Calculate the radial and tangential stresses at the interface between Material A and Material B$,$ as well as at the inner and outer surfaces of the vessel. Given:

$$

$$

$=$

$0\mathrm{.}5$

r

i

$=0\mathrm{.}5$ m

$$

$$

$=$

$1\mathrm{.}0$

r

o

$=1\mathrm{.}0$ m

$$

$$

$=$

$10$

p

i

$=10$ MPa

Material A:

$$

$$

$=$

$200$

E

A

$=200$ GPa,

$$

$$

$=$

$0\mathrm{.}3$

A

$=0\mathrm{.}3,$

$$

$$

$=$

$12$

$$

$1$

$0$

$$

$6$

A

$=1210$

$6$

$1/$C

Material B:

$$

$$

$=$

$150$

E

B

$=150$ GPa,

$$

$$

$=$

$0\mathrm{.}25$

B

$=0\mathrm{.}25,$

$$

$$

$=$

$10$

$$

$1$

$0$

$$

$6$

B

$=1010$

$6$

$1/$C

$$

$$

$=$

$50$

$$T$=50$C

c$.$ Determine the Thermal Stresses

Determine the thermal stresses induced in both materials due to the temperature change

$$

$$

$$T$.$ Use the equations derived in part $($a$)$ and consider the compatibility conditions at the interface of the two materials.

d$.$ Analyze the Combined Stress State

Analyze the combined stress state $($mechanical $+$ thermal$)$ at the inner surface, the interface, and the outer surface of the vessel. Identify the maximum stress locations and determine whether the vessel can withstand the applied internal pressure and temperature change without yielding or failing.

e$.$ Evaluate the Impact of Different Material Properties

Evaluate the impact of varying the thermal expansion coefficients and Young's moduli of the materials on the stress distribution and the structural integrity of the vessel. Discuss how changing these properties affects the overall performance and safety of the pressure vessel.

Detailed Solution Approach:

Derive the Equations for Radial and Tangential Stresses:

Use Lame's equations for thick$-$walled cylinders to derive the radial and tangential stresses.

Consider the boundary conditions and material properties for both layers.

Incorporate thermal expansion effects into the stress equations.

Calculate the Radial and Tangential Stresses:

Apply the derived equations to the given parameters.

Calculate the stresses at the inner surface, interface, and outer surface.

Use numerical methods if necessary to solve the resulting equations.

Determine the Thermal Stresses:

Use the thermal expansion coefficients and temperature change to calculate the thermal strains.

Determine the corresponding thermal stresses using Hooke's law for each material layer.

Analyze the Combined Stress State:

Combine the mechanical and thermal stresses at each critical location.

Use failure theories $($e$.$g$.,$ von Mises, maximum shear stress$)$ to assess the safety of the vessel.

Evaluate the Impact of Different Material Properties:

Perform a parametric study by varying the thermal expansion coefficients and Young's moduli.

Analyze how changes in these properties affect the stress distribution and safety margins.

Provide recommendations for optimal material selection based on the analysis.

This problem requires a deep understanding of solid mechanics, thermoelasticity, and numerical methods. It involves solving multiple coupled differential equations and considering complex boundary conditions.

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