Question

2. The composite body shown below is composed of two different materials, (1) and (2) (a) What are the boundary and interface conditions when these two materials are perfectly bonded? (b) How would the boundary and interface conditions change if the materials were in a slip interface condition? Write expressions for any conditions that are different from part (a).

Answer 1

Boundary and interface conditions

The traction continuity based on discontinuity of material in y directions.

Based on cauchy stress tensor, ny = 1, nx = nz = 0;

$T_x = \sigma_{xx}n_x+ \sigma_{xy}n_y+ \sigma_{xz}n_z$.....................Eq(1)

$T_y = \sigma_{xy}n_x+ \sigma_{yy}n_y+ \sigma_{yz}n_z$.......................Eq(2)

$T_z = \sigma_{xz}n_x+ \sigma_{yz}n_y+ \sigma_{zz}n_z$........................Eq(3)

From equations (1), (2) and (3)

$T_x = \sigma_{xy}, T_y = \sigma_{yy}, T_z = \sigma_{xz},$.....................Eq(4)

Therefor the stress components $\sigma_{xy}, \sigma_{yy}, \sigma_{xz}$ are uniform

i.e., $\sigma_{xy}^c = \sigma_{xy}^{(1)}= \sigma_{xy}^{(2)}$ ................................Eq(5)

$\sigma_{yy}^c = \sigma_{yy}^{(1)}= \sigma_{yy}^{(2)}$...................................Eq.(6)

where, superscript c, (1) and (2) denotes the composite, materials (1) and (2) respectively.

As the bonding is perfect and material is continuous in x - direction.

Therefore, the strain component

$\varepsilon _{xx}^c = \varepsilon_{xx}^{(1)}= \varepsilon_{xx}^{(2)}$...........................Eq(7)

The volume average stress/strain component i.e non-uniform component are

$\sigma_{xx}^c = V_{1}\sigma_{xx}^{(1)}+V_2 \sigma_{xx}^{(2)}$.......................Eq(8)

$\varepsilon _{yy}^c =V_1\varepsilon_{yy}^{(1)}+V_2 \varepsilon_{yy}^{(2)}$.........................Eq(9)

$\varepsilon _{xy}^c =V_1\varepsilon_{xy}^{(1)}+V_2 \varepsilon_{xy}^{(2)}$........................Eq(10)

(b) If there is a slip interface condition then Equation (1) - (3) will not be valid .

Uniform strain components

$\varepsilon _{xx}^c = \varepsilon_{xx}^{(1)}= \varepsilon_{xx}^{(2)}$.........................Eq.(7)

The non uniform components in x direction

$\sigma_{xx}^c = V_{1}\sigma_{xx}^{(1)}+V_2 \sigma_{xx}^{(2)}$............Eq.(8)