is to identify these adhesional strains by using the radius of curvature and to find the stress in each material. We assume that the total strain (£tot)at any point of the system represented in Fig. 2, is: (Navier-Bemouilli’s hypothesis). Therefore, the effect of transverse shear (rxy = 0) is neglected, (ii) the radius of curvature is large compared with transverse dimensions [width (b) and thickness (h) of the three-layer system], leading to R\ b\, hh (iii) longitudinal elements of the beam are subjected only to simple tension or compression inducing stresses in the x direction, (iv) Young’s modulii of the coating having bulk properties, the interphase and the substrate have the same value in both tension and compression (flexural modulus). Based on these assumptions, final uni-axial residual stresses (a), in the x -direction of the three-layer system (bulk coating/interphase/substrate) are given by: of the zero deformation (y0)- Therefore, we consider two equilibrium conditions for the force (N) and the moment (M) for any cross section (area S) of the coating/interphase/ substrate system: : with £mech is the mechanical strain. Considering the geometry and the size of the three-layer systems studied, the beam theory can be used. For the one-dimensional approach, without lateral (width- wise) stresses, the following assumptions are made: (i) transverse sections of the beam are planar before, during, and after bending where: yo is the position where total strains are equal to zero (£tot = 0), y is the coordinate distance of any longitudinal fiber, R\ is the radius of curvature and E is Young’s modulus. To determine the distribution of residual stresses in the tri-layer system from equation (10) requires a knowledge of the radius of curvature (R\) and the position

doi:10.1201/b11971-38

Chapter

$is to identify these adhesional strains by using the radius of curvature and to find the stress in each material. We assume that the total strain (£tot)at any point of the system represented in Fig. 2, is: (Navier-Bemouilli’s hypothesis). Therefore, the effect of transverse shear (rxy = 0) is neglected, (ii) the radius of curvature is large compared with transverse dimensions [width (b) and thickness (h) of the three-layer system], leading to R\ b\, hh (iii) longitudinal elements of the beam are subjected only to simple tension or compression inducing stresses in the x direction, (iv) Young’s modulii of the coating having bulk properties, the interphase and the substrate have the same value in both tension and compression (flexural modulus). Based on these assumptions, final uni-axial residual stresses (a), in the x -direction of the three-layer system (bulk coating/interphase/substrate) are given by: of the zero deformation (y0)- Therefore, we consider two equilibrium conditions for the force (N) and the moment (M) for any cross section (area S) of the coating/interphase/ substrate system:$

Chapter

is to identify these adhesional strains by using the radius of curvature and to find the stress in each material. We assume that the total strain (£tot)at any point of the system represented in Fig. 2, is: (Navier-Bemouilli’s hypothesis). Therefore, the effect of transverse shear (rxy = 0) is neglected, (ii) the radius of curvature is large compared with transverse dimensions [width (b) and thickness (h) of the three-layer system], leading to R\ b\, hh (iii) longitudinal elements of the beam are subjected only to simple tension or compression inducing stresses in the x direction, (iv) Young’s modulii of the coating having bulk properties, the interphase and the substrate have the same value in both tension and compression (flexural modulus). Based on these assumptions, final uni-axial residual stresses (a), in the x -direction of the three-layer system (bulk coating/interphase/substrate) are given by: of the zero deformation (y0)- Therefore, we consider two equilibrium conditions for the force (N) and the moment (M) for any cross section (area S) of the coating/interphase/ substrate system:

with £mech is the mechanical strain. Considering the geometry and the size of the three-layer systems studied, the beam theory can be used. For the one-dimensional approach, without lateral (width- wise) stresses, the following assumptions are made: (i) transverse sections of the beam are planar before, during, and after bending where: yo is the position where total strains are equal to zero (£tot = 0), y is the coordinate distance of any longitudinal fiber, R\ is the radius of curvature and E is Young’s modulus. To determine the distribution of residual stresses in the tri-layer system from equation (10) requires a knowledge of the radius of curvature (R\) and the position

ABSTRACT

To simplify writing we have adopted the following notation: the n order moment of a function / (y) is described as:

and

(13)

(14)

(15)

For the bending moment (M), the development and the rearrangement of equa tion ( 1 2 ), yields:

Then, equations (15) and (16) yield:

(16)

(17)