# Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

# Boundary Element Method for Composite Laminates

• Giuseppe Daví
• Alberto Milazzo
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_96-1

## Definition

The boundary element method (BEM) is a numerical technique to solve engineering/physical problems formulated in terms of boundary integral equations. Composite laminates are assemblages of stacked different materials layers, generally consisting of variously oriented fibrous composite materials.

## Generalized Plane Strain Problem

Let us consider an anisotropic, cylindrical, elastic body having cross section Ω with boundary Γ. The body is referred to the coordinate system xyz with the z-axis directed as the cylinder generatrices.

### Governing Equations

Under the hypothesis of generalized plain strain (ε zz  = 0), the elastic state of the body is described in terms of the displacement vector $$\boldsymbol {u}= \begin {Bmatrix} u_x & u_y & u_z \end {Bmatrix}^T$$ whose components depend on the x and y coordinates only. Introducing the strain vector $$\boldsymbol {\varepsilon }=\begin {Bmatrix} \varepsilon _{xx} & \varepsilon _{yy} & \varepsilon _{xy} & \varepsilon _{xz} & \varepsilon _{yz} \end {Bmatrix}^T$$, the geometrical relationships are written as
\displaystyle \begin{aligned} \boldsymbol{\varepsilon}=\boldsymbol{D} \boldsymbol{u} \end{aligned}
(1)
where D is the differential operator defined by
\displaystyle \begin{aligned} \boldsymbol{D}^T = \begin{bmatrix} \frac{\partial}{\partial x} & 0 & \frac{\partial}{\partial y} & 0 & 0\\ 0 & \frac{\partial}{\partial y} & \frac{\partial}{\partial x} & 0 & 0\\ 0 & 0 & 0 & \frac{\partial}{\partial x} &\frac{\partial}{\partial y} \end{bmatrix} \end{aligned}
(2)
Introducing accordingly the stress vector σ = { σ xx σ yy σ xy σ xz σ yz } T , the equilibrium equations are
\displaystyle \begin{aligned} \boldsymbol{D}^T \boldsymbol{\sigma} + \boldsymbol{f}= \boldsymbol{0} \quad \quad \text{in}\;\varOmega \end{aligned}
(3)
where f = f(x, y) is the vector containing the components of the applied body forces. The stresses and strains obey to the anisotropic Hooke’s law, which read as
\displaystyle \begin{aligned} \boldsymbol{\sigma}= \boldsymbol{E} \boldsymbol{\varepsilon} \end{aligned}
(4)
Eventually, substituting Eq. (1) into Eq. (4) and in turn into Eq. (3), the generalized plane strain problem governing equations are inferred:
\displaystyle \begin{aligned}\boldsymbol{D}^T \boldsymbol{E} \boldsymbol{D} \boldsymbol{u} +\boldsymbol{f}= \boldsymbol{0} \quad \quad \text{in} \;\varOmega \end{aligned}
(5)
Equation (5) is completed with the essential and natural boundary conditions, provided on the boundary parts Γ u and Γ t , respectively. They are expressed as
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{u} = \bar{\boldsymbol{u}} \quad \quad \text{on}\,\,\varGamma_u\\ \boldsymbol{t}=\boldsymbol{D}_n^T \boldsymbol{E} \boldsymbol{D} \boldsymbol{u} = \bar{\boldsymbol{t}} \quad \quad \text{on}\,\,\varGamma_t \end{gathered} \end{aligned}
(6)
where the overbar denotes prescribed quantities and t is the vector containing the boundary tractions components. In Eq. (6), D n is the boundary traction operator obtained from D by substituting the derivatives with the corresponding direction cosines of the boundary outer normal.

