graph of young's modulus experiment

Young's Modulus

This page discusses Young's Modulus, best described as the stiffness of a material, examples/problems involving this concept, and its numerous real-world applications. (Claimed by Sanjana Kapur; Fall 2017)

• 1.1 A Mathematical Model
• 1.2 A Computational Model
• 2.2 Middling
• 2.3 Difficult
• 3 Connectedness
• 5.1 Further reading
• 5.2 External links
• 6 References

The Main Idea

Young's Modulus is a macroscopic property of a material that measures the stiffness of a solid material. It is independent of size or weight of the material, and it will change depending on the type of material for which the Young's Modulus is being measured. The uses of Young's modulus extend to two main sets of relationships, macroscopic springs and microscopic springs. In the case of macroscopic springs, Young's modulus is a measure of the stretchiness of a solid material outside of considerations of size and shape. In microscopic strings, when a solid object is modeled as a system of balls (atoms) connected by springs (an image of which can be seen as the cover image of the physics textbook Matter and Interactions I 3rd Edition), the Young's modulus constant of the material can be used to determine the 'interatomic spring stiffness' constant Ksi of a material in order to determine the stiffness and stretchiness of interatomic bonds. The SI unit of Young's Modulus is the pascal (Pa or N/m^2 or kg. m. s^−2).

A Mathematical Model

Mathematically, the Young's Modulus is the ratio of the stress placed upon a material to the strain that it endures. The definition of Young's Modulus can be expressed as: [math]\displaystyle{ {Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}} }[/math] where [math]\displaystyle{ {F}_{T} }[/math] is equal to the tension force, [math]\displaystyle{ A }[/math] is equal to the cross sectional area, [math]\displaystyle{ ΔL }[/math] is equal to the change in length due to the tension force, and [math]\displaystyle{ L_o }[/math] is equal to the initial length of the material.

Young's Modulus can also be expressed as: [math]\displaystyle{ {Y = \frac{Ksi}{d}} }[/math] where Y is the Young's modulus of the material being examined, Ksi is the value that represents the interatomic bond stiffness (how much a bond between two atoms will stretch), and d is the distance from the center of one atom to the center of another atom.

A Computational Model

• The following program calculates the microscopic and macroscopic Young's Modulus of a material. The source of this model is: https://trinket.io/python/8f0b71cda5

Example 1. A cylinder of wood has a stress of 800 and a strain of [math]\displaystyle{ 8*10^-7 }[/math] . What is Young's modulus for wood?

First we lay out the equation for the problem:

[math]\displaystyle{ Y = \frac{stress}{strain} }[/math]

We then plug in using the numbers given to us.

[math]\displaystyle{ Y = \frac{800}{8*10^-7} }[/math]

and so Y = 10^9 N/m^2

The Young's Modulus of Collagen in Bone is about 6 GPa [1] . Given this, determine the force applied to a cylindrical segment of collagen of radius 1 cm and length 0.75 m to cause it to deform (stretch) 1 mm.

First, consider the equation of Young's modulus most useful for this problem:

[math]\displaystyle{ {Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}} }[/math]

Substitute the appropriate values:

[math]\displaystyle{ 6000000000 = \frac{\frac{{F}_{T}}{3.14*0.01^2}}{\frac{{0.001}}{0.75}} }[/math]

Solve for F:

[math]\displaystyle{ {F = 2.55*10^{10} Newtons} }[/math]

Does this seem reasonable? Yes! Collagen that makes up bone is a very hard material and does not stretch easily. Making bone stretch 1 mm longitudinally (using the bone as a vertical spring system) will take a lot of force.

A flan created by Dr. Schatz has a strawberry placed on it, stretching the flan from a length of 0.15 m to 0.2 m. The flan has a cross sectional area of .01. With the knowledge that flan has a Young’s modulus of ~ 1.6e4 in tension, what force was used to stretch the flan?

First we would write out the equation for Youngs modulus: [math]\displaystyle{ {Y = \frac{stress}{strain}} = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}} }[/math]

Next we would fill in the equation for Young's Modulus: [math]\displaystyle{ {1.6e4 = \frac{\frac{{F}_{T}}{.01}}{\frac{{.05}}{.15}}} }[/math]

We can then fill in every thing that we know: [math]\displaystyle{ {16000 = 300*F_T} }[/math]

And so, [math]\displaystyle{ {F_T} }[/math] is equal to 53.3!

A man of weight 100 kg gets onto a bungee jump ride at a carnival. He is suspended in air by one rubber band (Young's Modulus of 0.01 GPa [2] of diameter 5 cm and original length of 10 m. Calculate the stretch of the band when the man reaches the bottom of the ride.

First, use the appropriate Young's Modulus equation:

[math]\displaystyle{ Y = \frac{\frac{{F}_{T}}{A}}{\frac{{ΔL}}{L_o}} }[/math]

Then, substitute the appropriate values:

[math]\displaystyle{ {1e7 = \frac{{\frac{100*9.8}{3.14*0.05^2}}}{{\frac{ΔL}{10}}}} }[/math]

[math]\displaystyle{ {ΔL=.125 m} }[/math]

A new cylandrical flan is created by Dr. Schatz which has an orange placed on it which has a mass of .15 kg, compressing the flan a certain amount. Knowing that the initial length of the flan was .15 m and the flan has a diameter of .2 meters. With the knowledge that flan has a Young’s modulus of ~ 2e5 in tension, what length is the flan now?

First we need to write down the equation:

Next, let's solve for the unknowns: Cross sectional area:

[math]\displaystyle{ {A} = pi*r^2 }[/math] [math]\displaystyle{ {A} = pi*.1^2 }[/math] [math]\displaystyle{ {A} = .0314 meters^2 }[/math]

Force of the orange:

[math]\displaystyle{ {F}_{T} = m*g }[/math] [math]\displaystyle{ {F}_{T} = .15*-9.8 }[/math] [math]\displaystyle{ {F}_{T} = -1.47 Newtons }[/math]

Next we would write out the equation for Youngs modulus: [math]\displaystyle{ {2e5 = \frac{\frac{-1.47}{.0314}}{\frac{{ΔL}}{.15}}} }[/math]

We can solve the equation for : [math]\displaystyle{ {-7/ΔL = 200000} }[/math]

And so, [math]\displaystyle{ {ΔL } }[/math] is equal to -3.5e-5! It was compressed 3.5e-5 meters!

Using the Young's modulus of Tungsten ( [math]\displaystyle{ 4*10^11 }[/math] [3] ) determine the interatomic "spring" stiffness of Tungsten.

First, find the center-to-center distance between 2 tungsten atoms.

[math]\displaystyle{ \frac{\frac{184 grams/mole}{6.022*10^23 atoms/mol}}{d^3}=19.25 g/cm^3 }[/math] where d is the interatomic distance between Tungsten atoms. (19.25 g/cm^3 is the density of Tungsten [4] ).

Solving, we get:

[math]\displaystyle{ d = 2.51e-8 }[/math]

Using the appropriate formula:

[math]\displaystyle{ {Y = \frac{Ksi}{d}} }[/math]

[math]\displaystyle{ {4*10^11 = \frac{Ksi}{2.51e-8}} }[/math]

[math]\displaystyle{ {Ksi = 10040 N/m} }[/math]

Connectedness

• 1. How is this topic connected to something that you are interested in?

I am interested in engineering, and Young’s Modulus is directly relevant to this field. Young’s Modulus is used to calculate the stiffness of materials, which is useful in structural engineering applications. In high school, I partook in scientific competitions and had to construct certain structures with a partner in a time limit. The stiffness of different materials available determined whether or not they would adequately hold up other parts of the structure and allow the entire structure to remain intact. Therefore, the Young’s Modulus was directly relevant to my interests.

• 2. How is it connected to your major?

I am majoring in Computer Science. If I worked for an architectural firm, I could write code to calculate the Young’s Modulus of different building materials that the firm wanted to use to construct its buildings. The code could also determine whether or not the material was safe to use for a specific building design based on the value of the Young’s Modulus.

• 3. Is there an interesting industrial application?
• a. Engineers and architects use the Young’s Modulus to determine whether or not the material that they are using to construct a particular structure can withstand a certain amount of force (based on the structure they are building, i.e. a bridge vs. a skyscraper).
• b. Scientists (including physicists) use Young’s Modulus to see how strong the materials that they plan to use in experiments are. (This is especially important in experiments in which a lot of pressure or force is applied to the material in question.)
• c. Doctors and scientists, such as biomedical engineers, need to use Young’s Modulus to determine whether certain materials are appropriate to use to construct prosthetics for people (i.e. how rigid or flexible the materials will be).
• Young's modulus is connected to all solid material, and it highlights the slight impression given to everything showing that things do push down ever so slightly on stuff that seems stationary.
• Young's modulus is used all the time in civil engineering and it is often used to help determine structural integrity of certain materials when deciding on a building.
• Young's modulus has several biomedical applications in prosthetics and in human disease. It is used to determine the structural characteristics of prosthetic material used for implants. Young's modulus of tissues changes with aging and is being studied as a factor to evaluate mortality related to vascular stiffness from aging.

Young's Modulus was first developed in 1727 by the famous Leonhard Euler in Switzerland, but it was further expanded upon by Italian scientist Giordano Riccati in 1782. Finally, it was given a name by the British Scientist Thomas Young who finished work on it in the 1800s. It is used in order to discover the elasticity of solid materials and shows the stress per strain of a solid material.

Young’s Modulus is named for Thomas Young (1773-1829), an Englishman who worked as a medical practitioner. Young conducted a great deal of research on the workings of the human eye, writing papers such as “The Mechanism of the Eye” and “On the Theory of Light and Colors,” which laid the foundations for what we now know about how the eye works and processes light that it receives from its surroundings. He also translated a solid portion of the demotic script of the Rosetta Stone and found a connection between the script and Egyptian hieroglyphics. He later created a dictionary filled with hieroglyphic vocabulary.

He made his famous discovery of the Young’s Modulus while working as a lecturer at the Royal Institution. He explained that the stiffness of any elasticity could be expressed as a modulus of its elasticity. He correctly calculated the stress distribution upon a bar, leading to the discovery of the Young’s Modulus formula (which is “stress” divided by “strain”). His newly discovered equation did have practical applications, as London engineers (i.e. one company was Chapman and Buhagiar) used it to construct steel columns and other building designs.

Leonhard Euler

Thomas Young

Further reading

The VERY in-depth wiki page which goes for beyond applications in physics 1. [5]

That resource is especially useful if you are pursuing a degree in Materials Science and Engineering (MSE) or any other related fields.

External links

HyperPhysics [6] , which is a great tool for just about any entry-level physics curriculum.

History: [8]

Young's Modulus Tables = [9] [10]

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Young's modulus, or elastic modulus, describes the ability of materials to resist changes in length when they undergo tension or compression. This modulus  is very useful in engineering as it provides details about the elastic properties of materials , such as their tensile strength and stiffness.

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What is Young’s modulus?

Do all materials have the same Young’s modulus?

Does Young’s modulus depend on the area of a material?

How can we calculate Young’s modulus using a graph?

Is Young’s modulus the slope of a stress - strain graph?

What experiment can be used to estimate Young’s modulus?

What is the formula to find Young’s modulus experimentally?

What is the elastic region?

