机械学代写|MECHANICS MATHS2034 University of Glasgow Assignment

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Mechanics is the branch of physics that deals with the study of the motion, forces, and energy of objects. It is concerned with how objects move and the forces that cause this motion. The study of mechanics involves the development of mathematical models and theories to describe and explain the behavior of physical systems, such as machines, structures, and particles. Some of the key concepts in mechanics include Newton’s laws of motion, conservation of energy, and momentum. Mechanics is essential in many fields of science and engineering, including aerospace engineering, mechanical engineering, and materials science.

机械学代写|MECHANICS MATHS2034 University of Glasgow Assignment

问题 1.

Suppose the state of stress at a point in a $x, y, z$ coordinate system is given by
$$
\left[\begin{array}{ccc}
100 & 1 & 180 \
0 & 20 & 0 \
180 & 0 & 20
\end{array}\right]
$$
a. Calculate the three invariants of this stress tensor.

证明 .

a) Invariants of stress tensor
Recall: these values do not change no matter the coordinate system selected
$$
\begin{gathered}
I_1=\sigma_x+\sigma_y+\sigma_z=-100+20+20 \
I_2=\sigma_x \sigma_y+\sigma_y \sigma_z+\sigma_z \sigma_x-\tau_{x y}{ }^2-\tau_{y z}{ }^2-\tau_{z x}{ }^2 \
=(-100 \times 20)+(-100 \times 20)+20^2-0-0-80^2 \
I_1=-60 \
I_2=-10000 \
=\sigma_x \sigma_y \sigma_z+2 \tau_{x y} \tau_{y z} \tau_{z x}-\sigma_x \tau_{y z}{ }^2-\sigma_y \tau_{z x}{ }^2-\sigma_z \tau_{x y}{ }^2 \
=(-100 \times 20 \times 20)+2 \times 0-0-20 \times(-80)^2-0 \
I_3=-168000
\end{gathered}
$$


问题 2.

b. Determine the three principal stresses of this stress tensor.

证明 .

The eigenvalue problem for a stress tensor

$$
\left[\begin{array}{ccc}
-100 & 0 & -80 \
0 & 20 & 0 \
-80 & 0 & 20
\end{array}\right]
$$
is given by
$$
\operatorname{det}\left[\begin{array}{ccc}
-100-\lambda & 0 & -80 \
0 & 20-\lambda & 0 \
-80 & 0 & 20-\lambda
\end{array}\right]=0
$$
Solve for $\lambda$, we have three eigenvalues
$$
\begin{aligned}
& \lambda_1=-140 \
& \lambda_2=60 \
& \lambda_3=20
\end{aligned}
$$
The principal stresses are the three eigenvalues of the stress tensor
$$
\begin{aligned}
\sigma_{x p} & =-140 \
\sigma_{y p} & =60 \
\sigma_{z p} & =20
\end{aligned}
$$

问题 3.

Suppose the state of stress at a point relative to a $x, y, z$ coordinate system is given by:
$$
\left[\begin{array}{cc}
15 & -10 \
-10 & -5
\end{array}\right]
$$
Try to find a new coordinate system $\left(x^{\prime}, y^{\prime}\right)$ that corresponds to the principal directions of the stress tensor.
a. Find the principal stresses.

证明 .

a) Determine principal stresses
The eigenvalue problem for a stress tensor
$$
\left[\begin{array}{cc}
15 & -10 \
-10 & -5
\end{array}\right]
$$
is given by
$$
\operatorname{det}\left[\begin{array}{cc}
15-\lambda & -10 \
-10 & -5-\lambda
\end{array}\right]=0
$$
Solve for $\lambda$, we have two eigenvalues
$$
\begin{aligned}
& \lambda_1=19.14 \
& \lambda_2=-9.14
\end{aligned}
$$
The principal stresses are the two eigenvalues of the stress tensor
$$
\begin{aligned}
& \sigma_{x p}=19.14 \
& \sigma_{y p}=-9.14
\end{aligned}
$$

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