### Boundary Integral Representation and Equations

Let u, f, and t be the displacements, body forces, and boundary tractions constituting the solution of the investigated problem. Let also u j , f j , and t j be a fictitious system of displacements, body forces, and boundary tractions that satisfy Eq. (5). By applying Betti’s theorem, the following reciprocity statement is written (Betti, 1872).
\displaystyle \begin{aligned} \int_\varGamma \left( {\boldsymbol{u}_j}^T\boldsymbol{t} - {\boldsymbol{t}_j}^T \boldsymbol{u} \right) d\varGamma = \int_\varOmega \left( {\boldsymbol{f}_j}^T\boldsymbol{u} - {\boldsymbol{u}_j}^T \boldsymbol{f} \right) d\varOmega \end{aligned}
(7)
The boundary integral representation for the problem in hands is derived from Eq. (7) by taking the fictitious solution as the elastic response due to a concentrated line load f j , directed as j and uniformly distributed along a line parallel to the z-axis (see Fig. 1). Thus, it results
\displaystyle \begin{aligned} \boldsymbol{f}_j = \boldsymbol{c}_j \delta (\textbf{x}-\textbf{x}_0)\end{aligned}
(8)
where c j is the load intensity, $$\textbf {x}=\begin {Bmatrix} x,\;y\end {Bmatrix}^T$$ denotes cross section points, and δ(x −x 0) is the Dirac delta centered at the so-called source point x 0. The elastic state associated to such a load is the problem fundamental solution.
By using the fundamental solution, the reciprocity theorem provides the following relationship:
\displaystyle \begin{aligned}\boldsymbol{c}_j^T(\textbf{x}_0) \boldsymbol{u} (\textbf{x}_0) + \int_\varGamma \left[\boldsymbol{t}_j^T (\textbf{x},\textbf{x}_0) \boldsymbol{u}(\textbf{x}) -\boldsymbol{u}_j^T (\textbf{x},\textbf{x}_0) \boldsymbol{t}(\textbf{x})\right] d\varGamma = \int_\varOmega \boldsymbol{u}_j^T (\textbf{x},\textbf{x}_0) \boldsymbol{f} (\textbf{x}) d\varOmega \end{aligned}
(9)
which represents the Somigliana identity, Somigliana (1886), for the generalized plain strain elastic problem. By using three independent fundamental solutions, associated with body forces directed along the reference axes, the boundary integral representation for the displacement vector at the source point x 0 is written:
\displaystyle \begin{aligned} \boldsymbol{C}(\textbf{X}_0) \boldsymbol{u}(\textbf{x}_0) + \int_\varGamma \left[ \boldsymbol{T}(\textbf{x},\textbf{x}_0)\boldsymbol{u}(\textbf{x}) - \boldsymbol{U}(\textbf{x},\textbf{x}_0)\boldsymbol{t}(\textbf{x})\right]d\varGamma = \int_\varOmega \boldsymbol{U}(\textbf{x},\textbf{x}_0) \boldsymbol{f}(\textbf{x}) d\varOmega\end{aligned}
(10)
where U(x, x 0) and T(x, x 0) are the 3 × 3 fundamental solutions kernels whose components U ji and T ji are the i-th components of the displacements and tractions of the fundamental solution with the load applied along the j direction. When the source point x 0 belongs to the boundary Γ, the kernels become singular, and a limiting process is needed to evaluate Eq. (10), Banerjee and Butterfield (1981). Such a limiting process leads to the boundary integral equations for the problem, which writes as where the symbol Open image in new window indicates that the integrals are calculated as their Chauchy principal value and the free term matrix is

### Fundamental Solutions

The fundamental solutions satisfy the following equation in the unbounded domain:
\displaystyle \begin{aligned} \boldsymbol{D}^T \boldsymbol{E} \boldsymbol{D} \boldsymbol{u}_j + \boldsymbol{c}_j \delta (\textbf{x} - \textbf{x}_0 ) = \boldsymbol{0} \end{aligned}
(13)
Following Lekhnitskii (1963), Eq. (13) admits solutions of the form
\displaystyle \begin{aligned}\boldsymbol{v}_j = \lambda \boldsymbol{a} \ln (X + \mu Y ) \end{aligned}
(14)
where a, μ, and λ are complex constants, X = x(x) − x(x 0) and Y = y(x) − y(x 0). Substituting Eq. (14) into Eq. (13), the following eigenvalue problem is obtained:
\displaystyle \begin{aligned} \left[ \boldsymbol{I}_x^T \boldsymbol{E} \boldsymbol{I}_x + \mu \left( \boldsymbol{I}_x^T \boldsymbol{E} \boldsymbol{I}_y + \boldsymbol{I}_y^T \boldsymbol{E} \boldsymbol{I}_x \right) + \mu ^2\boldsymbol{I}_y^T \boldsymbol{E} \boldsymbol{I}_y \right]\boldsymbol{A} = \boldsymbol{0}\end{aligned}
(15)
where the matrices I α (α = x, y) are built from the operator D by setting the derivative with respect to α equal to one and replacing all the other terms with zeros. The solution of Eq. (15) provides six eigenvalues μ k and the relative eigenvectors a k , which form conjugate pairs for stable materials. Assuming distinct eigenvalues, the fundamental solutions are obtained by superposing the six eigensolutions of the form given in Eq. (14). If Im(μ k ) > 0 for k = 1, 2, 3, the fundamental solution displacements u j and tractions t j are
$$\displaystyle \begin{gathered} \boldsymbol{u}_j = 2\sum\limits_{k = 1}^3 {\mathop{Re}\nolimits} \left[ {\lambda _{kj} \boldsymbol{a}_k \ln \left( {X + \mu _k Y } \right)} \right] \end{gathered}$$
(16a)
$$\displaystyle \begin{gathered}\boldsymbol{t}_j = 2\sum\limits_{k = 1}^3 {{\mathop{Re}\nolimits} \left[ {\lambda _{kj} \boldsymbol{D}_n^T \boldsymbol{E} \boldsymbol{D}_{\mu k} \boldsymbol{a}_k \left(X {+} \mu _k Y\right)^{-1} } \right]} \end{gathered}$$
(16b)
where the matrix D μk is obtained from the operator D by replacing the derivative with respect to x with one and the derivative with respect to y with μ k . The constants λ kj are determined by enforcing the compatibility and equilibrium conditions. The vector $$\boldsymbol {\lambda }_j = \left \{ \begin {array}{*{20}c} \lambda _{1j} & \lambda _{2j} & \lambda _{3j} \end {array} \right \}^T$$ is determined by
\displaystyle \begin{aligned} \boldsymbol{\lambda}_j = \left( \boldsymbol{B} + \widetilde{\boldsymbol{B}} \widetilde{\boldsymbol{A}}^{ - 1} \boldsymbol{A} \right)^{ - 1} \boldsymbol{c}_j \end{aligned}
(17)
where the tilde denotes the complex conjugate, A is the eigenvectors matrix, and B is a matrix whose columns are defined as
\displaystyle \begin{aligned} \boldsymbol{b}_k = \bar{\boldsymbol{D}}_k \boldsymbol{E} \boldsymbol{D}_{\mu k} \boldsymbol{a}_k \end{aligned}
(18)
being
\displaystyle \begin{aligned} \bar{\boldsymbol{D}}_k{=} \pi \frac{{1 {+} \sqrt { - 1} \mu _k }}{{1 {+} \mu _k^2 }}\left[ {\begin{array}{*{20}c} { - 1} & 0 & 0 & 0 & {\sqrt { - 1} } \\ 0 & {\sqrt { - 1} } & 0 & 0 & { - 1} \\ 0 & 0 & {\sqrt { - 1} } & { - 1} & 0 \\ \end{array}} \right] \end{aligned}
(19)

