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# 物理代写|断裂力学代写Fracture mechanics代考|TMM4160 Mechanical behavior under tension

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## 物理代写|断裂力学代写Fracture mechanics代考|Mechanical behavior under tension

The simplest way to observe the mechanical behavior of a solid is by applying a tension force on a body of regular cross-section and record the load and elongation produced in the test specimen. The typical Load versus Elongation record of an engineering material is shown in Fig. 1.4. By using the definition of stress as $\sigma=F / A_0$, where $A_0$ is the initial cross-section area of the test specimen, and defining the elongation strain as $\varepsilon=\Delta l / l_0$, where $\Delta l$ is the elongation and $l_0$ is the initial length, the stress-strain curve in tension can be reconstructed from the original Load-Elongation curve. Since $A_0$ and $l_0$ are constants, the shape of the curve does not change, only the scale is modified, therefore the stress-strain curve in uniaxial tension has the same shape of the Load-Elongation curve.

As it can be observed on the stress-strain curve, at the onset of loading, the elongation is proportional to the stress, but if the load is removed, the body recovers its initial shape and length; this behavior is termed as elastic. In most materials, the strain is linearly proportional to the strain and the constant of proportionality is called the Young’s modulus, represented by the symbol $E$. When the stress surpasses a limit value the deformation becomes permanent, a condition termed as plastic strain, the stress at this point is termed as the yield strength, and it usually represented by the symbol $\sigma_0$. On further loading, the load has to be increased to sustain the plastic deformation, this behavior is called strain hardening and it makes the curve to take a parabolic-like shape, which maximum is referred as ultimate tensile strength, represented by the symbol $\sigma_{\text {uts. }}$ Just after the ultimate tensile strength is reached, the material experiences a local contraction known as necking, so the cross-section area is rapidly reduced and the load drops, leading to the final rupture. The maximum elongation in tension loading is referred as ductility, identified by the symbol $\varepsilon_f$. These essential features of the stressstrain curve in uniaxial tension allow determining the fundamental mechanical properties of engineering materials, which are:

• Young’s modulus (E): Proportionality constant between strain and stress in the elastic regime.
• Yield strength $\left(\sigma_0\right)$ : Tension stress at the onset of plastic deformation.
• Ultimate tensile strength $\left(\sigma_{u t s}\right)$ : Maximum tension stress that a material can withstand before failure.
• Rupture elongation or ductility $\left(\varepsilon_f\right)$ : Maximum plastic elongation measured after rupture in tension.

## 物理代写|断裂力学代写Fracture mechanics代考|The stress tensor

According to Cauchy’s stress theory, the state of stress at one point is described by the stress tensor which is determined by the stress components acting on the faces of an element of a differential volume (a cube) located at the origin of a Cartesian coordinate system of $x, y$, and $z$ axes, as shown in Fig. 1.8.

This results in nine stress components, three normal and six shear. An index notation identifies each stress component:
$$\sigma_{i j}$$
Where $i$ is the cube’s face where the stress is acting, and $j$ is the direction of the stress.

The nine stress components written in matrix form, become the stress tensor, since they are vectors, and it has the form:
$$\boldsymbol{\sigma}=\left(\begin{array}{lll} \sigma_{x x} & \tau_{x y} & \tau_{x z} \ \tau_{y x} & \sigma_{y y} & \tau_{y z} \ \tau_{z x} & \tau_{z y} & \sigma_{z z} \end{array}\right)$$

## 物理代写|断裂力学代写骨折力学代考|应力张量

$$\sigma_{i j}$$

$$\boldsymbol{\sigma}=\left(\begin{array}{lll} \sigma_{x x} & \tau_{x y} & \tau_{x z} \ \tau_{y x} & \sigma_{y y} & \tau_{y z} \ \tau_{z x} & \tau_{z y} & \sigma_{z z} \end{array}\right)$$

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