Elastic constants – volumetric response: Within the elastic range, the ratio of hydrostatic (all-round) stress to the corresponding volumetric strain of a material is called what?

Introduction / Context:Different elastic constants describe how materials respond to different types of loading. Under hydrostatic pressure (equal stress in all directions), a body changes volume without shear distortion. The constant linking pressure to volumetric strain is the bulk modulus, a fundamental property used in soil mechanics, fluid-structure interaction, and pressure vessel analysis.

Given Data / Assumptions:

• Linear-elastic, isotropic material behavior assumed in the small-strain range.
• Hydrostatic stress means σx = σy = σz = p (compressive positive by convention).
• Volumetric strain εv is the sum of principal strains.

Concept / Approach:Bulk modulus K quantifies resistance to volume change: K = hydrostatic stress / volumetric strain = p / εv. It is distinct from Young’s modulus E (uniaxial stress–strain) and shear modulus G (shear stress–shear strain). These constants are interrelated for isotropic materials by standard formulas involving Poisson’s ratio ν, such as K = E / (3 * (1 - 2ν)).

Step-by-Step Solution:Recognize loading as hydrostatic (equal principal stresses).Define volumetric strain εv as the algebraic sum of principal strains.Recall definition: K = p / εv.Identify the corresponding term among the options: bulk modulus of elasticity.

Verification / Alternative check:Cross-check relationships: with E and ν known, compute K and confirm consistent volumetric response under hydrostatic loading in textbook examples.

Why Other Options Are Wrong:

• Young’s modulus: relates axial stress to axial strain, not volumetric strain.
• Shear modulus: concerns shear deformation, not volume change.
• Tangent modulus: slope of the stress–strain curve at a point, not a fundamental isotropic constant for hydrostatic response.
• All of the above: incorrect since only bulk modulus fits the given definition.

Common Pitfalls:

• Confusing bulk modulus with compressibility (its reciprocal).
• Using engineering sign conventions inconsistently for compressive pressure.

Final Answer:bulk modulus of elasticity