Part coated with a composition for protection against CMAS with controlled cracking, and corresponding treatment method

A turbomachine part comprising:

• a substrate of metallic material, or of composite material;
• and
• a protective coating layer against the infiltration of compounds of the oxides of calcium, magnesium, aluminum or silicon type, referred to as CMAS type, the coating layer at least partially covering the surface of the substrate,
• wherein the protective coating layer comprises a plurality of elementary layers,
• wherein elementary layers of a first set of the plurality of elementary layers are interposed between elementary layers of a second set of the plurality of elementary layers, each elementary layer of the first set and each elementary layer of the second set comprising an anti-CMAS compound, and
• wherein each contact zone between an elementary layer of the first set and an elementary layer of the second set forms an interface promoting a propagation of cracks along said interface,
• wherein each elementary layer of the first set has a toughness which differs by at least 0.7 MPa·m1/2 from the toughnesses of all elementary layers of the second set, or
• wherein each elementary layer of the first set has a coefficient of thermal expansion which differs by at least 3.5 10−6 K−1 from the coefficients of thermal expansion of all elementary layers of the second set.