Within the elastic limit, the ratio of longitudinal (normal) stress to longitudinal strain is known as which modulus?

Introduction / Context:The linear stress–strain relation in the elastic range is a cornerstone of material mechanics. For uniaxial loading, the proportionality constant relating normal stress and normal strain characterizes stiffness and is essential for deflection, buckling, and vibration checks.

Given Data / Assumptions:

• Elastic behavior (Hooke’s law) is assumed.
• Longitudinal (normal) stress and strain are considered.
• We distinguish among several moduli used in mechanics.

Concept / Approach:Young’s modulus (modulus of elasticity, E) is defined by E = sigma / epsilon in the linear range for uniaxial loading. Shear modulus (G) relates shear stress to shear strain (tau / gamma). Bulk modulus (K) relates hydrostatic pressure to volumetric strain. Tangent modulus is the slope of the stress–strain curve at a specific point and is used in inelastic analysis, not the linear elastic proportionality constant for the entire elastic range.

Step-by-Step Solution:Identify the physical quantities: longitudinal stress sigma and longitudinal strain epsilon.Apply Hooke’s law in the elastic range: E = sigma / epsilon.Select the named modulus corresponding to this definition.

Verification / Alternative check:Dimensional check: E has units of stress (e.g., MPa). Typical values: structural steel ≈ 200,000 MPa; concrete (instantaneous) ≈ 20,000–35,000 MPa; wood depends on species and orientation.

Why Other Options Are Wrong:

• Shear modulus: pertains to shear response.
• Bulk modulus: pertains to volumetric response under hydrostatic loading.
• Tangent modulus: local slope, generally for plasticity/large-strain analysis.
• All the above: incorrect because only one modulus matches the stated ratio.

Common Pitfalls:

• Confusing E and G; they relate via Poisson’s ratio for isotropic materials (E = 2G(1 + nu)).

Final Answer:Modulus of elasticity.