But this curve is just an approximation.

但這條曲線只是一個近似值。

It doesn't represent the actual stress or strain in the test piece during the tensile test.

它並不代表拉伸試驗過程中試件的實際應力或應變。

In this video, we're going to talk about true stress and true strain.

在本視頻中，我們將討論真正的壓力和真正的應變。

In the typical stress-strain curve, stress is defined as the applied force divided by the initial cross-sectional area of the test specimen, and strain is defined as the change in specimen length divided by the initial length.

在典型的應力-應變曲線中，應力被定義為外力除以試樣的初始橫截面積，應變被定義為試樣長度的變化除以初始長度。

These stress and strain values are actually just approximations of the true stress and strain in the specimen.

這些應力和應變值實際上只是試樣中真實應力和應變的近似值。

We call them engineering stress and engineering strain, and denote them using the subscript

我們將其稱為工程應力和工程應變，並用下標表示

E. This curve is known as the engineering stress-strain curve.

E.這條曲線被稱為工程應力-應變曲線。

To determine the true stress and strain values, we would need to consider the fact that the dimensions of the specimen change throughout the duration of the test.

要確定真實的應力和應變值，我們需要考慮試樣的尺寸在整個試驗過程中會發生變化這一事實。

If we were to measure the true stress and strain, our stress-strain curve would look something like this.

如果我們要測量真實的應力和應變，我們的應力-應變曲線將如下所示。

You can differentiate a true curve from an engineering curve by noting that the engineering curve drops after necking, whereas the true curve is always increasing.

要區分真實曲線和工程曲線，可以注意到工程曲線在縮頸後會下降，而真實曲線始終在上升。

Remember that necking is the rapid reduction in the cross-sectional area of the test piece, which begins when the engineering curve reaches its maximum value, which is the ultimate tensile strength of the material.

請記住，縮頸是指試件橫截面積的快速減小，當工程曲線達到最大值（即材料的極限抗拉強度）時開始出現。

So if they do not accurately reflect the true stress and strain in the test specimen, why do engineers commonly use engineering curves instead of true curves?

那麼，如果它們不能準確反映試驗試樣中的真實應力和應變，為什麼工程師通常使用工程曲線而不是真實曲線呢？

Well, there are two main reasons.

主要原因有兩個。

Firstly, it's quite difficult to measure the instantaneous cross-sectional area during a tensile test, and so most of the time we just don't have the true stress-strain curves.

首先，在拉伸試驗中測量瞬時橫截面積相當困難，是以大多數情況下我們無法獲得真實的應力-應變曲線。

And secondly, most of the time we are only analyzing or designing things which deform within the elastic region.

其次，大多數情況下，我們只分析或設計在彈性區域內變形的物體。

As you can see here, the engineering and true curves are very similar for small strain values.

如圖所示，對於小應變值，工程曲線和真實曲線非常相似。

But for cases where we have large plastic deformation, the difference between the two curves becomes significant.

但在塑性變形較大的情況下，兩條曲線之間的差異就會變得很大。

This is in large part due to the sudden reduction in the cross-sectional area of the test piece when necking occurs.

這在很大程度上是由於發生縮頸時，試件的橫截面積會突然減小。

When assessing cases where we have significant plastic deformation, it becomes important to use the true stress-strain curves.

在評估有明顯塑性變形的情況時，使用真實的應力-應變曲線變得非常重要。

Examples of this might be the analysis of manufacturing processes, or performing finite element analysis which models large strains.

例如，對製造過程進行分析，或進行模擬大應變的有限元分析。

By making a few assumptions, we can calculate the true stress-strain curve based on the engineering curve, and so we can avoid having to measure the instantaneous cross-sectional area during the tensile test.

通過一些假設，我們可以根據工程曲線計算出真實的應力-應變曲線，從而避免在拉伸試驗中測量瞬時橫截面積。

Unlike engineering stress, which is calculated by dividing the applied force by the initial cross-sectional area of the test piece, true stress is calculated by dividing by the instantaneous cross-sectional area at each instant throughout the test.

工程應力的計算方法是用外力除以試驗件的初始橫截面積，而真實應力的計算方法則是用整個試驗過程中每個瞬間的瞬時橫截面積除以外力。

It accounts for the fact that the cross-sectional area of the test piece is changing as the test is performed.

在測試過程中，測試片的橫截面積會發生變化。

We can adjust this equation for true stress by assuming that the volume of the test piece remains constant.

我們可以假設試件的體積保持不變，從而根據真實應力調整該方程。

This assumption is valid in the elastic region of the stress-strain curve, because any volume changes in the elastic region will be small, and it is valid in the plastic region of the curve because materials are considered to be incompressible during plastic deformation.

這一假設在應力-應變曲線的彈性區域有效，因為彈性區域的任何體積變化都很小；在曲線的塑性區域有效，因為材料在塑性變形過程中被認為是不可壓縮的。

I talk more about material incompressibility in my video on Poisson's ratio.

我在關於泊松比的視頻中詳細介紹了材料的不可壓縮性。

If volume remains constant, the product of the instantaneous area and the instantaneous length is equal to the product of the original area and the original length.

如果體積不變，則瞬時面積與瞬時長度的乘積等於原始面積與原始長度的乘積。

But it is important to know that this assumption is not valid after necking has occurred, because of the associated change in the cross-sectional area.

但重要的是要知道，由於橫截面積的相關變化，這一假設在縮頸發生後就失效了。

Beyond necking, we would need to base the true stress on actual measurements of the cross-sectional area.

除了縮頸之外，我們還需要根據橫截面積的實際測量結果來確定真正的應力。

But anyway, we can rearrange this equation so that instantaneous area is on the left hand side, and then substitute it into our equation for true stress.

但無論如何，我們可以重新排列這個等式，使瞬時面積位於左側，然後將其代入真實應力的等式中。

The definition of engineering strain is that it is the change in length divided by the original length.

工程應變的定義是長度變化除以原始長度。

We can rearrange this equation to the form L divided by L0 minus 1, and we can use this to obtain an equation for true stress which is a function of the engineering stress and engineering strain, both of which can easily be obtained from a tensile test.

我們可以將該方程重新排列為 L 除以 L0 再減去 1 的形式，並由此得出真實應力方程，該方程是工程應力和工程應變的函數，而這兩者都可以通過拉伸試驗輕鬆獲得。

Now let's derive an equation for true strain.

現在我們來推導真實應變的方程。

True strain needs to consider the fact that the original length of the specimen is continuously changing at each instant throughout the duration of the tensile test.

真實應變需要考慮這樣一個事實，即在整個拉伸試驗過程中，試樣的原始長度在每個瞬間都在不斷變化。

We could calculate it by splitting the tensile test into increments, and calculating the change in strain at each increment based on the length at the start and end of the increment.

我們可以將拉伸試驗抽成若干個增量，然後根據增量開始和結束時的長度計算每個增量的應變變化。

In this example, I will consider 3 increments.

在這個例子中，我將考慮 3 個增量。

The increments can then be summed up to calculate the true strain at the end of increment 3, for example.

然後將增量相加即可計算出增量 3 結束時的真實應變。

Instead of doing this manually with large increment sizes, this approach can be defined mathematically using integration, like this.

這種方法無需手動操作，增量也不大，而是可以用數學方法進行整合，就像這樣。

By remembering that the integral of 1 over x is ln of x plus c, we end up with this equation, which can be rearranged to be a function of the engineering strain.

記住 1 對 x 的積分是 x 的 ln 加上 c，我們就得到了這個方程，它可以重新排列為工程應變的函數。

Because of the form of this equation, true strain is also known as logarithmic strain, or natural strain.

由於該方程的形式，真實應變也被稱為對數應變或自然應變。

I hope this has helped explain the differences between engineering and true stress-strain curves.

希望以上內容有助於解釋工程曲線和真實應力應變曲線之間的區別。

Thanks for watching, and remember to hit the subscribe button and the bell to be notified about future videos.

感謝您的收看，記得點擊訂閱按鈕和鈴鐺，以便收到未來視頻的通知。