Differences and relationships between strength, stiffness, and elastic modulus

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1. The ability of materials, mechanical parts, and components to resist external forces without failing. Strength includes both material strength and structural strength. The term \"strength problem\" has two meanings: narrow and broad. In a narrow sense, strength problems refer to various issues of excessive fracture and plastic deformation. The broad category of strength problems includes strength, stiffness, and stability issues, and sometimes also mechanical vibration problems. Strength requirements are a fundamental requirement in mechanical design. Material strength refers to the various mechanical property indicators of a material under different influencing factors. The influencing factors include the chemical composition of the material, processing techniques, heat treatment procedures, stress conditions, nature of the load, loading rate, temperature, and medium, among others. Based on the properties of materials, material strength is divided into the strength of brittle materials, the strength of plastic materials, and the strength of cracked materials. ①Strength of brittle materials: Brittle materials such as cast iron fracture suddenly under load, with almost no plastic deformation. Fragile materials use their ultimate strength as the standard for calculating strength. There are two types of strength limits: the maximum nominal stress that a tensile specimen can withstand before breaking is known as the material’s tensile strength limit, while the maximum nominal stress for a compressive specimen is referred to as the compressive strength limit. ②Strength of plastic materials: Plastic materials such as chin steel undergo significant plastic deformation before fracturing, and this deformation does not disappear after unloading; it is also known as residual deformation. Plastic materials use their yield limit as the standard for calculating strength. The yield limit of a material is the stress at which a tensile specimen begins to yield [the phenomenon of continuous increase in strain while stress remains constant]. For plastic materials that do not exhibit yield behavior, take 0. The stress corresponding to 2% plastic deformation is the nominal yield limit, denoted as σ0. 2 represents it. ③Strength of cracked materials: It is usually lower than the material’s strength limit; when calculating the strength, the material’s fracture toughness must be taken into account [see fracture mechanics analysis]. For the same material, when different heat treatment processes are used, the higher the strength, the lower the fracture toughness. Depending on the nature of the load, material strength includes static strength, impact strength, and fatigue strength. The strength of a material under static loads is determined, depending on the properties of the material, using either the yield limit or the strength limit as the standard for calculating strength. When the material is subjected to impact loads, both its yield limit and strength limit increase [see impact strength]. The strength of a material under cyclic stress is usually determined using the material’s fatigue limit as the standard for calculating strength [see fatigue strength design]. There is also contact strength [see contact stress]. Depending on environmental conditions, material strength includes high-temperature strength and corrosion resistance, among others. High-temperature strength includes creep strength and endurance strength. When the temperature of a metal under external load is higher than the recrystallization temperature [the lowest temperature required for deformed crystals to return to their undeformed state], the strain hardening resulting from plastic deformation is rapidly eliminated due to high-temperature annealing. As a result, the deformation continues to increase under constant load, a phenomenon known as creep. The material’s creep limit is used as the standard for calculating its strength. The fracture strength under sustained high-temperature loading may be lower than the tensile strength of the material at the same temperature; the material’s endurance limit is used as the standard for calculating its strength [see endurance strength]. In addition, there are material strength issues such as stress corrosion cracking and corrosion fatigue influenced by environmental media. Structural strength refers to the strength of mechanical parts and components. It involves the simplification of mechanical models, stress analysis methods, material strength, strength criteria, and safety factors. Based on their geometric shape, the strength issues of mechanical parts and components can be simplified by using mechanical models such as bars, bar systems, plates, shells, blocks, and infinite solids for analysis. Different mechanical models have various mechanical calculation methods for strength issues. Mechanics of materials generally studies the strength calculation of bars. Structural mechanics analyzes the internal forces and deformations of beam systems [trusses, rigid frames, etc.]