What is the plastic region?

What is the elastic point or yield point?

What is maximum tensile strength?

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Defining Young's modulus

The Young's modulus is equal to the longitudinal stress being applied divided by the strain. The stress and strain of an object undergoing tension can be also be expressed as follows: w hen a metal object is pulled by a force F at each end, the object will be stretched from the original length L 0 to a new stretched length L n .

Since the object is stretched the cross-sectional area will be decreased. The stress can be expressed as the tensile force applied (F) measured in newton (N) , divided by the cross-sectional area (A) measured in m 2 , as seen in the equation below. The resulting unit of stress is N/m 2 .

\[\sigma[N/m^2] = \frac{F}{A}\]

The strain, also known as relative deformation, is the change in length caused by tension or compression divided by the original length L 0 . Strain is dimensionless, as both terms of the fraction are measured in metres and can be calculated from the following equation.

\[\varepsilon = \frac{L_n[m]-L_0}{L_0[m]} = \frac{\Delta L}{L_0}\]

Young's modulus formula

The elastic modulus E can be expressed as the stress divided by the strain as shown in the formula below.

\[\text{Young's modulus }[N/m^2] = \frac{\text{stress}}{\text{strain}} = \frac{\sigma}{\varepsilon} = \frac{\frac{F}{A}}{\frac{\Delta L}{L_0}} = \frac{FL_0}{A \Delta L}\]

The units of Young's modulus units are the same as the stress, N/m 2 which is equivalent to Pa (pascal). S ince the elastic modulus is usually a very large number of the magnitude of 10 9 it is often expressed in Giga Pascals, shown as GPa .

\[\text{Young's modulus} = \frac{FL_0}{A \Delta L} = \frac{F[N] L_0[m]}{A[m^2]\Delta L[m]} = \frac{N}{m^2} = Pa\]

A metal bar experiences a load of 60 N applied at the end. The metal bar has a cylindrical shape and cross-sectional area of 0.02 mm 2 . The length of the bar increases by 0.30 percent. Find the elastic modulus of the bar.

Since Young's modulus is required, we need to find the stress and the strain first (remember, Young's modulus is the ratio of stress over strain). We apply the stress formula to find stress.

\[\sigma = \frac{F}{A} = \frac{60 N}{0.02 \cdot 10^{-6} m^2} = 3 \cdot 10^9 Pa\]

Then we apply the strain formula to find the strain:

\[\varepsilon = \frac{L_n - L_0}{L_0} = \frac{\Delta L}{L_0} = 0.3 \% = 0.003\]

Then finally we divide stress over strain to find Young's modulus.

\[\text{Young's modulus} = \frac{\text{stress}}{\text{strain}} = \frac{3 \cdot 10^9}{0.003} = 1 \cdot 10^{12} Pa\]

How is Young's modulus related to Hooke's Law?

Hooke's law states that the force acting on a body or spring creating a displacement Δx, is linear to the displacement created as shown by the equation below, where k is a constant relating to the spring's stiffness. Hooke's law can be applied to situations where a body deforms elastically.

\[F = k \Delta x\]

Similar to Hooke's law, where the extension or compression of a spring is linearly proportional to the force applied; the stress that is applied on a body is linearly proportional to Young's modulus E, as shown in the rearranged equation below.

\[\sigma = E \varepsilon\]

Graphically calculate Young's modulus

Since the elastic modulus of a material is linearly proportional to the stress applied divided by the strain, Young's modulus can also be calculated from a stress-strain graph, which describes a linear deformation as laid out by Hooke's law. Young's modulus is equal to the slope of the linear region of the stress vs strain curve, as shown in the figure below.

Experiments to show Young's modulus

In order to measure Young's modulus of a metal, several experiments can be constructed. Varying loads will be applied to a copper wire - the resulting extension will be measured to construct a stress-strain graph. According to the stress and strain equations, the required parameters will be measured with the following equipment.

Metre ruler

Wooden block

Methodology

Using the ruler, we measure the initial length of the wire. We use the micrometre to measure the diameter of the wire at three points along its length. This is for the average diameter to be used in the area calculation.

We connect one end of the wire to the pulley that is clamped to a bench, and the other end to a clamped wooden block.

The extension of the wire due to the force of the weights is measured and recorded. The difference between the new length and the original length prior to extension is used in the strain calculation.

Repeat the process to get 5 to 10 more readings. Various measurements with various weights are recommended to be taken to reduce errors.

Then plot stress vs strain and find the elastic modulus by taking the gradient of the line.

Results analysis

The aim of the experiment is to estimate Young's modulus. This can be found by estimating stress and strain. The following steps are required to find stress and strain.

Find the area of the wire using the measured diameter and the equation \(A= \pi r^2\) where \(r=\frac{R}{2}\).

Rearrange Young's modulus formula and solve it for F. This will give us \(F=\frac{(E\cdot A)}{L} \cdot \Delta L\).

The rearranged equation is similar to the equation of a line of the form y = ax, where y is F and the slope is the coefficient of ΔL.

A graph of Force vs ΔL is constructed from the recorded force and extension points so that the slope can be found. The slope \(\frac{\Delta F}{\Delta L}\) is found and multiplied by the original length L 0 and divided to area A, to estimate the value of Young's modulus.

Stress-strain graph and material characteristics

Some important characteristics of materials are shown in the figure below.

The red region indicates the elastic region where it deforms according to Hooke's law, and stress and strain are proportional to each other. The red point indicates the elastic limit or yield point. This is the point to which a material can still hold its original length after the load is applied.

The green region indicates the plastic region where the material is unable to return to its initial state and has undergone permanent deformation. The green point indicates the tensile strength point, until which the material can withstand the maximum load per unit without breaking.

The blue point indicates the breaking point or breaking stress, where the material will break.

Young's Modulus - Key takeaways

Young's modulus is a material's ability to resist change in length under tension or compression.

Young's modulus can be calculated graphically using a stress-strain graph.

Experimental calculation of Young's modulus is possible by constructing a load-length difference.

Using the stress and strain graph, a material's tensile strength and breaking point can be found.

Flashcards in Young's Modulus 13

It is a material's ability to retain its length under tension or compression.

By constructing a stress vs strain graph and using the slope.

Applying loads on a metal wire using a pulley, and measuring its elongation.

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Frequently Asked Questions about Young's Modulus

How do you determine Young's modulus experimentally?

You can determine Young's modulus experimentally by applying a load to a material, then creating a load vs change in length graph, and calculating the slope of the graph under study.

What is Young's modulus?

Young’s modulus is the ability of materials to retain their original length/shape under load.

Why does aluminium have a lower Young's modulus than steel?

Because steel is stiffer than aluminium, hence it will be more likely to retain its shape under load.

What does a higher Young's modulus indicate? 

A higher Young's modulus indicates that the material is stiffer and requires higher force to be deformed. 

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Young’s Modulus

Introduction

Young’s modulus is a numerical constant, named after the 18th-century English physician and physicist Thomas Young. Young’s modulus is a measure of the ability of a material to withstand changes in length under lengthwise tension or compression. Young’s modulus is also termed the modulus of elasticity.

A. Young’s modulus

Most materials under small strain obey Hooke’s law. Under this circumstance, the ratio between stress and strain is constant. This quantity is called Young’s modulus (E). Young’s modulus measures the resistance of a material to elastic deformation.

B. What Young’s Modulus Means

Young’s modulus describes the stiffness of a material. In simpler terms, it states how easy it is to bend or stretch the material. Young’s modulus of a material is its fundamental property that remains unchanged. But, it is dependent on material temperature. The stress-strain graph for two materials is illustrated in Figure 1 . Material A is stiffer than material B, as a higher stress is needed to produce the same amount of strain. So, Material A will have higher Young’s modulus than Material B.

Figure 1: Young’s Modulus Comparison

C. How to Calculate Young’s Modulus

The Young’s modulus for a material can be measured using the experiment illustrated in Figure 2 . The reference wire and test wire made of the selected material are hung from the ceiling. The reference wire supports a vernier scale, which will measure the extension of the test wire. The slotted masses can be used to vary the force acting on the test wire.

Figure 2: Experiment Set-up

D. Applications of Young’s Modulus

Young’s modulus is very important for doctors and scientists. This parameter can help them determine when a structural implant will deform. Hence, they can design a piece mechanically for use in the body. The following table shows Young’s modulus for different values and applications of the materials.

An Example of Application

A steel wire of diameter 4.0 mm (Area = 12.5 x 10^(-6) m^2) and length 30 m is suspended from an overhead crane and a load of 1500 N is suspended from its free end. The Young’s modulus of the material of the wire is 210 x 10^9 N. It is assumed that the proportionality limit of the wire is not exceeded. The stress, strain and the extension of the wire used in the overhead crane are calculated.

Young’s modulus is an important material property because of its relationship with stress and strain. Young’s modulus for almost all materials is established. In several engineering design calculations, Young’s modulus is used to find either thickness of the material to withstand a given load or stress level on the material for a given load.

The Young modulus may be measured for a material in the form of a wire using the apparatus shown in Figure 1. Two identical wires are hung from a beam; a scale is fixed to one wire and a mass hung on the end to remove kinks in it. This wire is used as a reference standard. The other wire has a small load placed on it to straighten it and a vernier scale which links with the scale on the reference wire. The original length (L) of the test wire is measured and its diameter is found for various points along its length and an average diameter calculated. Hence its mean radius r can be found. Loads are then placed gently on the wire and the extension of the wire found for each one. They should not be dropped, as this would subject the wire to a sudden shock. After each reading the load should be removed to check that the wire returns to its original length, showing that its elastic limit has not been exceeded. A graph is plotted of stress against strain and from this the value of the Young modulus may be found (this is the gradient of the line i.e. F/A divided by e/L). The wires should be long and thin to give as large an extension as possible for a given load while retaining its elastic properties. Two wires are used to eliminate errors due to changes of temperature and sagging of the beam.

Elasticity and Young’s Modulus (Theory, Examples, and Table of Values)

Young’s modulus–the most common type of elastic modulus, seems to be the most important material property for mechanical engineers. It’s pretty important for materials scientists, too, so in this article I’m going to explain what elasticity means, how to calculate Young’s modulus, and why stiffness is so important.

When a material is first exposed to force, it behaves elastically . Elastic behavior means that however the material moves while under load, it returns to its original position when the load is removed. This is true for every material, although sometimes the elastic regime might be really small.

Within the elastic region, a material has stiffness . Stiffness refers to how much force is required for elastic deformation. The inverse of stiffness is called “compliance” (stiffness and compliance have the same relationship as conductivity and resistivity).  Different measures of stiffness are called elastic moduli , and the most common elastic modulus is Young’s modulus . (Yes, named after Thomas Young, the guy who developed the double slit experiment).

If you have ever heard of Hooke’s law, you might already know about elasticity. 

Hooke’s Law as You Learned in High School

Hooke’s law with stress and strain, identifying young’s modulus from the stress-strain curve, calculating young’s modulus, elastic behavior, values of young’s modulus for common materials, applications of young’s modulus, load distribution, final thoughts, references and further reading.

Hooke’s law relates force on a spring to the spring’s displacement.

You could say that applying a force causes elastic deformation in the material. “Deformation” means that the shape is changing, and “elastic” means that when the force is removed, the material returns to its original shape.

