19.2: Equilibrium differential equations

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A.2.1 Equilibrium differential equations

Consider the forces acting on a rectangular parallelepiped at point P. The free body diagram is shown in figure A.7. The vector sum of forces is

For a rectangular parallelepiped of arbitrary size around interior point cap P, an external body force vector cap B times delta x sub 1, delta x sub 2, and delta x sub 3, is applied. The body force results in each surface having a corresponding surface force vector cap T, with a subscript corresponding to whichever direction it is pointed, multiplied by the delta x values for the corresponding face. Each direction has two faces, one evaluated at initial location x sub 1, x sub 2, or x sub 3, and an equal and opposite force evaluated on the x plus delta x face for each corresponding direction. — Fig. A.7 Surface forces and a body force acting on a rectangular parallelepiped $Δ x_{1} Δ x_{2} Δ x_{3}$ $math-4581.png$ .

Fig. A.7 Surface forces and a body force acting on a rectangular parallelepiped $Δ x_{1} Δ x_{2} Δ x_{3}$ $math-4581.png$ .

$\begin{array}{l} \overset{⇀}{T} 1 Δ x_{2} Δ x_{3} |_{x_{1}} + Δ x_{1} - \overset{⇀}{T} 1 Δ x_{2} Δ x_{3} |_{x_{1}} + \overset{⇀}{T} 2 Δ x_{1} Δ x_{3} |_{x_{2}} + Δ x_{2} - \overset{⇀}{T} 2 Δ x_{1} Δ x_{3} |_{x_{2}} + \overset{⇀}{T} 3 Δ x_{1} Δ x_{2} |_{x_{3}} + Δ x_{3} \\ - \overset{⇀}{T} 3 Δ x_{1} Δ x_{2} |_{x_{3}} + \overset{⇀}{B} Δ x_{1} Δ x_{2} Δ x_{3} . \end{array}$ $math-4582.png$

For small increments in Δx_i, $i = 1, 2, 3$ $math-4583.png$ , use the Taylor series representation of surface forces results in the equilibrium equation to get eq. (A.78) below.

$\begin{align} \frac{\partial}{\partial x_{1}} ({\overset{⇀}{T}}_{1} Δ x_{2} Δ x_{3}) Δ x_{1} + \frac{\partial}{\partial x_{2}} ({\overset{⇀}{T}}_{2} Δ x_{1} Δ x_{3}) Δ x_{2} + \frac{\partial}{\partial x_{3}} ({\overset{⇀}{T}}_{3} Δ x_{1} Δ x_{2}) Δ x_{3} + \overset{⇀}{B} Δ x_{1} Δ x_{2} Δ x_{3} + O ({(Δ x_{i})}^{4}) = 0 . & (A.78) \end{align}$ $math-4584.png$

Arrange the terms in eq. (A.78) to the form

$\begin{align} (\frac{\partial {\overset{⇀}{T}}_{1}}{\partial x_{1}} + \frac{\partial {\overset{⇀}{T}}_{2}}{\partial x_{2}} + \frac{\partial {\overset{⇀}{T}}_{3}}{\partial x_{3}} + \overset{⇀}{B}) Δ x_{1} Δ x_{2} Δ x_{3} + O ({(Δ x_{i})}^{4}) = 0 & (A.79) \end{align}$ $math-4585.png$

Divide eq. (A.79) by the volume followed by the limit as Δx₁Δx₂Δx₃→0 to get the vector differential equation of force equilibrium at point P as

$\begin{align} \frac{\partial {\overset{⇀}{T}}_{1}}{\partial x_{1}} + \frac{\partial {\overset{⇀}{T}}_{2}}{\partial x_{2}} + \frac{\partial {\overset{⇀}{T}}_{3}}{\partial x_{3}} + \overset{⇀}{B} = 0 . & (A.80) \end{align}$ $math-4586.png$

Substitute eq. (A.70) for the traction vectors in eq. (A.80) to write the equilibrium differential equations in the x_i-coordinate directions. In the order of x₁, x₂, x₃ coordinate directions these equations are

$\begin{align} \begin{aligned} \frac{\partial σ_{11}}{\partial x_{1}} + \frac{\partial σ_{21}}{\partial x_{2}} + \frac{\partial σ_{31}}{\partial x_{3}} + B_{1} = 0 \\ \frac{\partial σ_{12}}{\partial x_{1}} + \frac{\partial σ_{22}}{\partial x_{2}} + \frac{\partial σ_{32}}{\partial x_{3}} + B_{2} = 0 \\ \frac{\partial σ_{13}}{\partial x_{1}} + \frac{\partial σ_{23}}{\partial x_{2}} + \frac{\partial σ_{33}}{\partial x_{3}} + B_{3} = 0 \end{aligned} . & (A.81) \end{align}$ $math-4587.png$

