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Statics Definition

Statics is the branch of mechanics dealing with bodies in equilibrium, analyzing forces and moments without considering motion.
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The statement of the theorem

Let BB be a continuous, rigid body occupying a domain Ω⊂R3\Omega \subset \mathbb{R}^3. The state of equilibrium is defined by the condition that the body is subjected only to external forces and moments, and these effects balance out. Mathematically, this requires satisfying the following system of partial differential equations and integral constraints:\n\n1. **Force Equilibrium (Cauchy's Equation):** The balance of linear momentum requires that the divergence of the stress tensor σ\sigma plus the body force density b⃗\vec{b} must vanish everywhere within the domain Ω\Omega:\n∇⋅σ+b⃗=0⃗in Ω\nabla \cdot \sigma + \vec{b} = \vec{0} \quad \text{in } \Omega\n\n2. **Moment Equilibrium (Rotational Balance):** The balance of angular momentum requires that the moment generated by the stress tensor and body forces relative to any point must vanish. This is often expressed as the equilibrium of the moment tensor, which simplifies to the condition that the stress tensor must be symmetric (σij=σji\sigma_{ij} = \sigma_{ji}) and that the body forces must be conservative (if b⃗\vec{b} is derived from a potential). For a general body, the condition is:\n∑j=13(xi∂σij∂xj+bi)=0for i=1,2,3\sum_{j=1}^{3} (x_i \frac{\partial \sigma_{ij}}{\partial x_j} + b_i) = 0 \quad \text{for } i=1, 2, 3 \n\n3. **Boundary Conditions:** The equilibrium must also hold on the boundary ∂Ω\partial\Omega. The traction vector t⃗\vec{t} acting on the boundary must satisfy the force balance:\n∫∂Ωt⃗⋅n⃗ dS=0⃗\int_{\partial\Omega} \vec{t} \cdot \vec{n} \, dS = \vec{0} \n\nWhere n⃗\vec{n} is the outward unit normal vector, and σij\sigma_{ij} are the components of the Cauchy stress tensor.