Basic equations for ELASTCITY

Assumptions of linear elasticity

1. The Body is Continuous. The whole volume of the body is considered to be filled with continuous matter, without any void, and keep continuous during the whole deformation process. Only under this assumption, can the physical quantities in the body, such as stresses, strains and displacements, be continuously distributed and thereby expressed by continuous functions of coordinates in space.

2. The Body is Perfectly Elastic. The body is considered to wholly obey Hooke’s law of elasticity, which shows the linear relations between the stress components and strain components. Under this assumption, the elastic constants will be independent of the magnitudes of stress and strain components.

3. The Body is Homogenous. In other words, the elastic properties are the same throughout the body. Thus, the elastic constants will be independent of the location in the body. Under this assumption, one can analyse an elementary volume isolated from the body and then apply the results of analysis to the entire body.

4. The Body is Isotropic. The elastic properties in a body are the same in all directions. Hence, the elastic constants will be independent of the orientation of coordinate axes.

5. The Displacements and Strains are Small. ¹ The displacement components of all points of the body during deformation are very small in comparison with its original dimensions and the strain components and the rotations of all line elements are much smaller than unity. Hence, when formulating the equilibrium equations relevant to the deformed state, the lengths and angles of the body before deformation are used. In addition, when geometrical equations involving strains and displacements are formulated, the squares and products of the small quantities are neglected. Therefore, these two measures are necessary to linearize the algebraic and differential equations in elasticity for their easier solution.

The bodies meet the assumption 1-4 are called perfect elastic bodies.

Basic Equations

Equilibrium equations

Written in scalar notation:

$$ \begin{aligned} &\frac{\partial \sigma_{x x}}{\partial x}+\frac{\partial \sigma_{y x}}{\partial y}+\frac{\partial \sigma_{z x}}{\partial z}+f_{x}=0 \\ &\frac{\partial \sigma_{x y}}{\partial x}+\frac{\partial \sigma_{y y}}{\partial y}+\frac{\partial \sigma_{z y}}{\partial z}+f_{y}=0 \\ &\frac{\partial \sigma_{x z}}{\partial x}+\frac{\partial \sigma_{y z}}{\partial y}+\frac{\partial \sigma_{z z}}{\partial z}+f_{z}=0 \end{aligned} $$

Written in tensor notation:

$$\sigma_{j i, j}+f_{i}=0 $$ $$\nabla \cdot \sigma+\boldsymbol{f}=0$$

Dynamic equilibrium equations

For elastodynamic problems, the inertial force can be regarded as the body force, so the differential equation of motion can be derived from the balance equation.

Written in scalar notation: $$ \begin{aligned} &\frac{\partial \sigma_{x x}}{\partial x}+\frac{\partial \sigma_{y x}}{\partial y}+\frac{\partial \sigma_{z x}}{\partial z}+f_{x}=\rho a_{x}=\rho \frac{\partial^{2} u_{x}}{\partial t^{2}} \\ &\frac{\partial \sigma_{x y}}{\partial x}+\frac{\partial \sigma_{y y}}{\partial y}+\frac{\partial \sigma_{z y}}{\partial z}+f_{y}=\rho a_{y}=\rho \frac{\partial^{2} u_{y}}{\partial t^{2}} \\ &\frac{\partial \sigma_{x z}}{\partial x}+\frac{\partial \sigma_{y z}}{\partial y}+\frac{\partial \sigma_{z z}}{\partial z}+f_{z}=\rho a_{z}=\rho \frac{\partial^{2} u_{z}}{\partial t^{2}} \end{aligned} $$

Written in tensor notation:

$$\sigma_{j i, j}+f_{i}=\rho \frac{\partial^{2} u_{i}}{\partial t^{2}} $$ $$\nabla \cdot \sigma+f=\rho \ddot{\boldsymbol{u}}$$

Geometric equations

In the Cartesian coordinate system, the strain-displacement relationship or geometric equation can be written as following.

Written in scalar notation: $$ \begin{array}{ll} \varepsilon_{11}=\frac{\partial u_{1}}{\partial x_{1}}, & \varepsilon_{12}=\varepsilon_{21}=\frac{1}{2}\left(\frac{\partial u_{1}}{\partial x_{2}}+\frac{\partial u_{2}}{\partial x_{1}}\right) \\ \varepsilon_{22}=\frac{\partial u_{2}}{\partial x_{2}}, & \varepsilon_{23}=\varepsilon_{32}=\frac{1}{2}\left(\frac{\partial u_{2}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{2}}\right) \\ \varepsilon_{33}=\frac{\partial u_{3}}{\partial x_{3}}, & \varepsilon_{31}=\varepsilon_{13}=\frac{1}{2}\left(\frac{\partial u_{3}}{\partial x_{1}}+\frac{\partial u_{1}}{\partial x_{3}}\right) \end{array} $$

Written in tensor notation:

$$ \varepsilon_{i j}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right) $$ $$ \boldsymbol{\varepsilon}=\frac{1}{2}(\boldsymbol{u} \nabla+\nabla \boldsymbol{u}) $$

Strain compatibility

$$ \begin{aligned} &\frac{\partial^{2} \varepsilon_{x}}{\partial y^{2}}+\frac{\partial^{2} \varepsilon_{y}}{\partial x^{2}}=\frac{\partial^{2} \gamma_{x y}}{\partial x \partial y} ; \quad \frac{\partial}{\partial z}\left(\frac{\partial \gamma_{y z}}{\partial x}+\frac{\partial \gamma_{z x}}{\partial y}-\frac{\partial \gamma_{x y}}{\partial z}\right)=2 \frac{\partial^{2} \varepsilon_{z}}{\partial x \partial y} \\ &\frac{\partial^{2} \varepsilon_{y}}{\partial z^{2}}+\frac{\partial^{2} \varepsilon_{z}}{\partial y^{2}}=\frac{\partial^{2} \gamma_{y z}}{\partial y \partial z} ; \quad \frac{\partial}{\partial x}\left(\frac{\partial \gamma_{z x}}{\partial y}+\frac{\partial \gamma_{x y}}{\partial z}-\frac{\partial \gamma_{y z}}{\partial x}\right)=2 \frac{\partial^{2} \varepsilon_{x}}{\partial y \partial z} \\ &\frac{\partial^{2} \varepsilon_{z}}{\partial x^{2}}+\frac{\partial^{2} \varepsilon_{x}}{\partial z^{2}}=\frac{\partial^{2} \gamma_{z x}}{\partial z \partial x} ; \quad \frac{\partial}{\partial y}\left(\frac{\partial \gamma_{x y}}{\partial z}+\frac{\partial \gamma_{y z}}{\partial x}-\frac{\partial \gamma_{z x}}{\partial y}\right)=2 \frac{\partial^{2} \varepsilon_{y}}{\partial z \partial x} \end{aligned} $$