Assume that the bodies
\(\,{\mathcal {B}}_k\) are welded together and there is no separation of material during deformation. The conditions that the displacement and the stress vector fields are continuous in passing from one material to another have to be adjoin to the plane strain boundary-value problems (
12), (
13), in the case of composite materials. Thus, we consider the extended plane strain boundary-value problems
\(\,{\mathcal {R}}^{(1)}\,\),
\(\,{\mathcal {R}}^{(2)}\,\) and
\(\,{\mathcal {R}}^{(3)}\,\) given by
$$\begin{aligned}&{\mathcal {R}}^{(\gamma )}:\left\{ \begin{array}{l} t_{\beta \alpha ,\,\beta }=-(c_{\alpha 3}\, x_\gamma )_{,\,\alpha }\quad \mathrm {in}\,\,\, \,S_k\,,\qquad \quad (\gamma =1,2)\\ t_{\beta \alpha }n_\beta =-c_{\alpha 3}\, x_\gamma n_{\alpha } \quad \mathrm {on}\,\,\, \partial \varSigma ,\\ \big [u_\alpha \big ]_{-}^{+}=0,\quad n_\beta \big [ t_{\beta \alpha }+c_{\alpha 3}\, x_\gamma \delta _{\alpha \beta } \big ]_{-}^{+}=0\quad \mathrm {on}\,\,\,\, {\mathcal {C}}_k \,\,\,(k=1,...,n), \end{array}\right. \nonumber \\&{\mathcal {R}}^{(3)}:\left\{ \begin{array}{l} t_{\beta \alpha ,\,\beta }=-c_{\alpha 3,\,\alpha }\quad \mathrm {in}\,\,\, \,S_k\,\,\,(k=1,...,n),\\ t_{\beta \alpha }n_\beta =-c_{\alpha 3}\, n_{\alpha } \quad \mathrm {on}\,\,\, \partial \varSigma ,\\ \big [u_\alpha \big ]_{-}^{+}=0,\quad n_\beta \big [ t_{\beta \alpha }+c_{\alpha 3} \,\delta _{\alpha \beta } \big ]_{-}^{+}=0\quad \mathrm {on}\,\,\,\, {\mathcal {C}}_k \,\,\,(k=1,...,n), \end{array}\right. \end{aligned}$$
(15)
where the subscript
\(\,\alpha =1,2\,\) is not summed. The solutions of the problems
\({\mathcal {R}}^{(s)}\) will be denoted by
\(u_\alpha ^{(s)}(x_1,x_2)\),
\(s=1,2,3\). The torsion function
\(\varphi (x_1,x_2)\) for composite rods is determined from the extended boundary-value problem
$$\begin{aligned}&\left\{ \begin{array}{l} (c_{55}\,\varphi _{,1})_{,1}+(c_{44}\,\varphi _{,2})_{,2}= c_{55,1}\,x_2-c_{44,2}\,x_1 \quad \mathrm {in}\,\,\,\,\, S_k\,\,\,(k=1,...,n),\\ c_{55}\,\varphi _{,1}\,n_1+c_{44}\,\varphi _{,2}\,n_2= c_{55}\,x_2\,n_1- c_{44}\,x_1\,n_2 \quad \mathrm {on}\,\,\, \partial \varSigma ,\\ \big [\varphi \big ]_{-}^{+}=0,\qquad \big [ c_{55}\,\varphi _{,1}\,n_1+c_{44}\,\varphi _{,2}\,n_2- c_{55}\,x_2\,n_1+ c_{44}\,x_1\,n_2 \big ]_{-}^{+}=0 \quad \mathrm {on}\,\,\,\, {\mathcal {C}}_k\,.\end{array}\right. \end{aligned}$$
(16)
If we generalize the analysis made in [
4, Sect. 6] to the case of
\(\,n\,\) orthotropic materials, then we find the following expressions for the effective stiffness coefficients in terms of the solutions
