ANALYSIS OF THERMOELASTIC STRESSES IN LAYERED PLATES

⎫B E z z E hnn* *= ∑ i ( i−i−1) = ∑ i i⎪ i= 1 i=1 ⎪⎪n1 * 2 2= ∑ i ( i−i−1) ⎬2 i=1⎪⎪n1 * 3 3= ∑ i ( i−i−1) ⎪3 i=1⎭C E z zD E z z. (22)For the constants (10) we havea = aΔT ; b= bΔT ,whereCNT− BMT⎫a =2C − BD⎪ ⎬ . (23)CMT− DNTb =⎪2C − BD ⎭⎪Herenn* * ⎫NT = ∑Eiαi( zi − zi−1)= ∑Eiαihi⎪ i= 1 i=1 ⎪⎬⎪. (24)n1 * 2 2MT = ∑Eiαi( zi −zi−1)2 i=1⎪⎭In accordance with equation (8) stresses inthe ith layer are*σ = E ΔT − α + az+b . (25)i i i( )Deformation parameters areε ⎫0= bΔT⎪ ⎬ . (26)æ0=−aΔT⎪⎭3.2. Plate with sliding edgesIn this case a = 0 , andNb = TB . (27)Stresses in the ith layer are*σ = E ΔT − α + b . (28)i i i( )Strain on the reference surfaceε = bΔT(29)0.3.3. Plate with fixed edgesFor this case a = b = 0 , and stresses in theith layer are*σ =−Eα ΔT.(30)i i i4. APPLICATIONS: THERMALSTRESSES DUE TO UNIFORMTEMPERATURE CHANGE IN ACOPPER PLATE WITH A GALVANICSTEEL COATINGUsually, the coefficients of thermal expansionof the substrate and coating are differentand the temperature of the coatingprocess differs from room temperature(+20°C). Therefore, thermal stresses aregenerated in coated parts. In experimentalanalysis of residual stresses it may turn outthat coating temperature differs from thetemperature at which the deformation parametersof the substrate are measured.In such cases measurement results have tobe corrected taking into account thermalstresses.As a numerical example, thermal stressesand displacements for a quadratic copperplate 30×30×1 mm with galvanic steelcoating deposited at 95°C are calculated.Fig. 5. Copper plate with a steel coatingIn addition, the following data is used: h 1 == 1.00 mm; E 1 = 110 GPa; μ 1 = 0.34; α 1 == 17.5 ⋅ 10 -6 1/°C for the substrate and h 2 == 0.26 mm; E 2 = 202 GPa; μ 2 = 0.28; α 2 == 14.3 ⋅ 10 -6 1/°C for the coating.