## Beam-Type Composite Laminates

Consider now that the elastic body is a beam-type composite laminate with general lay-up that consists of N anisotropic plies, perfectly bonded along the interfaces. The laminate is subjected to a combined load characterized by the uniform extension e, the bending curvatures κ x and κ y , the twisting curvature 𝜗, and the shear/bending loading parameters γ x and γ y . Assuming Saint Venant’s principle is satisfied, sufficiently far from the laminate ends, the displacement field s, associated with the loading above-described, can be expressed as (Lekhnitskii 1963)
\displaystyle \begin{aligned}\boldsymbol{s}=\boldsymbol{w}_0+z\, \boldsymbol{w}_1 + z\, \bar{\boldsymbol{X}}_1\, \boldsymbol{\kappa} - \frac{1}{2} z^2\, \bar{\boldsymbol{X}}_2\, \boldsymbol{\kappa} + \frac{1}{2} z^2\, \bar{\boldsymbol{X}}_3\, \boldsymbol{\gamma} - \frac{1}{6} z^3\, \bar{\boldsymbol{X}}_4\, \boldsymbol{\gamma}\end{aligned}
(20)
where $$\boldsymbol {\kappa }= \begin {Bmatrix} e & \kappa _x & \kappa _y & \vartheta \end {Bmatrix}^T$$ and $$\boldsymbol {\gamma }= \begin {Bmatrix} \gamma _x & \gamma _y \end {Bmatrix}^T$$. In Eq. (20), w 0 and w 1 are 3 × 1 vectors of displacement functions that depend on x and y only, and
\displaystyle \begin{aligned} \bar{\boldsymbol{X}}_1=\begin{bmatrix} 0 & 0 & 0 & -y \\ 0 & 0 & 0 & x \\ 1 & x & y & 0 \\ \end{bmatrix}, \quad \bar{\boldsymbol{X}}_2=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ \end{bmatrix}, \quad \bar{\boldsymbol{X}}_3=\begin{bmatrix} 0 & 0 \\ 0 & 0 \\ x & y\\ \end{bmatrix}, \quad \bar{\boldsymbol{X}}_4=\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0\\ \end{bmatrix}\end{aligned}
(21)
The strain field associated with the displacements of Eq. (20) is given by
\displaystyle \begin{aligned} \boldsymbol{\varepsilon}= \boldsymbol{D}\boldsymbol{w}_0 +\boldsymbol{Z}_1 \boldsymbol{w}_1 + \boldsymbol{X}_1\boldsymbol{\kappa} + z \boldsymbol{D}\boldsymbol{w}_1= \boldsymbol{\varepsilon}_0 + z \boldsymbol{\varepsilon}_1 \end{aligned}
(22a)
\displaystyle \begin{aligned} \varepsilon_{zz}= \boldsymbol{Z}_2 \boldsymbol{w}_1 + \boldsymbol{X}_2 \boldsymbol{\kappa} + z \boldsymbol{X}_3 \boldsymbol{\gamma} \end{aligned}
(22b)
where
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{Z}_1 =\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{bmatrix}, \,\, \boldsymbol{Z}_2=\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}^T, \,\, \boldsymbol{X}_1=\begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -y \\ 0 & 0 & 0 & x \\ \end{bmatrix}, \,\, \boldsymbol{X}_2=\begin{bmatrix} 1 \\ x \\ y \\ 0 \end{bmatrix}^T, \,\, \boldsymbol{X}_3=\begin{bmatrix} x \\ y \end{bmatrix}^T \end{gathered} \end{aligned}
(23)
The stress field in each ply of the laminate is expressed by using the constitutive equations, which are conveniently partitioned as
$$\displaystyle \begin{gathered}\begin{Bmatrix} \boldsymbol{\sigma} \\ \sigma_{zz} \end{Bmatrix}= \begin{bmatrix} \boldsymbol{E}_{\sigma\varepsilon} &\boldsymbol{E}_{\sigma{\varepsilon_{zz}}} \\ \boldsymbol{E}_{{\sigma_{zz}}\varepsilon} & E_{{\sigma_{zz}}{\varepsilon_{zz}}} \end{bmatrix} \begin{Bmatrix} \boldsymbol{\varepsilon} \\ \varepsilon_{zz}\\ \end{Bmatrix}= \begin{Bmatrix} \boldsymbol{\sigma}_0 \\ \sigma_{zz0} \end{Bmatrix} + z \begin{Bmatrix} \boldsymbol{\sigma}_1 \\ \sigma_{zz1}\\ \end{Bmatrix} \end{gathered}$$
(24a)
$$\displaystyle \begin{gathered} \boldsymbol{\tau}= \begin{Bmatrix} \sigma_{xz}\\ \sigma_{yz}\\ \sigma_{zz}\\ \end{Bmatrix}= \begin{bmatrix} \boldsymbol{E}_{\tau\varepsilon} & \boldsymbol{E}_{\tau{\varepsilon_{zz}}} \end{bmatrix} \begin{Bmatrix} \boldsymbol{\varepsilon} \\ \varepsilon_{zz} \end{Bmatrix} = \boldsymbol{\tau}_0 + z \boldsymbol{\tau}_1 \end{gathered}$$
(24b)
With the notation introduced above, the equilibrium equations are rewritten as
\displaystyle \begin{aligned} \boldsymbol{D}^T\boldsymbol{\sigma}+ \frac{\partial\boldsymbol{\tau}}{\partial z} =\boldsymbol{0} \end{aligned}
(25)
Focusing on the k-th ply having cross section Ω k with boundary Γ k (see Fig. 2), upon substitution of σ and τ from Eqs. (24) into Eq. (25) and observing that $$\boldsymbol {E}_{\sigma \varepsilon }\boldsymbol {Z}_1 \boldsymbol {w}_1+\boldsymbol {E}_{\sigma {\varepsilon _{zz}}}\boldsymbol {Z}_2 \boldsymbol {w}_1= {\boldsymbol {E}_{\tau \varepsilon }}^T \boldsymbol {w}_1$$, one obtains (Daví 1997; Milazzo 2000)
