Young's modulus

Imagine pushing on a wall—it barely budges. Now imagine tugging a rubber band—it stretches with ease. Why do materials respond so differently to force? The answer lies in a fundamental property called Young's modulus, a crucial concept for understanding how solids deform, and ultimately, how structures stand strong or flexible objects bend without breaking. Young's modulus quantifies a material's stiffness, revealing how much it resists deformation under tension or compression. This concept is rooted in the linear elastic region where stress is directly proportional to strain, following Hooke's Law. Engineers use Young's modulus to predict how materials will behave in real-world applications, from designing bridges to creating medical implants.

Source: Wikipedia

AI Summary

Imagine pushing on a wall—it barely budges. Now imagine tugging a rubber band—it stretches with ease. Why do materials respond so differently to force? The answer lies in a fundamental property called Young's modulus, a crucial concept for understanding how solids deform, and ultimately, how structures stand strong or flexible objects bend without breaking.

What is Young's Modulus?

At its core, Young's modulus is a measure of a solid material's stiffness—specifically, how much it resists being stretched or compressed along its length. Think of it as a material's inherent reluctance to change its shape when pulled or pushed. A high Young's modulus means a very stiff material, while a low value indicates a more flexible one.

This fundamental mechanical property helps engineers and scientists predict how a material will behave under stress. It's the elastic modulus specifically for tensile (stretching) or axial compressive (squishing) loads, making it indispensable for everything from designing skyscraper beams to crafting tiny micro-components.

A Curious History

While we call it Young's modulus, named after the brilliant 19th-century British scientist Thomas Young, the concept itself has a much older lineage. The mathematical framework for what we now understand as elasticity was first developed by the Swiss polymath Leonhard Euler in 1727.

Even more surprisingly, the very first experiments to measure this property in its modern form were conducted by the Italian scientist Giordano Riccati in 1782—a full 25 years before Young's own significant contributions. So, while Young popularized and formalized its application, the underlying ideas were explored by others long before.

The Definition: Stress and Strain

Young's modulus, often represented by the letter 'E', defines the relationship between two crucial concepts: stress and strain. Stress is the force applied per unit area on a material, while strain is the resulting proportional deformation—how much the material stretches or compresses relative to its original length.

In simple terms, Young's modulus (E) is the ratio of stress (σ) to strain (ε) in the material's linear elastic region. This elegant equation is the cornerstone of understanding material stiffness.

E = \frac{\sigma}{\epsilon}

When a material is subjected to a force (F) over a certain cross-sectional area (A), that's stress. The resulting change in length (ΔL) compared to its original length (L₀) is the strain. Putting it all together, the modulus can be expressed as:

The units for Young's modulus are typically expressed in Pascals (Pa) within the International System of Units (SI). Given the immense forces involved in materials science, values are commonly seen in gigapascals (GPa)—billions of Pascals.

E = \frac{F/A}{\Delta L/L_0} = \frac{F L_0}{A \Delta L}

Hooke's Law and Linear Elasticity

When a small load is applied to a solid, it undergoes elastic deformation. This means the change is reversible—remove the load, and the material springs back to its original shape. This behavior is incredibly useful for engineers!

Within this 'linear elastic region,' the relationship between stress and strain is directly proportional, a principle known as Hooke's Law. Young's modulus is precisely this constant of proportionality. Imagine stretching a spring: the more force you apply, the more it stretches, and it returns to normal once released. That's Hookean behavior.

A higher Young's modulus signifies a material that requires more stress to achieve the same amount of strain. An idealized, perfectly rigid body, for example, would theoretically have an infinite Young's modulus, meaning it wouldn't deform at all. Conversely, a fluid, which deforms with almost no force, would have a Young's modulus approaching zero.

Real-World Applications

Young's modulus is incredibly practical for engineers. It allows them to calculate precisely how much a material will stretch or shorten under a given tensile or compressive load. This is vital for ensuring the integrity of structures like bridges, buildings, and aircraft.

For instance, it can predict how much a beam will sag under a certain weight. While Young's modulus directly applies to 'uniaxial stress'—force in one direction—other elastic properties like shear modulus (G), bulk modulus (K), and Poisson's ratio (ν) are often needed for more complex, multi-directional scenarios.

