| For a graded elastic layer subjected to an in-plane compression parallel to the freesurface, when the compressive strain exceeds its critical value, the surface will becomeunstable and the buckling or wrinkling can occur. By using the second-order variationof the strain energy, we obtain an equilibrium equation expressed by incrementaldisplacements, a stress boundary condition, and differential relations betweenincremental stresses and displacements. The equilibrium equation for homogeneouselastic layers is determined by reducing that for graded elastic layers.According to the equilibrium equation for homogeneous isotropic elastic layers andthe boundary conditions, we derive an analytical solution for critical condition ofsurface instability for an isotropic homogeneous layer on a rigid support. Byconsidering the continuity conditions of the displacements and stresses at the interfacebetween the surface layer and substrate layer, the analytical solution for criticalcondition of surface instability for elastic bilayers is obtained. The analytical results ofhomogeneous layer show that the critical strain is related to Poisson’s ratio andindependent of the elastic modulus. For a fixed modulus ratio, the critical strain ofsurface instability for elastic bilayers decreases with the increasing thickness ratiobetween the substrate layer and the surface layer; for a fixed thickness ratio, the criticalstrain decreases with the increasing modulus ratio between the surface layer and thesubstrate layer. The analytical solutions obtained in this paper agree well with the resultsin the related existing literature.By dividing the graded elastic layer into a number of sub-layers and in accordance tothe differential relations between incremental stresses and displacements, a state spacesolution is developed to analyze the critical condition of surface instability for gradedelastic layers. For the elastic bilayer with dissimilar material properties and the elasticlayer with material properties varying continuously in the thickness direction, thecritical strain and the corresponding critical wave number (wavelength) are presented bythe state space solution. The results show that the state space solution is in completeagreement with the analytical solution. According to the analyses of a linearly gradedlayer and a locally linearly graded layer, the relation between the number of sub-layers and convergence of state space solution is obtained. On this basis, the present paperfurther acquires the state space solutions for the graded layers with modulus varying inthe thickness direction as an exponential function and a complementary error function.And the results are compared to the relevant literature. Moreover, this paper alsoprovides the relation between critical thermal expansion strain of surface instability andmaterial properties as the temperature of elastic layer increases, and the comparison ofthe critical strain between the present analytical solutions and the numerical results byABAQUS for surface instability of elastic bilayers. |