Springs are very elastic. You can push them or pull them and the spring displaces, but when the force is removed, the spring returns to its original shape. All materials behave like a spring for at least some small displacement. Hooke’s law applies when a material behaves elastically. 

The point of Hooke’s law is that the elastic deformation is proportional to the force applied. The spring constant determines that proportionality, and if you double the force you get double the displacement.

When we want to measure the elastic behavior of a material–which is important, because every material behaves elastically for some portion–we need to rewrite Hooke’s law to depend on intrinsic properties.

Intrinsic properties are properties that depend only on the material–not how much of the material there is. For example, the mass and volume of steel will be different for every single piece of steel, so they are extrinsic properties.

On the other hand, the ratio of mass and volume–or density–is constant. Whether you have a steel ball bearing or a piece of a skyscraper, the material density is the same, so it’s an intrinsic property.

The intrinsic analogues of force and displacement are stress and strain . 

Stress is just the force divided by the cross-sectional area.

Strain is just the change in displacement divided by the original length.

Young’s modulus specifically applies to tension, or pulling forces. The way we test this is with a tensile test , which basically just applies a strain and measures the stress.

If you want to know why strain is the independent variable (instead of stress) or you have any other questions about the stress-strain curve, I suggest you read this article .

I’ll also provide a quick recap in collapsable text here:

Strain, or how far the material is stretched, is graphed on the x-axis. Stress, or the force applied, is graphed on the y-axis. As the material is stretched, at first the force required to stretch it increases linearly. The slope of this linear line is Young’s Modulus .

At some point (we call this point the yield point ) the relationship is no longer linear. The force continues to increase because of strain hardening , but at a less-than-linear rate. Eventually, the bar becomes thin because of necking and the force required to continue displacement actually decreases.

Again, a more in-depth explanation of this behavior is explained in my other article , but for now we are going to focus on the linear portion of the graph, also called the elastic regime .

Young’s modulus tells you exactly how much force will produce a certain displacement, as long as the material is still in the elastic region.

Young’s modulus is just the slope of the linear portion of the stress-strain curve. Slope is 

So just pick any two points on the linear portion, divide the difference in y-values by the difference in x-values, and you have your modulus of elasticity! 

Remember, this modulus is called “Young’s modulus” when the stress-strain graph shows pure tension, but “modulus of elasticity” is a broad term that refers to stiffness in any direction.

In the elastic regime, atomic bonds are being stretched . It turns out that atomic bonds behave similarly to springs, which is why there is a linear relationship between stress and strain here (and yes stretching atomic bonds means that volume is NOT conserved in the elastic regime).

Atomic bonds can stretch and perfectly return to their original shape, which is why this kind of deformation is called elastic deformation.

The opposite of “elastic deformation” is “plastic deformation,” which means that the material does not return to its original shape.

You can see this in the image below.

This stress-strain curve shows the force on the y-axis, and the deformation on the x-axis. The dotted line shows the relationship between force and deformation all the way until the material breaks, but imagine that we didn’t want to break the material.

The graph on the left shows that we add some force and then remove it. Since we stay in the elastic region, atomic bonds simply stretch and return to their original position.

The graph on the right extends the stress past the yield point , which is when the atoms have to move past each other to continue deformation. If you remove the stress, the bonds will relax and some deformation will reverse, but atoms that have moved past each other are now stuck.

Specifically, the yield point is where the stress-strain curve has a 0.2% offset from the Young’s modulus.

Engineers use this 0.2% offset to have an easily-identifiable point on the stress-strain curve. Technically what I’ve been describing so far is called the “proportionality limit” which is the exact point that stress and strain are no longer perfectly linear, but in practice this point is usually impossible to determine.

Before we talk about applications of stiffness, take a look at this table that shows values of Young’s modulus for common materials.

Here are also two Ashby charts that show elastic modulus on one axis, and other properties on another axis.

Since stiffness is a good approximation of bond strength, it is closely related to melting point. Stiffness is not closely related to strength, since bond strength is only one factor in a material’s overall strength.

When selecting materials, engineers need to control a variety of properties. Cost, operating temperature range, strength, corrosion resistance, and more. For many applications, the most important property is stiffness.

Analogy to Strength

Young’s modulus–or stiffness– is NOT strength. However, it does relate to strength. In most engineering applications, “strength” means yield strength–or the point where elasticity breaks down. Assuming similar yield stresses, higher Young’s modulus will result in higher yield strengths. (But yield stresses can vary quite dramatically).

When you design a part which is limited by strength, what you really mean is that the part must be able to survive a specific force without suffering damage.

If you made a pull up bar, it needs to have enough strength to withstand a person’s bodyweight. As far as engineering applications go, human bodyweight is not that demanding. Nylon is about ⅓ as strong as steel, so from a strength perspective either one would work.

However, the bar should also not move when a person gets on it. Nylon is about 1/100 as stiff as steel, which is why you don’t make pullup bars out of nylon. They would bend too much!

Young’s modulus describes how far something deforms elastically per given force, not how much force it can withstand.

So now you can see that one of the biggest applications of Young’s modulus is to calculate small elastic deformations.

Calculation of Deformation

Although people talk about how “strong” something is, in many cases they are actually interested in the stiffness.

You can use elastic modulus to calculate how far something will deform elastically. For example, imagine a door on a hinge.

The metal hinge needs to keep the door straight enough that it doesn’t bend and touch the floor. If you know the tolerance that the door has, an expected weight on the door (+ a huge safety factor), and the elastic modulus of your hinge material, you can calculate how thick the material needs to be!

Stiffness-Limited Design

Stiffness-limited design refers to applications where stiffness is the main property of interest. Examples of stiffness-limited design applications are support beams (or shafts, or struts), columns, panels, and pressure vessels. These can also be strength-limited–it depends on the other circumstances.

For example, the situation with the door hinge would be stiffness-limited because the wooden door will fail in strength before the hinge does. If you kept putting force on the door, it would either fail because the wood cracked and the screws ripped out, or it would fail because the hinge bent enough that the door touched the floor.

Elastic energy stored

Most engineering applications of elastic energy storage are based on springs, but now you know which materials will work best! You can also think about elastic energy storage if you were making a bow for archery.

If you have multiple materials that support a load together, the materials with the highest elastic modulus will also bear the highest load.

This has implications that go far beyond architectural column design. For example, this phenomenon is one reason why steel is such a bad material for prosthetic implants.

When doctors first began replacing body parts with steel prosthetics, the stiff steel would support most of the patient’s weight. Since the surrounding bones were not be required to support much load anymore, they became very weak and the patient developed further problems.

Today, prosthetics are carefully engineered so that the prosthetic’s elastic modulus matches the elastic modulus of human bone.

Modulus of Resilience

Modulus of Resilience is like toughness, but just for the elastic regime. It tells you how much energy can be absorbed before the material has permanent deformation.

Modulus of Resilience:

Assuming this area is a right triangle (which introduces slight error), then

Speed of Sound

You might not have expected this one! The speed of sound is related to a material’s stiffness and density. Acoustic Engineers would need this information when designing auditoriums.

I don’t have much more to say about this, but it’s just one example of where stiffness comes up that you might not expect!

Stiffness is one of the most important mechanical properties. Stiffness can be determined by calculating the slope of a stress-strain diagram, and it tells you how far a material will bend under a given force.

Ceramics usually have very high stiffness, and polymers have very low stiffness.

Very stiff materials are useful for a variety of applications where engineers don’t want materials to bend. Although strength is the first mechanical property most people think about, stiffness-limited design is just as common as strength-limited design.

If you want to get familiar with the concepts of stress and strain in materials, don’t forget to read this post !

Brandon Ohl

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Here’s How to Calculate Young’s Modulus

And why it’s important to know this equation

Whether you’re fresh out of college or it’s been some time since you’ve taken math classes, it’s always helpful to review equations that frequently show face in the work that you do. One example of this that we’d like to cover in our engineering fundamental series is Young’s modulus, which we’ll define below and then break down how to calculate it.

.css-2xf3ee{font-size:0.6em;margin-left:-2em;position:absolute;color:#22445F;} .css-14nvrlq{display:inline-block;line-height:1;height:1em;background-color:currentColor;-webkit-mask:url(https://assets.xometry.com/fontawesome-pro/v6/svgs/light/link.svg) no-repeat center/contain content-box;mask:url(https://assets.xometry.com/fontawesome-pro/v6/svgs/light/link.svg) no-repeat center/contain content-box;-webkit-mask:url(https://assets.xometry.com/fontawesome-pro/v6/svgs/light/link.svg) no-repeat center/contain content-box;aspect-ratio:640/512;vertical-align:-15%;}.css-14nvrlq:before{content:"";} Young’s Modulus — A Quick Refresher

Young’s modulus points to the slope of a stress strain curve and can help show when a material is simply stretching and holding up okay under stress versus when it’s bound to deform or break. It’s an essential insight into the behavior of a material, and can help determine which materials are best to use based on this number and other factors. 

As you can see in this image, this style of graph can help show Young’s modulus, the ultimate strength point of a material, and at what point you can expect deformation or breakage to occur.

Steps to Calculate Young’s Modulus

To calculate Young’s modulus of elasticity, follow these steps:

1. Measure the Original Length of Your Material with a Micrometer

Use a micrometer to measure the original length (L0) of the material you’re working with, but do this before any force is applied or stretching occurs so that you can get the most accurate answer.

2. Use the Micrometer to Measure the Cross-Sectional Area and Other Diameters

Measure the width and height of your material, then measure out any other diameters it has and note those down. Doing so will help you clock any irregularities and get the most accurate number for the cross-sectional area, which you’ll need to calculate Young’s modulus.

3. Get Slotted Masses Into Place and Apply Force to the Material

Set up several slotted masses and attach them to your material. These will help you better control the tension and deformation process.

4. Use a Vernier Scale to Measure the Different Lengths

Once you start applying force and tension with the slotted masses, you can measure the various lengths of your material as it extends with a vernier scale.

5. Graph Your Measurements

Using a graphing calculator or a computer, create a graph using your measurements so you have a visual representation of your experimental data and can accurately show the important points on the graph.

6. Calculate Young’s Modulus 

Now that you have all the necessary information, you can complete the calculation to find your material’s stiffness and elasticity. You’ll get it by dividing the tensile stress by the tensile strain and then adding in other important information, which you’ll see examples of below. 

Use this equation to do find Young’s modulus: E = Tensile Stress / Tensile Strain = (FL) / (A * Change in L)

7. Analyze Your Graph and Note the Most Important Points

Once you have the graph and equation calculated, double-check your graph and ensure that you can see the clear linear region where elastic behavior for your specific material is shown. If a material has already been stretched and deformed, it’s not likely that you’ll be able to correctly and accurately complete this equation. 