\displaystyle \begin{aligned}\begin{gathered} \boldsymbol{D}^T\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{D}\boldsymbol{w}_0^{\langle k\rangle} + \left( \boldsymbol{D}^T {\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T +\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle} \boldsymbol{D}\right)\boldsymbol{w}_1^{\langle k\rangle}+ \boldsymbol{D}^T \left(\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{X}_{1}+\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{2}\right)\boldsymbol{\kappa}+\\ \boldsymbol{E}_{\tau{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma}+ z \left[\boldsymbol{D}^T \boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{D}\boldsymbol{w}_1^{\langle k\rangle} + \boldsymbol{D}^T \boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma} \right]=\boldsymbol{0} \end{gathered} \end{aligned}
(26)
where the superscript 〈k〉 is used to denote quantities related to the k-th ply of the laminate. Equation (26) is satisfied for eachz by simultaneously fulfilling the following expressions:
$$\displaystyle \begin{gathered} \boldsymbol{D}^T\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{D}\boldsymbol{w}_1^{\langle k\rangle} + \boldsymbol{D}^T\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma}=\boldsymbol{0} \end{gathered}$$
(27a)
\displaystyle \begin{gathered}\begin{aligned} \boldsymbol{D}^T\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{D}\boldsymbol{w}_0^{\langle k\rangle} + &\left( \boldsymbol{D}^T {\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T +\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle} \boldsymbol{D}\right)\boldsymbol{w}_1^{\langle k\rangle}+\\ &\boldsymbol{D}^T \left(\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{X}_{1}+\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{2}\right)\boldsymbol{\kappa}+\boldsymbol{E}_{\tau{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma}=\boldsymbol{0} \end{aligned} \end{gathered}
(27b)
The appropriate essential and natural boundary conditions need to be associated to the ply governing equations, namely, Eqs. (27a) and (27b). The essential boundary conditions are given in terms of prescribed displacement functions w 1 and w 0, respectively, whereas the natural boundary conditions are supplied in terms of assigned values of the tractions. According to the mathematical structure evidenced by the elastic response, the boundary tractions are expressed as t = t 0 + z t 1 where $$\boldsymbol {t}_0=\boldsymbol {D}_n\boldsymbol {E}_{\sigma \varepsilon } \boldsymbol {D}\boldsymbol {w}_0+\boldsymbol {D}_n\boldsymbol {E}_{\tau \varepsilon }^T\boldsymbol {w}_1+ \boldsymbol {D}_n \left (\boldsymbol {E}_{\sigma \varepsilon }\boldsymbol {X}_{1}+\boldsymbol {E}_{\sigma {\varepsilon _{zz}}}\boldsymbol {X}_{2}\right )\boldsymbol {\kappa }$$ and $$\boldsymbol {t}_1=\boldsymbol {D}_n\boldsymbol {E}_{\sigma \varepsilon } \boldsymbol {D}\boldsymbol {w}_1+\boldsymbol {D}_n\boldsymbol {E}_{\sigma {\varepsilon _{zz}}} \boldsymbol {X}_{3}\boldsymbol {\gamma }$$. Thus, the natural boundary conditions for Eqs. (27) consist of assigned values of t 1 and t 0, respectively.
It is worth to note that Eq. (27) constitutes a system of decoupable partial differential equations: the first equation can be solved for w 1 and, in turn, the second equation can be solved for w 0 . These equations govern the behavior of each ply as a single entity with unknown displacement and tractions at the laminate interfaces, which actually belong to the ply boundaries. The whole laminate model is built enforcing the interface continuity, namely, displacements continuity and tractions equilibrium, at the laminate interfaces. It consists of Eq. (27) written for all of the laminate plate and the interface continuity conditions written for all of the laminate interfaces. For the interface Γ k between the plies k and k + 1, these read as
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{w}_0^{\langle k\rangle} =\boldsymbol{w}_0^{\langle k+1\rangle},\quad \boldsymbol{w}_1^{\langle k\rangle} =\boldsymbol{w}_1^{\langle k+1\rangle}\\ \boldsymbol{t}_0^{\langle k\rangle} +\boldsymbol{t}_0^{\langle k+1\rangle}=\boldsymbol{0},\quad \boldsymbol{t}_1^{\langle k\rangle} +\boldsymbol{t}_1^{\langle k+1\rangle}=\boldsymbol{0} \end{gathered} \end{aligned}
(28)

