An elastic modulus (also known as modulus of elasticity) is a quantity that measures an object or substance's resistance to being deformed elastically (i.e., non-permanently) when a stress is applied to it. The elastic modulus of an object is defined as the slope of its stress–strain curve in the elastic deformation region: A stiffer material will have a higher elastic modulus. An elastic modulus has the form:
δ
=
def
stress
strain
{\displaystyle \delta \ {\stackrel {\text{def}}{=}}\ {\frac {\text{stress}}{\text{strain}}}}
where stress is the force causing the deformation divided by the area to which the force is applied and strain is the ratio of the change in some parameter caused by the deformation to the original value of the parameter. Since strain is a dimensionless quantity, the units of
δ
{\displaystyle \delta }
will be the same as the units of stress.Specifying how stress and strain are to be measured, including directions, allows for many types of elastic moduli to be defined. The three primary ones are:
Young's modulus (E) describes tensile elasticity, or the tendency of an object to deform along an axis when opposing forces are applied along that axis; it is defined as the ratio of tensile stress to tensile strain. It is often referred to simply as the elastic modulus.
The shear modulus or modulus of rigidity (G or
μ
{\displaystyle \mu \,}
Lamé second parameter) describes an object's tendency to shear (the deformation of shape at constant volume) when acted upon by opposing forces; it is defined as shear stress over shear strain. The shear modulus is part of the derivation of viscosity.
The bulk modulus (K) describes volumetric elasticity, or the tendency of an object to deform in all directions when uniformly loaded in all directions; it is defined as volumetric stress over volumetric strain, and is the inverse of compressibility. The bulk modulus is an extension of Young's modulus to three dimensions.Two other elastic moduli are Lamé's first parameter, λ, and P-wave modulus, M, as used in table of modulus comparisons given below references.
Homogeneous and isotropic (similar in all directions) materials (solids) have their (linear) elastic properties fully described by two elastic moduli, and one may choose any pair. Given a pair of elastic moduli, all other elastic moduli can be calculated according to formulas in the table below at the end of page.
Inviscid fluids are special in that they cannot support shear stress, meaning that the shear modulus is always zero. This also implies that Young's modulus for this group is always zero.
In some texts, the modulus of elasticity is referred to as the elastic constant, while the inverse quantity is referred to as elastic modulus.
Question 1: Find the modulus and argument of ##z=-\sin \frac {\pi}{8}-i\cos \frac {\pi}{8}##.
The modulus is obviously 1. I can't prove that the argument is ##\frac {-5\pi} {8}##. I think ##\frac {-5\pi} {8}## is not correct ...
What I've done:
$$\tan \theta=\cot \frac {\pi}{8}$$$$\tan...
It is claimed that the modulus of ##(\log(z))^jz^\lambda##, where ##j## is a positive integer (or ##0##) and ##\lambda## a complex number, can be bounded above by ##c|z|^l## for some integer ##l## and constant ##c##. Assume we are on the branch ##0\leq \mathrm{arg}(z)<2\pi## (yes, ##0##...
π
My take; i multiplied by the conjugate of the denominator...
$$\dfrac{z_1}{z_2}=\dfrac{2(\cos\dfrac{π}{3}+i \sin \dfrac{π}{3})}{3(\cos\dfrac{π}{6}+i \sin \dfrac{π}{6})}⋅\dfrac{3(\cos\dfrac{π}{6}-i \sin \dfrac{π}{6})}{3(\cos\dfrac{π}{6}-i \sin \dfrac{π}{6})}=\dfrac{2(\cos\dfrac{π}{3}+i \sin...
Hello,
Everything is in th summary :
I have a set of distance modulus and I would like to know if this is possible to convert numerically these values into a Hubble vs redshift H(z) data ( with the standard deviation).
Any relation between distance modulus and H(z) are welcome.
Regards
Hello,
I am trying and failing to derive that the shear modulus ##G## is equal to the Lame parameter ##\mu##. I start with the linear, symmetric, isotropic stress-strain formula: $$\sigma = \lambda \mathrm{tr}(\epsilon) \mathrm{I} + 2\mu \epsilon$$ I then substitute a simple (symmetric) shear...
Hi,
I have a problem, using the 2 different equations for calculating the Modulus of Resilience, I get 2 different answers. It seems simple, but I’m a little perplexed.
For the following values:
Yield Stress = 1880 MPa
Yield Strain = 0.0068
E = 401 GPa
[U(r) = (Yield Stress x Yield Strain) /...
We need to find a single wire supporting the load. But, when I think about it there can so many different types of materials we could use for the single wire and each of these single wires would have a different Young's Modulus. So, I don't get what equivalent means here?
