= Minimum potential energy principle in elasticity
Let a convex <strain energy density> have derivative equal to the actual <stress tensor>. Under mixed displacement and <traction> data, any equilibrated <stress> $\sigma_0$ with the prescribed <body force> and boundary <tractions> represents the external work. For any compatible trial <strain> $\varepsilon'$, the convex supporting-plane inequality gives $W(\varepsilon')-W(\varepsilon)\ge\sigma:(\varepsilon'-\varepsilon)$. The integral of $(\sigma-\sigma_0):(\varepsilon'-\varepsilon)$ vanishes by <integration by parts>. Thus the actual <strain> minimizes $\int(W-\sigma_0:\varepsilon)$.
Back to article page