. Objects of other shapes fall within the scope of study of elastoplastic mechanics. A rod is an object whose two dimensional dimensions of the cross-section are much smaller than its length dimension, including tensioned rods, compressed columns, bent beams, and torsionally stressed shafts. Plates and shells are characterized by a thickness that is much smaller than their dimensions in the other two directions; those that are flat are called plates, while those that are curved are called shells. To address the issue of structural strength, in addition to stress analysis, it is also necessary to consider material strength and strength criteria, as well as study the relationship between them. The fatigue strength of parts and components under cyclic stress is related not only to the fatigue strength of the material, but also to factors such as the size of the parts and components, the stress concentration factor, and their surface condition. When the cyclic load varies irregularly, the influence of the load sequence included in the load spectrum must also be considered. Strength theory is required for the case of combined stresses. In the case of macroscopic cracks, fracture mechanics analysis is required. For certain components, it is often necessary to consider several strength criteria simultaneously and compare them in order to determine the most likely mode of failure. For most structural strength issues, the structural form is typically determined first, followed by stress analysis and strength verification based on the external loads. With the application of computer methods, optimal design has become a feasible task; one can first establish specific design objectives [such as minimizing the structure’s weight], and then seek the most appropriate structural form. 2. The ability of metal materials to resist permanent deformation and fracture under external forces is known as 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 a component’s ability 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 ability to bear load [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 classified into engineering properties such as static strength, fatigue strength [including bending fatigue and contact fatigue], fracture strength, impact strength, strength at high and low temperatures, strength under corrosive conditions, creep strength, and adhesion strength. Experimental studies on strength are comprehensive investigations that primarily use the stress state to examine the loading conditions of components and to predict the conditions and timing of failure. Strength refers to a material’s ability to withstand external forces without being damaged (irreversible deformation also constitutes damage). Based on the type of stress applied, they are classified as follows: (1) Compressive strength -- the ability of a material to withstand pressure. (2) Tensile strength -- the ability of a material to withstand tensile forces. (3) Bending strength -- the material’s ability to withstand bending forces. (4) Shear strength -- the ability of a material to withstand shear forces. 3. Strength is the ability of a material to resist deformation and failure under external forces. Depending on the manner in which external forces act, there are various strength indicators, such as tensile strength, bending strength, shear strength, etc. When a material is subjected to tensile forces, the key strength properties are yield strength and tensile strength. Note that strength and hardness are fundamentally different concepts. Hard and brittle materials such as glass have high hardness [a low ratio of deformation to external force], but low strength [a low total external force they can withstand before breaking]. For metals in the same series, there can be a certain corresponding relationship between these two. Strength measurement often requires the complete destruction of the material, whereas hardness testing causes less damage or no damage at all. Therefore, the calibrated hardness-to-strength conversion relationship is used to estimate strength from hardness. The strength of a metal material is its ability to resist permanent deformation and fracture under external forces. In engineering, the yield strength and tensile strength are commonly used indicators to represent the strength of metal materials. Yield strength is the yield limit of a metal material at the point when it begins to yield, that is, the stress required to resist slight plastic deformation. σS = Fs/AO, where Fs is the maximum external force exerted on the specimen at the point of yield (in N), and AO is the original cross-sectional area of the specimen (in mm2). σS represents the yield strength (in Mpa). Tensile strength refers to the maximum stress that a metal material can withstand before it breaks; it is calculated as σb = FO/AO, where FO is the maximum external force acting on the specimen before it breaks (in N), and AO is again the original cross-sectional area of the specimen (in mm2). σb represents the tensile strength (in Mpa). Stiffness and its definition: Stiffness is the ability of a material, component, or structure to resist deformation when subjected to external forces. 