Intriguingly, any two of these elastic parameters are sufficient to fully characterize the elastic behavior of an 'isotropic' material—one whose properties are the same in all directions. Scientists are even exploring these properties in biological tissues; for example, the Young's modulus of cancerous skin tissue has been measured, offering potential new diagnostic tools.

Linear vs. Non-linear Materials

Hooke's Law, and thus a constant Young's modulus, is strictly valid only for an 'elastic and linear' response. This means that for small enough forces, most solid materials behave predictably and reversibly. However, push any material too far, and it will eventually deform permanently or even break.

Some materials, like steel or carbon fiber, maintain this linear relationship over a wide range of stresses and are considered 'linear materials' for most engineering purposes. Others, such as rubber or many soils, exhibit non-linear behavior much sooner, meaning their stiffness changes significantly as they deform.

Even 'linear' materials will become non-linear under extreme loads. You wouldn't use linear theory to describe a steel bridge collapsing, for example, because the material would have long passed its linear elastic limit. For non-linear materials, the 'tangent modulus'—the slope of the stress-strain curve at any given point—is used to describe their changing stiffness.

Directional Anisotropy

For many materials, like most metals and ceramics, Young's modulus is the same regardless of the direction the force is applied—they are 'isotropic.' But not all materials are created equal. Some, like wood or carbon fiber composites, are 'anisotropic'—their properties depend on the direction.

Imagine wood: it's far stiffer when you push or pull along the grain than across it. Similarly, carbon fiber is significantly stiffer when loaded parallel to its fibers. Engineers cleverly exploit this directional stiffness to design lighter, stronger structures, orienting materials precisely where they'll offer the most resistance to anticipated forces.

Elastic Potential Energy

When a linear elastic material is stretched or compressed, it stores energy, much like a spring. This is called elastic potential energy. The amount of energy stored per unit volume can be calculated by integrating Hooke's Law over the strain.

For a linear elastic material, this stored energy (ue) is proportional to one-half times Young's modulus (E) and the square of the strain (ε). This quadratic relationship means a small increase in deformation can lead to a significant increase in stored energy, a critical factor in understanding material failure or energy absorption.

u_e = \frac{1}{2} E \epsilon^2

It's important to remember that this formula applies specifically to linear elastic materials. In non-linear materials, Young's modulus itself changes with strain, making the energy calculation more complex and no longer a simple quadratic function.

One final note: the elasticity of coiled springs doesn't primarily come from Young's modulus. Instead, it's governed by the material's 'shear modulus,' as the spring wire itself undergoes twisting or shearing deformation, not significant changes in its length.

Factors Affecting Modulus Values

The reported values for Young's modulus are often approximate because they can vary based on several factors. Differences in a sample's exact chemical composition, manufacturing process, and even the specific test method used can influence the result. For polymers, in particular, the rate at which the material is deformed has a significant impact on its measured stiffness.

Article

Young's modulus

Young's modulus (or the Young modulus) is a mechanical property of solid materials that measures the tensile or compressive stiffness when the force is applied lengthwise. It is the elastic modulus for tension or axial compression. Young's modulus is defined as the quotient of the stress (force per unit area) applied to the object and the resulting axial strain (a dimensionless quantity that quantifies relative deformation) in the linear elastic region of the material. As such, Young's modulus is similar to and proportional to the spring constant in Hooke's law, but with dimensions of pressure instead of force per distance.

Although Young's modulus is named after the 19th-century British scientist Thomas Young, the concept was developed in 1727 by Leonhard Euler. The first experiments that used the concept of Young's modulus in its modern form were performed by the Italian scientist Giordano Riccati in 1782, pre-dating Young's work by 25 years. The term modulus is derived from the Latin root term modus, which means measure.

Definition

Young's modulus

Young's modulus, ${\displaystyle E}$, quantifies the relationship between tensile or compressive stress ${\displaystyle \sigma }$ (force per unit area) and axial strain ${\displaystyle \varepsilon }$ (proportional deformation) in the linear elastic region of a material: ${\displaystyle E={\frac {\sigma }{\varepsilon }}}$Young's modulus is commonly measured in the International System of Units (SI) in multiples of the pascal (Pa) and common values are in the range of gigapascals (GPa).

• Rubber (increasing pressure: large length increase, meaning low ${\displaystyle E}$) • Aluminium (increasing pressure: small length increase, meaning high ${\displaystyle E}$)

Linear elasticity

A solid material undergoes elastic deformation when a small load is applied to it in compression or extension. Elastic deformation is reversible, meaning that the material returns to its original shape after the load is removed.

At near-zero stress and strain, the stress–strain curve is linear, and the relationship between stress and strain is described by Hooke's law that states stress is proportional to strain. The coefficient of proportionality is Young's modulus. The higher the modulus, the more stress is needed to create the same amount of strain; an idealized rigid body would have an infinite Young's modulus. Conversely, a very soft material (such as a fluid) would deform without force, and would have zero Young's modulus.

Related but distinct properties

Material stiffness is a distinct property from the following:

• Strength: maximum amount of stress that material can withstand while staying in the elastic (reversible) deformation regime; • Geometric stiffness: a global characteristic of the body that depends on its shape, and not only on the local properties of the material; for instance, an I-beam has a higher bending stiffness than a rod of the same material for a given mass per length; • Hardness: relative resistance of the material's surface to penetration by a harder body; • Toughness: amount of energy that a material can absorb before fracture. • The point E is the elastic limit or the yield point of the material within which the stress is proportional to strain and the material regains its original shape after removal of the external force.

Usage

Young's modulus

Young's modulus enables the calculation of the change in the dimension of a bar made of an isotropic elastic material under tensile or compressive loads. For instance, it predicts how much a material sample extends under tension or shortens under compression. The Young's modulus directly applies to cases of uniaxial stress; that is, tensile or compressive stress in one direction and no stress in the other directions. Young's modulus is also used in order to predict the deflection that will occur in a statically determinate beam when a load is applied at a point in between the beam's supports.

Other elastic calculations usually require the use of one additional elastic property, such as the shear modulus ${\displaystyle G}$, bulk modulus ${\displaystyle K}$, and Poisson's ratio ${\displaystyle \nu }$. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. For example, calculating physical properties of cancerous skin tissue, has been measured and found to be a Poisson's ratio of 0.43±0.12 and an average Young's modulus of 52 KPa. Defining the elastic properties of skin may become the first step in turning elasticity into a clinical tool. For homogeneous isotropic materials simple relations exist between elastic constants that allow calculating them all as long as two are known: ${\displaystyle {\begin{aligned}E&=G(2+2\nu )&&=K(3-6\nu )&&={\frac {9KG}{3K+G}}\\G&={\frac {E}{2+2\nu }}&&={\frac {K(3-6\nu )}{2+2\nu }}&&={\frac {E}{3-{\frac {E}{3K}}}}\\K&={\frac {G(2+2\nu )}{3-6\nu }}&&={\frac {E}{3-6\nu }}&&={\frac {EG}{9G-3E}}\\\nu &={\frac {E}{2G}}-1&&={\frac {1}{2}}-{\frac {E}{6K}}&&={\frac {3K-2G}{2(3K+G)}}\\\end{aligned}}}$

Linear versus non-linear

Young's modulus represents the factor of proportionality in Hooke's law, which relates the stress and the strain. However, Hooke's law is only valid under the assumption of an elastic and linear response. Any real material will eventually fail and break when stretched over a very large distance or with a very large force; however, all solid materials exhibit nearly Hookean behavior for small enough strains or stresses. If the range over which Hooke's law is valid is large enough compared to the typical stress that one expects to apply to the material, the material is said to be linear. Otherwise (if the typical stress one would apply is outside the linear range), the material is said to be non-linear.

Steel, carbon fiber and glass among others are usually considered linear materials, while other materials such as rubber and soils are non-linear. However, this is not an absolute classification: if very small stresses or strains are applied to a non-linear material, the response will be linear, but if very high stress or strain is applied to a linear material, the linear theory will not be enough. For example, as the linear theory implies reversibility, it would be absurd to use the linear theory to describe the failure of a steel bridge under a high load; although steel is a linear material for most applications, it is not in such a case of catastrophic failure.

In solid mechanics, the slope of the stress–strain curve at any point is called the tangent modulus. It can be experimentally determined from the slope of a stress–strain curve created during tensile tests conducted on a sample of the material.

