The relationship between elastic modulus and stiffness

4 hits 2025/8/13 10:13:20

Elastic Modulus

1. Definition

Elastic modulus is the ratio of normal stress to corresponding normal strain within the elastic deformation stage of a material.

Within the elastic deformation range, a material’s stress and strain follow a proportional relationship (Hooke’s Law), and the proportional coefficient is called the elastic modulus.

"Elastic modulus" is a general term describing material elasticity, encompassing "Young’s modulus," "shear modulus," "bulk modulus," etc. Thus, "elastic modulus" and "bulk modulus" have an inclusive relationship.

Generally, when an external force (referred to as "stress") is applied to an elastic body, the body undergoes shape changes (referred to as "strain"). The general definition of "elastic modulus" is: stress divided by strain. For example:

Linear strain: When a tensile force F is applied to a thin rod, the force divided by the rod’s cross-sectional area S is called "linear stress." The rod’s elongation dL divided by its original length L is called "linear strain." The ratio of linear stress to linear strain equals Young’s modulus: \(E = (F/S) / (dL/L)\).
Shear strain: When a lateral force f (usually friction) is applied to an elastic body, the body deforms from a square to a rhombus. The deformation angle \(\alpha\) is called "shear strain," and the corresponding force f divided by the Stress area S is called "shear stress." The ratio of shear stress to shear strain equals the shear modulus: \(G = (f/S) / \alpha\).
Volumetric strain: When overall pressure p is applied to an elastic body, this pressure is called "volumetric stress." The volume reduction \((-dV)\) divided by the original volume V is called "volumetric strain." The ratio of volumetric stress to volumetric strain equals the bulk modulus: \(K = P / (-dV/V)\).

To avoid confusion, the elastic modulus of metallic materials generally refers to Young’s modulus, i.e., the normal elastic modulus. Unit: E (elastic modulus) in gigapascals (GPa).

2. Influencing Factors

The elastic modulus is a key performance parameter of engineering materials. Macroscopically, it measures a material’s resistance to elastic deformation; microscopically, it reflects the bonding strength between atoms, ions, or molecules.

Any factor affecting bonding strength can influence the elastic modulus, such as bonding type, crystal structure, chemical composition, microstructure, and temperature. Due to differences in alloy composition, heat treatment, and cold plastic deformation, the Young’s modulus of metallic materials can fluctuate by 5% or more.

However, the elastic modulus of metals is generally insensitive to microstructure. Alloying, heat treatment (fiber structure), and cold plastic deformation have minimal effects. External factors like temperature and loading rate also have little impact, so the elastic modulus is treated as a constant in most engineering applications.

3. Significance

The elastic modulus can be regarded as an indicator of a material’s resistance to elastic deformation. A higher value means greater stress is required to achieve a given elastic deformation, i.e., the material has higher stiffness and undergoes less elastic deformation under a given stress.

The elastic modulus E represents the stress required to produce unit elastic deformation in a material. It reflects the material’s resistance to elastic deformation, analogous to the stiffness of a regular spring.

Stiffness

1. Definition

Stiffness is a structure or component’s ability to resist elastic deformation, measured by the force or torque required to produce unit strain.

Rotational stiffness (k): \(k = M/\theta\), where M is the applied torque and \(\theta\) is the rotation angle.
Other types of stiffness include:
- Tension and compression stiffness: axial force divided by axial linear strain (EA)
- Shear stiffness: shear force divided by shear strain (GA)
- Torsional stiffness: torque divided by torsional strain (GI)
- Bending stiffness: bending moment divided by curvature (EI)

2. Calculation Methods

Stiffness calculation theories are divided into small displacement theory and large displacement theory:

Large displacement theory establishes equilibrium equations based on the deformed position of the structure after loading, yielding accurate results but with complex calculations.
Small displacement theory temporarily assumes the structure is undeformed when establishing equilibrium equations, derives internal forces from external loads, and then calculates deformation.

Most mechanical designs use small displacement theory. For example, in beam bending calculations, due to small actual deformations, the first derivative of deflection in the curvature formula is neglected, and the second derivative is used to approximate the beam axis curvature. This linearizes the differential equation, simplifying the solution. When multiple loads act simultaneously, bending deformations from each load can be calculated separately and superimposed.

3. Classification and Significance

Static stiffness: Resistance to deformation under static loads.
Dynamic stiffness: Resistance to deformation under dynamic loads, i.e., the dynamic force required to produce unit amplitude. If the interference force changes very slowly (its frequency is much lower than the structure’s natural frequency), dynamic stiffness is approximately equal to static stiffness. If the interference force changes extremely rapidly (frequency much higher than the natural frequency), deformation is small, meaning high dynamic stiffness. When the interference frequency is close to the natural frequency, resonance occurs, resulting in minimal dynamic stiffness (maximum deformation), which can be several to over ten times the static deformation.

Component deformation often affects performance: excessive deformation of a gear shaft impairs gear meshing, and excessive machine tool deformation reduces machining accuracy. Stiffness depends on the material’s elastic modulus and structural design; modifying the structure significantly impacts stiffness.

Stiffness calculation is fundamental to vibration theory and structural stability analysis. For a given mass, higher stiffness corresponds to a higher natural frequency. Stress distribution in statically indeterminate structures depends on stiffness ratios. In fracture mechanics, the stress intensity factor of cracked components can be derived from compliance.

Relationship Between Elastic Modulus and Stiffness

In general, stiffness and elastic modulus are distinct: the elastic modulus is a property of the material’s composition, while stiffness is a property of the solid structure. In other words, the elastic modulus describes microscopic material behavior, and stiffness describes macroscopic structural behavior.

In material mechanics, the product of the elastic modulus and a beam’s cross-sectional moment of inertia represents various stiffnesses, such as GI (torsional stiffness) and EI (bending stiffness).

Additional Notes on Stiffness and Strength

Stiffness: Refers to a component’s ability to resist elastic deformation under load. Stiffness (or rigidity) is often expressed as the force or torque required to produce unit deformation. It depends on the component’s geometry and material (elastic modulus). For isotropic materials, stiffness depends on the elastic modulus E and shear modulus G (see Hooke’s Law). Structural stiffness depends not only on the material’s elastic modulus but also on geometry, boundary conditions, and load type.

Stiffness is critical for components where excessive elastic deformation impairs performance, such as machine tool spindles, guide rails, and lead screws.

Analyzing material and structural stiffness is vital in engineering design. For structures requiring strict deformation control (e.g., aircraft wings, high-precision assemblies), stiffness analysis ensures deformation stays within limits. Many structures (buildings, machinery) rely on stiffness control to prevent vibration, flutter, or instability. Additionally, devices like spring scales and ring dynamometers require controlled stiffness to function properly. In structural mechanics, stiffness analysis of components is essential for determining deformation and stress using the displacement method.
Strength: Refers to a metallic material’s ability to resist permanent deformation and fracture under external forces. Based on the type of external force, key metrics include yield strength, tensile strength, compressive strength, and bending strength. Engineering applications commonly use yield strength and tensile strength, measured via tensile tests.

Strength is a key indicator of a component’s load-bearing capacity (resistance to failure) and a primary requirement for mechanical parts. Strength categories include static strength, fatigue strength (bending, contact), fracture strength, impact strength, high/low-temperature strength, corrosion-resistant strength, creep strength, and adhesion strength. Strength research is comprehensive, focusing on stress states to predict failure conditions.

Strength describes a material’s ability to withstand external forces without failure (permanent deformation is also considered failure). It is categorized by the type of force:
- Compressive strength: resistance to pressure
- Tensile strength: resistance to tension
- Bending strength: resistance to bending forces
- Shear strength: resistance to shear forces