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 applied. Analyzing the stiffness of materials and structures is an important task in engineering design. For some structures where deformation must be strictly restricted [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 components as well. Stiffness refers to a component’s ability to resist elastic deformation under 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 threshold, as this can affect the performance of the machine; examples include the spindle, guide rails, and lead screws in machine tools. Stiffness of the process system 1. Basic concept The general concept of stiffness refers to the ability of an object or system to resist deformation. It is expressed as the ratio of the force applied to the object to the amount of deformation that occurs in the direction of that force. 　 During the cutting process, under the action of various external forces, the different components of the machining system will deform in corresponding directions depending on the forces applied. For the stress-induced deformation of the process system, the deformation amount in the error-sensitive direction is primarily studied. Therefore, the stiffness of the process system is defined as the ratio of the cutting force acting in the normal direction of the surface being machined to the displacement of the tool in that normal direction due to the cutting force. In the definition of process system stiffness, the forces and deformations are measured under static conditions, which represents the static stiffness of the process system ； The deformation amount is the result of the combined effect of the total cutting force; when the displacement caused by this force exceeds the displacement caused by other factors, the total displacement is in the opposite direction to the Y-axis and takes a negative value. At this point, the tool holder is in a state of negative stiffness. Negative stiffness causes the cutting tip to pierce into the surface of the workpiece [penetration], and it also induces vibration in the workpiece, which should be avoided as much as possible. 2. Calculation of the stiffness of the process system: The total deformation of the process system should be the sum of the normal deformations at the same point in each of its constituent components. By knowing the stiffness of each component, the stiffness of the entire process system can be determined. For workpieces and cutting tools, they are generally simple components that can be approximated using formulas from material mechanics. For example, the stiffness of a turning tool can be calculated as that of a cantilever beam; when a workpiece is held in a three-jaw chuck, its stiffness can also be calculated as that of a cantilever beam. When machining long and slender shafts with center points, the stiffness of the workpiece can be considered as that of a simply supported beam, and so on ； For machine tools and fixtures, whose structure is relatively complex, their stiffness is usually determined by experimental methods. The difference between strength and stiffness: From the perspective of engineering mechanics, strength refers to a material’s ability to resist failure, that is, the stress required for the material to fail. It generally applies only to materials. Its size is related to the properties of the material itself and the mode of stress application. For example, the tensile strength and shear strength of a material refer to the maximum tensile and shear forces that the material can withstand per unit area, and these values are independent of the shape of the material. Stiffness refers to a component’s or structure’s ability to resist deformation, that is, the stress required to cause a unit of deformation. It generally applies to components or structures. Its size depends not only on the properties of the material itself, but also on the cross-section and shape of the component or structure. The expressions for different types of stiffness vary as well. For example, section stiffness refers to the ability of a section to resist deformation, and its expression is the product of the material’s elastic modulus or shear modulus and the corresponding section moment of inertia or section area. Among them, the expression for the shear [compression] stiffness of the cross-section is the product of the material’s elastic modulus and the cross-sectional area ； The bending stiffness of a cross-section is the product of the material’s elastic modulus and the cross-sectional moment of inertia, among other things. Component stiffness refers to a component’s ability to resist deformation, and it is expressed as the ratio of the internal forces generated by the forces applied to the component to the corresponding deformation of that component. The expression for the bending stiffness of a member is the ratio of the moment applied to the bending member to the change in curvature resulting from that moment ； The shear stiffness of a member is the ratio of the shear force applied to the shear-member to the change in the orthogonal angle between this force and the deformation it