Detailed explanation of the relationship between elastic modulus and stiffness

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Image 1: Elastic modulus: As defined in Image [1], the elastic modulus is the ratio of normal stress to the corresponding normal strain in a material during its elastic deformation phase. During the elastic deformation stage, the stress and strain of a material are in direct proportion to each other [that is, it obeys Hooke’s law], and the coefficient of this proportionality is known as the elastic modulus. “\"Young’s modulus\" is a physical quantity that describes the elasticity of a material; it is an umbrella term that includes concepts such as \"Young’s modulus\", \"shear modulus\", and \"bulk modulus\". Therefore, \"elastic modulus\" and \"volume modulus\" are in an inclusive relationship. Generally speaking, when an external force [referred to as \"stress\"] is applied to an elastomer, the elastomer undergoes a change in shape [referred to as \"strain\"]. The elastic modulus is generally defined as stress divided by strain. For example: Axial strain – When a pulling force F is applied to a thin rod, this force divided by the rod’s cross-sectional area S is called \"axial stress\", while the increase in the rod’s length dL divided by its original length L is called \"axial strain\". Linear stress divided by linear strain equals Young’s modulus E = (F/S) / (dL/L). Shear strain – when a lateral force f [usually frictional force] is applied to an elastic body, the shape of the body changes from square to rhombus; the angle a of this deformation is called “shear strain,” and the corresponding force f divided by the area over which the force acts, S, is called “shear stress.” Shear stress divided by shear strain equals the shear modulus G = (f/S)/a. Volume strain – when a uniform pressure p is applied to an elastic material, this pressure is referred to as \"volume stress\"; the decrease in the volume of the elastic material, denoted as -dV, divided by its original volume V, is called \"volume strain\". Volume stress divided by volume strain gives the bulk modulus: K = P/(-dV/V). When confusion is unlikely, the elastic modulus of most metal materials refers to Young’s modulus, that is, the positive elastic modulus. 　　Unit: E [elastic modulus] in gigapascals [GPa]. Image [2]: Influencing factors. The elastic modulus is an important property parameter of engineering materials; from a macroscopic perspective, it serves as a measure of a material’s ability to resist elastic deformation, while from a microscopic perspective, it reflects the strength of the bonds between atoms, ions, or molecules. Any factor that affects the bonding strength can also affect the elastic modulus of the material, such as the bonding method, crystal structure, chemical composition, microstructure, temperature, etc. Due to differences in alloy composition, heat treatment conditions, cold plastic deformation, etc., the Young’s modulus values of metallic materials can vary by 5% or more. Generally speaking, the elastic modulus of metallic materials is a mechanical property that is not sensitive to the microstructure; processes such as alloying, heat treatment [fiber structure], and cold plastic deformation have little effect on it. External factors such as temperature and loading rate also have little influence on it. Therefore, in most engineering applications, the elastic modulus is treated as a constant. Image [3] Meaning: The elastic modulus can be regarded as an indicator of the ease with which a material undergoes elastic deformation. The higher this value, the greater the stress required to cause a certain degree of elastic deformation in the material; in other words, the greater the stiffness of the material, and the less elastic deformation it will experience under a given stress. The elastic modulus E refers to the stress required for a material to undergo a unit of elastic deformation under an external force. It is an indicator of a material’s ability to resist elastic deformation, equivalent to the stiffness in ordinary springs. Image 2: Stiffness. Image [1] Definition: Stiffness is the ability of a structure or component to resist elastic deformation, measured by the force or torque required to produce a unit strain. . Rotational stiffness [k]: ——k=M/θ, where M is the applied torque and θ is the angle of rotation. Other types of stiffness include: tension and compression stiffness, axial stress to axial strain ratio (EA), shear stiffness, shear force to shear strain ratio (GA), torsional stiffness, torque to torsional strain ratio (GI), bending stiffness, and bending moment to curvature ratio (EI). Figure [2] shows the calculation methods; the theories used for calculating stiffness are divided into small-displacement theory and large-displacement theory. The large-displacement theory establishes equilibrium equations based on the deformation positions of the structure under stress, yielding accurate results but with relatively complex calculations. In the theory of small displacements, it is temporarily assumed that the structure remains undeformed when establishing the equilibrium equations. After determining the internal forces within the structure from external loads, the issue of deformation calculation is then considered. Most mechanical designs employ the small-displacement theory. For example, in the calculation of the bending deformation of a beam, since the actual deformation is very small, the first derivative of the deflection in the curvature formula is generally ignored, and the second derivative of the deflection is used as an approximation to represent the curvature of the beam’s axis. The purpose of doing this is to linearize the differential equation in order to **simplify the solution process ； When several loads act simultaneously, the bending deformation caused by each load can be calculated separately and then combined. Image [3] Classification and Significance: The ability to resist deformation under static loads is referred to as static stiffness. The ability to resist deformation under dynamic loads is