Example Calculations with Different Materials

To really help this concept stick, take a look at the following calculations that find the elasticity modulus of different materials:

In this example, we’ll look at a steel rod and find Young’s modulus for it. Consider the equation if the following factors are true:

• The rod has an original length (L0) of 2 meters
• It has a final length (Ln) of 2.04 meters
• The cross-sectional area (A) is 1 square centimeter (0.0001 square meters)
• To get the elongation you’ll subtract L0 from Ln (which will be 4 cm in this case)
• The force (F) applied is 1,000 Newtons (N)

Here’s how you would calculate this: 

Equation: Young’s modulus = (F x L0) / (A x (Ln - L0))

With numbers inputted: Young’s modulus = (1,000 N x 2 m) / (0.0001 m2 x 0.04 m)

Answer: Young’s modulus ≈ 5 x 108 Pascal (Pa)

2. Aluminum

If you’d like to calculate the same equation but with an aluminum rod, here’s how to do so using sample numbers and with less force:

• The cross sectional area (A) is 1 square centimeter (0.0001 square meters)
• To get the elongation you’ll subtract L0 from Ln (which will be 4 cm)
• The force (F) applied is 800 Newtons (N)

With numbers inputted: Young’s modulus = (800 N x 2 m) / (0.0001 m2 x 0.04 m)

Answer: Young’s modulus ≈ 4 x 108 Pascal (Pa)

Here is how this calculation could be determined if we use a rubber band and possible measurements for this type of material: 

• The rubber band has an L0 of 10 centimeters
• It has an Ln of 15 centimeters
• The elongation is 5 centimeters (Ln-L0)
• The force applied is 20 Newtons

With numbers inputted: Young’s modulus = (20 N x 0.1 m) / (0.0001 m2 x 0.05 m)

Answer: Young’s modulus ≈ 4 x 105 Pascal (Pa)

When you compare all of these different materials, you’ll notice that the Young’s moduli for rubber is a much lower magnitude than steel and aluminum, meaning that the former material is more flexible and elastic, whereas the latter materials are stiffer.

Why It’s Useful to Calculate Young’s Modulus

Engineers use this equation in real-world scenarios for several reasons: 

• Understanding materials: Knowing the Young’s modulus for a certain material will help you get to know it better and help predict what environments and situations it can be put through with success.
• Design considerations : When a structure is relying on a specific material or a human needs a particular implant, calculating Young’s modulus will help determine which materials to use, as you’ll have a better understanding of which ones can withstand the stresses. 
• Stress calculation: You’ll need to know Young’s modulus in order to calculate the stress in a material — another vital factor in engineering and design. 
• Predicting material failure: With the modulus of elasticity, you can have a much better and more accurate prediction of the point at which a material will break or deform for good under pressure. 

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Do Graph Neural Networks Work for High Entropy Alloys?

Graph neural networks (GNNs) have excelled in predictive modeling for both crystals and molecules, owing to the expressiveness of graph representations. High-entropy alloys (HEAs), however, lack chemical long-range order, limiting the applicability of current graph representations. To overcome this challenge, we propose a representation of HEAs as a collection of local environment (LE) graphs. Based on this representation, we introduce the LESets machine learning model, an accurate, interpretable GNN for HEA property prediction. We demonstrate the accuracy of LESets in modeling the mechanical properties of quaternary HEAs. Through analyses and interpretation, we further extract insights into the modeling and design of HEAs. In a broader sense, LESets extends the potential applicability of GNNs to disordered materials with combinatorial complexity formed by diverse constituents and their flexible configurations.

1 Introduction

Advances in machine learning (ML) have greatly facilitated the understanding and design of materials. In particular, graph neural networks (GNNs) have undergone extensive development and shown outstanding performance across various tasks for both crystalline and molecular materials [ gnnrev-comm-22 ] , mainly benefiting from the versatility of graph representation in capturing local chemical environments. For molecules, a graph representation is straightforwardly defined by encoding atoms as nodes and chemical bonds as edges [ neurmp-icml-17 ] . This method has then been adapted to crystals [ cgcnn-prl-18 ] , whose repeating units are represented as graphs, with atoms as nodes and nearest neighbors connected by edges. Moreover, graphs are flexible to incorporate physics information, such as electrical charge [ chgnet-nmi-23 ] , spin [ spingnn-prb-24 ] , or symmetry [ e3nn-icml-21 ] , which is highly desirable for materials science applications.

Adopting GNNs to high-entropy alloys (HEAs) has been deemed challenging [ mlhea-prog-23 ] . Single-phase HEAs are chemically disordered alloys formed by multiple metal elements often with near-equimolar concentrations that can lead to short-range correlations [ taheri-mrsb-23 ] . They form a promising candidate space for materials with desired properties ranging from thermal [ mlhea-science-22 ] , mechanical [ toughhea-science-22 ] to catalysis [ heacat-jourle-19 ] . Owing to the lack of periodic (long-range) chemical order of the multiple species, atomistic representations of HEAs with graphs are not as easy to construct as they are for molecules or crystals. Existing ML models for HEAs mainly use tabular descriptors [ nnhea-matter-20 , mlhea-co2-22 , mlelast-acta-22 ] or graph representation of the composition [ ecnet-npj-22 ] , whereas local environments are not considered.

Our recent work [ molsets-prxe-24 ] proposes a “graph set” representation for molecular mixtures, along with an ML model architecture named MolSets to learn molecular chemistry and make predictions of mixture properties. We recognize that HEAs have similar characteristics as molecular mixtures: both are unordered collections of local chemical environments (molecules or atomic sites). With this analogy, we propose a graph set representation of HEAs and an extended ML model LESets to predict the properties of HEAs represented thereby. The remaining sections are organized as follows: Section 2 reviews the literature of related works. In Section 3 , we describe the proposed representation and model. We then present results including benchmark, sensitivity analysis, and model interpretation in Section 4 . Section 5 discusses the benefits, outlook, and limitations.

2 Related Works

In this section, we briefly review the literature in two relevant directions: (1) GNNs for materials property prediction and (2) computational modeling of HEAs at the atomic scale.

GNNs for materials.

GNNs have become an essential method in materials structure–property modeling. The key to building such models is to represent the structure of a material as a graph, a data structure with nodes and edges. Intuitively, graphs are formulated with atoms as nodes, and atoms that share electrons (bonded or are nearest neighbors) are connected by edges, with atomic/bonding descriptors as node/edge features. On graph data, GNNs perform “message passing” to fuse node/edge information and predict graph-level outputs, such as materials properties. Following this approach, GNNs have been applied to molecules [ neurmp-icml-17 ] and crystals [ cgcnn-prl-18 ] . Yet, the applicability is limited to materials with well-defined structures or repeating units, whereas for unordered, combinatorial materials systems, such as HEAs, a graph representation is hard to formulate.

Beyond the simple formulation, graph representation and GNNs are flexible to incorporate various physical information or knowledge. Chemprop [ chemprop-jcim-24 ] employs bi-directional message passing to attain high accuracy in molecular property prediction. ALIGNN [ alignn-npj-21 ] represents bonds and angles by nodes and edges, respectively, thus incorporating angular information in crystals. M3GNet [ m3gnet-ncs-22 ] and MACE [ mace-nips-22 ] utilize many-body interactions; CHGNet [ chgnet-nmi-23 ] and SpinGNN [ spingnn-prb-24 ] encode electrical charge and spin on atoms, respectively. Another broad category of physical information is symmetry, which is incorporated in GNNs in the form of invariance or equivariance [ e3nn-icml-21 , painn-icml-21 ] . As the model proposed in this work is a generic framework agnostic of the exact form of GNN, these are orthogonal yet complementary directions that could be combined with the framework we propose.

Modeling HEAs.

One key challenge to the computational modeling of HEAs is the randomness, or lack of order, in their structures. The atomic scale physics-based modeling techniques, such as density functional theory and molecular dynamics, require well-defined structures as input. Due to the lack of order in HEAs, such studies generally use (special quasi) random structures given composition, construct large supercells to account for the randomness, or construct cluster expansions [ mdhea-ijms-23 , dfthea-jpcc-23 , widom-jmr-18 ] .

Machine learning models take various approaches to handle or bypass such randomness. One way is to form a tabular input data structure from summary statistics of elemental descriptors and model it using conventional ML methods, such as gradient boosting and random forest [ gbrf-md-24 ] , or neural networks. Other ways include constructing input representations from multiple random realizations of structures or frequency counting [ dnnhea-acscat-24 ] . There are also GNN methods, such as ECNet [ ecnet-npj-22 ] , working on graph representations of composition alone. However, the capability of GNNs to capture connectivity and interactions in local chemical environments has not yet been unleashed in HEAs.

3.1 Representing HEA by local environments

mole 1 {\mathrm{cm}}^{3}\text{\,}{\mathrm{mol}}^{-1} start_ARG power start_ARG roman_cm end_ARG start_ARG 3 end_ARG end_ARG start_ARG times end_ARG start_ARG power start_ARG roman_mol end_ARG start_ARG - 1 end_ARG end_ARG ). The weight fraction of a non-center atom is used as an edge feature. Every type of LE is thus represented as a graph.

The HEA as a whole is an unordered collection of different local environments. For example, the quaternary HEA shown here consists of four types of LEs, centered by Fe, Co, Cr, and Ni, respectively. We represent the HEA as a set of four LE graphs x 𝑥 x italic_x , each associated with a weight fraction w 𝑤 w italic_w that is the fraction of the center element in the HEA’s composition, formally, X = { ( x i , w i ) } 𝑋 subscript 𝑥 𝑖 subscript 𝑤 𝑖 X=\{(x_{i},w_{i})\} italic_X = { ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } . This graph set representation captures the atomic interactions of LEs, as well as the flexibility of relations between LEs, matching the unordered configuration of LEs in the HEA structure.

3.2 LESets model

In [ molsets-prxe-24 ] , we present MolSets that integrates GNN with the permutation invariant Deep Sets architecture [ deepstes-nips-17 ] and attention mechanism [ attention-nips-17 ] for molecular mixture property modeling. In MolSets, each molecule is represented as a graph, and a mixture is represented as a weighted set of graphs. Noticing the analogy of this data structure to the LE graph-based representation of HEAs, we extend MolSets to HEAs, named LESets.

LESets consists of three modules, as shown in Figure 2 . Taking an input datapoint X = { ( x i , w i ) } 𝑋 subscript 𝑥 𝑖 subscript 𝑤 𝑖 X=\{(x_{i},w_{i})\} italic_X = { ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } , a GNN module ϕ italic-ϕ \phi italic_ϕ maps each local environment graph x i subscript 𝑥 𝑖 x_{i} italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT to its intermediate representation vector z i subscript 𝑧 𝑖 z_{i} italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , i.e., z i = ϕ ⁢ ( x i ) subscript 𝑧 𝑖 italic-ϕ subscript 𝑥 𝑖 z_{i}=\phi(x_{i}) italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = italic_ϕ ( italic_x start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) . Then, { ( z i , w i ) } subscript 𝑧 𝑖 subscript 𝑤 𝑖 \{(z_{i},w_{i})\} { ( italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) } is aggregated into a global representation vector Z 𝑍 Z italic_Z , which is mapped to the target response y 𝑦 y italic_y (HEA property) by a multilayer perceptron ρ 𝜌 \rho italic_ρ . Overall, the model learns an input–response function of the following form:

Specifically, for the aggregation ⊕ direct-sum \oplus ⊕ , we take two approaches:

Weighted summation of LE representations, i.e.,

Adjust LE representations using attention mechanism:

taking into account the disproportionate importance and interactions between LEs within a HEA, before weighted summation ⊕ { z i , w i } = ∑ w i ⁢ z i direct-sum subscript 𝑧 𝑖 subscript 𝑤 𝑖 subscript 𝑤 𝑖 subscript 𝑧 𝑖 \oplus{\{z_{i},w_{i}\}=\sum w_{i}z_{i}} ⊕ { italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT } = ∑ italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT .

We name the model using each approach “LESets-att” and “LESets-ws”, respectively. A formal description of the model is provided in Algorithm 1 within Appendix A .

4.1 Data and experimental setup

To test and demonstrate the performance of the LESets model, we conduct experiments on a previously reported DFT-calculated HEA property dataset [ deepsets-hea-22 ] . The dataset contains bulk modulus B 𝐵 B italic_B and Wigner-Seitz radius r ws subscript 𝑟 ws r_{\rm ws} italic_r start_POSTSUBSCRIPT roman_ws end_POSTSUBSCRIPT for 7086 HEAs, 1909 of which also have Young’s modulus E 𝐸 E italic_E reported. We test our and other ML models in predicting these properties and measure their performances using two metrics, coefficient of determination R 2 superscript 𝑅 2 R^{2} italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and mean absolute error (MAE).

We compare the LESets model with several previous methods: (1) The Deep Sets model with HEAs represented as sets of elements, as presented in [ deepsets-hea-22 ] . Note that the major distinction of LESets from this model is incorporating local environment information encoded in graphs. (2) Conventional statistical learning methods, including gradient boosting decision trees, random forest, k-nearest neighbors (kNN), support vector machine (SVM), Lasso, and ridge regression, using elemental property descriptors, all implemented in sklearn [ sklearn-jmlr ] . For LESets, we also investigate the effect of using attention in the ⊕ direct-sum \oplus ⊕ module.

Experiments of the deep learning models (Deep Sets and LESets) are conducted with the following procedure. First, the hyperparameters of every model and target property are tuned using grid search on one random split of the dataset. The optimal hyperparameters found by tuning are listed in Table 1 within Appendix B . Then, the dataset undergoes 30 random splits into training, validation, and test data in a 3:1:1 ratio. In each split, the model is trained by minimizing mean squared error (MSE) loss on the training dataset using the AdamW optimizer [ adamw ] , with a weight decay coefficient of 10 − 4 superscript 10 4 10^{-4} 10 start_POSTSUPERSCRIPT - 4 end_POSTSUPERSCRIPT for regularization. The model’s loss on the validation dataset is monitored for learning rate scheduling (learning rate is reduced by half after 10 epochs of no improvement) and early stopping (training is terminated after 20 epochs of no improvement). After training, the model makes predictions on the testing dataset, and its performance metrics are recorded. The models are implemented in PyTorch and PyTorch Geometric (PyG).

For the conventional ML models, experiments are conducted similarly: the hyperparameters are tuned using grid search and 5-fold cross-validation on 40% of the dataset, then their performances are assessed on 30 random splits of the remaining data into training and testing in a 2:1 ratio. Computational details are provided in Appendix A .

4.2 Benchmark test results

In Figure 3 , we show the performance metrics on predicting bulk modulus B 𝐵 B italic_B and Young’s modulus E 𝐸 E italic_E for LESets, Deep Sets, and gradient boosting, which is the best among conventional ML models. As all models attain high R 2 superscript 𝑅 2 R^{2} italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and low error for r ws subscript 𝑟 ws r_{\rm ws} italic_r start_POSTSUBSCRIPT roman_ws end_POSTSUBSCRIPT , we exclude it from comparison here. We list the average performance metrics of all tested models and properties in Table 2 within Appendix B . To visualize different models’ predictions, we show their predicted values vs. true values in one of the random data splits in Figure 4 .

From the results, we find that LESets attains higher prediction accuracy than its counterparts. Notably, compared to Deep Sets, R 2 superscript 𝑅 2 R^{2} italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and MAE of LESets vary less across random data splits, which indicates better robustness to data variations. The improvement in accuracy could be because local chemical environment information, such as bonding, plays an important role in determining elastic properties [ mech-book-08 ] .

A more interesting comparison is between two versions of LESets with different aggregation mechanisms. The simpler LESets-ws outperforms LESets-att in predicting B 𝐵 B italic_B , suggesting that attention-based aggregation is unnecessary in this task. Whereas in predicting E 𝐸 E italic_E , LESets-att regains the advantage. This difference indicates that elements or local environments may have disproportionate importance in determining E 𝐸 E italic_E but not for B 𝐵 B italic_B . Further investigations using electronic structure methods could find physical insights and guide the design principles of HEAs’ elastic properties.

4.3 Sensitivity analysis

As a deep learning model, LESets requires sufficient data to be trained. Although high-throughput computation and data platforms have made materials data more available than before, understanding the data requirement guides model usage as well as data collection efforts [ redundancy-nc-23 ] . To this end, we conduct sensitivity analyses of LESets’ performance with respect to data size.

The analyses are conducted on the performant version of LESets on bulk and Young’s moduli, respectively, viz. LESets-ws for B 𝐵 B italic_B and LESets-att for E 𝐸 E italic_E . For each property, we leave out 20% of the dataset to be the testing dataset and use a fraction of the remaining data in model training. This fraction of data is split into training and validation datasets in a 3:1 ratio. We repeat this experiment 10 times with different random states for every fraction. Figure 5 shows the relation between performance metrics and data fraction. For B 𝐵 B italic_B , LESets’ performance begins saturating when the data fraction is high, whereas for E 𝐸 E italic_E , it maintains a trend of increasing within the range of analysis. This can be partly explained by the much larger number of datapoints for B 𝐵 B italic_B (7086) than that for E 𝐸 E italic_E (1909). Note that LESets’ number of trainable parameters typically ranges from 5000–8000 (varies depending on hyperparameters). The sensitivity analysis results suggest that the number of parameters might be a rough guideline for the required amount of data.

4.4 Interpretation

One benefit of LESets-att is its interpretability provided by the attention-based aggregation. Following the approach in [ molsets-prxe-24 ] , we investigate the Euclidean norm change of a local environment’s representation vector after attention:

as a score that reflects its relative importance in the HEA. For simplicity, we refer to the importance of a local environment as the importance of its center element. With this importance score, we use two criteria to find out which elements contribute more to a HEA’s Young’s modulus E 𝐸 E italic_E : (1) elements whose importance score is at least three times the lowest in the HEA; (2) elements with the largest score in the HEA. We plot the frequency of elements satisfying criterion (1) in Figure 6 (a), and of those satisfying both criteria in Figure 6 (b). The identified elements, such as Cr, Co, Mo, Hf, and W, appear to be frequent choices in previous studies designing high-stiffness HEAs [ mechhea-book-21 ] .

Besides the importance of single elements, we are also interested in the interactions between elements. In Figure 6 (c), we show how an element’s importance score is affected by the presence of another element. Though not identified as “important” by its own I ⁢ m ⁢ p 𝐼 𝑚 𝑝 Imp italic_I italic_m italic_p , elements such as Mn, Nb, and Ta strongly correlate with other elements’ contributions to E 𝐸 E italic_E , and are thus beneficial as additives. Theories behind the additives’ effects, such as mixing enthalpy, have been explored [ mixing-nature-24 ] and remain an active research topic. We envision these and further interpretations enabled by LESets can help derive useful design principles for HEAs and guide materials discovery.

5 Discussions

The methodological advances of LESets are two-fold. Firstly, the graph set representation effectively captures the chemistry of the local environment while maintaining flexibility to cope with the lack of long-range orders in HEAs. Secondly, the use of attention mechanism in aggregation enables finding disproportionate contributions of certain ingredients (LEs in this case). It also allows a qualitative assessment of the ingredients’ importance. This interpretability is useful for guiding materials design and is desired for scientific ML models.

Note that the focus of this work is to highlight the applicability of graph representation and GNNs to HEAs, while accuracy is less emphasized. Hence, the accuracy of LESets shown in Section 4 potentially has room for improvement. Here, we suggest two possible approaches to further optimizing its accuracy: (1) Revisit the selection of elemental descriptors for constructing node features; (2) Fine-tune the layers’ numbers and hidden dimensions, with a focus on trainable parameter numbers in each module of LESets.

The LESets model architecture has demonstrated general applicability for molecular mixtures in [ molsets-prxe-24 ] and HEAs in this work. These are both instances of combinatorial materials systems, whose diverse constituents (LEs, molecules, etc.) and various configurations of constituents form an expansive space with multilevel complexity. Many other materials families, such as heteroanionic materials [ heteroan-am-19 ] , heterostructures [ superlattice-acs-23 ] and defected functional materials [ delibrate-matter-19 ] , are combinatorial materials systems or possess combinatorial complexity. LESets can potentially provide a generic and extensible framework for ML method development for these materials systems.

Limitations.

At last, we discuss the limitations of LESets and future work directions toward overcoming them. (1) LESets’ representation of local environment information is based on an assumption that all elements could appear in an atom’s nearest neighbors, and simplified by modeling their appearance frequencies rather than exact locations. With emerging materials characterization techniques [ atomic-am-24 ] , the atomic-scale structures of HEAs can be obtained. When more exact structural data become available, LESets could utilize them to represent HEAs in a more informative way. (2) In addition to LEs and their configurations at the atomic scale, microstructure also impacts the mechanical properties of HEAs [ hea-micro-22 ] . LESets currently does not take into account microstructure. A future work direction toward more accurate modeling of HEAs’ mechanical properties is extending the representation and model to incorporate multiscale information.

6 Conclusion

In this work, we present a representation of high-entropy alloys (HEAs) as a collection of local environments, and the LESets machine learning model to accurately predict HEAs’ properties using this representation. Through benchmark tests, we demonstrate the advantage of LESets in predicting HEA mechanical properties over existing ML models. Furthermore, we conduct sensitivity analyses to provide guidelines for LESets’ data requirements. Finally, utilizing LESets’ interpretability, we investigate elemental contributions to Young’s modulus of HEAs, deriving insights to guide HEA design. The code will be available at https://github.com/Henrium/LESets .

Acknowledgments and Disclosure of Funding

This work was supported in part by the U.S. National Science Foundation (Awards DMR-2324173 and DMR-2219489). H.Z. was also supported by the Ryan Graduate Fellowship. J.M.R. acknowledges support from the U.S. Department of Navy, Office of Naval Research (Award N000014-16-12280). The authors thank Wangshu Zheng, Xiao-Yan Li, and Pengfei Ou for their valuable insights.

Appendix A Algorithm and computing details

Computing details..

Experiments are performed on a Linux workstation with Intel Xeon W-2295 CPU (18 cores, 3.0 GHz), NVIDIA Quadro RTX 5000 GPU (16 GB memory), and 256 GB of RAM. Training of LESets typically takes 10–20 minutes on one GPU, with variations depending on data size, hyperparameters, and epochs before termination.

Appendix B Supplemental results

Figure 7 shows a typical learning curve of LESets.

Hyperparameter tuning and experiment tracking are performed using the weights and biases platform [ wandb ] ; the tuning results are listed in Table 1 .

Table 2 lists the complete results of model comparisons.

