## 2 Basic Equation  
### The Governing Equation and Boundary Conditions of Dipole Strain Gradient Elasticity  
Based on the Toupin-Mindlin basic elastodynamic equation, a continuum with microstructure can be considered as a collection of sub-particles (or micro-media) with a unit (cube) form, then in the Cartesian coordinate system of \(\text{Ox}_{1\times 2\times 3}\), the expression for kinetic energy density \(T\) is given by:  
\[T=\frac{1}{2}\rho\dot{u}_{j}\dot{u}_{j}+\frac{1}{6}\rho d^{2}\dot{u}_{k,j}\dot{ u}_{k,j}. \tag{1}\]  
where \(\dot{u}\) denotes the derivative of the function \(u\) with respect to time, \(u_{j}\) denotes the component of the function \(u\) in the \(j\)-direction, and \(u_{k,j}\) denotes the partial derivative of the function \(u_{k}\) in the \(j\)-direction.  
The expression for the potential energy density can be represented as follows:  
\[W=\frac{1}{2}\big{(}\lambda\varepsilon_{ii}\varepsilon_{jj}+2\mu\varepsilon_{ ij}\varepsilon_{ij}\big{)}+\frac{1}{2}c\big{(}\lambda\varepsilon_{ii} \varepsilon_{jj}+2\mu\varepsilon_{ij}\varepsilon_{ij}\big{)}_{k}, \tag{2}\]  
where the gradient coefficient \(c\) has dimensions of L\({}^{2}\), and the coefficient of inertia \(d\), has dimensions of L, along with the Lame constants \(\lambda\) and \(\mu\). where \(\rho\) denotes the density, \(\varepsilon_{ij}\) denotes the stress component, and \(i,j,k=1,2,3\).  
The constitutive relationship is defined as follows:  
\[\tau_{ij}=\frac{\partial W}{\partial\epsilon_{ij}}=\lambda\delta_{ij}\epsilon_{ pp}+2\mu\epsilon_{ij}, \tag{3}\]  
\[\mu_{ijk}=\frac{\partial W}{\partial\Big{(}\epsilon_{jk,i}\Big{)}}=c\Big{(} \lambda\delta_{jk}\epsilon_{pp,j}+2\mu\epsilon_{jk,i}\Big{)}, \tag{4}\]  
\(\delta_{jk}\) denotes the Kronecker delta, \(\tau_{ij}\) represents the Cauchy stress or unipolar stress, and \(\mu_{ijk}\) symbolizes the dipole stress with dimensions of \(\mathrm{NL}^{-1}\). It is noteworthy that there exists a symmetry between the Cauchy stress and dipole stress, expressed as \(\tau_{ij}=\tau_{ji}\) and \(\mu_{ijk}=\mu_{kij}\).  
By employing Equations (1) and (2) and applying [PERSON]'s principle, we can derive the variational expression for the work of external forces as follows:  
\[\delta W_{1}=\int_{V}f_{j}\delta u_{j}dV+\int_{V}F_{j}\delta\psi_{jk}dV+\int_{ S}t_{j}\delta u_{j}dS+\int_{S}T_{jk}\delta\psi_{jk}dS, \tag{5}\]  
where \(V\) refers to the volume, \(S\) refers to the surface of the object, \(f_{j}\) stands for the body force per unit macroscopic volume, \(F_{j}\) indicates the micro body force per unit macroscopic volume, \(t_{j}\) signifies the surface force per unit macroscopic area on the boundary, and \(T_{jk}\) is the micro surface force per unit macroscopic area.  
Hamilton variational principle:  
\[\int_{t_{1}}^{t_{2}}\biggl{\{}\int_{V}(\delta T-\delta W)\biggr{\}}dVdt+\int_{ t_{1}}^{t_{2}}\int_{S}\delta W_{1}dSdt=0, \tag{6}\]  
Ignoring the body and boundary forces, we can derive the governing equation and boundary conditions as follows:  
\[\Big{(}\tau_{jk}-\mu_{ijk,i}\Big{)}_{,j}=\rho\ddot{u}_{k}-\frac{\rho d^{2}}{3} \Big{(}\ddot{u}_{k,j}\Big{)}\,,\,\text{in the volume }V \tag{7a}\] \[P_{k}=n_{j}\Big{(}\tau_{jk}-\mu_{ijk,i}\Big{)}-D_{j}\Big{(}n_{j}\mu_{ijk}\Big{)} +(D_{l}n_{l})n_{i}n_{j}\mu_{ijk}+\frac{\rho d^{2}}{3}n_{j}\Big{(}\ddot{u}_{k, j}\Big{)},\,\text{on the surface }S\] (7b) \[R_{k}=n_{i}n_{j}\mu_{ijk},\,\text{on the surface }S \tag{7c}\]  
where \(P_{k}\) represents the monopole force (related to the Cauchy stress), while \(R_{k}\) represents the dipole force (related to higher-order stress), and \(D_{j}=\Big{(}\delta_{jl}-n_{j}n_{l}\Big{)}\partial_{l},\,\,D_{l}=n_{l} \partial_{l}\). \(n_{j}\) is the unit vector of the external normal of the surface of an object.