known as dynamic stiffness, which is the dynamic force required to induce a unit amplitude of vibration. If the perturbation force changes slowly [i.e., its frequency is much lower than the natural frequency of the structure], the dynamic stiffness is exactly the same as the static stiffness. When the disturbance force changes very rapidly [that is, when the frequency of the disturbance force is much higher than the natural frequency of the structure], the structural deformation is relatively small, meaning that the dynamic stiffness is relatively high. When the frequency of the disturbing force is close to the natural frequency of the structure, resonance occurs. At this time, the dynamic stiffness is at its minimum, meaning the structure is most prone to deformation; the dynamic deformation can be several times or even a dozen times greater than the deformation under static load. Deformation of components often affects their performance; for example, excessive deformation of gear shafts can impact the meshing condition of gears, while significant deformation in machine tools can reduce machining accuracy. The factors affecting stiffness are the elastic modulus of the material and the structural configuration; changing the structural configuration has a significant impact on stiffness. Stiffness calculation is the foundation of vibration theory and structural stability analysis. With constant mass, greater stiffness results in a higher natural frequency. The stress distribution in a statically indeterminate structure is related to the ratio of the stiffnesses of its various parts. In fracture mechanics analysis, the stress intensity factor of a cracked component can be determined from its flexibility. Image 3: Relationship between elastic modulus and stiffness. Generally speaking, stiffness and elastic modulus are not the same thing. Elastic modulus is a property of the material components ； And stiffness is a property of solids. In other words, the elastic modulus is a microscopic property of a material, whereas stiffness is a macroscopic property. In mechanics of materials, the product of the elastic modulus and the moment of inertia of the cross-section of a beam is expressed as various types of stiffness; for example, GI represents torsional stiffness, while EI represents bending stiffness. Stiffness refers to the ability of a material, component, or structure under external forces to resist deformation. The stiffness of a material is measured by the amount of external force required to cause a unit deformation in it. The stiffness of isotropic materials depends on their elastic modulus E and shear modulus G (see Hooke’s law). The stiffness of a structure depends not only on the elastic modulus of its constituent materials but also on factors such as its geometric shape and boundary conditions, as well as the nature of the external forces acting on it. Analyzing the stiffness of materials and structures is an important task in engineering design. For some structures whose deformation must be strictly limited [such as wings, high-precision assemblies, etc.], deformation must be controlled through stiffness analysis. Many structures [such as buildings, machinery, etc.] also need to have their stiffness controlled in order to prevent vibration, flutter, or instability. Furthermore, devices such as spring scales and ring dynamometers must have their stiffness controlled to a certain reasonable value in order to ensure their specific functions. In the displacement method analysis of structural mechanics, to determine the deformation and stress of a structure, it is usually necessary to analyze the stiffness of its various parts as well. Stiffness refers to a part’s ability to resist elastic deformation under the action of a load. The stiffness of a part is usually expressed in terms of the force or torque required to cause a unit amount of deformation. The degree of stiffness depends on the geometric shape of the part and the type of material used, that is, the elastic modulus of the material. Stiffness requirements are particularly important for those components whose elastic deformation exceeds a certain level, as this can affect the performance of the machine; examples include the spindle, guide rails, and lead screws in machine tools. Image Image Strength The ability of metal materials to resist permanent deformation and fracture under external forces is called strength. Depending on the nature of the external forces applied, there are mainly yield strength, tensile strength, compressive strength, and bending strength. In engineering, yield strength and tensile strength are commonly used, and these two strength parameters can be determined through tensile testing. Strength refers to the ability of a part to resist fracture or excessive deformation beyond acceptable limits when subjected to loads. In other words, strength is an important indicator for measuring a part’s own load-bearing capacity [that is, its ability to resist failure]. Strength is the fundamental requirement that mechanical components must meet first and foremost. The strength of mechanical parts can generally be categorized into static strength, fatigue strength [such as bending fatigue and contact fatigue], fracture strength, impact strength, high- and low-temperature strength, strength under corrosive conditions, creep, adhesion strength, etc. Experimental studies on strength are comprehensive investigations that primarily examine the stress conditions of components in order to understand their loading conditions as well as the conditions and timing of failure. Strength refers to a material’s ability to withstand external forces without being damaged (irreversible deformation is also considered damage). Depending on the type of force applied, it can be classified as follows: (1) Compressive strength – the ability of a material to resist compressive forces. (2) Tensile strength – the ability of a material to resist tensile forces. (3) Flexural strength – the ability of a material to resist bending forces. (4) Shear strength – the ability of a material to resist shear forces.