causes. The lateral displacement stiffness of a structure refers to its ability to resist lateral deformation, and it is equal to the ratio of the horizontal force applied to the structure to the resulting horizontal displacement, among other things. Of course, the elastic modulus or strain modulus of a material can also be understood as the stiffness of that material. Strength: Its legal unit is newtons per square millimeter [N/mm^2], which represents the amount of force that a metal can withstand per unit area. It refers to the ability of metal materials to resist damage caused by external forces. It can be divided into: tensile strength, compressive strength, bending strength, and shear strength. Stiffness: also known as hardness, it refers to a material’s ability to resist the penetration of hard objects into its surface. Depending on the testing method, it can be measured using Rockwell [HR] hardness, Surface Rockwell [HR] hardness, Vickers [HV] hardness, or Brinell [HB] hardness; however, none of these values have units. Hardness is an important performance parameter for measuring the hardness of metal materials. It can be understood as the material’s ability to resist elastic deformation, plastic deformation, or failure, or it can also be described as the material’s ability to resist residual deformation and reverse failure. Hardness is not a simple physical concept; rather, it is a comprehensive indicator of a material’s mechanical properties such as elasticity, plasticity, strength, and toughness. Hardness tests can be classified into various methods based on their testing approaches: static pressure methods (such as Brinell hardness, Rockwell hardness, Vickers hardness, etc.), scratch methods (such as Mohs hardness), rebound methods (such as Shore hardness), as well as microhardness and high-temperature hardness tests. Strength refers to a component’s ability 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 ability to bear load (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 classified into static strength, fatigue strength (such as bending fatigue and contact fatigue), fracture strength, impact strength, strength at high and low temperatures, strength under corrosive conditions, creep strength, and adhesion strength. Experimental studies on strength are comprehensive investigations that primarily use the stress state to examine the loading conditions of components and to predict the conditions and timing of failure. Stiffness refers to a component’s ability to resist elastic deformation under load. The stiffness of a part (also known as rigidity) is commonly expressed in terms of the force or torque required to cause a unit amount of deformation. The level of stiffness depends on the geometric shape of the part and the type of material used (i.e., the material’s elastic modulus). Stiffness requirements are particularly important for those components whose elastic deformation exceeds a certain threshold, as this can affect the performance of the machine; examples include the spindle, guide rails, and lead screws in machine tools. Strength is the ability to resist plastic deformation, while stiffness indicates the ease with which a material undergoes elastic deformation. Young’s modulus, elastic modulus, shear modulus, bulk modulus, strength, and stiffness – these \"moduli\" can be understood as standard quantities or indicators. The “modulus” of a material usually requires a descriptive term before it, such as elastic modulus, compressive modulus, shear modulus, section modulus, etc. These are all indicators related to deformation. Young’s Modulus: Young’s Modulus is also known as the elastic modulus; it is a concept in material mechanics. For linearly elastic materials, the formula σ (normal stress) = Eε (normal strain) holds true, where σ is the normal stress, ε is the normal strain, and E is the elastic modulus – a constant that depends on the material and is related to the properties of that material. Thomas Young [1773–1829] studied shear deformation in the field of mechanics of materials, considering shear stress to be a type of elastic deformation. In 1807, the definition of elastic modulus was proposed, and for this reason it was later referred to as Young’s modulus. The Young’s modulus of steel is approximately 2×1011 N·m-2, while that of copper is 1.1×1011 N·m-2. Elastic Modulus [Elastic Modulus] E: The elastic modulus E is a constant of proportionality between the longitudinal stress applied to a material and its longitudinal strain, within the range of elastic deformation (that is, within the proportional limit). It also often refers to the ratio of the stress applied to a material, such as tension, compression, bending, torsion, shear, etc., to the corresponding strain produced in the material. The elastic modulus is a physical quantity that indicates the strength of the bonding forces between atoms in a crystal, and therefore it is a parameter that is not sensitive to the structural organization. In engineering, the elastic modulus is a measure of a material’s stiffness, and it indicates the ease with which an object deforms. The elastic modulus E, within the proportional