Anyways, I get the...
Hello,
I an clear on how the modulo operator ##%##, also called the remainder operator, is used in programming. It gives the remainder of the integer division. For example, ##5 % 2 = 1## because the divisor 2 fits 2 wholly times into the dividend 5 leaving behind a reminder of 1.
## 6% 3 =0##...
(I could solve the problem but could not make sense of the solution given in the text. Let me put my own solutions below first).
1. Problem Statement : I copy and paste the problem to the right as it appears in the text.
2. My attempt : There are three "regions" where ##x## can lie.
(1)...
How did Voigt derive the average shear modulus of an anisotropic material, G=1/5 (A-B+3C), where 3A=c11+c22+c33, 3B=c12+c13+c23, 3C=c44+c55+c66?
The original text is published in German about 100 years ago. I looked for other papers explaining this, but none has explained the derivation. They...
My interest is on question 9. b(i)
Find the question and solution here;
I understand that ##a## should be less than ##2## because when ##a=2##, the two equations shall have same gradients which implies that the two lines are parrallel to each other. Now to my question, this solution does not...
I am trying to go through my old notes ...i came across this question,i do not have the solution.
Solve ##2|3x+4y-2|##+##3####\sqrt{25-5x+2y}##=##0##
Ok my approach on this;
##3####\sqrt{25-5x+2y}##=##-2|3x+4y-2|##
##9(25-5x+2y)=4(3x+4y-2)^2##
##9(25-5x+2y)=4(9x^2+16y^2+24xy-12x-16y+4)##...
This is the question * consider the highlighted question only *with its solution shown (from textbook);
My approach is as follows (alternative method),
##|\frac {5}{2x-3}| ##< ## 1##
Let, ##\frac {5}{2x-3} ##⋅##\frac {5}{2x-3}##=##1##
→##x^2-3x-4=0##
##(x+1)(x-4)=0##
it...
Could I please ask for help with the following question?
Four uniform rods of equal length l and weight w are freely jointed to form a framework ABCD. The joints A and C are connected by a light elastic string of natural length a. The framework is freely suspended from A and takes up the shape...
In this question, how does the step marked with 1 become the step marked with 2? I can see that the transitivity property of congruence is used, but I don’t know what exactly is going on here. Can someone please explain? Also at which step is Congruence Add and Multiply used?
Thanks...
Hello, I am learning trying to set up a code that searches for a certain pattern in the text by implementing Rabin-Karp Algorithm.
When setting up the Rabin Karp hash function, I have been told that I should set a value q in order to lower the time complexity of the function, and that q should...
Hi. I am in the IB and am looking at doing a physics IA (internal assessment) on Young's modulus on an elastic material. I was thinking of doing it on the the stretchy snakes like the candy. What would I need to cover in this?
I'm trying to figure out how to relate expected thermal expansion of a uv cured polymer within a rigid cylinder to a modulus specification. The issue is the expected change in refractive index due to thermal expansion. The expansion coefficients are not available. Anybody have an idea. Do...
We can rewrite |x-3|<10 in the following way.
-10<x-3<10
But can rewrite |x-3|+|x+1|+|x|<10 in the following way?
-10<x-3+x+1+x<10.
If we cannot, will anybody please explain why we cannot?
1) -|2x-3|+|5-x|+|x-10|=|3-x|
2) |2x-3|-|5-x|-|x-10|-|3-x|=28
3) -|2x-3|+|5-x|+|x-10|≥|3-x|
How can we solve these problems?
The method I know is to plug in the critical values to see which modulus becomes positive and which one becomes negative. Then find out the values of x for which the...
In a video, a person discussed how to solve modulus problems with a negative sign. This is the link of that video lecture.
He showed two methods to solve the problem. The first method is commonly used. Later he showed another method where he used a number line and a graph.
Unfortunately, I...
Summary:: What is Young's modulus of banana leaf in MPa?
What is Young's modulus of banana leaf in MPa?
[Thread moved from the technical forums, so no Schoolwork Template is shown]
In what cases it is better to call a thing "modulus" and in what cases "determinant"? In my algebra "determinant" is not a norm, discontinuous, positive for non-zero elements, not abiding triangle inequality. Should I better call it "modulus"?
Hi all,
I'm a little confused about something.
Force-extension graphs and stress-strain graphs are always both straight lines up until the limit of proportionality, implying both the spring constant and the Young modulus are constant up until then.
For a force-extension graph, Hooke's Law...
##|2x+3|-x=1##
i am getting ##x=-2## and ##x=\frac {-4}{3}## of which none satisfies the original equation, therefore we do not have a solution, right?
