Final answer:
The volumetric stress induced in the cube due to thermal expansion while being constrained is calculated using the formula for volumetric stress. With the given parameters, this stress is found to be 120 MPa.
Step-by-step explanation:
Calculating Volumetric Thermal Stress
To estimate the volumetric stress induced due to heating, we start with understanding that stress is caused by the restraint of thermal expansion. In the given scenario, a cube with a side of 1 meter is uniformly heated from 32°C to 52°C. Since the material is constrained and cannot expand, thermal stress will develop within the material. The coefficient of linear expansion (α) for the material is given as 1 x 10⁻⁵°C⁻¹, and the bulk modulus (B) is 200 GPa or 2 x 10¹ N/m².
The formula to calculate the change in length (δL) due to thermal expansion is δL = α * L * ΔT, where L is the original length and ΔT is the change in temperature. However, as the cube is fully constrained, this expansion is not realized physically, but it leads to stress. Volumetric stress (σv) is given by the formula σv = 3 * B * α * ΔT. Plugging in the values, we get σv = 3 * (2 x 10¹) * (1 x 10⁻⁵) * (52 - 32), which results in a volumetric stress of 120 MPa. This stress is theoretically the pressure the material would exert internally if it were allowed to expand freely.