σ (s, \cdot) : W \to M_{3} (R)$ with $t (s, y, n) = σ (s, y) n$ on $W \times S^{2}$ , and $t (s, y, - n) = - t (s, y, n)$ holds there.

3.1step 1.1step 2.1F1F2F4algebra

Symmetry: let $D$ be any bounded $C^{1}$ domain with $\overline{D} \subseteq W$ . Then $D$ is admissible, and the force balance $\int_{\partial D} σ ν d S + \int_{D} g d y = 0$ holds by steps 1.1 and 2.1; the torque balance (T) of [F2] with the same volume density $g$ reads $\int_{\partial D} y \times (σ ν) d S + \int_{D} y \times g d y = 0$ . These are the two hypotheses of [F4] with $b_{0} : = g$ , provided $σ (s, \cdot)$ is of class $C^{1}$ on $W$ : by step 2.1 the columns of $σ (s, y)$ are $σ (s, y) e_{j} = t (s, y, e_{j})$ , and by [F1] $y \mapsto t (s, y, e_{j})$ is of class $C^{1}$ for $j = 1, 2, 3$ , so $σ (s, \cdot)$ is of class $C^{1}$ . Applying [F4] gives $σ (s, y)^{T} = σ (s, y)$ for every $y \in W$ .

4.1step 2.1step 3.1∎

Conclusion: steps 2.1 and 3.1 give the existence, uniqueness, reversed-normal identity and symmetry of the stress matrix on every bounded open $W$ with closure in the field domain at every time $s$ ; no positivity, constitutive relation, boundary condition or measured value is asserted.

Remarks

Where symmetry comes from. Symmetry is a consequence of the couple-free torque balance with the same volume force density as the force balance, not of the definition of traction; a model with couple stresses has a non-symmetric stress matrix and is excluded by Adopted balance principles of the continuum model: force and couple-free torque balance on every admissible region. The constitutive items below produce symmetric matrices by construction, which is consistent with the present conclusion.
Uniqueness is per admissible region. The matrix is unique on each $W$ because the columns $σ (y) e_{j}$ are exactly $t (y, e_{j})$ ; different bounded regions give consistent restrictions because both represent the same traction field.

Existence, uniqueness and symmetry of the stress matrix from the adopted balance principles

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources