04 Atomic Definition of Young's Modulus
Summary
TLDR本视频讲解了弹性行为和弹性变形的概念。通过弹性带的例子,说明了材料在弹性区域内加载后卸载能够恢复原状的特性。视频中探讨了原子层面的解释,将原子模拟为硬球体并通过弹簧连接,展示了原子间的净力与原子间距的关系。强调了杨氏模量与原子间力-间距曲线在平衡间距处的一阶导数成正比,表明杨氏模量仅取决于原子类型,与材料结构无关,是材料固有的属性。
Takeaways
- 🔍 弹性行为指的是材料在外力作用下发生形变,移除外力后能够恢复到原始形状的特性。
- 📏 胡克定律描述了材料在弹性范围内,应力与应变之间的线性关系。
- 🔵 弹性带是一个展示弹性行为的直观例子,它在被拉伸后能够恢复到原始形状。
- 🔬 弹性变形可以被理解为材料在卸载后能够恢复到原始几何形状的现象。
- 🧠 弹性应变也被称为可恢复应变,因为材料在弹性区域内卸载后应变会恢复到零。
- 🌐 原子层面上,弹性行为意味着当外力移除后,原子能够回到它们原始的位置。
- 🎾 通过将原子模拟为硬球体,并用弹簧连接,可以建立一个机械模型来模拟原子间的相互作用。
- 📉 原子间的净力可以通过绘制原子间力与原子间距的关系曲线来表示,其中包括吸引力和排斥力。
- 🔄 当原子处于平衡状态时,净力为零,此时的原子间距被称为平衡原子间距。
- 🔗 杨氏模量与原子间力-间距曲线在平衡原子间距处的一阶导数成正比,表明它仅取决于原子类型,与材料的结构无关。
Q & A
什么是弹性行为?
-弹性行为指的是材料在外力作用下发生形变,当外力移除后,材料能够恢复到原始几何形状的特性。
胡克定律是什么?
-胡克定律是描述材料在弹性范围内,应力与应变成正比关系的定律。
弹性变形和弹性区域在材料科学中意味着什么?
-弹性变形是指材料在卸载后能够恢复到原始几何形状的变形。弹性区域是指材料在受到应力作用时,应力与应变呈线性关系,且卸载后能够完全恢复的区域。
什么是可恢复应变?
-可恢复应变,也称为弹性应变,是指材料在卸载后能够恢复到原始形状的应变,通常与弹性行为相关联。
原子层面上,弹性变形意味着什么?
-在原子层面上,弹性变形意味着当外力移除后,原子能够回到它们原始的位置。
如何用机械模型来模拟原子间的相互作用?
-可以通过将原子视为硬球体,并在它们之间放置一个代表原子间净力的弹簧来模拟原子间的相互作用。
什么是原子间的平衡距离?
-原子间的平衡距离是指在没有外力作用下,原子之间自然保持的距离,也称为平衡原子间距。
原子间的净力与原子间距有什么关系?
-原子间的净力与原子间距的关系可以通过一个曲线来描述,其中净力是吸引力和排斥力的总和,且在平衡间距附近近似为线性关系。
杨氏模量与原子间力-分离曲线有什么关系?
-杨氏模量与原子间力-分离曲线的关系是直接成比例的,具体来说,杨氏模量是该曲线在平衡原子间距处的一阶导数。
为什么说杨氏模量是结构独立的?
-杨氏模量是结构独立的,因为它只取决于材料中原子的类型,而与材料的宏观结构、化学组成或微观结构无关。
材料的弹性行为如何影响其杨氏模量?
-材料的弹性行为直接影响其杨氏模量,因为杨氏模量是描述材料在弹性范围内应力与应变关系的物理量。
Outlines
🔍 弹性行为与弹性变形
第一段主要探讨了弹性行为和弹性变形的概念。通过弹性带的例子,说明了物体在外力作用下形变后能够恢复原状的特性。引入了应力-应变曲线,解释了材料在弹性区域内的线性行为,并提出了弹性变形即物体在卸载后能够恢复到原始几何形状的观点。进一步探讨了弹性变形的原子层面解释,即原子在外力作用下移动后能够在卸载后返回到原始位置。最后,通过简化模型,将原子视为硬球并通过弹簧连接,来模拟原子间的相互作用力,为后续深入探讨弹性行为的微观机制奠定了基础。
📏 原子间距与原子间力的关系
第二段深入讨论了原子间距(R)的概念,即原子核之间的距离,并引入了原子间力的概念。通过简化模型,展示了原子间的吸引力和排斥力,并解释了这两种力的合力如何影响原子间的平衡状态。重点介绍了净原子间力的概念,并指出在平衡状态下,净原子间力为零。此外,还讨论了在平衡点附近,原子间力与原子间距的关系近似为线性,从而引出了原子间的弹簧常数概念。这一部分为理解材料的弹性行为提供了微观层面的解释,并为后续讨论材料的宏观力学性质打下了基础。
🔗 杨氏模量与原子间力分离曲线的关系
第三段进一步探讨了杨氏模量与原子间力分离曲线之间的关系。通过分析小扰动下的原子位移与所施加的力之间的关系,得出了杨氏模量与原子间力分离曲线在平衡间距处的一阶导数成正比的结论。强调了杨氏模量仅依赖于原子类型,而与材料的结构无关,即它是结构独立的。这一发现对于理解不同材料的弹性特性具有重要意义,因为它表明改变材料的微观结构,如合金成分或强化处理,不会改变其杨氏模量,除非改变了原子类型。
Mindmap
Keywords
💡弹性行为
💡胡克定律
💡应力-应变曲线
💡弹性变形
💡原子解释
💡原子间距离
💡吸引力与排斥力
💡净原子间力
💡杨氏模量
💡结构独立性
Highlights
弹性行为的基本概念,即物体在受到拉伸后释放能够恢复原状。
胡克定律的介绍,即材料在弹性区域内的应力与应变成正比。
弹性带的非线性弹性行为示例。
弹性变形的定义,即样品在卸载后能够恢复到原始几何形状。
弹性应变与可恢复应变的等同性。
原子层面上弹性变形的解释,即原子在卸载后返回到原始位置。
硬球模型用于模拟原子间的相互作用。
原子间净力的概念,包括吸引和排斥力。
平衡状态下原子间的净力为零,定义了平衡原子间距R。
原子间力-位移关系的微观解释,与宏观的弹簧常数类比。
杨氏模量与原子间力-位移曲线一阶导数的关系。
杨氏模量仅取决于原子类型,与材料结构无关。
材料的微观结构变化不影响杨氏模量。
杨氏模量与材料强度的关系,即强度可以改变但杨氏模量不变。
原子间相互作用力的曲线图解释,包括吸引和排斥力的平衡。
弹性变形的微观机制,即原子间距的微小变化。
弹性带的弹性行为与原子间力-位移曲线的联系。
Transcripts
okay so last CL uh last lecture or video
I mean we had uh looked at um elastic
behavior and you know we said that you
stretch something out and you let it go
it goes back in fact what we had looked
at was hooks law um and what I want to
explore a little B it's more what that
elasticity means so in fact this elastic
band is a good example you stretch it
out and you let it go and it returns to
its original geometry
so what we had seen was this we looked
at a stress strin
curve and we said that most
materials have a linear region initially
now this elastic band we'll explore
later is actually more nonlinear but it
it illustrates the point of elasticity
so what I want to explore is what
elasticity really
means
Okay so so what
is elastic
deformation and I think that the
explanation that we had right here is
pretty reasonable it's intuitive we
could say that the
sample returns to its original
geometry uh when we unload it
right uh upon
unloading that's fairly intuitive the
other thing we could do is we could look
at this uh stress Trin behavior and say
well if we load the sample up to a point
and it's still in the elastic region if
we then unload we would expect to get
back to zero stress and zero strain so
we could say then that
the
strain is recovered in fact you know
what I'm going to do I'm going to write
that in Orange
tell you why in a second
recoverable because often elastic strain
is referred to as recoverable strain
they're used synonymously okay so that's
a fairly good explanation I think it's
intuitive the final thing I want to look
at though
is could we give an atomic explanation
for this we got a couple of atoms you
know what does it mean if we apply a
stress to um this this paper CL
right and I you know I bend it or I I I
put this paper clip onto some paper and
I take it off and it looks to be the
same geometry as when it started what
does that tell us that has happened to
the atoms inside that
paperclip I think that it's not a large
stretch of the imagination to to say
that it must mean the
atoms also return to their original
positions
upon
unloading uh upon unloading there we
go and in fact it's that explanation
there that I'd like to explore in a
little bit more detail and i' like to do
that
by considering we could modeling atoms
as um a couple of hard spheres so what
I'm trying my best to uh do here is to
sketch out a sphere so I'm going to try
to shade this in as you know I like to
do so it looks like it's popping over
the page so there you go that's not too
shabby he I'm not an artist but that's
uh looks like it you could believe
perhaps that's a a circle I mean a
sphere not a circle it's a sphere and so
I'll duplicate that and there we go
we've now
got um
a couple of hard spheres and I'm going
to draw a little spring between them so
that is a
spring holding together these
two hard
spheres and so what this is of course is
it's a model it's a mechanical model of
it's a mechanical
model give myself a little bit of space
here a mechanical model of oops I leave
that um a mechanical model of the atoms
undergoing
elastic uh Behavior elastic
Behavior okay so this the mechanical
model and
and what the spring is doing is it's
modeling whatever the net force
is between atoms
okay the last thing I'm going to do with
this little sketch is I'm going to
actually Dimension
here the space between the
atoms and that is going to be called for
historical reasons R okay that is the
I'll give you the formal name it's
called the interatomic spacing
I appreciate it can be frustrating
because we're using an R here for
spacing between two atoms and the atoms
have a rad have their own radi right so
you might get confused between radius
and interatomic spacing but remember
from Context if you know we're talking
about interatomic spacing R is going to
refer to the distance between the nuclei
and it's it's an unfortunate bit of
History I guess um it comes from from
physics so you know blame the physicists
but don't blame the physicists my
father's a physicist but that's what's
used and so we got to deal with it um
for this course I'm gonna I'm gonna
promise you or try my very best to use
capital r for radius but from Context I
do hope that you will know the
difference so that's interatomic spacing
and what we can now do that we've got
this little crude mechanical model is we
can we can
plot
now the
Force the interatomic force I'll write
that in so it's clear interatomic
force and we can plot that against the
interatomic spacing
R and what we'll find is there's there's
some force that has to hold things
together we're going to explore later in
the course what that is but the curve
for that looks something like this and
so this is the attractive
Force so attract is positive
here and this is repulsive pushing them
apart so you know there has to be
something that opposes that otherwise
everything would just collapse down
infinitely close so there has to be
something that's pushing them apart and
thankfully there is and it's this force
that picks up very rapidly at really
close spacing when the atoms get really
close together they get close enough
that the electrons around the atom start
to see each other and repel so what
we're actually most interested in is the
sum of the attractive and the repulsive
forces let me label that
repulsive so what we are interested in
here is the sum of these two and the sum
of these two looks something like this
try my best to sketch
this follows that fa closely so that is
the net interatomic
force which is of course what we are
most interested in because that's what's
telling us really what the atoms are
feeling in this little model here right
these two atoms at rest the net force
should be zero in fact let's look at
that right here when they're at rest or
they're at
equilibrium the net
force equals zero right that force is
zero well what does that also tell us
well it tells us there must be a special
value of R in fact that value of R has
to be the distance between the atoms at
rest or at equilibrium and so we in fact
define r r as this special value of r r
KN which we call the U
equilibrium interatomic
spacing okay now why is all this
important well here it is this is all
important because you can
see now
that if we look closely at the curve
here in this region here very close to r
equal to R
KN the curve looks almost like a
straight line so if we take the tangent
to that
curve or the slope there close to r
equal R KN we can also write that
mathematically as the first derivative
of that curve DF by
Dr well that slope is something that
should be kind of familiar to us
it's essentially we're looking at the
relationship between force and
displacement and we did that
macroscopically and then we came up with
the spring constant so it's almost like
we've now got a little a tiny little
spring constant for these hard spheres
connected by a spring so that makes a
bit of sense but what we're also doing
is we're saying well this was our model
for how a material behaves we know if I
stretch this elastic
band this not elastic this paperclip if
I if I deform this a little bit and move
it just a little bit I must be moving
atoms in there okay we're just talking
about small disturbances we're not
talking about you know I take it and I
put it on a book and you know it that
that's something else we're going to
explore that in another video but right
now we're just talking about little
disturbances when it's still
elastic
so when they're still elastic there's
some kind of a relationship between
force in fact we could apply a certain
Force force and observe the resulting
displacement so the conclusion which I
hope is not a big stretch of your your
imagination or your understanding is
that in fact there is a relationship
then between the Young's
modulus this is the Young's
modulus okay
and this interatomic Force separation
curve and the relationship is in fact
this it's that yung's modulus is
directly proportional to the first
derivative of the interatomic force
separation curve at R equals to R KN and
that is actually an important
result why because I'm going to give it
a little box because it's kind of an
important conclusion that is important
because it tells
us it tells us that the Young's modulus
then depends only on the type of
atoms okay it depends only on the type
of atoms you have not what you've done
to the
material only
on type of atoms there's another way of
stating that that you you may come
across and it's this that the Young's
modulus is structure independent
structure in fact if you can permit me
I'm GNA because it's so commonly used
I'm actually going to write it out in
this my orange color so you realize this
is very specific usage um oh my let me
correct this structure independent I got
excited
independent let me tell you just what
that means so it's structure independent
and that
means we can make changes to the
um
ex example um small
changes to say the even the chemistry or
the
composition okay of an
alloy what you would probably in high
school call concentration which is
really equivalent
here do not change the Young's
modulus nor does other stuff that we'll
talk about later like strengthening so
we'll see that you can dramatically
increase the strength of of a metal
alloy but you will not change the
Young's modulus why because it's
structure independent because at the end
of the day the Young's modulus only
depends on the type of atoms that you
have and if you haven't changed the type
of atoms you won't change the young
mindist oh it's so beautiful all right
thank you very much
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