Problem 2 – Composites
A composite component (such as shown in Fig.2) is required for an aerospace application. The specification for the component stipulates that it must have an Elastic Modulus in the fibre direction of at least 320 GN/m2, and the transverse direction modulus must not be less...
Given the equation : ##|y| x = x##.
Two conditions are possible :
(1) ##\underline{y\geq 0}## : ##xy = x\Rightarrow \boxed{y = 1}\; (x \neq 0)##. We note that except for zero, ##-\infty<x<+\infty## for this case.
(2) ##\underline{y < 0}## : ##-xy = x\Rightarrow \boxed{y = -1}\; (x \neq 0)##...
I can't seem to figure out which chemical properties govern the physical property that is young's modulus. For example, any linear (or with a low degree of branching) polyethylene with no crosslinking is still a somewhat rigid and solid substance (higher ym), whereas the most linear possible...
The answer to the above problem is baffling, despite its straightforward nature. I will post the answer later, but here is my solution first.
Solution :
(1) ##2x+1 > x## : In this case, we have ##2x - x > -1 \Rightarrow \boxed{x > - 1}##
(2) ##2x+1 < -x## : In this case, we have ##2x+x < -1...
Hi. I'm a physician trying to understand the micromechanics of lung injury due to overdistension. The basic idea is that overstretching of the plasma membrane of the lung epithelial cell causes "stress failure" --> i.e. plasma membrane rupture --> cell death. The concepts of stress, strain, and...
1. I think the question is asking where is the graph of |3x-2| below the graph of 1/x.
To sketch the graph of y= |3x-2| draw the line of y=3x-2 and reflect the section with negative y-values in the x-axis. Alternatively, I could set 3x-2 ≥0, meaning |3x-2|=3x-2 so draw the line of y=3x-2. Then...
Hi. I've been given these sets of value. How do I calculate Young's modulus and Poisson Ratio from these set of value. I tried to create a stress vs strain graph but the graph does not look like a common stress vs strain graph but instead more of a y=x graph.
The diameter of the rod is 10mm and...
I just found a definition to the Young modulus as:
Is this a plausible representation of Y? That is, i know the definition , i don't think we can say this definition and the first definition is equal.
See this figure:
The rod is put at first between A and B shims without comprehension, suddenly a force is applied at R from an axis as the figure points. Find Fa and Fb. (the figure is a look from above)
This can be solved by consider the "constant elastic" of the shims equal, but my...
there are two materials:
Material A
Material B
surface area A/surface area B= 30
bulk modulus B/bulk modulus A= 11
How much more pressure is exerted through material B than material A in a collision, all else being equal?
I think the right solution is c). I'll pass on my reasoning to you:
R=6\, \textrm{cm}=0'06\, \textrm{m}
\sigma =\dfrac{10}{\pi} \, \textrm{nC/m}^2=\dfrac{1\cdot 10^{-8}}{\pi}\, \textrm{C/m}^2
P=0'03\, \textrm{m}
P'=10\, \textrm{cm}=0,1\, \textrm{m}
Point P:
\left.
\phi =\oint E\cdot...
In the equation for minimum link length here
https://www.defproc.co.uk/analysis/lattice-hinge-design-choosing-torsional-stress/
he uses a quantity Torsional Modulus, denoted as G, which I cannot seem to find either a definition for or any kind of expected value. Shear modulus (also commonly...
I am trying to calculate Young's Modulus on the graph below.
I have run the test through some software, now want to calculate Young's Modulus to ensure my analysis can be relied upon to be scientifically sound.
This is what i have calculated from learning online. Is this correct? I have 2 more...
As all attempts to get it right but without success this is one of the problems with my workout . Where i did wrong calculations ?
The questions got the answers in brackets.
I'm attempting to understand fully the distance ladder we use in astronomy to determine the distance to stars that are too far away for parallax to work. I understand we calibrate to a standard candle data of period vs luminosity for the cepheid variable stars in a group. Then from knowing the...
Summary:: why this answer ?
I have the result of a young s modulus that is 358280256.25 and the answer the teacher gave us is 3.58 x 10 to the power of 8 .
why not 358.3 x 10 to the power of 6 ?
how she s deciding how many steps goes back with the point , the answer of a sum before this...
given
$|y+3|\le 4$
we don't know if y is plus or negative so
$y+3\le 4 \Rightarrow y\le 1$
and
$-(y+3)\le 4$
reverse the inequality
$ y+3 \ge -4$
then isolate y
$y \ge -7$
the interval is
$-7 \le y \le 1$
A steel bar 6.00 m long and with rectangular cross section of 5.00 cm x 2.50 cm supports a mass of 2000 kg.
How much is the bar stretched ? (the young s modulus of steel is 20 x 10 n\m squared)