Directional materials

Young's modulus is not always the same in all orientations of a material. Most metals and ceramics, along with many other materials, are isotropic, and their mechanical properties are the same in all orientations. However, metals and ceramics can be treated with certain impurities, and metals can be mechanically worked to make their grain structures directional. These materials then become anisotropic, and Young's modulus will change depending on the direction of the force vector. Anisotropy can be seen in many composites as well. For example, carbon fiber has a much higher Young's modulus (is much stiffer) when force is loaded parallel to the fibers (along the grain). Other such materials include wood and reinforced concrete. Engineers can use this directional phenomenon to their advantage in creating structures.

Temperature dependence

The Young's modulus of metals varies with the temperature and can be realized through the change in the interatomic bonding of the atoms, and hence its change is found to be dependent on the change in the work function of the metal. Although classically, this change is predicted through fitting and without a clear underlying mechanism (for example, the Watchman's formula), the Rahemi-Li model demonstrates how the change in the electron work function leads to change in the Young's modulus of metals and predicts this variation with calculable parameters, using the generalization of the Lennard-Jones potential to solids. In general, as the temperature increases, the Young's modulus decreases via ${\displaystyle E(T)=\beta (\varphi (T))^{6}}$ where the electron work function varies with the temperature as ${\displaystyle \varphi (T)=\varphi {0}-\gamma {\frac {(k{B}T)^{2}}{\varphi {0}}}}$ and ${\displaystyle \gamma }$ is a calculable material property which is dependent on the crystal structure (for example, BCC, FCC). ${\displaystyle \varphi {0}}$ is the electron work function at T=0 and ${\displaystyle \beta }$ is constant throughout the change. Additionally, the Rahemi–Li model links the temperature dependence of Young’s modulus in metals to changes in the electron work function, providing a theoretical basis using a generalized Lennard-Jones potential.

Calculation

Young's modulus

Young's modulus is calculated by dividing the tensile stress, ${\displaystyle \sigma (\varepsilon )}$, by the engineering extensional strain, ${\displaystyle \varepsilon }$, in the elastic (initial, linear) portion of the physical stress–strain curve:

${\displaystyle E\equiv {\frac {\sigma (\varepsilon )}{\varepsilon }}={\frac {F/A}{\Delta L/L{0}}}={\frac {FL{0}}{A\,\Delta L}}}$ where

• ${\displaystyle E}$ is the Young's modulus (modulus of elasticity); • ${\displaystyle F}$ is the force exerted on an object under tension; • ${\displaystyle A}$ is the actual cross-sectional area, which equals the area of the cross-section perpendicular to the applied force; • ${\displaystyle \Delta L}$ is the amount by which the length of the object changes (${\displaystyle \Delta L}$ is positive if the material is stretched, and negative when the material is compressed); • ${\displaystyle L{0}}$ is the original length of the object.

An equivalent definition of Young's modulus is ${\displaystyle E={\frac {L{0}}{A}}\left.{\frac {d^{2}U}{dL^{2}}}\right|{L{0}}}$ where ${\displaystyle U}$ is material's energy.

Force exerted by stretched or contracted material

Young's modulus of a material can be used to calculate the force it exerts under specific strain.

${\displaystyle F={\frac {EA\,\Delta L}{L{0}}}}$

where ${\displaystyle F}$ is the force exerted by the material when contracted or stretched by ${\displaystyle \Delta L}$.

Hooke's law for a stretched wire can be derived from this formula:

${\displaystyle F=\left({\frac {EA}{L{0}}}\right)\,\Delta L=kx}$

${\displaystyle k\equiv {\frac {EA}{L{0}}}\,}$ and ${\displaystyle x\equiv \Delta L.}$

Note that the elasticity of coiled springs comes from shear modulus, not Young's modulus. When a spring is stretched, its wire's length doesn't change, but its shape does. This is why only the shear modulus of elasticity is involved in the stretching of a spring.