Nano-silica and Ground Granulated Blast Furnace Slag Blended Concrete: Impact of Temperature on Stress–Strain Constitutive Model

• Research Paper
• Published: 29 August 2024

Cite this article

• Harpreet Singh 1 &
• Aditya Kumar Tiwary 1  

This research aims to advance the construction industry’s progression by examining the complicated dynamics of concrete combined with nano-silica (NS) and ground granulated blast furnace slag (GGBFS), with the fundamental goal of establishing a reliable stress–strain constitutive correlation. The potential of blended concrete with NS (0–5%) and GGBFS (0–25%) as partial cement replacements at temperatures ranging from 27 to 1000 °C was investigated to address critical issues such as fire damage and durability aspects. The results showed an impactful improvement in the stress–strain characteristics within blended concrete by selectively evaluating stress–strain behaviour together with thorough evaluations of compressive strength, elastic modulus, water sorptivity, sulphate resistance, and water absorption. The results appear at 4% NS and 20% GGBFS, yielding better mechanical, resilient, and micro-structural performance at high temperatures. Amidst deterioration, the blended concrete outperformed the control sample, demonstrating the synergistic benefits of NS and GGBFS in creating a more waterproof and long-lasting concrete structure. In the last phase, the correlation between mechanical properties at ambient (27 °C) and increased temperatures was presented to develop a strong stress–strain constitutive model. This model relates the experimental data well, confirming the intricacies of the created concrete blend. This study not only improves the clarity of the observations into concrete performance but also strengthens the application of this study in real-world circumstances, laying the framework for future construction improvements.

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Data Availability

No datasets were generated or analysed during the current study.

Abbreviations

Reference concrete samples

Blended concrete samples having 1% NS and 5% GGBFS

Blended concrete samples having 2% NS and 10% GGBFS

Blended concrete samples having 3% NS and 15% GGBFS

Blended concrete samples having 4% NS and 20% GGBFS

Blended concrete samples having 5% NS and 25% GGBFS

Ordinary Portland cement

Ground granulated blast furnace slag

Nano-silica

Calcium hydroxide

Calcium silicate hydrate

Calcium aluminate silicate hydrate

Fineness modulus

Specific gravity

Scanning electron microscopy

X-ray diffraction

Water/cement ratio

Depth of water absorbed by concrete specimen through capillary action \((mm)\)

Variation in mass of concrete sample in sorptivity test \((gm)\)

Exposed area of concrete specimen to water in sorptivity test \(({mm}^{2})\)

Density of water \((gm/m{m}^{3})\)

Weight loss \((\%)\)

Weight of concrete specimen before acid resistance test \((kg)\)

Weight of concrete specimen after acid resistance test \((kg)\)

Water absorption of the concrete specimens \((\%)\)

Weight of oven-dry concrete specimen \((kg)\)

Weight of saturated surface dry concrete specimen after immersion in water for 24 h \((kg)\)

Compression strength of concrete specimen \((MPa)\)

Modulus of elasticity of concrete specimen \((MPa)\)

Stress value analogous with axial strain value of 0.00005 in concrete specimen \((MPa)\)

Stress value analogous with 40% of peak load in concrete specimen \((MPa)\)

Axial strain analogous with stress value of \({S}_{2}\) in concrete specimen

Ultimate load in compression experiments on concrete specimen \((N)\)

Diameter of the cylindrical concrete sample for compression test \((mm)\)

Residual compression strength of cylindrical concrete specimens at raised temperature \(\left(MPa\right)\)

Compression strength of cylindrical concrete specimens \(\left(MPa\right)\)

Residual modulus of elasticity of concrete at raised temperature \(\left(MPa\right)\)

Residual strain analogous with ultimate stress at elevated temperature

Strain analogous with ultimate stress

Modification parameter for compression strength of concrete depending on exposure temperature

Modification factor for compression strength of concrete based on amount of nano-silica

Modification factor for compression strength of concrete based on amount of GGBFS

Modification factor for elastic modulus of concrete depending on exposure temperature

Modification factor for elastic modulus of concrete depending on proportion of nano-silica

Modification factor for elastic modulus of concrete depending on proportion of GGBFS

Modification factor for strain corresponding to peak stress depending on exposure temperature

Modification factor for strain corresponding to peak stress depending on proportion of nano-silica

Modification factor for strain corresponding to peak stress depending on proportion of GGBFS

Modification factor for stress–strain constitutive model depending of exposure temperature

Proportion of nano-silica in decimals \((N\%/100)\)

Proportion of GGBFS in decimals \((G\%/100)\)

Exposure temperature

Secant modulus of elasticity at peak stress level

Stress value in stress–strain constitutive model

Strain value in stress–strain constitutive model

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Singh, H., Tiwary, A.K. Nano-silica and Ground Granulated Blast Furnace Slag Blended Concrete: Impact of Temperature on Stress–Strain Constitutive Model. Iran J Sci Technol Trans Civ Eng (2024). https://doi.org/10.1007/s40996-024-01580-w

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Core Practical 5: Investigating Young Modulus ( Edexcel A Level Physics )

Revision note.

Core Practical 5: Investigating Young Modulus of a Material

• To measure the Young Modulus of a metal in the form of a wire requires a clamped horizontal wire over a pulley (or vertical wire attached to the ceiling with a mass attached) as shown in the diagram below

• A reference marker is needed on the wire. This is used to accurately measure the extension with the applied load
• The independent variable is the load
• The dependent variable is the extension
• Measure the original length of the wire using a metre ruler and mark this reference point with tape
• Measure the diameter of the wire with micrometer screw gauge or digital calipers
• Measure or record the mass or weight used for the extension e.g. 300 g
• Record initial reading on the ruler where the reference point is
• Add mass and record the new scale reading from the metre ruler
• Record final reading from the new position of the reference point on the ruler
• Add another mass and repeat method

Reducing Uncertainty

• Take pairs of readings of the diameter right angles to each other, to ensure the wire is circular
• Six to ten readings altogether is enough to get an average value
• Remove the load and check the wire returns to the original limit after each reading. A little 'creep' is acceptable but a large amount indicates that the elastic limit has been exceeded
• Take several readings with different loads and find average
• Use a Vernier scale to measure the extension of the wire

Measurements to Determine the Young Modulus

1. Determine extension x from final and initial readings

Example table of results:

Table with additional data

2. Plot a graph of force against extension and draw line of best fit

3. Determine gradient of the force v extension graph

4. Calculate cross-sectional area from:

5. Calculate the Young modulus from:

Safety Considerations

• Safety glasses should be worn in case of the wire snapping
• Protect feet and the floor from falling weights by cushioning the area underneath the weights

Although every care should be taken to make the experiment as reliable as possible, you will be expected to suggest improvements in producing more accurate and reliable results

Good examples of improvements in any experiment are:

• Take repeat readings and take an average to improve accuracy
• Measure longer distances, such as using a longer length of wire, to reduce percentage error

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Author: Lindsay Gilmour

Lindsay graduated with First Class Honours from the University of Greenwich and earned her Science Communication MSc at Imperial College London. Now with many years’ experience as a Head of Physics and Examiner for A Level and IGCSE Physics (and Biology!), her love of communicating, educating and Physics has brought her to Save My Exams where she hopes to help as many students as possible on their next steps.

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Wd-1d-vgg19-fea: an efficient wood defect elastic modulus predictive model.

1. Introduction

2. materials and methods, 2.1. near-infrared spectroscopy data acquisition, 2.2. d-vgg19 and 3d finite element modeling.

• Input layer:

3. Experiments and Analysis

4. results and discussion, 4.1. nonlinear sheet shape inversion, 4.2. establishment of finite element analysis model and prediction of elastic modulus of larch lumber, 5. conclusions, author contributions, data availability statement, acknowledgments, conflicts of interest.

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Pan, S.; Chang, Z. WD-1D-VGG19-FEA: An Efficient Wood Defect Elastic Modulus Predictive Model. Sensors 2024 , 24 , 5572. https://doi.org/10.3390/s24175572

Pan S, Chang Z. WD-1D-VGG19-FEA: An Efficient Wood Defect Elastic Modulus Predictive Model. Sensors . 2024; 24(17):5572. https://doi.org/10.3390/s24175572

Pan, Shen, and Zhanyuan Chang. 2024. "WD-1D-VGG19-FEA: An Efficient Wood Defect Elastic Modulus Predictive Model" Sensors 24, no. 17: 5572. https://doi.org/10.3390/s24175572

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• Published: 28 August 2024

Calcium-rich seawater affects the mechanical properties of echinoderm skeleton

• Przemysław Gorzelak   ORCID: orcid.org/0000-0001-5706-1881 1 ,
• Jarosław Stolarski   ORCID: orcid.org/0000-0003-0994-6823 1 ,
• Paweł Bącal   ORCID: orcid.org/0000-0001-7935-639X 1 ,
• Philippe Dubois 2 &
• Dorota Kołbuk   ORCID: orcid.org/0000-0003-4547-6531 3  

Communications Earth & Environment volume  5 , Article number:  467 ( 2024 ) Cite this article

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• Geochemistry
• Palaeontology

Shifts in the magnesium to calcium ratio of seawater in the geological history are thought to have profoundly affected biomineralization of marine invertebrates, including some echinoderms, which changed their skeletal mineralogy from high-magnesium to low-magnesium calcite and vice versa. Here we report on experiments that aimed to investigate the effect of ambient seawater magnesium to calcium ratio on magnesium to calcium ratio and nanomechanical properties in the spines of two echinoid species ( Arbacia lixula and Paracentrotus lividus ). We found that echinoids cultured in seawater with a low magnesium to calcium ratio produced a skeleton with lower both magnesium content and nanohardness than those of the control specimens incubated under normal (high) magnesium to calcium ratio conditions. These results may suggest that at certain times in the geological past (during the so-called calcite seas) sea urchins with decreased skeletal magnesium contents were more susceptible to damage due to physical disturbances, predation and post-mortem taphonomic processes. Increased skeletal hardness of echinoids from the so-called aragonite seas is expected to enhance their taphonomic potential, thus, to some extent, mitigates the preservation bias related to increased solubility of high-magnesium calcite.

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The role of aspartic acid in reducing coral calcification under ocean acidification conditions

Introduction.

It has been argued that skeletal Mg/Ca ratio in echinoderms can be affected by Mg 2+ and Ca 2+ concentrations in the ambient seawater 1 , 2 . This has been recently confirmed by a number of laboratory experiments that showed that echinoderms incubated under low Mg 2+ /Ca 2+ molar ratio produced a skeleton with decreased Mg/Ca ratio 2 , 3 , 4 , 5 . Accordingly, well-preserved fossil echinoderms, though with some exceptions 6 , preserve a record of global changes in seawater Mg 2+ /Ca 2+ ratios (calcite-aragonite sea transitions) 1 , 7 , 8 . For example, Pennsylvanian-Triassic echinoderms are typically preserved as high-Mg calcite whereas Jurassic-Cretaceous echinoderms are commonly preserved as low-Mg calcite 9 . This pattern matches the change between high Mg 2+ /Ca 2+ ratio of late Paleozoic-early Mesozoic aragonite sea and low Mg 2+ /Ca 2+ ratio of the mid-late Mesozoic calcite sea 10 , 11 , .