\(u_\alpha ^{(s)} \) and
\(\varphi \,\):
$$\begin{aligned} A_3= & {} \!\displaystyle {\sum _{k=1}^{n}\int _{S_k}} \big (c_{33}^{(k)}+c_{13}^{(k)} u_{1,1}^{(3)}+c_{23}^{(k)} u_{2,2}^{(3)}\big )\mathrm {d}x_1\mathrm {d}x_2\,,\qquad A_{12}=0, \nonumber \\ B_{31}= & {} \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_2\big (c_{33}^{(k)}+c_{13}^{(k)} u_{1,1}^{(3)}+c_{23}^{(k)} u_{2,2}^{(3)}\big )\mathrm {d}x_1\mathrm {d}x_2\,,\qquad B_{13}=0,\nonumber \\ B_{32}= & {} - \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_1\big (c_{33}^{(k)} +c_{13}^{(k)} u_{1,1}^{(3)}+c_{23}^{(k)} u_{2,2}^{(3)}\big )\mathrm {d}x_1\mathrm {d}x_2\,, \qquad B_{23}=0,\nonumber \\ C_1= & {} \!\!\displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_2\big ( c_{33}^{(k)}x_2 +c_{13}^{(k)} u_{1,1}^{(2)}+c_{23}^{(k)} u_{2,2}^{(2)}\big )\mathrm {d}x_1\mathrm {d}x_2,\nonumber \\ C_2= & {} \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_1(c_{33}^{(k)}x_1 +c_{13}^{(k)} u_{1,1}^{(1)}+c_{23}^{(k)} u_{2,2}^{(1)})\mathrm {d}x_1\mathrm {d}x_2\,,\nonumber \\ C_{12}= & {} - \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_1\big ( c_{33}^{(k)}x_2 +c_{13}^{(k)} u_{1,1}^{(2)}+ c_{23}^{(k)} u_{2,2}^{(2)}\big )\mathrm {d}x_1\mathrm {d}x_2\,,\nonumber \\ C_3= & {} \displaystyle {\sum _{k=1}^{n}\int _{S_k}} \big [c_{44}^{(k)}x_1(x_1+\varphi _{,2})+ c_{55}^{(k)}x_2(x_2-\varphi _{,1})\big ]\mathrm {d}x_1\mathrm {d}x_2\,, \end{aligned}$$
(17)
and for the transverse shear stiffness we obtain
$$\begin{aligned} A_1= & {} \dfrac{\kappa \, \mathrm {A}(\varSigma )}{\langle \,x_1^2\,\rangle } \Big ( \displaystyle {\sum _{k=1}^{n}\!\int _{S_k}}\!\! c_{55}^{(k)}\mathrm {d}x_1\mathrm {d}x_2\Big ) \Big (\displaystyle {\sum _{k=1}^{n}\!\int _{S_k}}\!\! \rho ^{(k)}x_1^2\mathrm {d}x_1\mathrm {d}x_2\Big )\Big (\displaystyle { \sum _{k=1}^{n}\! \int _{S_k}}\!\! \rho ^{(k)}\mathrm {d}x_1\mathrm {d}x_2\Big )^{-1}\!,\nonumber \\ A_2= & {} \dfrac{\kappa \, \mathrm {A}(\varSigma )}{\langle \,x_2^2\,\rangle } \Big ( \displaystyle {\sum _{k=1}^{n}\!\int _{S_k}}\!\! c_{44}^{(k)}\mathrm {d}x_1\mathrm {d}x_2\Big ) \Big (\displaystyle {\sum _{k=1}^{n}\!\int _{S_k}}\!\! \rho ^{(k)}x_2^2\mathrm {d}x_1\mathrm {d}x_2\Big )\Big (\displaystyle { \sum _{k=1}^{n}\!\int _{S_k}} \!\!\rho ^{(k)}\mathrm {d}x_1\mathrm {d}x_2\Big )^{-1}\!. \end{aligned}$$
(18)
where
\(\,\kappa =\dfrac{\pi ^2}{12}\,\,\) is the shear correction factor and
\( \mathrm {A}(\varSigma ) \) denotes the area of
\(\varSigma \).
