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# 物理代写|弹性力学代写Elasticity代考|ME340 Spherical and deviatoric strains

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## 物理代写|弹性力学代写Elasticity代考|Spherical and deviatoric strains

In particular applications it is convenient to decompose the strain tensor into two parts called spherical and deviatoric strain tensors. The spherical strain is defined by
$$\tilde{e}{i j}=\frac{1}{3} e{k k} \delta_{i j}=\frac{1}{3} \vartheta \delta_{i j}$$
while the deviatoric strain is specified as
$$\widehat{e}{i j}=e{i j}-\frac{1}{3} e_{k k} \delta_{i j}$$
Note that the total strain is then simply the sum
$$e_{i j}=\tilde{e}{i j}+\widehat{e}{i j}$$
The spherical strain represents only volumetric deformation and is an isotropic tensor, being the same in all coordinate systems (as per the discussion in Section 1.5). The deviatoric strain tensor then accounts for changes in shape of material elements. It can be shown that the principal directions of the deviatoric strain are the same as those of the strain tensor.

## 物理代写|弹性力学代写Elasticity代考|Strain compatibility

We now investigate in more detail the nature of the strain-displacement relations $(2.2 .5)$, and this will lead to the development of some additional relations necessary to ensure continuous, single-valued displacement field solutions. Relations (2.2.5), or the index notation form (2.2.6), represent six equations for the six strain components in terms of three displacements. If we specify continuous, single-valued displacements $u, v, w$, then through differentiation the resulting strain field will be equally well behaved. However, the converse is not necessarily true; given the six strain components, integration of the strain-displacement relations (2.2.5) does not necessarily produce continuous, single-valued displacements. This should not be totally surprising since we are trying to solve six equations for only three unknown displacement components. In order to ensure continuous, singlevalued displacements, the strains must satisfy additional relations called integrability or compatibility equations.

Before we proceed with the mathematics to develop these equations, it is instructive to consider a geometric interpretation of this concept. A two-dimensional example is shown in Fig. $2.8$ whereby an elastic solid is first divided into a series of elements in case (a). For simple visualization, consider only four such elements. In the undeformed configuration shown in case (b), these elements of course fit together perfectly. Next, let us arbitrarily specify the strain of each of the four elements and attempt to reconstruct the solid. For case (c), the elements have been carefully strained, taking into consideration neighboring elements so that the system fits together yielding continuous, single-valued displacements. However, for case (d), the elements have been individually deformed without any concern for neighboring deformations. It is observed for this case that the system will not fit together without voids and gaps, and this situation produces a discontinuous displacement field. So, we again conclude that the strain components must be somehow related to yield continuous, single-valued displacements. We now nursue these narticular relations.

## 物理代写|弹性力学代写弹性代考|球形和偏应变

$$\tilde{e}{i j}=\frac{1}{3} e{k k} \delta_{i j}=\frac{1}{3} \vartheta \delta_{i j}$$
，而偏应变指定为
$$\widehat{e}{i j}=e{i j}-\frac{1}{3} e_{k k} \delta_{i j}$$

$$e_{i j}=\tilde{e}{i j}+\widehat{e}{i j}$$

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