### Boundary Integral Representation and Equations

It can be recognized that Eq. (27a) resembles the structure of Eq. (5), provided that E = E σε and the system of body forces is specified as $$\boldsymbol {f}=\boldsymbol {D}^T\boldsymbol {E}_{\sigma {\varepsilon _{zz}}}\boldsymbol {X}_{3}\boldsymbol {\gamma }$$. Therefore, accounting for the expression of t 1, the following boundary integral representation holds for the displacement function $$\boldsymbol {w}_1^{\langle k\rangle }$$ of the k-th ply
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{C}^{\langle k\rangle}\boldsymbol{w}_1^{\langle k\rangle}(\textbf{x}_0) + \int_{\varGamma^{\langle k \rangle}}\boldsymbol{T}^{\langle k\rangle}\boldsymbol{w}_1^{\langle k\rangle} d\varGamma- \int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle}\boldsymbol{t}_1^{\langle k\rangle} d\varGamma=\\ \int_{\varOmega^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle} \boldsymbol{D}^T\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma} d\varOmega - \int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle} \boldsymbol{D}_n\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma}d\varGamma \end{gathered} \end{aligned}
(29)
which, applying the divergence theorem to the right-hand side, provides
\displaystyle \begin{aligned} \begin{aligned}{} \boldsymbol{C}^{\langle k\rangle}\boldsymbol{w}_1^{\langle k\rangle}(\textbf{x}_0) + & \int_{\varGamma^{\langle k \rangle}}\boldsymbol{T}^{\langle k\rangle}\boldsymbol{w}_1^{\langle k\rangle} d\varGamma-\\ &\int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle}\boldsymbol{t}_1^{\langle k\rangle} d\varGamma= - \int_{\varOmega^{\langle k \rangle}} \left( \boldsymbol{D}\boldsymbol{U}^{\langle k\rangle}\right)^T\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma}d\varOmega \end{aligned} \end{aligned}
(30)
Analogously, the boundary integral representation for Eq. (27b) is obtained from Eq. (5) assuming that $$\boldsymbol {w}_1^{\langle k\rangle }$$ is known and provided that $$\boldsymbol {f}= \left ( \boldsymbol {D}^T\boldsymbol {E}_{\tau \varepsilon }^T +\boldsymbol {E}_{\tau \varepsilon } \boldsymbol {D}\right )\boldsymbol {w}_1+ \boldsymbol {D}^T \left (\boldsymbol {E}_{\sigma \varepsilon } \boldsymbol {X}_{1}+\boldsymbol {E}_{\sigma {\varepsilon _{zz}}} \boldsymbol {X}_{2}\right )\boldsymbol {\kappa }+\boldsymbol {E}_{\tau {\varepsilon _{zz}}} \boldsymbol {X}_{3}\boldsymbol {\gamma }$$. By using the definition of t 0, one has
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{C}^{\langle k\rangle}\boldsymbol{w}_0^{\langle k\rangle}(\textbf{x}_0) + \int_{\varGamma^{\langle k \rangle}}\boldsymbol{T}^{\langle k\rangle}\boldsymbol{w}_0^{\langle k\rangle} d\varGamma- \int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle}\boldsymbol{t}_0^{\langle k\rangle} d\varGamma=\\ \int_{\varOmega^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle} \left[ \left( \boldsymbol{D}^T{\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T +\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle} \boldsymbol{D}\right)\boldsymbol{w}_1^{\langle k\rangle}+ \boldsymbol{D}^T \left(\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{X}_{1}+\boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{2}\right)\boldsymbol{\kappa}+ \boldsymbol{E}_{\tau{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma} \right] d\varOmega -\\ \int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle} \boldsymbol{D}_n \left[{\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T\boldsymbol{w}_1^{\langle k\rangle}+ \left(\boldsymbol{E}_{\sigma\varepsilon}^{\langle k\rangle} \boldsymbol{X}_{1}+ \boldsymbol{E}_{\sigma{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{2}\right)\boldsymbol{\kappa} \right] d\varGamma \end{gathered} \end{aligned}
(31)
Observing that $$\boldsymbol {E}_{\sigma \varepsilon } \boldsymbol {X}_{1}+\boldsymbol {E}_{\sigma {\varepsilon _{zz}}} \boldsymbol {X}_{2} =\boldsymbol {E}_{\tau \varepsilon } ^T {\bar {\boldsymbol {X}}}_{1}$$ and applying the divergence theorem, from Eq. (31), one obtains
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{C}^{\langle k\rangle}\boldsymbol{w}_0^{\langle k\rangle}(\textbf{x}_0) + \int_{\varGamma^{\langle k \rangle}}\boldsymbol{T}^{\langle k\rangle}\boldsymbol{w}_0^{\langle k\rangle} d\varGamma- \int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle}\boldsymbol{t}_0^{\langle k\rangle} d\varGamma=\\ \int_{\varOmega^{\langle k \rangle}} \left[\boldsymbol{U}^{\langle k\rangle}\left(\boldsymbol{D}^T{\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T +\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle} \boldsymbol{D}\right)\boldsymbol{w}_1^{\langle k\rangle}\right] d\varOmega - \int_{\varGamma^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle}\boldsymbol{D}_n{\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T\boldsymbol{w}_1^{\langle k\rangle} d\varGamma -\\ \int_{\varOmega^{\langle k \rangle}} \left( \boldsymbol{D}\boldsymbol{U}^{\langle k\rangle}\right)^T{\boldsymbol{E}_{\tau\varepsilon}^{\langle k\rangle}}^T\bar{\boldsymbol{X}}_1\boldsymbol{\kappa} d\varOmega+ \int_{\varOmega^{\langle k \rangle}}\boldsymbol{U}^{\langle k\rangle}\boldsymbol{E}_{\tau{\varepsilon_{zz}}}^{\langle k\rangle} \boldsymbol{X}_{3}\boldsymbol{\gamma} d\varOmega + \end{gathered} \end{aligned}
(32)
Equations (30) and (32) can be regarded as the form of the beam-type Somigliana identity for each single ply within the laminate. They provide a direct link between the displacement functions $$\boldsymbol {w}_0^{\langle k\rangle }$$ and $$\boldsymbol {w}_1^{\langle k\rangle }$$ at the field point x 0 and the characteristics of the elastic response on the boundary, namely, displacements and tractions. Therefore, they give the boundary integral representation of the ply displacements field. Applying the usual limit procedure, Banerjee and Butterfield (1981), Eqs. (30) and (32) can be written for boundary points obtaining the boundary integral equations (BIE) for the problem solution. The BIEs take the same form of Eqs. (30) and (32) with the appropriate value of the free term and the adequate treatment of the singular integrals (Aliabadi 2002; Sladek and Sladek 1998).