limit, is the ratio of stress to the corresponding strain of the material. For some materials whose stress-strain curves do not follow a linear relationship within the elastic range, the tangent modulus of elasticity, secant modulus of elasticity, or other artificially defined values can be used to replace their actual modulus of elasticity values as needed. Depending on the type of stress applied, there are corresponding modulus of elasticity for tension (Young’s modulus), shear modulus of elasticity (rigidity modulus), bulk modulus, compression modulus, and so on. Shear Modulus G: The shear modulus refers to the ratio of shear stress to shear strain. Shear modulus G = shear elastic modulus G = shear modulus. Young’s modulus, elastic modulus, shear modulus, bulk modulus, strength, stiffness – the shear elastic modulus G is one of the fundamental physical properties of materials. Along with Young’s (compression, tension) elastic modulus E and Poisson’s ratio ν, it constitutes the three key physical properties of materials, and it is widely used in material mechanics and elasticity theory. It is defined as: G=τ/γ, where G (Mpa) is the shear elastic modulus ； τ is the shear stress (Mpa) ； γ is the shear strain (radians). Bulk Modulus K: The bulk modulus describes the elasticity of homogeneous, isotropic solids; it can be expressed as a force per unit area and indicates the degree of incompressibility. The formula is as follows: K = E / (3 × (1 – 2 × v)), where E is the elastic modulus and v is the Poisson’s ratio. For more details, refer to any textbook on elasticity mechanics available in universities. Property: The volume of an object at pressure p0 is V0 ； Assuming the pressure increases (p0→p0+dP), then the volume decreases to (V0-dV). Then K=(p0+dP)/(V0-dV) is known as the modulus of volume elasticity of that object. If it is within the elastic range, it is specifically referred to as the volume elastic modulus. The bulk modulus is a relatively stable material constant. Since the volume of the material always decreases under isotropic pressure, the K value is always positive, with the unit being MPa. The reciprocal of the bulk modulus is called the bulk compliance. There is a relationship between the bulk modulus and tensile modulus, as well as Poisson’s ratio: E=3K(1-2μ). Compression Modulus: The compression modulus refers to the ratio of compressive stress to compressive strain. Storage modulus E': The storage modulus E' is essentially the Young’s modulus, indicating the material’s ability to store elastic deformation energy. The storage modulus is a parameter that indicates the resilience of a material after deformation. The storage modulus E' refers to the ability of viscoelastic materials to store energy over one cycle under alternating stress, and it generally denotes elasticity ； Energy dissipation modulus E'': The energy dissipation modulus E'' is the component of the modulus in which stress and deformation are asynchronous ； The ability to represent the energy dissipated due to deformation in a material reflects its viscous nature. The energy dissipation modulus E'' refers to the capacity to dissipate energy over one cycle. It usually refers to the viscous tangent modulus [Tangent Modulus]: The tangent modulus is the slope of the curve between the yield limit and the strength limit in the plastic phase. It is the first derivative of stress with respect to strain on the stress-strain curve. Its size is related to the degree of stress and is not a fixed value. Tangent modulus is generally used in incremental finite element calculations. The units for tangent modulus and yield stress are both N/m2. Section modulus: The section modulus is a mechanical property of the cross-section of a component. It is an indicator that represents a component’s cross-sectional capacity to resist certain types of deformation, such as the flexural section modulus and the torsional section modulus. It depends only on the shape of the cross-section and the position of the neutral axis, and not on the properties of the material itself. In some books, the section modulus is also referred to as the section coefficient or section moment of resistance, among other names. Strength: Strength refers to a material’s ability to resist destruction, that is, its ability to resist deformation (elastic or plastic) and fracture (stress). It generally applies only to materials. Its size is related to the properties of the material itself and the mode of stress application. It can be divided into: yield strength, tensile strength, compressive strength, flexural strength, shear strength, etc. For example, the tensile strength and shear strength of a material refer to the maximum tensile and shear forces that the material can withstand per unit area, and these values are independent of the shape of the material. For example, the comparison of tensile strength and tensile modulus: their units are both MPa or GPa. Tensile strength refers to the maximum stress that a material can withstand during stretching, while tensile modulus indicates the elasticity of the material under tension. For steel, such as 45 steel, the tensile