An interesting, yet largely unexplored, consequence of calcite-aragonite sea transitions relates to their effects on mechanical properties of the skeleton. Theoretically, incorporation of magnesium into calcite should enhance its mechanical strength 12 . This is due to the fact that impurities of magnesium, the atomic radius of which is smaller than calcium, induce lattice distortion in mineral, which in turn increases both the sliding resistance of dislocations and deformation resistance in the crystal 13 . Surprisingly, very few studies investigated the effect of Mg impurities on mechanical properties in biominerals 14 . The most striking example refers to the high-Mg calcite teeth of sea urchins, which are able to chew even basaltic substrate (hardness of ca. 10 GPa) 15 . The magnesium-enriched mineral cementing plates and needles, in addition to amino acid inclusions 16 , is thought to increase tooth hardness 17 . Furthermore, comparative data obtained from field-collected echinoderm specimens with contrasting Mg contents indicate a possible correlation between hardness and Mg content 18 , but the relationship is not so straightforward, especially if differences in Mg content are relatively minor 19 . Studies experimentally testing the impact of ambient seawater Mg 2+ /Ca 2+ ratio on skeletal chemistry and mechanical properties of echinoderm skeleton are scarce. Lemloh et al. 20 found that incubation of sea urchins larvae under extremely low Mg 2+ /Ca 2+ ratio (0.71 mol/mol) negatively affected not only their physiological and developmental aspects but also the mechanical properties of spicules. Following up, Kanold et al. 21 cultured sea urchin larvae under extremely high Mg 2+ /Ca 2+ ratio (11.4 mol/mol) and found that seawater Mg 2+ enrichment increased both the reduced modulus (the material stiffness) and the nanohardness (the material resistance to permanent or plastic deformation at the nano-micro level) of the newly grown spicules (the latter not being statistically significant though). These studies, however, were only performed on larvae skeletons and using seawater with Mg 2+ /Ca 2+ ratios much lower or higher than those thought to have existed in the geological history.

In this paper we report on experiments which aimed at investigating the effect of decreased ambient Mg 2+ /Ca 2+ ratios (corresponding to geologically realistic Mg 2+ /Ca 2+ levels that existed in the Mesozoic 10 , 11 ) on mechanical properties of the newly grown spine skeleton with decreased Mg content in the adult sea urchins Arbacia lixula and Paracentrotus lividus . These studies are supplemented by the analyses of the ossicles of field-collected echinoid specimens displaying contrasting Mg contents (Supplementary Material). Our data can be applicable to many calcite-secreting marine invertebrates that are sensitive to seawater Mg 2+ /Ca 2+ ratio, and thus carry broader paleobiological, paleoecological, taphonomic and sedimentological implications.

Prior to the experiment, tips of the primary spines from each individual were cut-off to initiate regeneration. Next, the sea urchins were grown in seawater with three different Mg 2+ /Ca 2+ ratios [mol/mol]: ~5.1 (control; representative of modern seawater Mg 2+ /Ca 2+ ratio), ~2.5, and ~ 1.5. Newly grown spine tips were subjected to geochemical (Mg/Ca) and nanomechanical analyses (measurements of nanohardness and reduced modulus).

Results of the experiment (Fig.  1 ; Tables  S2 , S4 in Supplementary Material) showed that spine regenerates of sea urchins cultured under lower seawater Mg 2+ /Ca 2+ ratios displayed significantly decreased mean skeletal Mg/Ca ratios compared to the control specimens (ANOVA, F(2,9) = 25.04, p  = 0.00021 for Paracentrotus lividus ; F(2,9) = 94.94, p  = 9 × 10 −7 for Arbacia lixula ). As for the nanomechanical properties, spine regenerates of Paracentrotus lividus and Arbacia lixula grown under the lowest Mg 2+ /Ca 2+ seawater ratio exhibited significantly lower nanohardness (ANOVA, F(2,9) = 4.51, p  = 0.044 for P. lividus , F(2,9) = 4.802, p  = 0.038 for A. lixula ). In the case of reduced modulus, although the means in control specimens appeared to be higher, the detected differences were not statistically significant (ANOVA, F(2,9) = 2.184, p  = 0.17 for P. lividus , F(2,9) = 1.807, p  = 0.22 for A. lixula ). Likewise, the local stereom porosities do not differ statistically between treatments (ANOVA, F(2,9) = 0.035, p  = 0.996 for P. lividus , F(2,9) = 0.143, p  = 0.868 for A. lixula ), and consequently do not correlate with any variables (nanohardness, reduced modulus, Mg/Ca ratio (Table  S6 in Supplementary Materials)).

Paracentrotus lividus a and Arbacia lixula f during experiment with enlargement of regenerated spines ( b , g ; BSE SEM images). c , h Bar charts showing means of skeletal Mg/Ca ratios ( ± SD), d , i hardness and e , j reduced modulus in regenerated spines. Means sharing the same superscript ( a , b , c ) are not significantly different ( p  ≥ 0.05, post hoc Tukey’s HSD test).

Test plates of Paracentrotus lividus field-collected in the Atlantic and the Meditarranean displayed higher skeletal Mg/Ca ratios [mmol/mol] and nanohardness (but not reduced modulus) than spines from the same individuals (linear mixed-effects model, t(4) = −22.17, p  = 2 × 10 −5 for Mg/Ca, t(4) = −6.37, p  = 0.003 for nanohardness and t(4) = −1.442, p  = 0.223 for reduced modulus), while neither skeletal Mg/Ca, stereom porosities, nor nanomechanical properties differed between the two sampling locations (linear mixed-effects model, t4) = −1.377, p  = 0.24 for Mg/Ca; t(4) =−0.674, p  = 0.537 for nanohardness; t(4) = 0.012, p  = 0.991 for reduced modulus; t(4) = −1.31, p  = 0.259 for stereom porosity; see Figs.  2 – 4 and Supplementary Material, Tables  S3 , S5 ).

Mg/Ca ratios [mol/mol] in spines and test plates of field-collected specimens of Paracentrotus lividus from two localities (ANT1-3: Antiparos, Greece; and MOR1-3: Morgat, France).

Nanohardness [GPa] in spines and test plates of field-collected specimens of Paracentrotus lividus from two localities (ANT1-3: Antiparos, Greece; and MOR1-3: Morgat, France).

Reduced modulus [GPa] in spines and test plates of field-collected specimens of Paracentrotus lividus from two localities (ANT1-3: Antiparos, Greece; and MOR1-3: Morgat, France).

Our experimental studies confirmed that the skeletal Mg/Ca ratios in echinoderms may be strongly controlled by seawater Mg 2+ /Ca 2+ ratios. More specifically, we showed that sea urchins grown in experimental seawaters formulated at low Mg 2+ /Ca 2+ ratios (corresponding to conditions which are thought to have occurred throughout the geological history) produced regenerating spines with decreased Mg/Ca ratio. These results expand the number of echinoderm species in which this pattern has been documented 2 , 3 , 4 , 5 . However, a novel aspect of the impact of changes in the Phanerozoic ocean chemistry (specifically decreased Mg 2+ /Ca 2+ ratio), explored herein, is their effects on the mechanical strength of calcareous skeletons. We showed that a notable consequence of significantly reduced skeletal Mg/Ca ratios is decreased nanohardness. This is consistent with results obtained from ossicles displaying contrasting Mg contents collected in the field. A common pattern emerged from the analyses of these field-collected samples: ossicles with significantly enriched Mg contents displayed increased nanohardness (Figs.  2 – 4 and Supplementary Material). Given the fact that a number of carbonate-secreting marine calcifiers are thought to have been sensitive to changes in seawater Mg 2+ /Ca 2+ , our data can be potentially applicable also to these invertebrates (Fig.  5 ). In the case of reduced modulus, the pattern is not so obvious and statistically insignificant.

The uppermost horizontal bar shows intervals of ‘aragonite’ and ‘calcite’ seas. Green curve is molar Mg 2+ /Ca 2+ ratio of seawater (after 10 , 52 ). The graph at the upper right shows mean values of nanohardness of spines of two euechinoid taxa grown under modern seawater conditions (Mg 2+ /Ca 2+ sw  = 5.2) (gray and black stars), and of inner (aragonite) [blue dots] and outer (calcite) [red dots] shell layers of two Recent mollusc species (after 39 ). Gray and black stars in the major central graph correspond to hypothetical nanohardness values of spines displayed by fossil euechinoids during the Late Triassic (Mg 2+ /Ca 2+ sw  = 2.5) and the Early Jurassic (Mg 2+ /Ca 2+ sw  = 1.5) times.

Our experimentally validated findings, consistent with theoretical predictions, may have a number of far-reaching and previously unnoticed implications. For instance, if mechanical properties of the echinoderm biomineral are to some extent related to its Mg content, latitudinal pattern in skeletal hardness may be observed given the fact that Mg content in echinoderms is strongly related with latitudinal temperature gradient 22 . Interestingly, our results suggest that at certain times in the past (during the so-called calcite seas with low Mg 2+ /Ca 2+ ratio of seawater) sea urchins with significantly decreased skeletal Mg contents were probably more susceptible to damages due to predation, physical disturbances (such as storms and water-bourne particles), wedging themselves in rock holes, and post-mortem taphonomic processes (e.g., abrasion-induced wearing).

Although decreased hardness (down to about 20%, Fig.  1 ) of spines appears not to be so important with regard to larger-bodied durophagous vertebrate predators 23 (due to usually high biting forces of the latter and specific design of echinoid spines to break and remain in the predator tissues 24 ), it should be emphasized that echinoderms are also preys of a number of predatory groups displaying different hunting techniques (i.e. shearing and grasping fishes, grazing and peeling arthropods, and drilling molluscs) 25 . Though the enhanced properties of the echinoderm skeleton, in general, are mostly due to highly organized skeletal microstructure 26 , 27 , increased Mg content in their skeletons, from the compositional point of view, should also be considered as an important factor contributing to their increased hardness. Perhaps most notably, changes in Mg content in the test plates may be of greater functional significance with regard to protection from grazing, peeling, and mechanical boring predators.

Given the above, one may speculate that secular trends in the ocean chemistry throughout the Phanerozoic might have some evolutionary consequences. Whereas the changes in Mg 2+ /Ca 2+ ratio of seawater appear not to have significantly impacted the diversification dynamics of the entire clade of echinoderms 28 , 29 and also the echinoid diversity trends 30 , it is noteworthy that certain major evolutionary events in specific echinoderm clades (such as the infaunalization of echinoids), coincide roughly with the transition from the high Mg 2+ /Ca 2+ ratio of the aragonite sea to low Mg 2+ /Ca 2+ ratio of the calcite sea. This ecological innovation among irregular echinoids, which occurred in the early Jurassic, was attributed to expansion into unoccupied niches 31 , 32 and/or to increased predation pressure 33 . Interestingly, increased infaunalization at around this time has been also observed in some molluscs 34 , some of which are also known to change their skeletal Mg content and/or switch their mineralogy in response to changes in Mg 2+ /Ca 2+ sw ratio 35 (Fig.  5 ). Nonetheless, a recent study 36 exploring the trends of drill holes in Meso-Cenozoic echinoids concluded that either infaunalization was not driven by increased predation pressure by carnivorous cassid gastropods or that other echinoid predators, such as crustaceans or fish, were responsible. We point out that the secular changes in the ocean chemistry at around this time induced dramatic shift in echinoid mineralogy from high to low-Mg calcite (down to about 30–50% decrease in MgCO 3 4 ), which to some extent might have increased their vulnerability to predation. The question whether these environmental circumstances could have contributed to this macroecological change remains open.

