## I. **Modulus of Elasticity**

**1. Definition**

Modulus of Elasticity: The ratio of normal stress to corresponding normal strain in the elastic deformation stage of a material.

In the elastic deformation stage, a material’s stress and strain are proportional, in accordance with Hooke’s Law, and the coefficient of proportionality is referred to as the elastic modulus.

The term “modulus of elasticity” is a general description of a material’s elasticity. It encompasses several specific moduli, including Young’s modulus, shear modulus, and bulk modulus, among others.

Therefore, “elastic modulus” and “bulk modulus” are inclusive terms.

When an external force (known as “stress”) is applied to an elastomer, it will change its shape (known as “strain”). The elastic modulus is defined as the ratio of stress to strain.

For example:

**Linear Strain:**

When a tensile force F is applied to a thin rod, the linear stress is calculated as the tensile force divided by the cross-sectional area S of the rod.

The linear strain is calculated as the elongation of the rod (dL) divided by its original length (L).

The linear stress divided by the linear strain is equal to Young’s modulus, E = (F / S) / (dL / L).

**Shear Strain:**

When a lateral force (usually a friction force) f is applied to an elastomer, it changes from a square to a diamond shape.

This deformation angle is known as the “shear strain,” and the corresponding force divided by the stress area is called “shear stress.”

The shear stress divided by the shear strain is equal to the shear modulus, G = (f / S) / a.

**Volume Strain:**

When an overall pressure P is applied to the elastomer, it is known as “volume stress.”

The reduction in volume of the elastomer (-dV) divided by its original volume (V) is called “volume strain.”

The volume stress divided by the volume strain is equal to the bulk modulus, K = P / (-dV / V).

In general, when there is no confusion, the elastic modulus of metal materials refers to Young’s modulus, also known as the positive elastic modulus.

Unit: E (modulus of elasticity) is measured in GPa.

**2. Influencing factors**

Elastic modulus is a crucial performance parameter of engineering materials.

From a macro perspective, it measures an object’s ability to resist elastic deformation, while from a micro viewpoint, it reflects the bonding strength between atoms, ions, or molecules.

Factors that affect bonding strength can also impact the elastic modulus of a material, such as bonding mode, crystal structure, chemical composition, microstructure, temperature, and others.

The Young’s modulus of metal materials can fluctuate by over 5% due to different alloy compositions, heat treatment states, and cold plastic deformations.

However, generally speaking, the elastic modulus of metal materials is a mechanical property index that is insensitive to structure.

Alloying, heat treatment (fiber structure), and cold plastic deformation have limited effect on the elastic modulus, and external factors such as temperature and loading rate have a negligible impact on it.

Therefore, in general engineering applications, the elastic modulus is considered a constant.

Unit: GPa (gigapascals) for elastic modulus.

**3. Meaning**

The Elastic Modulus is a measure of a material’s resistance to elastic deformation.

The higher its value, the greater the stress required to produce a certain amount of elastic deformation, meaning that the material is stiffer and experiences less elastic deformation under a given stress.

The Elastic Modulus, represented by E, is a measure of the amount of stress required for a material to undergo unit elastic deformation under an external force.

It represents the material’s ability to resist elastic deformation and can be compared to the stiffness of a spring.

## II. **Stiffness**

**1. Definition**

Stiffness” refers to the ability of a structure or component to resist elastic deformation. It is determined by the force or moment required to produce a unit of strain.

In terms of rotational stiffness, it is represented by “k” and can be calculated as “k = M / θ”, where “M” is the applied torque and “θ” is the rotation angle.

Other stiffnesses include:

- tension and compression stiffness
- axial force ratio axial linear strain (EA)
- shear stiffness
- shear force ratio shear strain (GA)
- torsional stiffness
- torque ratio torsional strain (GI)
- bending stiffness
- bending moment ratio curvature (EI).

**2. Calculation method**

The method of calculating stiffness can be divided into two approaches: the small displacement theory and the large displacement theory.

The large displacement theory takes into account the deformation of the structure after stress and forms the equilibrium equation accordingly, providing accurate results but with a more complex calculation process.

In contrast, the small displacement theory assumes that the structure is not significantly deformed, so the internal force can be obtained from the external load and then used to calculate the deformation.

This approach is widely used in most mechanical design applications, as it is much simpler to solve.

For example, in the calculation of beam bending deformation, the small displacement theory is often employed because the actual deformation is very small.