Elastic potential energy

The elastic potential energy stored in a linear elastic material is given by the integral of the Hooke's law:

${\displaystyle U{e}=\int {kx}\,dx={\frac {1}{2}}kx^{2}.}$

${\displaystyle U{e}=\int {\frac {EA\,\Delta L}{L{0}}}\,d\Delta L={\frac {EA}{L{0}}}\int \Delta L\,d\Delta L={\frac {EA\,{\Delta L}^{2}}{2L{0}}}}$

This means that the elastic potential energy density (that is, per unit volume) is given by:

${\displaystyle {\frac {U{e}}{AL{0}}}={\frac {E\,{\Delta L}^{2}}{2L{0}^{2}}}={\frac {1}{2}}\times {\frac {E\,{\Delta L}}{L{0}}}\times {\frac {\Delta L}{L{0}}}={\frac {1}{2}}\times \sigma (\varepsilon )\times \varepsilon }$

or, in simple notation, for a linear elastic material: ${\textstyle u{e}(\varepsilon )=\int {E\,\varepsilon }\,d\varepsilon ={\frac {1}{2}}E{\varepsilon }^{2}}$, since the strain is defined ${\textstyle \varepsilon \equiv {\frac {\Delta L}{L{0}}}}$.

In a nonlinear elastic material the Young's modulus is a function of the strain, so the second equivalence no longer holds, and the elastic energy is not a quadratic function of the strain:

${\displaystyle u{e}(\varepsilon )=\int E(\varepsilon )\,\varepsilon \,d\varepsilon \neq {\frac {1}{2}}E\varepsilon ^{2}}$

Examples

Young's modulus

Influences of selected glass component additions on Young's modulus of a specific base glass

Young's modulus can vary somewhat due to differences in sample composition and test method. The rate of deformation has the greatest impact on the data collected, especially in polymers. The values here are approximate and only meant for relative comparison.

