Elasticity is defined by the Young's modulus, which measures the ratio of stress to strain. A material is more elastic if it requires a greater force to produce a unit of strain. Steel requires a much larger force to deform than rubber, making it more elastic.

The modulus of rigidity, also known as the shear modulus, is defined as the ratio of shear stress to shear strain within the elastic limit of a material. It measures a material's resistance to shape deformation when subjected to a shearing force. This property is fundamental in solid mechanics and depends entirely on the material's internal structure; stiffer materials exhibit a higher modulus of rigidity.

Young's modulus, or the modulus of elasticity, quantifies the relationship between tensile stress and tensile strain in a material. It is a fundamental property of solid materials that undergo elastic deformation. While fluids (liquids and gases) exhibit bulk modulus (resistance to compression), they do not possess Young's modulus because they cannot support shear or tensile stress in a static state, as they flow under such forces.

Density is a fundamental physical property defined as the ratio of an object's mass to its volume. It represents how much matter is contained within a specific amount of space. The SI unit for density is kilograms per cubic meter (kg/m³).

Elastic modulus is defined as the ratio of stress to strain. Since strain is a dimensionless quantity, the dimensions of the elastic modulus are identical to those of stress. Stress is defined as force per unit area (Force/Area). Given that Force has dimensions MLT^-2 and Area has dimensions L^2, the resulting dimension for stress and elastic modulus is ML^-1T^-2.

Young's modulus (Y) is defined as stress divided by strain. Stress is force/area (10,000 / 5 * 10^-5 = 2 * 10^8 Pa). Strain is change in length/original length (0.001 m / 1 m = 0.001). Dividing stress by strain gives Y = 2 * 10^8 / 0.001 = 2 * 10^11 N/m^2.

Young's modulus, or the modulus of elasticity, quantifies the relationship between tensile stress and tensile strain in a material. This mechanical property is specifically defined for solids, which maintain a definite shape and resist deformation. Fluids (liquids and gases) do not possess a Young's modulus because they cannot support shear stress or tensile stress in a static state, instead exhibiting bulk modulus properties.

The energy density in a deformed material is defined as 0.5 * stress * strain. Since stress = Y * strain, substituting this into the formula yields 0.5 * Y * (strain)^2. The provided answer D represents the proportional relationship to the square of the strain, although it omits the factor of 0.5 commonly found in the standard derivation.

The bulk modulus (K) measures a substance's resistance to uniform compression. It is defined as the ratio of the change in pressure (normal stress) applied to a body to the resulting fractional change in its volume (volumetric strain). Mathematically, it is expressed as K = -ΔP / (ΔV/V), where the negative sign indicates volume decrease with pressure increase.

Young's modulus is defined as the ratio of stress to strain. Since strain is a dimensionless quantity, the unit of Young's modulus is the same as stress. Stress is defined as force per unit area, which is measured in Newtons per square meter (N/m² or Nm⁻²).