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One of the most classical examples in the mechanics of materials is that engineering strain is not tensor. I want to use Mathematica to show why it doesn't meet the tensor requirements.

Cauchystrain = {{εx, γxy/2, γxz/2}, {γxy/2, εy, γyz/2}, {γxz/2, γyz/2, εz}};
Engineeringstrain = {{εx, γxy, γxz}, {γxy, εy, γyz}, {γxz, γyz, εz}};
EulerMatrix[{α, β, γ}].Engineeringstrain.(EulerMatrix[{α, β, \
γ}]\[Transpose])
EulerMatrix[{α, β, γ}].Cauchystrain.(EulerMatrix[{α, β, γ}]\
\[Transpose])

In the above formula, $\frac{1}{2} γxy = εxy, \frac{1}{2} γxz = εxz, \frac{1}{2} γyz = εyz$.

I want to know why engineering strain is not a tensor. It is better to demonstrate the specific difference between engineering strain and Cauchy strain with Mathematica.

Question notes: It may be a difficult mathematical problem to prove that engineering strain is not tensor according to the definition of the tensor.

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    $\begingroup$ "One of the most classical examples in the mechanics of materials is that engineering strain is not tensor. " I never heard of that, and fails to find any reference with a quick googling, can you add one? "I want to know why engineering strain is not a tensor. " If this is your real question i.e. you only happens to know the statement above but don't know why, then I'm afraid the question is off-topic (or at least in the gray zone) here. Better to ask in physics.stackexchange.com or engineering.stackexchange.com $\endgroup$ – xzczd Mar 24 at 4:09
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    $\begingroup$ An english post would be helpful....i atleast cannot read whatever language that is....(possibly japanese?) and on top of that wiki even suggest strain is a tensor... $\endgroup$ – morbo Mar 24 at 10:39
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    $\begingroup$ In your reference it seems that there is no difference between the *engineering strain" and the "Cauchy strain". In this article, the definition is given on a very elementary level, such that tensorial properties are hidden. $\endgroup$ – Alexei Boulbitch Mar 24 at 11:00
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    $\begingroup$ @morbo The text in the above post is Chinese. I'm sorry that my professional terms of English are insufficient to explain my problems clearly. $\endgroup$ – A little mouse on the pampas Mar 24 at 11:15
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    $\begingroup$ In the "reference" given by OP: “张量是可以用 $3×3×…$ 的矩阵来表达的,但是这二者并不等同。你要说什么样子的矩阵是张量,这个我所知的没有一个很好的判据。” Translation: "A tensor can be expressed with a $3×3×…$ matrix, but a matrix is not equivalent to a tensor. Then what kind of matrix is tensor? There's no good criterion, AFAIK. " I'm voting to close this question because it's unclear. $\endgroup$ – xzczd Mar 24 at 13:13
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The following demonstration process is not a strict proof. The main meaning is that the invariance of strain in the rotation of the coordinate system is for the derivation of the displacement field, not for the isolated strain matrix.

u[x_,y_,z_]:=x
v[x_,y_,z_]:=-y


RotationTransform[-Pi/4][{u[x,y,z],v[x,y,z]}]
RotationMatrix[-Pi/4].{x,y}

(Eliminate[{#,x1==Sqrt[2]/2 (x+y),y1==Sqrt[2]/2 (y-x)},{x,y}]&/@{u1==Sqrt[2]/2 x-Sqrt[2]/2 y,v1==-(Sqrt[2]/2)x-Sqrt[2]/2 y})//FullSimplify

u1[x_,y_,z_]:=-y
v1[x_,y_,z_]:=-x

({{D[u[x,y,z],x],(D[u[x,y,z],y]+D[v[x,y,z],x])},{(D[u[x,y,z],y]+D[v[x,y,z],x]),D[v[x,y,z],y]}}//Evaluate)(*The initial strain state is selected cleverly,excluding the controversial angular strain*)

{{D[u1[x,y,z],x],(D[u1[x,y,z],y]+D[v1[x,y,z],x])},{(D[u1[x,y,z],y]+D[v1[x,y,z],x]),D[v1[x,y,z],y]}}//Evaluate(*After rotating the coordinate system 45 degrees,a new expression of strain matrix expressed by displacement is obtained*)

RotationMatrix[-Pi/4].{{1,0},{0,-1}}.(RotationMatrix[-Pi/4]\[Transpose])
(*When the strain is rotated 45 degrees,the difference between the real strain state and the engineering strain matrix expressed by displacement is twice*)

enter image description here

Since there are two times differences between the engineering strain matrix expressed by displacement and the actual strain state when the coordinate system is rotated 45 degrees. The engineering strain expressed by displacement does not satisfy the invariance of coordinate rotation. However, the Cauchy strain obtained by dividing the engineering strain expressed by displacement by 2 can keep the coordinate rotation invariance.

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