Abstract Two kindred problems on the compression of an elastic layer by a local load applied symmetrically to its surfaces are considered. In one case the layer has an annular crack with inner radius a and outer radius b on the middle plane. The quantities a and b (0 /s> a /s> b) are selected from the condition that the annular crack subjected to a load would be opened up and a normal tensile stress concentration would originate on the circumferential contours r = a and r = b. In the other case, the layer has a circular crack of radius b on the middle plane. Under the effect of a load in a circular domain of radius a (a /s> 6) the crack edges will be in contact, and will separate from each other in the annular region a /s> r /s> b. The quantity a is unknown and to be determined from the condition that the contact pressure on the circumferential contour r= a is zero; the quantity b is selected from the condition that a normal tensile stress concentration would originate on the contour r = b. In both cases the crack lips are assumed smooth. The crack is a mathematical slit in the unloaded layer. In the general case, the layer is compressed under the effect of an arbitrary local load applied to its upper and lower boundary planes symmetrically relative to the axis and the middle plane. As an illustration, the particular case of compression of the layer by two normal concentrated forces directed along the axis of symmetry of the problem is considered (Fig. 1). The problems of annular and circular cracks in an infinite layer were considered in another formulation in [1–5], It was assumed therein that the layer is stretched under the effect of a load applied directly to the crack edges.