<table><thead><tr><th>Material</th><th>Young's modulus</th><th>Ref.</th></tr></thead><tbody><tr><td>GPa</td><td>psi</td></tr><tr><td>Aluminium (13Al)</td><td>68</td><td>9.9×106</td><td></td></tr><tr><td>Amino-acid molecular crystals</td><td>21–44</td><td>3.0×106–6.4×106</td><td></td></tr><tr><td>Aramid (for example, Kevlar)</td><td>70.5–112.4</td><td>10.23×106–16.30×106</td><td></td></tr><tr><td>Aromatic peptide-nanospheres</td><td>230–275</td><td>33.4×106–39.9×106</td><td></td></tr><tr><td>Aromatic peptide-nanotubes</td><td>19–27</td><td>2.8×106–3.9×106</td><td></td></tr><tr><td>Bacteriophage capsids</td><td>1–3</td><td>0.15×106–0.44×106</td><td></td></tr><tr><td>Beryllium (4Be)</td><td>287</td><td>41.6×106</td><td></td></tr><tr><td>Bone, human cortical</td><td>14</td><td>2.0×106</td><td></td></tr><tr><td>Brass</td><td>106</td><td>15.4×106</td><td></td></tr><tr><td>Bronze</td><td>112</td><td>16.2×106</td><td></td></tr><tr><td>Carbon nitride (CN2)</td><td>822</td><td>119.2×106</td><td></td></tr><tr><td>Carbon-fiber-reinforced plastic (CFRP), 50/50 fibre/matrix, biaxial fabric</td><td>30–50</td><td>4.4×106–7.3×106</td><td></td></tr><tr><td>Carbon-fiber-reinforced plastic (CFRP), 70/30 fibre/matrix, unidirectional, along fibre</td><td>181</td><td>26.3×106</td><td></td></tr><tr><td>Cobalt-chrome (CoCr)</td><td>230</td><td>33×106</td><td></td></tr><tr><td>Copper (Cu), annealed</td><td>110</td><td>16×106</td><td></td></tr><tr><td>Diamond (C), synthetic</td><td>1,050–1,210</td><td>152×106–175×106</td><td></td></tr><tr><td>Diatom frustules, largely silicic acid</td><td>0.35–2.77</td><td>0.051×106–0.402×106</td><td></td></tr><tr><td>Flax fiber</td><td>58</td><td>8.4×106</td><td></td></tr><tr><td>Float glass</td><td>47.7–83.6</td><td>6.92×106–12.13×106</td><td></td></tr><tr><td>Glass-reinforced polyester (GRP)</td><td>17.2</td><td>2.49×106</td><td></td></tr><tr><td>Gold</td><td>77.2</td><td>11.20×106</td><td></td></tr><tr><td>Graphene</td><td>1,050</td><td>152×106</td><td></td></tr><tr><td>Hemp fiber</td><td>35</td><td>5.1×106</td><td></td></tr><tr><td>High-density polyethylene (HDPE)</td><td>0.97–1.38</td><td>0.141×106–0.200×106</td><td></td></tr><tr><td>High-strength concrete</td><td>30</td><td>4.4×106</td><td></td></tr><tr><td>Lead (82Pb), chemical</td><td>13</td><td>1.9×106</td><td></td></tr><tr><td>Low-density polyethylene (LDPE), molded</td><td>0.228</td><td>0.0331×106</td><td></td></tr><tr><td>Magnesium alloy</td><td>45.2</td><td>6.56×106</td><td></td></tr><tr><td>Medium-density fiberboard (MDF)</td><td>4</td><td>0.58×106</td><td></td></tr><tr><td>Molybdenum (Mo), annealed</td><td>330</td><td>48×106</td><td></td></tr><tr><td>Monel</td><td>180</td><td>26×106</td><td></td></tr><tr><td>Mother-of-pearl (largely calcium carbonate)</td><td>70</td><td>10×106</td><td></td></tr><tr><td>Nickel (28Ni), commercial</td><td>200</td><td>29×106</td><td></td></tr><tr><td>Nylon 66</td><td>2.93</td><td>0.425×106</td><td></td></tr><tr><td>Osmium (76Os)</td><td>525–562</td><td>76.1×106–81.5×106</td><td></td></tr><tr><td>Osmium nitride (OsN2)</td><td>194.99–396.44</td><td>28.281×106–57.499×106</td><td></td></tr><tr><td>Polycarbonate (PC)</td><td>2.2</td><td>0.32×106</td><td></td></tr><tr><td>Polyethylene terephthalate (PET), unreinforced</td><td>3.14</td><td>0.455×106</td><td></td></tr><tr><td>Polypropylene (PP), molded</td><td>1.68</td><td>0.244×106</td><td></td></tr><tr><td>Polystyrene, crystal</td><td>2.5–3.5</td><td>0.36×106–0.51×106</td><td></td></tr><tr><td>Polystyrene, foam</td><td>0.0025–0.007</td><td>0.00036×106–0.00102×106</td><td></td></tr><tr><td>Polytetrafluoroethylene (PTFE), molded</td><td>0.564</td><td>0.0818×106</td><td></td></tr><tr><td>Rubber, small strain</td><td>0.01–0.1</td><td>0.0015×106–0.0145×106</td><td></td></tr><tr><td>Silicon, single crystal, different directions</td><td>130–185</td><td>18.9×106–26.8×106</td><td></td></tr><tr><td>Silicon carbide (SiC)</td><td>90–137</td><td>13.1×106–19.9×106</td><td></td></tr><tr><td>Single-walled carbon nanotube</td><td>1,000</td><td>150×106</td><td></td></tr><tr><td>Steel, A36</td><td>200</td><td>29×106</td><td></td></tr><tr><td>Stinging nettle fiber</td><td>87</td><td>12.6×106</td><td></td></tr><tr><td>Titanium (22Ti)</td><td>116</td><td>16.8×106</td><td></td></tr><tr><td>Titanium alloy, Grade 5</td><td>114</td><td>16.5×106</td><td></td></tr><tr><td>Tooth enamel, largely calcium phosphate</td><td>83</td><td>12.0×106</td><td></td></tr><tr><td>Tungsten carbide (WC)</td><td>600–686</td><td>87.0×106–99.5×106</td><td></td></tr><tr><td>Wood, American beech</td><td>9.5–11.9</td><td>1.38×106–1.73×106</td><td></td></tr><tr><td>Wood, black cherry</td><td>9–10.3</td><td>1.31×106–1.49×106</td><td></td></tr><tr><td>Wood, red maple</td><td>9.6–11.3</td><td>1.39×106–1.64×106</td><td></td></tr><tr><td>Wrought iron</td><td>193</td><td>28.0×106</td><td></td></tr><tr><td>Yttrium iron garnet (YIG), polycrystalline</td><td>193</td><td>28.0×106</td><td></td></tr><tr><td>Yttrium iron garnet (YIG), single-crystal</td><td>200</td><td>29×106</td><td></td></tr><tr><td>Zinc (30Zn)</td><td>108</td><td>15.7×106</td><td></td></tr><tr><td>Zirconium (40Zr), commercial</td><td>95</td><td>13.8×106</td><td></td></tr></tbody></table>