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# 物理代写|弹性力学代写Elasticity代考|MECH_ENG495 Principal stresses

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## 物理代写|弹性力学代写Elasticity代考|Principal stresses

We can again use the previous developments from Section $1.6$ to discuss the issues of principal stresses and directions. It is shown later in the chapter that the stress is a symmetric tensor. Using this fact, appropriate theory has been developed to identify and determine principal axes and values for the stress. For any given stress tensor we can establish the principal value problem and solve the characteristic equation to explicitly determine the principal values and directions. The general characteristic equation for the stress tensor becomes
$$\operatorname{det}\left[\sigma_{i j}-\sigma \delta_{i j}\right]=-\sigma^3+I_1 \sigma^2-I_2 \sigma+I_3=0$$
where $\sigma$ are the principal stresses and the fundamental invariants of the stress tensor can be expressed in terms of the three principal stresses $\sigma_1, \sigma_2, \sigma_3$ as
\begin{aligned} &I_1=\sigma_1+\sigma_2+\sigma_3 \ &I_2=\sigma_1 \sigma_2+\sigma_2 \sigma_3+\sigma_3 \sigma_1 \ &I_3=\sigma_1 \sigma_2 \sigma_3 \end{aligned}
In the principal coordinate system, the stress matrix takes the special diagonal form
$$\sigma_{i j}=\left[\begin{array}{ccc} \sigma_1 & 0 & 0 \ 0 & \sigma_2 & 0 \ 0 & 0 & \sigma_3 \end{array}\right]$$
A comparison of the general and principal stress states is shown in Fig. 3.5. Notice that for the principal coordinate system, all shearing stresses vanish and thus the state includes only normal

## 物理代写|弹性力学代写Elasticity代考|Spherical, deviatoric, octahedral, and von Mises stresses

As mentioned in our previous discussion on strain, it is often convenient to decompose the stress into two parts called the spherical and deviatoric stress tensors. Analogous to relations (2.5.1) and (2.5.2), the spherical stress is defined by
$$\tilde{\sigma}{i j}=\frac{1}{3} \sigma{k k} \delta_{i j}$$
while the deviatoric stress becomes
$$\widehat{\sigma}{i j}=\sigma{i j}-\frac{1}{3} \sigma_{k k} \delta_{i j}$$

Note that the total stress is then simply the sum
$$\sigma_{i j}=\tilde{\sigma}{i j}+\widehat{\sigma}{i j}$$
The spherical stress is an isotropic tensor, being the same in all coordinate systems (as per the discussion in Section 1.5). It can be shown that the principal directions of the deviatoric stress are the same as those of the stress tensor itself (see Exercise 3.15).

We next briefly explore a couple of particular stress components or combinations that have been defined in the literature and are commonly used in formulating failure theories related to inelastic deformation. It has been found that ductile materials normally exhibit inelastic yielding failures that can be characterized by these particular stresses.

Consider first the normal and shear stresses (tractions) that act on a special plane whose normal makes equal angles with the three principal axes. This plane is commonly referred to as the octahedral plane. Determination of these normal and shear stresses is straightforward if we use the principal axes of stress. Since the unit normal vector to the octahedral plane makes equal angles with the principal axes, its components are given by $n_i=\pm(1,1,1) / \sqrt{3}$. Referring to Fig. $3.6$ and using the results of the previous section, relations (3.4.7) give the desired normal and shear stresses as
\begin{aligned} N=\sigma_{o c t} &=\frac{1}{3}\left(\sigma_1+\sigma_2+\sigma_3\right)=\frac{1}{3} \sigma_{k k}=\frac{1}{3} I_1 \ S=\tau_{o c t} &=\frac{1}{3}\left[\left(\sigma_1-\sigma_2\right)^2+\left(\sigma_2-\sigma_3\right)^2+\left(\sigma_3-\sigma_1\right)^2\right]^{1 / 2} \ &=\frac{1}{3}\left(2 I_1^2-6 I_2\right)^{1 / 2} \end{aligned}

## 物理代写|弹性力学代写弹性代考|主应力

$$\operatorname{det}\left[\sigma_{i j}-\sigma \delta_{i j}\right]=-\sigma^3+I_1 \sigma^2-I_2 \sigma+I_3=0$$
，其中$\sigma$为主应力，应力张量的基本不变量可以用三个主应力$\sigma_1, \sigma_2, \sigma_3$表示为
\begin{aligned} &I_1=\sigma_1+\sigma_2+\sigma_3 \ &I_2=\sigma_1 \sigma_2+\sigma_2 \sigma_3+\sigma_3 \sigma_1 \ &I_3=\sigma_1 \sigma_2 \sigma_3 \end{aligned}

$$\sigma_{i j}=\left[\begin{array}{ccc} \sigma_1 & 0 & 0 \ 0 & \sigma_2 & 0 \ 0 & 0 & \sigma_3 \end{array}\right]$$

## 物理代写|弹性力学代写弹性代考|球形，偏离，八面体，和冯米塞斯应力

$$\tilde{\sigma}{i j}=\frac{1}{3} \sigma{k k} \delta_{i j}$$
，而偏应力定义为
$$\widehat{\sigma}{i j}=\sigma{i j}-\frac{1}{3} \sigma_{k k} \delta_{i j}$$

$$\sigma_{i j}=\tilde{\sigma}{i j}+\widehat{\sigma}{i j}$$

\begin{aligned} N=\sigma_{o c t} &=\frac{1}{3}\left(\sigma_1+\sigma_2+\sigma_3\right)=\frac{1}{3} \sigma_{k k}=\frac{1}{3} I_1 \ S=\tau_{o c t} &=\frac{1}{3}\left[\left(\sigma_1-\sigma_2\right)^2+\left(\sigma_2-\sigma_3\right)^2+\left(\sigma_3-\sigma_1\right)^2\right]^{1 / 2} \ &=\frac{1}{3}\left(2 I_1^2-6 I_2\right)^{1 / 2} \end{aligned}

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