Notwithstanding these putative temporal changes in the vulnerability to predation, increased incidences of regeneration events due to physical disturbances (e.g., storms), wedging themselves in rock holes, and/or sublethal predation, might also have had some consequences in terms of energy allocation and higher energetic costs of calcification 24 , which, as we know from experimental studies on living echinoids, can affect the overall reproduction 37 . However, as demonstrated by Cole et al 28 . lack of correspondence between any metrics in the echinoderm diversification dynamics with calcite-aragonite sea transitions suggests that echinoderms, in general, were little affected (from the biodiversity standpoint) by seawater chemistry, although its impact on the life habit and development of ecological (functional) innovations (e.g., enabling infaunalization), as stressed above, cannot be ruled out.

Secular variation in seawater chemistry and skeletal mineralogy has also been suggested to impart a temporally variable taphonomic overprint on the fossil record, resulting in megabiases, particularly because of increased solubility of aragonite and high-Mg calcite 38 . Indeed, it has been commonly argued that calcite skeletons have a preservational advantage over high-Mg calcite and aragonitic ones. However, increased skeletal hardness of high-Mg calcite (and also aragonite 39 ) relative to that of low-Mg calcite, to some extent, may enhance their preservation potential, thus ameliorating the preservation bias imposed by increased solubility of high-Mg calcite and aragonite (Fig.  5 ). Accordingly, a number of studies found that individual echinoderm plates can withstand considerable physical disturbance 40 and extensive temporal mixing 41 . Interestingly, some experiments showed notable differences in tumbling-induced damages between the tests of some echinoid genera displaying the interlocking of stereom trabeculae across plate suture faces, namely Echinometra and Tripneustes 42 . This pattern was attributed by the latter author to their different rigidities (possibly related to thickness and microstructural organization). However, it should be pointed that the test plates of Echinometra , which suffered less breakage, are much more enriched in Mg (ca. 15% MgCO 3 ) than those of Tripneustes (with ca. 10% MgCO 3 ) 43 . Thus, Mg enrichment may be an additional factor contributing to the greater resistance of the echinometrid corona. Accordingly, other tumbling experiments 44 performed on foraminifers displaying contrasting MgCO 3 levels, namely 4–5% in Asterigerina and Amphistegina vs. 14–16% in Archaias , showed surprisingly no differences in their solubilities, but revealed greatly increased rates of abrasion of low-Mg calcite taxa. Overall, these data may explain why there is no obvious relationship between ‘calcite-aragonite seas’ and extent of taphonomic loss 45 .

Experimental setup and seawater parameters

18 specimens of each species were randomly allocated to independent 1 L beakers filled with seawater with three different Mg 2+ /Ca 2+ ratios [mol/mol]: ~5.1, ~2.5, and ~ 1.5 (6 specimens of each species per treatment). The molar Mg 2+ /Ca 2+ ratio in seawater was decreased by lowering Mg 2+ and increasing Ca 2+ concentration prepared by mixing natural seawater from the English Channel (North Sea) with artificial Ca 2+ -enriched seawater without Mg 2+ (prepared by dissolving NaCl, Na 2 SO 4 , CaCl 2 , KCl, KBr, SrCl 2 , NaF, H 3 BO 3 , and NaHCO 3 (SIGMA) in MilliQ water).

Sea urchins, each with a few tips of the primary spines (from the interambulacra near the ambitus) cut-off to initiate regereration, were incubated for 14 ( Arbacia lixula ) or 21 ( Paracentrotus lividus ) days under constant temperature ( ~ 20.1 °C), salinity ( ~ 35 psu) and pH T ( ~ 8.0), and Mg 2+ /Ca 2+ ( ~ 5.1, ~2.5, ~1.5; Table  S1 ). Water parameters were measured three times a day with a WTW Multi 340i multimeter equipped with a conductivity cell, and integrated temperature sensor, and a Metrohm pH-meter. The electrode was calibrated every day with Merck CertiPUR buffer solutions pH 4.00 and 7.00. pH measurements were converted to total scale according to DelValls and Dickson’s 46 method using TRIS/AMP buffers. Seawater in each beaker was renewed every day.

Seawater samples were collected daily and Mg 2+ and Ca 2+ concentrations in each sample were analysed with inductively coupled plasma-optical emission spectrometry (Thermo Scientific iCAP 6500 Duo ICP-OES spectrometer; detection limits for Mg and Ca: 0.1 mg/l) at the Central Chemical Laboratory of the Polish Geological Institute–National Research Institute in Warsaw. The stability of parameters including temperature, salinity, pH, and three levels of Mg 2+ /Ca 2+ was verified using one-way ANOVA and post-hoc Tukey’s HSD tests.

After the experiment, regenerated spines were removed from the urchins, treated with 2.5% NaOCl solution and rinsed in ultrapure water, then embedded in epoxy resin, cut along the spine length, and gently polished through a series of diamond suspensions.

Field-collected specimens

Apart from the spines collected in the experiment, we also analyzed spines and ambital plates from echinoids grown in their natural environment. More specifically, the specimens of Paracentrotus lividus were collected from two geographic locations: Morgat, Brittany, France (48°13′14′′N, 4°29′38′′W) and Antiparos Island, Greece (36° 59’ 47”, 25° 03’ 04”). Similarly to the procedure applied for the experimental specimens, ossicles (2 populations × 3 specimens × [3 spines + 3 ambital plates] = 36 ossicles total) were bleached, embedded in epoxy resin, and polished.

SEM observations

Microstructural observations of the stereom, which allowed for the selection of appropriate locations of the indents, were performed using backscattered electrons (BSE) mode on uncoated spines (low vacuum mode, accelerating voltage = 10 kV, beam current = 0.74 nA, working distance = approx. 10 mm), with the aid of a Thermo Fisher Quattro S Environmental SEM at the Institute of Paleobiology of the Polish Academy of Sciences in Warsaw, Poland. After nanoindentation, the surfaces of ossicles were coated with carbon and re-examined under SEM (same operating conditions; Fig. S 1 in Supplementary Materials) to investigate whether the indents were affected by local defects or cracks in which case such artefacts were excluded from further analysis. The local stereom porosity for each sample, which may potentially affect reduced modulus, was determined following Presser et al. 47 approach using ImageJ Fiji software.

Nanoindentation analyses

Nanoindentation analyses were conducted on the galleried stereom of the regenerated spine tips of the experimental specimens of Arbacia lixula (12 specimens [4 + 5 + 3 per treatment] × 3 spines) and Paracentrotus lividus (12 specimens [3 + 4 + 5 per treatment] × 3 spines). The spines from remaining specimens were not considered in the analyses either due to: (i) insufficiently preserved (i.e., broken/fractured) and/or very limited extension of regenerated tips (making it impossible to collect an adequate amount of data points) (ii) growth inhibition, and/or (iii) inappropriately oriented (oblique) cut surfaces. This resulted in unequal number of actual individuals per treatment. Galleried sterom in the spines and ambital plates of six field specimens of Paracentrotus lividus from two locations, in French Brittany and Antiparos island in Greece were also analysed (6 specimens × 6 ossicles). The nanoindentation analyses were carried out at the Faculty of Materials Science and Engineering at Warsaw University of Technology, using a nanoindenter (NanoTest Vantage, Micro Materials, UK) equipped with a Berkovitch tip and using a maximum load of 2 mN and a maximum depth penetration of approximately 200 nm (Fig. S2 in Supplementary Materials). Approximately ten to forty indents were obtained from each gently polished and uncoated ossicle. Nanohardness (H) and reduced modulus (E) values (Tables  S2 and S3 ) were determined from the unloading curve of the indentation test following Oliver and Pharr 47 , 48 .

Geochemical analyses

Mg/Ca ratios (mmol/mol) of the newly grown open stereom in Arbacia lixula and Paracentrotus lividus spine tips, as well as galleried stereom in spines and ambital plates of field specimens of Paracentrotus lividus were determined using Wavelength Dispersive Spectroscopy (WDS) on CAMECA SX100 electron microprobe at the Micro-Area Analysis Laboratory, Polish Geological Institute - National Research Institute in Warsaw (accelerating voltage 15 kV, beam current 5 nA for calcium, 20 nA for magnesium, beam diameter ~ 5 μm; standard used: ‘NIST’ (serial number: 12570)) (Tables  S2 and S3 ). These analyses were carried out on the same stereom bars as those used for the nanoindentation tests.

Statistical analysis

Skeletal Mg/Ca ratio, nanohardness (H), reduced modulus (E) and local stereom porosity in Arbacia lixula ( n  = 12) and Paracentrotus lividus ( n  = 12) [mean values pooled from the measurements of 3 spines of each individual] were analysed through one-way ANOVA with seawater Mg 2+ /Ca 2+ ratio level as a fixed factor. Differences in skeletal Mg/Ca ratio, nanohardness (H), reduced modulus (E) and stereom porosity between spines and plates in field specimens of Paracentrotus lividus as well as comparison between sampling locations were assessed through linear mixed effects model with specimen as a nested random factor and ossicle and location as a fixed factor (Supplementary Material, Tables  S4 - S5 ). Analyses were performed in RStudio (R version 1.2.5033; RStudio Team, 2020), with significance level set to 0.05.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

Data recorded during this study are included in this published article and its Supplementary information files (Supplementary Material linked to this article and Additional Data deposited in RepOD repository https://doi.org/10.18150/MWKIBH ).

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Acknowledgements

The project was funded by NCN grant 2020/37/B/ST10/01460. Ph.D. is a Research Director of the National Fund for Scientific Research (Belgium). We thank members of the laboratory of Marine Biology at ULB for help in performing experiments and Michał Gloc and Krzysztof Kulikowski (both from Warsaw University of Technology, Faculty of Materials Science and Engineering) for help in performing nanomechanical measurements. We thank three anonymous reviewers for numerous useful comments.

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Institute of Paleobiology, Polish Academy of Sciences, Warszawa, Poland

Przemysław Gorzelak, Jarosław Stolarski & Paweł Bącal

Laboratoire de Biologie Marine, Faculté des Sciences, Université Libre de Bruxelles, Bruxelles, Belgium

Philippe Dubois

UCD Earth Institute and School of Biology and Environmental Science, University College Dublin, Belfield, Dublin, Ireland

Dorota Kołbuk

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P.G. designed the study. P.G. and D.K. conducted the experiment, performed analyses and drafted the manuscript. Ph.D. provided field specimens of P. lividus and assisted in the experiments. P.B. performed SEM imaging. P.G., D.K., Ph.D., P.B., J.S participated in data interpretation and contributed to the writing, editing, and revision of the manuscript.

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Correspondence to Dorota Kołbuk .

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Communications Earth & Environment thanks Valentina Perricone, Tobias Grun and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: Nadine Schubert and Carolina Ortiz Guerrero. A peer review file is available.

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Gorzelak, P., Stolarski, J., Bącal, P. et al. Calcium-rich seawater affects the mechanical properties of echinoderm skeleton. Commun Earth Environ 5 , 467 (2024). https://doi.org/10.1038/s43247-024-01609-y

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Received : 23 February 2024

Accepted : 06 August 2024

Published : 28 August 2024

DOI : https://doi.org/10.1038/s43247-024-01609-y

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