The delamination fracture in four-point bending beams made of adhesively bonded lengthwise vertical layers is studied assuming that each layer exhibits smooth material inhomogeneity along the width and length of the layer. The study aims at determining the strain energy release rate with applying the Ramberg-Osgood equation for modeling the non-linear mechanical behavior of the material in each layer. Cosine laws are used to describe the continuous variation of the modulus of elasticity in width and length directions of layers. Beams made of an arbitrary number of vertical layers which have individual widths and material properties are considered. Besides, the delamination crack is located arbitrary between layers, i.e. the two crack arms have different widths. The J-integral is applied for verification of the non-linear solution to the strain energy release rate derived in the present paper. The solution is used to investigate the influence of material inhomogeneity in width and length directions of layers, the crack location along the beam width, the non-linear mechanical behavior of the material and the crack length on the delamination fracture behavior. The approach developed is expected to be useful in structural design of multilayered inhomogeneous beams with considering the delamination fracture behavior.