This theory involves ignoring the first derivative of deflection in the curvature formula and using the second derivative of deflection to approximate the curvature of the beam axis, which helps simplify the solution process by linearizing the differential equation.

When multiple loads are acting simultaneously, the bending deformation caused by each load can be calculated separately and then combined.

**3. Classification and significance**

The resistance to deformation under a static load is known as static stiffness, while the resistance to deformation under a dynamic load is referred to as dynamic stiffness, meaning the amount of dynamic force required for unit amplitude.

When the interfering force changes slowly (i.e., the frequency of the interfering force is much less than the natural frequency of the structure), the dynamic stiffness is essentially equal to the static stiffness.

However, if the interfering force changes rapidly (i.e., the frequency of the interfering force is much greater than the natural frequency of the structure), the structural deformation will be relatively small, and thus the dynamic stiffness will be relatively large.

If the frequency of the interfering force is close to the natural frequency of the structure, resonance occurs, and the dynamic stiffness will be at its minimum, making the structure the easiest to deform, with its dynamic deformation capable of reaching several times or even more than ten times that of the static load deformation.

Excessive deformation of components can have an impact on their operation.

For instance, excessive deformation of a gear shaft can affect gear meshing, and excessive deformation of a machine tool can reduce machining accuracy.

The factors affecting stiffness include the elastic modulus of materials and the structural form. Changing the structural form can have a significant impact on stiffness.

Stiffness calculation is the foundation of vibration theory and structural stability analysis. When the mass remains constant, high stiffness results in a high natural frequency.

The stress distribution in a statically indeterminate structure is related to the stiffness ratio of each part.

In fracture mechanics analysis, the stress intensity factor of a cracked member can be determined based on its flexibility.

## III. **Relationship Between Elastic Modulus and Stiffness**

Generally speaking, stiffness and elastic modulus are different concepts.

Elastic modulus is a property of material components, while stiffness is a property of solids.

In other words, elastic modulus refers to the microscopic property of a material, while stiffness refers to the macroscopic property of a material.

In material mechanics, the product of the elastic modulus and the moment of inertia of the cross-section of a beam is expressed as various stiffnesses.

For example, “GI” represents the torsional stiffness and “EI” represents the flexural stiffness.

**1. Stiffness**

Stiffness refers to a part’s resistance to elastic deformation under load.

The stiffness of a part is typically expressed as the force or moment required for a unit deformation.

This property is determined by both the material’s elastic modulus and its geometry.

In the case of isotropic materials, the stiffness also depends on its shear modulus (according to Hooke’s law).

External forces and other factors, such as boundary conditions and geometry, also play a role in determining the stiffness of a structure.

In engineering design, analyzing the stiffness of materials and structures is crucial, especially for parts that are sensitive to elastic deformation, such as spindles, guide rails, and lead screws.

Stiffness analysis is also critical for structures that require strict deformation control, such as wings and high-precision assemblies.

It is important for many structures, such as buildings and machinery, to control stiffness to prevent vibrations, flutter, and instability.

Devices such as spring scales and ring dynamometers also require controlling their stiffness for proper functioning.

In the displacement analysis of structural mechanics, the stiffness of each part must be analyzed to determine its deformation and stress.

**2. Intensity**

The ability of metal materials to resist permanent deformation and fracture under the action of external force is known as strength.

It mainly includes yield strength, tensile strength, compressive strength, bending strength, among others.

Yield strength and tensile strength are frequently used in engineering, and these two strength indexes can be determined through a tensile test.

Strength is a crucial index for measuring the bearing capacity of parts and their ability to resist failure, and it is a fundamental requirement for mechanical parts.

The strength of mechanical parts can typically be divided into static strength, fatigue strength (bending fatigue and contact fatigue), fracture strength, impact strength, high and low temperature strength, strength under corrosive conditions, bonding strength, and other factors.

The study of strength is a comprehensive examination, primarily focusing on the stress state of parts and components and predicting the conditions and timing of failure through the stress state.

Strength refers to the ability of materials to withstand external forces without being damaged, which also includes unrecoverable deformation.

It can be categorized into the following types based on the types of forces:

- Compressive strength — the ability of a material to withstand pressure;
- Tensile strength — the ability of a material to withstand tensile force;
- Bending strength — the bearing capacity of the material to the external bending force;
- Shear strength – the ability of a material to withstand shear force.