Chapter 1 Preliminary Concepts

Last updated: 09 May 2026

1.4 Mechanics of Continuous Bodies

Three fundamental laws of mechanics:

Conservation of Mass （can be easily satisfied for a Lagrangian description）
Conservation of Angular Momentum \(\rightarrow\) Symmetry of Cauchy stress tensor \(\sigma\).
Conservation of Linear Momentum \(\rightarrow\) a differential equation of force equilibrium

1.4.1 Boundary-Valued Problem

The balance of linear momentum at each point in the interested domain \(\Omega\) can be expressed as:

\[ \nabla \cdot \boldsymbol{\sigma} + \boldsymbol{f}^b = \boldsymbol{0}, \quad \boldsymbol{x}\in \Omega \]

The boundary-valued problem is to find \(\boldsymbol{u}\) such that

\[\begin{aligned} \nabla \cdot \boldsymbol{\sigma}(\boldsymbol{u}) + \boldsymbol{f}^b = \boldsymbol{0}, & \quad \boldsymbol{x}\in \Omega \\ \boldsymbol{u} = \boldsymbol{0}, & \quad\boldsymbol{x}\in \Gamma^h \\ \boldsymbol{\sigma} \cdot \boldsymbol{n} = \boldsymbol{t}, & \quad\boldsymbol{x}\in \Gamma^s \end{aligned} \]

where \(\Gamma^h\) is the essential boundary and \(\Gamma^s\) is the natural boundary. \(\Gamma = \Gamma^h \cup \Gamma^s\) and \(\Gamma^h \cap \Gamma^s = \emptyset\). The above problem is also called the strong form of the BVP, because the differential equation is satisfied at every point in the domain.

Generally, if the order of the differential equation is \(2m\), the BCs that contains derivatives of order \(m-1\) or lower are called essential BCs, and those that contains derivatives of a higher order than \(m-1\) are called natural BCs.

1.4.2 Principle of Minimum Potential Energy

The principle of minimum potential energy states that: for all kinematically admissible displacements, those that satisfy the above BVP make the total potential energy:

\[\Pi(\boldsymbol{u}) = \underbrace{\frac{1}{2} \int_\Omega \boldsymbol{\sigma}(\boldsymbol{u}) : \boldsymbol{\varepsilon}(\boldsymbol{u}) \, d\Omega}_{\text{Strain Energy } U(\boldsymbol{u})} - \underbrace{\int_\Omega \boldsymbol{u} \cdot \boldsymbol{f}^b\, d\Omega - \int_{\Gamma^s} \boldsymbol{u} \cdot \boldsymbol{t} \, d\Gamma}_{\text{External Work } W(\boldsymbol{u})} \]

stationary on the solution space:

\[ \mathbb{Z} = \{\boldsymbol{u} \in [H^1(\Omega)]^3 | \boldsymbol{u} = \boldsymbol{0} \text{ on } \boldsymbol{x} \in \Gamma^h\} \]

where \(H^1(\Omega)\) is the Sobolev space of order 1.

The virtual displacement or variation of \(\boldsymbol{u}\), denoted as \(\overline{\boldsymbol{u}}\), is defined by considering an perturbation \(\boldsymbol{\eta}(\boldsymbol{x})\) in the solution space \(\mathbb{Z}\) and a small scalar \(\tau\):

\[ \overline{\boldsymbol{u}} = \lim_{\tau \to 0} \frac{1}{\tau} [(\boldsymbol{u} + \tau \boldsymbol{\eta}) - (\boldsymbol{u})] = \frac{d}{d\tau} \left. (\boldsymbol{u} + \tau \boldsymbol{\eta}) \right|_{\tau = 0} =\boldsymbol{\eta} \]

An important property of \(\overline{\boldsymbol{u}}\) is that it is independent of differentiation w.r.t. spatial coordinates:

\[ \frac{\partial \overline{\boldsymbol{u}}}{\partial x_i} = \overline{\left( \frac{\partial \boldsymbol{u}}{\partial x_i} \right)} \]

The principle of minimum potential energy states that the true displacement field \(\boldsymbol{u}\) uniquely minimizes the potential energy functional \(\Pi(\boldsymbol{u})\). To find this minimum, we seek a stationary condition.

For any kinematically admissible virtual displacement \(\overline{\boldsymbol{u}}\), we consider a perturbed configuration \(\boldsymbol{u} + \tau \overline{\boldsymbol{u}}\). If \(\boldsymbol{u}\) is the true minimizer, then the real-valued function \(g(\tau) = \Pi(\boldsymbol{u} + \tau \overline{\boldsymbol{u}})\) must achieve its minimum at \(\tau = 0\).

Consequently, the first variation of \(\Pi\) at \(\boldsymbol{u}\) in the direction of \(\overline{\boldsymbol{u}}\), defined as the derivative of this function:

\[ \delta \Pi(\boldsymbol{u}; \overline{\boldsymbol{u}}) = \frac{d}{d\tau} \Pi(\boldsymbol{u} + \tau \overline{\boldsymbol{u}}) \big|_{\tau=0} \]

must vanish for all arbitrary \(\overline{\boldsymbol{u}}\). This leads to the variational equation:

\[ \delta \Pi(\boldsymbol{u}; \overline{\boldsymbol{u}}) = 0, \quad \forall \overline{\boldsymbol{u}} \in \mathbb{Z} \]

This equation serves as the necessary condition for a minimum of \(\Pi\) at \(\boldsymbol{u}\). The variational equation can be further written as:

\[ \begin{aligned} \delta\Pi(\boldsymbol{u}; \overline{\boldsymbol{u}}) &= \delta U(\boldsymbol{u}; \overline{\boldsymbol{u}}) - \delta W(\boldsymbol{u}; \overline{\boldsymbol{u}})\\ &= \underbrace{\int_\Omega \boldsymbol{\varepsilon}(\overline{\boldsymbol{u}}):\mathbb{D}:\boldsymbol{\varepsilon}(\boldsymbol{u}) \, d\Omega}_{a(\boldsymbol{u}, \overline{\boldsymbol{u}})} - \underbrace{(\int_\Omega \overline{\boldsymbol{u}} \cdot \boldsymbol{f}^b\, d\Omega + \int_{\Gamma^s} \overline{\boldsymbol{u}} \cdot \boldsymbol{t} \, d\Gamma)}_{l(\overline{\boldsymbol{u}})}\\ &= a(\boldsymbol{u}, \overline{\boldsymbol{u}}) - l(\overline{\boldsymbol{u}})=0 \end{aligned} \]

where \(a(\boldsymbol{u}, \overline{\boldsymbol{u}})\) is called the energy bilinear form and \(l(\overline{\boldsymbol{u}})\) is called the load linear form (only conservative loads are considered here). The variational equation can be rewritten as:

\[ a(\boldsymbol{u}, \overline{\boldsymbol{u}}) = l(\overline{\boldsymbol{u}}), \quad \forall \overline{\boldsymbol{u}} \in \mathbb{Z} \]

For some specific conditions, the above variational equation has a unique solution. (See revelant contents and other useful discussion on P45).

1.4.3 Principle of Virtual Work

The principle of minimum potential energy only works for conservative systems, usually the elastic problmes. A more general principle is the principle of virtual work, which states that: for all kinematically admissible virtual displacements, the internal virtual work equals to the external virtual work. The equation remains the same as the variational equation but it is unnecessary to require the point-wise satisfaction of the differential equation.