## 物理代写|力学代写mechanics代考|Stress-Optical Equations

Consider a plane transparent specimen of a birefringent material. According to the Neuman-Maxwell stress-optical law, when a light beam is incident on the specimen it is divided into two linearly polarized beams along the principal stress directions and the index of refraction changes according to the following equation
\begin{aligned} &\Delta n_1=n_1-n=b_1 \varepsilon_1+b_2\left(\varepsilon_2+\varepsilon_3\right) \ &\Delta n_2=n_2-n=b_1 \varepsilon_2+b_2\left(\varepsilon_1+\varepsilon_3\right) \end{aligned}
where
For linear elastic materials under conditions of plane stress $\left(\sigma_3=0\right)$ we have according to Hooke’s law
$$\varepsilon_1=\frac{1}{E}\left(\sigma_1-v \sigma_2\right), \quad \varepsilon_2=\frac{1}{E}\left(\sigma_2-v \sigma_1\right), \quad \varepsilon_3=-\frac{v}{E}\left(\sigma_1+\sigma_2\right)$$
where $\sigma_1$ and $\sigma_2$ are the principal stresses, and $E$ and $v$ are the modulus of elasticity and Poisson’s ratio.
When Eq. (6.29) is expressed in terms of stresses it takes the form

\begin{aligned} &\Delta n_1=n_1-n=A \sigma_1+B \sigma_2 \ &\Delta n_2=n_2-n=B \sigma_1+A \sigma_2 \end{aligned}
where
$$A=\frac{b_1-2 v b_2}{E}, \quad B=\frac{b_2-v\left(b_1+b_2\right)}{E}$$
Consider a linearly polarized light beam normally traversing and passing through the specimen along the direction of the principal stress $\sigma_1$. Let $\mathrm{A}$ and $\mathrm{B}$ be two reference points along the light ray on opposite sides of the specimen (Fig. 6.6). The optical length (product of geometrical length times the index of refraction) between points $\mathrm{A}$ and $\mathrm{B}, s_A=(A B)$, is given by
$$S_A=L n_0+d\left(n-n_0\right)$$

## 物理代写|力学代写mechanics代考|Principle of the Method

In the method of caustics, a transparent or opaque specimen with a crack is illuminated by a light beam and the obtained optical effect is observed on a screen placed at some distance from the specimen. The reflected or transmitted rays undergo a change of their optical path due to the variation of the refractive index and the thickness of the specimen when it is loaded. The reflected or transmitted rays near the crack tip, due to the existing stress singularity, deviate and generate a highly illuminated three-dimensional surface in space (Fig. 6.8). When this surface is intersected by a reference screen, a highly illuminated curve, the so-called caustic is formed. The caustic surrounds the crack tip. It is the image on the reference screen of the initial curve on the specimen. The caustic separates an inside dark area from an outside gray area. The dark area is a result of the deflection of light rays. For transparent materials, three caustics are formed by the light rays reflected from the front and rear surfaces and those transmitted through the specimen. For opaque materials, one caustic is formed by the light rays reflected from the front surface of the specimen. The dimensions of the caustic are related to the state of stress in the neighborhood of the crack tip. The stress intensity factor, which governs the stress field near the crack tip, can be determined by measuring characteristic dimensions of the caustic, usually, its diameter perpendicular to the crack plane.

In the following we develop the equations of caustics for two-dimensional crack problems under conditions of generalized plane stress and relate the dimensions of the caustics to the stress intensity factors.

## 物理代写|力学代写mechanics代考|Stress-Optical Equations

$$\Delta n_1=n_1-n=b_1 \varepsilon_1+b_2\left(\varepsilon_2+\varepsilon_3\right) \quad \Delta n_2=n_2-n=b_1 \varepsilon_2+b_2\left(\varepsilon_1+\varepsilon_3\right)$$

$$\varepsilon_1=\frac{1}{E}\left(\sigma_1-v \sigma_2\right), \quad \varepsilon_2=\frac{1}{E}\left(\sigma_2-v \sigma_1\right), \quad \varepsilon_3=-\frac{v}{E}\left(\sigma_1+\sigma_2\right)$$

$$\Delta n_1=n_1-n=A \sigma_1+B \sigma_2 \quad \Delta n_2=n_2-n=B \sigma_1+A \sigma_2$$

$$A=\frac{b_1-2 v b_2}{E}, \quad B=\frac{b_2-v\left(b_1+b_2\right)}{E}$$

$$S_A=L n_0+d\left(n-n_0\right)$$

## 物理代写|力学代写mechanics代考|Principle of the Method

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