4.1 Composite beams made of isotropic materials
Let us consider the special case when the composite beams consist of
\(\,n\,\) isotropic materials. If we denote by
\( \lambda ^{(k)} \) and
\( \mu ^{(k)} \) the Lamé moduli of each constituent material (
\( k=1,\dots ,n \)), then the constitutive relations (
11) hold with
$$\begin{aligned} c_{11}^{(k)}= & {} c_{22}^{(k)} =c^{(k)}_{33}= \lambda ^{(k)}+ 2\mu ^{(k)}, \nonumber \\ c_{12}^{(k)}= & {} c_{13}^{(k)} =c^{(k)}_{23}= \lambda ^{(k)},\qquad c_{44}^{(k)}=c_{55}^{(k)} =c^{(k)}_{66}= \mu ^{(k)}. \end{aligned}$$
(19)
Substituting the relations (
19) into the equations (
15)-(
18), we obtain the corresponding expressions of the effective stiffness coefficients in the isotropic case. For instance, for the torsional rigidity
\(C_3\) and shear effective stiffness
\(A_1\,,\,A_2\) we find the formulas
$$\begin{aligned} C_3= & {} \displaystyle {\sum _{k=1}^{n}\int _{S_k}} \mu ^{(k)}\big [x_1(x_1+\varphi _{,2})+x_2(x_2-\varphi _{,1})\big ] \mathrm {d}x_1\mathrm {d}x_2\,,\nonumber \\ A_\alpha= & {} \dfrac{\kappa \, \mathrm {A}(\varSigma )}{\langle \,x_\alpha ^2\,\rangle } \Big ( \displaystyle {\sum _{k=1}^{n}\!\int _{S_k}}\!\! \mu ^{(k)}\mathrm {d}x_1\mathrm {d}x_2\Big ) \Big (\displaystyle {\sum _{k=1}^{n}\!\int _{S_k}}\!\! \rho ^{(k)}x_\alpha ^2\mathrm {d}x_1\mathrm {d}x_2\Big )\Big (\displaystyle { \sum _{k=1}^{n}\! \int _{S_k}}\!\! \rho ^{(k)}\mathrm {d}x_1\mathrm {d}x_2\Big )^{-1}\!. \end{aligned}$$
(20)
These results for composite beams made of isotropic materials have been obtained previously in the paper [
5].
These formulas for the effective stiffness properties of composite beams made of several non-homogeneous materials are very general and are applicable to a variety of problems. The difficulty resides in finding the solutions to the plane strain boundary-value problems (
15) and (
16). In some special cases, the boundary-value problems (
15), (
16) can be solved analytically and the formulas for the effective stiffness coefficients (
17), (
18) can be simplified.
In the case of composite beams made of isotropic materials with constant Poisson ratio
\(\,\nu \,\) (the same constant for all material constituents), the plane strain boundary-value problems
\({\mathcal {R}}^{(s)}\) written in (
15) admit the following simple solutions
$$\begin{aligned} u_1^{(1)}= & {} -\dfrac{1}{2}\,\nu (x_1^2-x_2^2)\,,\qquad u_2^{(1)}=-\nu \, x_1x_2\,, \qquad u_1^{(2)}=-\nu \, x_1x_2\,, \nonumber \\ u_2^{(2)}= & {} \dfrac{1}{2}\,\nu (x_1^2-x_2^2)\,,\qquad u_1^{(3)}= -\nu \, x_1\,,\qquad u_2^{(3)}=-\nu \, x_2\,. \end{aligned}$$
(21)
We mention that the case of non-homogeneous materials with constant Poisson ratio has been investigated within the classical three-dimensional elasticity theory, e.g., in [
21]. Let us denote by
\(\,E^{(k)}(x_1,x_2)\) the Young modulus of the non-homogeneous material occupying the domain
\({\mathcal {B}}_k\,\). As mentioned above, the Poisson ratio is the same for all materials, i.e., we assume that
\(\nu ^{(k)}(x_1,x_2)=\nu \) (constant).
Inserting the relations (
19) and (
21) into the formulas (
17), we obtain in this case the following effective stiffness coefficients [
5]
$$\begin{aligned} A_3= & {} \displaystyle {\sum _{k=1}^{n}\int _{S_k}} E^{(k)} \mathrm {d}x_1\mathrm {d}x_2, \qquad A_{12}=0, \qquad B_{31}=\displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_2E^{(k)}\mathrm {d}x_1\mathrm {d}x_2, \nonumber \\ B_{32}= & {} - \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_1E^{(k)}\mathrm {d}x_1\mathrm {d}x_2 , \qquad B_{\alpha 3}=0, \qquad C_1= \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_2^2\,E^{(k)}\mathrm {d}x_1\mathrm {d}x_2,\nonumber \\ C_2= & {} \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_1^2\,E^{(k)}\mathrm {d}x_1\mathrm {d}x_2\,,\qquad C_{12}=- \displaystyle {\sum _{k=1}^{n}\int _{S_k}} x_1x_2\,E^{(k)}\mathrm {d}x_1\mathrm {d}x_2\,. \end{aligned}$$
(22)
The above relations will be employed in the next sections to determine the effective stiffness properties of various sandwich beam structures.