modulus is in the order of 100 MPa, usually ranging from 200 to 500 MPa; whereas the tensile modulus in the order of 100 GPa is generally between 180 and 210 GPa. Stiffness: Stiffness (i.e., hardness) refers to a component’s or structure’s ability to resist deformation; it is an indicator of how difficult it is for a material to undergo elastic deformation, and it mainly denotes the stress required to cause a unit amount of deformation. It generally applies to components or structures. Its size depends not only on the properties of the material itself, but also on the cross-section and shape of the component or structure. The higher the stiffness, the more “hard” the object appears to be. For different things, there are various ways to express stiffness, such as static stiffness, dynamic stiffness, ring stiffness, etc. Generally, the unit of stiffness is Newtons per meter, or Newtons per millimeter, which indicates the force required to produce a unit length of deformation. The units for normal stiffness and shear stiffness are also N/m or N/mm; the difference lies in the direction of the force. It is generally expressed by the value of the elastic modulus, E. The magnitude of E is generally only related to the interatomic forces, and has little to do with the state of the material. Generally, the elastic moduli of steel and cast iron differ very little; in other words, their stiffness is almost the same. However, their strengths differ significantly. “\"Elastic 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 \"bulk 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 general definition of \"elastic modulus\" is stress divided by strain. For example: Linear 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 \"linear stress\", while the increase in the rod’s length dL divided by its original length L is called \"linear strain\". Linear stress divided by linear strain equals Young’s modulus E: F/S = E(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 is applied, S, is called “shear stress.” Shear stress divided by shear strain equals the shear modulus G: f/S = G*a. Volume strain – When a uniform pressure p is applied to an elastic body, this pressure is referred to as \"volume stress.\" The decrease in the volume of the elastic body, denoted as (-dV), divided by its original volume V, is called \"volume strain.\" Volume stress divided by volume strain gives the bulk modulus: p = K(-dV/V). Note: Liquids have only a bulk modulus; all other elastic moduli are zero, so the term elastic modulus is used to refer to the bulk modulus. The strain of ordinary elastomers is very small; that is, the change in volume compared to the original volume is a very small value. In this case, the relative change in volume and the relative change in density are simply opposite in sign but have the same magnitude; for example, the volume decreases by 0 percent. 01: The density increases by 0 percent. 01. The bulk modulus is not a negative value [as can be seen from the previous definition], and it is not only gases that have a bulk modulus; all solids, liquids, and gases possess a bulk modulus. However, liquids and gases do not have Young’s modulus or shear modulus. Poisson’s ratio is named after the French mathematician Simeon Denis Poisson. Within the proportional limit of the material, it is the absolute value of the ratio of the transverse strain caused by uniformly distributed longitudinal stress to the corresponding longitudinal strain. For example, when a rod is stretched, its axial elongation is accompanied by transverse contraction (and vice versa), and the ratio of the transverse strain e' to the axial strain e is called Poisson’s ratio V. The Poisson’s ratio of materials is generally determined by experimental methods. It can be remembered this way: the Poisson’s ratio of air is 0, that of water is 0.5, and those for intermediate materials can be deduced. The difference between the major and minor Poisson’s ratios: The major Poisson’s ratio, PRXY, refers to the compressive (or tensile) strain in the Y direction resulting from a unit tensile [or compressive] strain in the X direction under uniaxial stress. The minor Poisson’s ratio, NUYX, represents the Poisson’s ratio in the direction orthogonal to PRXY; it is the compressive (or tensile) strain in the X direction resulting from a unit tensile [or compressive] strain in the Y direction under uniaxial stress. PRXY and NUYX are related: PRXY/NUXY = EX/EY. For orthotropic materials, it is necessary to enter the principal Poisson’s ratios separately based on the material data; however, for isotropic materials, there is no difference between using PRXY or NUYX to enter the Poisson’s ratio – either one can be used. This can be simplified as follows: Assuming under uniaxial loading: (1) A unit tensile (or compressive) strain in the X direction causes a compressive (or tensile) strain of b in the Y direction; (2) A unit tensile (or compressive) strain in the Y direction causes a compressive (or tensile) strain of a in the X direction. Then, according to Hooke’s law, σ = EX × a = EY × b → EX/EY = b/a. And since PRXY/NUXY = b/a, it follows that PRXY/NUXY = EX/EY