Assuming the material considered in each statement is homogeneous, isotropic, linear elastic, and the deformations are in the elastic range, which one or more of the following statement(s) is/are TRUE?

The correct statement(s) are:
"A body subjected to hydrostatic pressure has no shear stress"
"If a long solid steel rod is subjected to tensile load, then its volume increases"

Let us evaluate each of the statements in detail:

1. Analysis of Statement: "A body subjected to hydrostatic pressure has no shear stress" (TRUE)
Under hydrostatic loading, a body is subjected to equal compressive stress in all directions. The state of stress at any point in the body is given by:

$${\sigma }_x={\sigma }_y={\sigma }_z=-p$$

where $p$ is the hydrostatic pressure, and the shear stress components are zero:

$${\tau }_{\mathrm{xy}}={\tau }_{\mathrm{yz}}={\tau }_{\mathrm{zx}}=0$$

The maximum shear stress on any plane is determined using the principal stresses:

$${\tau }_{\max }=\frac{{\sigma }_{\max }-{\sigma }_{\min }}{2}$$

Since ${\sigma }_{\max }={\sigma }_{\min }=-p$, we have:

$${\tau }_{\max }=\frac{-p-(-p)}{2}=0$$

Hence, there is no shear stress on any plane within the body. Thus, this statement is TRUE.

2. Analysis of Statement: "If a long solid steel rod is subjected to tensile load, then its volume increases" (TRUE)
When a solid rod is subjected to a uniaxial tensile load along its longitudinal axis (say, the x-axis), the stress components are:

$${\sigma }_x>0\text{ (tensile stress), and }{\sigma }_y={\sigma }_z=0$$

The volumetric strain ${\epsilon }_v$, which represents the fractional change in volume, is given by the sum of the linear strains in three perpendicular directions:

$${\epsilon }_v=\frac{dV}{V}={\epsilon }_x+{\epsilon }_y+{\epsilon }_z$$

Using Hooke's law in terms of Young's modulus ($E$) and Poisson's ratio ($\nu$):

$${\epsilon }_x=\frac{{\sigma }_x}{E}$$

$${\epsilon }_y=-\nu \frac{{\sigma }_x}{E}$$

$${\epsilon }_z=-\nu \frac{{\sigma }_x}{E}$$

Substituting these into the volumetric strain equation:

$${\epsilon }_v=\frac{{\sigma }_x}{E}-\nu \frac{{\sigma }_x}{E}-\nu \frac{{\sigma }_x}{E}=\frac{{\sigma }_x}{E}(1-2\nu )$$

For steel, Poisson's ratio $\nu$ typically ranges between 0.27 and 0.30. Since $\nu <0.5$, the factor $(1-2\nu )$ is positive. Since ${\sigma }_x>0$ under tension, the volumetric strain ${\epsilon }_v$ is greater than zero (${\epsilon }_v>0$), indicating that the volume increases. Thus, this statement is TRUE.

3. Analysis of Statement: "Maximum shear stress theory is suitable for failure analysis of brittle materials" (FALSE)
The Maximum Shear Stress Theory (also known as Tresca's yield criterion) is used for ductile materials, which primarily fail in shear. Brittle materials, on the other hand, are weak in tension and fail due to normal tensile stress. Therefore, the Maximum Principal Stress Theory (Rankine's theory) is suitable for brittle materials. Thus, this statement is FALSE.

4. Analysis of Statement: "If a portion of a beam has zero shear force, then the corresponding portion of the elastic curve of the beam is always straight" (FALSE)
The relationship between shear force ($V$) and bending moment ($M$) along a beam is:

$$V=\frac{dM}{dx}$$

If the shear force is zero over a portion of the beam, it implies that the bending moment in that region is constant ($M=\text{constant}$). The curvature of the elastic curve of the beam is defined by the Euler-Bernoulli equation:

$$EI\frac{d^{2}y}{d{x}^2}=M$$

If the constant bending moment $M$ is non-zero (pure bending), the curvature $\frac{d^{2}y}{d{x}^2}$ is constant and non-zero, representing a circular arc rather than a straight line. The elastic curve is straight only if both shear force and bending moment are zero in that section. Thus, this statement is FALSE.