Table of Contents

**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 of a material, its stress and stress become positively proportional (i.e. in accordance with Hooke’s law), and its proportional coefficient is called elastic modulus.

“Modulus of elasticity” is a physical quantity that describes the elasticity of matter.

It is a general term, including “Young’s modulus”, “shear modulus”, “bulk modulus”, etc.

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

Generally speaking, after an external action (called “stress”) is applied to the elastomer, the elasticity will change its shape (called “strain”).

The general definition of “elastic modulus” is: stress divided by strain.

For example:

**Linear strain: **apply a tensile force F to a thin rod.

This tensile force divided by the cross-sectional area S of the rod is called “linear stress”.

The elongation of the rod DL divided by the original length L is called “linear strain”.

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 f (usually friction force) is applied to an elastomer, the elasticity will change from square to diamond.

This deformation angle a is called “shear strain”, and the corresponding force f divided by the stress area s is called “shear stress”.

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

**Volume strain: **apply an overall pressure P to the elastomer, which is called “volume stress”.

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

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

When it is not easy to cause confusion, the elastic modulus of general metal materials refers to Young’s modulus, that is, positive elastic modulus.

Unit: E (modulus of elasticity) GPa.

**2. Influencing factors**

Elastic modulus is an important performance parameter of engineering materials.

From a macro point of view, elastic modulus is a measure of the ability of an object to resist elastic deformation, and from a micro point of view, it is a reflection of the bonding strength between atoms, ions or molecules.

All factors affecting the bonding strength can affect the elastic modulus of the material, such as bonding mode, crystal structure, chemical composition, microstructure, temperature and so on.

Due to different alloy composition, different heat treatment state and different cold plastic deformation, the young’s modulus of metal materials will fluctuate by 5% or more.

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

Alloying, heat treatment (fiber structure) and cold plastic deformation have little influence on the elastic modulus, and external factors such as temperature and loading rate have little influence on it.

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

**3. Meaning**

The elastic modulus can be regarded as an index to measure the difficulty of elastic deformation of materials.

The greater its value, the greater the stress that causes certain elastic deformation of materials, that is, the greater the stiffness of materials, that is, the smaller the elastic deformation under the action of certain stress.

Elastic modulus E refers to the stress required by the material to produce unit elastic deformation under the action of external force.

It is an index reflecting the ability of materials to resist elastic deformation, which is equivalent to the stiffness of ordinary springs.

**Stiffness**

**1. Definition**

Stiffness is the ability of a structure or member to resist elastic deformation.

It is measured by the force or moment required to produce unit strain.

Rotational stiffness (k): k = m/ θ。

Where M is the applied torque, 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 theory of calculating stiffness is divided into small displacement theory and large displacement theory.

The large displacement theory establishes the equilibrium equation according to the deformation position of the structure after stress.

The result is accurate, but the calculation is complex.

When establishing the equilibrium equation, the small displacement theory temporarily assumes that the structure is not deformed, so the internal force of the structure can be obtained from the external load, and then the deformation calculation problem can be considered.

Small displacement theory is adopted in most mechanical design.

For example, in the calculation of bending deformation of beam, because the actual deformation is very small, the first derivative of deflection in the curvature formula is generally ignored, and the second derivative of deflection is used to approximate the curvature of beam axis.

The purpose of this is to linearize the differential equation to greatly simplify the solution process;

When several loads act at the same time, the bending deformation caused by each load can be calculated separately and then superimposed.

**3. Classification and significance**

The ability to resist deformation under static load is called static stiffness;

The ability to resist deformation under dynamic load is called dynamic stiffness, that is, the dynamic force required to cause unit amplitude.

If the interference force changes very slowly (i.e. the frequency of the interference force is far less than the natural frequency of the structure), the dynamic stiffness is basically the same as the static stiffness.

The interference force changes very fast (that is, when the frequency of the interference force is much greater than the natural frequency of the structure), the structural deformation is relatively small, that is, the dynamic stiffness is relatively large.

When the frequency of the interference force is close to the natural frequency of the structure, there is resonance.

At this time, the dynamic stiffness is the smallest, that is, it is the easiest to deform, and its dynamic deformation can reach several times or even more than ten times of the static load deformation.

Component deformation often affects the work of components.

For example, the excessive deformation of gear shaft will affect the gear meshing condition, and the excessive deformation of machine tool will reduce the machining accuracy.

The factors affecting the stiffness are the elastic modulus of materials and structural form. Changing the structural form has a significant impact on the stiffness.

Stiffness calculation is the basis of vibration theory and structural stability analysis.

When the mass remains unchanged, the natural frequency is high when the stiffness is large.

The stress distribution of 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 obtained according to the flexibility.

**Relationship between elastic modulus and stiffness**

Generally speaking, stiffness and elastic modulus are different. Elastic modulus is the property of material components;

Stiffness is the property of solids.

In other words, the elastic modulus is the microscopic property of the material, while the stiffness is the macroscopic property of the material.

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

For example, GI is the torsional stiffness and EI is the flexural stiffness.

**1. Stiffness**

Stiffness refers to the ability of parts to resist elastic deformation under load.

The stiffness (or rigidity) of a part is usually expressed by the force or moment required for unit deformation.

The stiffness depends on the geometry of the part and the type of material (i.e. the elastic modulus of the material).

The stiffness of an isotropic material depends on its elastic modulus E and shear modulus G (see Hooke’s law).

In addition to the elastic modulus of the constituent materials, the stiffness of the structure is also related to its geometry, boundary conditions and other factors, as well as the action form of external forces.

Stiffness requirements are particularly important for some parts that will affect the working quality of the machine after the elastic deformation exceeds a certain value, such as the spindle, guide rail, lead screw, etc.

Analyzing the stiffness of materials and structures is an important work in engineering design.

For some structures whose deformation must be strictly limited (such as wings, high-precision assemblies, etc.), the deformation must be controlled through stiffness analysis.

Many structures (such as buildings, machinery, etc.) also need to control the stiffness to prevent vibration, flutter or instability.

In addition, such as spring scale and ring dynamometer, its specific function must be ensured by controlling its stiffness to a reasonable value.

In the displacement analysis of structural mechanics, in order to determine the deformation and stress of the structure, it is usually necessary to analyze the stiffness of each part.

**2. Intensity**

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

According to the nature of external force, it mainly includes yield strength, tensile strength, compressive strength, bending strength, etc. yield strength and tensile strength are commonly used in engineering.

These two strength indexes can be measured by tensile test.

Strength is an important index to measure the bearing capacity of parts (i.e. the ability to resist failure), and it is the basic requirement that mechanical parts should first meet.

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

The experimental study of strength is a comprehensive study, which mainly studies the stress status of parts and components and predicts the conditions and timing of failure through its stress state.

Strength refers to the ability of materials to withstand external forces without being damaged (unrecoverable deformation also belongs to being damaged).

**It can be divided into the following types according to 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.