### Numerical Solution

The laminate solution through the boundary integral equations is numerically obtained by using the multidomain boundary element method, Aliabadi (2002). The boundary Γ k of each ply is discretized into n k boundary elements $$\varGamma _q^{\langle k\rangle }$$ and its cross section Ω k into m k internal cells with domain $$\varOmega _r^{\langle k\rangle }$$. On each boundary element $$\varGamma _q^{\langle k\rangle }$$, the displacement functions $$\boldsymbol {w}_1^{\langle k\rangle }$$ and the tractions $$\boldsymbol {t}_1^{\langle k\rangle }$$ are expressed through their nodal values $$\boldsymbol {\delta }_1^{\;\varGamma _q^{\langle k\rangle }}$$ and $$\boldsymbol {p}_1^{\;\varGamma _q^{\langle k\rangle }}$$ by using suitable shape functions matrices N
\displaystyle \begin{aligned} \boldsymbol{w}_1^{\langle k\rangle}=\boldsymbol{N} \boldsymbol{\delta}_1^{\;\varGamma_q^{\langle k\rangle}},\quad \boldsymbol{t}_1^{\langle k\rangle}=\boldsymbol{N} \boldsymbol{p}_1^{\;\varGamma_q^{\langle k\rangle}} \end{aligned}
(33)
The discretized version of Eq. (30) for the point x i of the k-th ply is
\displaystyle \begin{aligned} \boldsymbol{C}\boldsymbol{w}_1^{\langle k\rangle} (\textbf{x}_i)+\sum_{q=1}^{n^{\langle k\rangle}} \hat{\boldsymbol{H}}_{iq}^{\langle k\rangle} \boldsymbol{\delta}_{1}^{\;\varGamma_q^{\langle k\rangle}}+ \sum_{q=1}^{n^{\langle k\rangle}} \boldsymbol{G}_{iq}^{\langle k\rangle} \boldsymbol{p}_{1}^{\;\varGamma_q^{\langle k\rangle}} =\sum_{r=1}^{m^{\langle k\rangle}} \boldsymbol{B}_{ir}^{\langle k\rangle}\end{aligned}
(34)
where the influence coefficients and the right-hand side are defined by
\displaystyle \begin{aligned} \begin{gathered} \hat{\boldsymbol{H}}_{iq}^{\langle k\rangle}= \int_{\varGamma_q^{\langle k\rangle}} \boldsymbol{T}^{\langle k\rangle} \boldsymbol{N} d\varGamma, \quad \boldsymbol{G}_{iq}^{\langle k\rangle} = - \int_{\varGamma_q^{\langle k\rangle}} \boldsymbol{U}^{\langle k\rangle} \boldsymbol{N} d\varGamma, \\ \boldsymbol{B}_{ir}^{\langle k\rangle}= - \int_{\varOmega_r^{\langle k\rangle}} \left( \boldsymbol{D} \boldsymbol{U}^{\langle k\rangle}\right)^T \boldsymbol{E}_{\sigma \varepsilon_{zz}}^{\langle k\rangle} {\boldsymbol{X}}_{3} \boldsymbol{\gamma} \, d\varOmega \end{gathered} \end{aligned}
(35)
Analogously, on the boundary element $$\varGamma _q^{\langle k\rangle }$$, the displacement functions $$\boldsymbol {w}_0^{\langle k\rangle }$$ and the tractions $$\boldsymbol {t}_0^{\langle k\rangle }$$ are expressed in terms of their boundary nodal values $$\boldsymbol {\delta }_0^{\;\varGamma _q^{\langle k\rangle }}$$ and $$\boldsymbol {p}_0^{\;\varGamma _q^{\langle k\rangle }}$$
\displaystyle \begin{aligned} \boldsymbol{w}_0^{\langle k\rangle}=\boldsymbol{N} \boldsymbol{\delta}_0^{\;\varGamma_q^{\langle k\rangle}},\quad \boldsymbol{t}_0^{\langle k\rangle}=\boldsymbol{N} \boldsymbol{p}_0^{\;\varGamma_q^{\langle k\rangle}} \end{aligned}
(36)
Moreover, let us assume that on the r-th cell, the displacement function $$\boldsymbol {w}_1^{\langle k\rangle }$$ can be interpolated as
\displaystyle \begin{aligned} \boldsymbol{w}_1^{\langle k\rangle}=\boldsymbol{M} \boldsymbol{\delta}_1^{\;\varOmega_r^{\langle k\rangle}} \end{aligned}
(37)
where $$\boldsymbol {\delta }_1^{\;\varOmega _r^{\langle k\rangle }}$$ is the vector collecting the cell nodal values of $$\boldsymbol {w}_1^{\langle k\rangle }$$ and M is the matrix of shape functions. The discretized version of Eq. (32) is
\displaystyle \begin{aligned} \boldsymbol{C}\boldsymbol{w}_0^{\langle k\rangle} (\textbf{x}_i)+\sum_{q=1}^{n^{\langle k\rangle}} \hat{\boldsymbol{H}}_{iq}^{\langle k\rangle} \boldsymbol{\delta}_0^{\;\varGamma_q^{\langle k\rangle}}+ \sum_{q=1}^{n^{\langle k\rangle}} \boldsymbol{G}^{\langle k\rangle}_{iq}\boldsymbol{p}_0^{\;\varGamma_q^{\langle k\rangle}} = \sum_{q=1}^{n^{\langle k\rangle}} \hat{\boldsymbol{J}}_{iq}^{\langle k\rangle} \boldsymbol{\delta}_{1}^{\;\varGamma_q^{\langle k\rangle}}+ \sum_{r=1}^{m^{\langle k\rangle}} \hat{\boldsymbol{W}}_{ir}^{\langle k\rangle} \boldsymbol{\delta}_{1}^{\;\varOmega_r^{\langle k\rangle}}+ \sum_{r=1}^{m^{\langle k\rangle}} \boldsymbol{Y}_{ir}^{\langle k\rangle} \end{aligned}
(38)
where
