Stroh-like Formalism Research Articles

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

Green’s function for the composite laminates with bending extension coupling

By employing the recently developed Stroh-like formalism for the composite laminates with bending extension coupling, the explicit closed form solutions for an infinite composite laminate subjected to concentrated forces and moments at any arbitrary points have been found analytically. Although Stroh-like formalism is a complex variable formulation, through the use of some identities developed in the literature real form solutions of the present problem are also obtained in this paper. Because the concentrated forces and moments will influence the mechanical behavior of the composite in different ways when they are applied in different directions, three different loading cases are discussed. The results show that the solutions corresponding to the inplane concentrated forces f ̂ 1 , f ̂ 2 and out-of-plane concentrated moments m ̂ 1 , m ̂ 2 have exactly the same form as that of the corresponding two-dimensional problems. While for the cases under the concentrated force f ̂ 3 and pure torsion m ̂ 3 , new types of solutions are obtained explicitly.

Explicit solutions for the coupled stretching–bending problems of holes in composite laminates

Although the classical lamination theory was developed long time ago, it is still not easy to apply this theory to find the analytical solutions for the curvilinear boundary value problems especially when the stretching and bending are coupled each other. To overcome the difficulties, recently we developed a Stroh-like formalism for the general composite laminates. By using this formalism, most of the relations for the coupled stretching–bending problems can be organized into the forms of Stroh formalism for two-dimensional anisotropic elasticity problems. With this newly developed Stroh-like formalism, it becomes easier to obtain an analytical solution for the coupled stretching–bending problems of holes in composite laminates. Because the Stroh-like formalism is a complex variable formalism, the analytical solutions for the whole field are expressed in complex form. Through the use of some identities derived in this paper, the resultant forces and moments around the hole boundary are obtained explicitly in real form. Due to the lack of analytical solutions for the general cases, the comparison is made with the existing analytical solutions for some special cases. In addition, to show the generality of our analytical solutions, several numerical examples are presented to discuss the coupling effect of the laminates and the shape effect of the holes.

Stroh-like formalism for the coupled stretching–bending analysis of composite laminates

Due to the two-dimensional nature of thin plates, the lamination theory considering the composite laminates with in-plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation. By following the steps of Stroh formalism for two-dimensional linear anisotropic elasticity, a displacement complex variable formalism developed by the other researchers was introduced and re-derived in a different but more Stroh-like way. In addition, a brand-new mixed formalism (mixed use of displacements and stresses as basic functions) is established to compensate the displacement formalism. In order to transfer all the related formulae and mathematical techniques of the Stroh formalism to these two formalisms, the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of Stroh formalism. Moreover, by using the presently developed mixed formalism, the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates. Furthermore, letting the coupling stiffness vanish, the formalism has been reduced to the case of symmetric laminates and checked by a recently developed Stroh-like formalism for the plate bending problems. The comparison between Stroh formalism for two-dimensional problem, Stroh-like formalism for plate bending problem, displacement formalism and mixed formalism is then made at the end of this paper, and through their connection some useful relations are obtained.

Anisotropic elastic plates with holes/cracks/inclusions subjected to out-of-plane bending moments

The problems of anisotropic plates containing holes, cracks and/or inclusions have been studied extensively for two-dimensional deformations. Although the correspondence between the two-dimensional problems and the plate bending problems has been observed long time ago, without clarifying the involved mathematical details one still cannot get the solutions of the plate bending problems directly from the solutions of the corresponding two-dimensional problems. Based upon the correspondence relation, recently we developed a Stroh-like formalism for the bending theory of anisotropic plates. By this newly developed formalism, most of the relations for bending problems can be organized into the forms for two-dimensional problems. Thus, by using the Stroh-like formalism, the analytical solutions for problems of anisotropic plates with holes/cracks/inclusions subjected to out-of-plane bending moments can be obtained directly from the solutions of the corresponding two-dimensional problems. The stresses (or resultant bending moments) around the hole/crack/inclusion boundaries are also given explicitly in this paper. Since the explicit closed-form solutions for the present cases have not been found in the literature, comparison is made with some special cases of which the analytical solutions exist, which shows that our solutions are exact, simple and general.

Explicit Expressions of the Fundamental Elasticity Matrices of Stroh-Like Formalism for Symmetric/Unsymmetric Laminates

AbstractBased upon our recent development of Stroh-like forma lism for symmetric/unsymmetric laminates, most of the relations for bending problems can be organized into the forms of Stroh formalism for two-dimensional problems. Through the use of Stroh-like formalism, the fundamental elasticity matrices Ni, S, H and L appear frequently in the real form solutions of plate bending problems. Therefore, the determination of these matrices becomes important in the analysis of plate bending problems. In this paper, by following the approach for two-dimensional problems, we obtain the explicit expressions of the fundamental elasticity matrices for symmetric and unsymmetric laminates, which are all expressed in terms of the extensional, bending and coupling stiffnesses of the composite laminates.