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{J}_{iq}^{\langle k\rangle}= -\int_{\varGamma_q^{\langle k\rangle}} \boldsymbol{U}^{\langle k\rangle} \boldsymbol{D}_n {\boldsymbol{E}_{\tau \varepsilon}^{\langle k\rangle}}^T \boldsymbol{N} d\varGamma, \quad \boldsymbol{Y}_{ir}^{\langle k\rangle}= \int_{\varOmega_r^{\langle k\rangle}} \left[\boldsymbol{U}^{\langle k\rangle} \boldsymbol{E}_{\tau \varepsilon_{zz}}^{\langle k\rangle} \boldsymbol{X}_{3} \boldsymbol{\gamma} -\left(\boldsymbol{DU}^{\langle k\rangle}\right)^T {\boldsymbol{E}_{\tau \varepsilon}^{\langle k\rangle}}^T \bar{X}_1 \boldsymbol{\kappa}\right]d\varOmega\\ \boldsymbol{W}_{ir}^{\langle k\rangle}= \int_{\varOmega_r^{\langle k\rangle}} \boldsymbol{U}^{\langle k\rangle} \left(\boldsymbol{D}^T {\boldsymbol{E}_{\tau \varepsilon}^{\langle k\rangle}}^T +\boldsymbol{E}_{\tau \varepsilon}^{\langle k\rangle} \boldsymbol{D}\right)\boldsymbol{M} d\varOmega \end{gathered} \end{aligned}
(39)
The boundary discretized model governing the behavior of the k-th ply within the laminate is obtained by: (i) collocating Eqs. (34) and (38) at the boundary nodes and (ii) collocating Eq. (30) at the internal cell nodes. By so doing one obtains
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{H}^{\langle k\rangle} \boldsymbol{\Delta}_1^{\langle k\rangle}+\boldsymbol{G}^{\langle k\rangle} \boldsymbol{P}_1^{\langle k\rangle}=\boldsymbol{B}^{\langle k\rangle}\\ \boldsymbol{H}^{\langle k\rangle} \boldsymbol{\Delta}_0^{\langle k\rangle}+\boldsymbol{G}^{\langle k\rangle} \boldsymbol{P}_0^{\langle k\rangle}=\boldsymbol{J}^{\langle k\rangle} \boldsymbol{\Delta}_1^{\langle k\rangle}+\boldsymbol{W}^{\langle k\rangle} \bar{\boldsymbol{\Delta}}_1^{\langle k\rangle}+\boldsymbol{Y}^{\langle k\rangle}\\ \bar{\boldsymbol{\Delta}}_1^{\langle k\rangle} = \bar{\boldsymbol{H}}^{\langle k\rangle} \boldsymbol{\Delta}_1^{\langle k\rangle} +\bar{\boldsymbol{G}}^{\langle k\rangle} \boldsymbol{P}_1^{\langle k\rangle} = \bar{\boldsymbol{B}}^{\langle k\rangle} \end{gathered} \end{aligned}
(40)
where $$\boldsymbol {\Delta }_1^{\langle k\rangle }$$ and $$\boldsymbol {\Delta }_0^{\langle k\rangle }$$ contain the boundary nodal values of the displacement functions $$\boldsymbol {w}_1^{\langle k\rangle }$$ and $$\boldsymbol {w}_0^{\langle k\rangle }$$, respectively, $$\boldsymbol {P}_1^{\langle k\rangle }$$ and $$\boldsymbol {P}_0^{\langle k\rangle }$$ contain the nodal values of the corresponding boundary tractions, and $$\bar {\boldsymbol {\Delta }}_1^{\langle k\rangle }$$ is the vector of the cells nodal displacements.
The boundary element model for the whole laminate is obtained by writing the discretized integral equation for all of the plies of the laminate, namely, Eq. (40) for k = 1, 2, …, N, and imposing the interface continuity conditions and the external boundary conditions in terms of variables nodal values. In particular, for k = 1, 2, …, N − 1, the interface continuity conditions expressed in terms of nodal values read as
\displaystyle \begin{aligned} \begin{gathered} \boldsymbol{Q}_k^{k+1} \boldsymbol{\Delta}_0^{\langle k\rangle} = \boldsymbol{Q}_{k+1}^{k} \boldsymbol{\Delta}_0^{\langle k+1\rangle} \quad \boldsymbol{Q}_k^{k+1} \boldsymbol{\Delta}_1^{\langle k\rangle} = \boldsymbol{Q}_{k+1}^{k} \boldsymbol{\Delta}_1^{\langle k+1\rangle} \\ \boldsymbol{Q}_k^{k+1} \boldsymbol{P}_0^{\langle k\rangle} =- \boldsymbol{Q}_{k+1}^{k} \boldsymbol{P}_0^{\langle k+1\rangle}, \quad \boldsymbol{Q}_k^{k+1} \boldsymbol{P}_1^{\langle k\rangle} =- \boldsymbol{Q}_{k+1}^{k} \boldsymbol{P}_1^{\langle k+1\rangle} \end{gathered} \end{aligned}
(41)
where $$\boldsymbol {Q}_k^{k+1}$$ and $$\boldsymbol {Q}_{k+1}^{k}$$ are Boolean matrices selecting the corresponding nodal displacements that belong to the interface between the plies k and k + 1. The external boundary conditions, for k = 1, 2, …, N, are given by
\displaystyle \begin{aligned} \boldsymbol{Q}_k^{k} \boldsymbol{\Delta}_0^{\langle k\rangle} =\boldsymbol{0}, \quad \boldsymbol{Q}_k^{k} \boldsymbol{\Delta}_1^{\langle k\rangle} =\boldsymbol{0} \end{aligned}
(42)
where $$\boldsymbol {Q}_k^{k}$$ is a Boolean matrix which selects displacements of the nodes belonging to the ply external boundary.

The system of Eqs. (40), together with the interface continuity conditions Eqs. (41) and external boundary conditions Eqs. (42), evidences the boundary nature of the model due to the involvement of boundary unknowns only. The mathematical structure of the resolving system allows to solve first for $$\boldsymbol {\Delta }_1^{\langle k\rangle }$$ and $$\boldsymbol {P}_1^{\langle k\rangle }$$ and, in turn, for $$\boldsymbol {\Delta }_0^{\langle k\rangle }$$ and $$\boldsymbol {P}_0^{\langle k\rangle }$$; suitable solution strategies can be adopted for a computationally efficient solution. Once the boundary solution is known in terms of displacement functions $$\boldsymbol {\Delta }_1^{\langle k\rangle }$$ and $$\boldsymbol {\Delta }_0^{\langle k\rangle }$$ and tractions $$\boldsymbol {P}_0^{\langle k\rangle }$$ and $$\boldsymbol {P}_1^{\langle k\rangle }$$, the stress field is calculated in a pointwise fashion by using the boundary integral representation for the stresses. More details can be found in Daví (1996, 1997), Daví and Milazzo (1997a,b, 1999), and Milazzo (2000).

## References

1. Aliabadi MH (2002) The boundary element methods. Volume 2: applications in solids and structures. Wiley, ChichesterGoogle Scholar
2. Banerjee PK, Butterfield R (1981) Boundary element methods in engineering science. McGraw-Hill, New York
3. Betti E (1872) Teoria dell’elasticit. Il Nuovo Cimento 12:7–10Google Scholar
4. Daví G (1996) Stress fields in general composite laminates. AIAA J 34(12):2604–2608
5. Daví G (1997) General theory for cross-ply laminated beams. AIAA J 35(8):1334–1340
6. Daví G, Milazzo A (1997a) Boundary element solution for free edge stresses in composite laminates. J Appl Mech Trans ASME 64(4):877–884
7. Daví G, Milazzo A (1997b) Boundary integral formulation for composite laminates in torsion. AIAA J 35(10):1660–1666
8. Daví G, Milazzo A (1999) Bending stress fields in composite laminate beams by a boundary integral formulation. Comput Struct 71(3):267–276
9. Lekhnitskii SG (1963) Theory of elasticity of an anisotropic body. Holden-Day, San Francisco
10. Milazzo A (2000) Interlaminar stresses in laminated composite beam-type structures under shear/bending. AIAA J 38(4):687–694
11. Sladek V, Sladek J (1998) Singular integrals in boundary element methods. Comput Mech Pub
12. Somigliana C (1886) Sopra l’equilibrio di un corpo elastico isotropo. Il Nuovo Cimento 3:17–19Google Scholar