Now consider moment equilibrium about the coordinate axes of the rectangular parallelepiped at point P. For moment equilibrium about the x₁-axis refer to the free body diagram in figure A.8. The moment arm from point P to the line of action of the normal force ${(σ_{22} Δ x_{1} Δ x_{3})}_{x_{2} + Δ x_{2}}$ $math-4590.png$ acting on the positive x₂-face is denoted by εΔx₃, where ε is a small numerical value. Parameter ε is not known, but this will not matter in the end result. The moment arm from point P to the line of action of the shear force ${(σ_{23} Δ x_{1} Δ x_{3})}_{x_{2} + Δ x_{2}}$ $math-4591.png$ acting on the positive x₂ face is Δx₂∕2. Including all the forces shown in figure A.8, the sum of moments about the x₁-axis through point P, counterclockwise positive, is

$\begin{align} - ε Δ x_{3} {(σ_{22} Δ x_{1} Δ x_{3})}_{x_{2} + Δ x_{2}} + ε Δ x_{3} {(σ_{22} Δ x_{1} Δ x_{3})}_{x_{2}} + \frac{Δ x_{2}}{2} {(σ_{23} Δ x_{1} Δ x_{3})}_{x_{2} + Δ x_{2}} + \frac{Δ x_{2}}{2} {(σ_{23} Δ x_{1} Δ x_{3})}_{x_{2}} \\ + ε Δ x_{2} {(σ_{33} Δ x_{1} Δ x_{2})}_{x_{3} + Δ x_{3}} - ε Δ_{2} {(σ_{33} Δ x_{1} Δ x_{2})}_{x_{3}} - \frac{Δ x_{3}}{2} {(σ_{32} Δ x_{1} Δ x_{2})}_{x_{3} + Δ x_{3}} - \frac{Δ x_{3}}{2} {(σ_{32} Δ x_{1} Δ x_{2})}_{x_{3}} = 0 . & (A.82) \end{align}$ $math-4592.png$

A slice of a rectangular parallelepiped around point cap P is shown in the x sub 2 x sub 3 plane. Each face has a normal stress sigma sub x x, and shear stress sigma sub x y, with each stress applied on the x face in the y direction. The left and bottom faces correspond to x sub 2 and x sub 3 locations, repsectively, while the right and top faces are x sub 2 plus delta x sub 2 and x sub 3 plus delta x sub 3, respectively. Each stress is multiplied by the width and depth of each face to give the corresponding surface force, such as sigma sub 2 3 on the left face being multiplied by delta x sub 1 and delta x sub 3. — Fig. A.8 A free body diagram of the parallelepiped at point P for moment equilibrium about the $x_{1}$ $math-4588.png$ -axis. The $x_{1}$ $math-4589.png$ -axis points normal to the page toward the reader.

Fig. A.8 A free body diagram of the parallelepiped at point P for moment equilibrium about the $x_{1}$ $math-4588.png$ -axis. The $x_{1}$ $math-4589.png$ -axis points normal to the page toward the reader.

Use the Taylor series to expand the forces acting on the positive coordinate faces with respect to the forces acting on the negative coordinate faces to get

$\begin{align} - ε Δ x_{3} [\frac{\partial σ_{22}}{\partial x_{2}} Δ x_{2} + O (Δ x_{2}^{2})] Δ x 1 Δ x_{3} + \frac{Δ x_{2}}{2} [2 σ_{23} + \frac{\partial σ_{23}}{\partial x_{2}} Δ x_{2} + O (Δ x_{2}^{2})] Δ x 1 Δ x_{3} \\ + ε Δ x_{2} [\frac{\partial σ_{33}}{\partial x_{3}} Δ x_{3} + O (Δ x_{3}^{2})] Δ x 1 Δ x_{2} - \frac{Δ x_{3}}{2} [2 σ_{32} + \frac{\partial σ_{32}}{\partial x_{3}} Δ x_{3} + O (Δ x_{3}^{2})] Δ x 1 Δ x_{2} = 0 . & (A.83) \end{align}$ $math-4593.png$

Expand eq. (A.83) in powers of Δx_i to write it as

$\begin{align} (σ_{23} - σ_{32}) Δ x_{1} Δ x_{2} Δ x_{3} + ε [- \frac{\partial σ_{22}}{\partial x_{2}} Δ x_{1} Δ x_{2} Δ x_{3}^{2} + \frac{\partial σ_{33}}{\partial x_{3}} Δ x_{1} Δ x_{2}^{2} Δ x_{3}] + H.O.T. = 0, & (A.84) \end{align}$ $math-4594.png$

where H.O.T. means higher order terms, that is, terms of quartic powers and higher in the increments in the coordinates. Notice the terms multiplied by ε are quartic powers of the increments in the coordinates. Division of eq. (A.84) by Δx₁Δx₂Δx₃, followed by the limit of Δx_i→0 leads the condition of moment equilibrium about the x₁-axis that $σ_{23} - σ_{32} = 0$ $math-4595.png$ . Moment equilibrium about the x₂-axis leads to $σ_{31} - σ_{13} = 0$ $math-4596.png$ , and moment equilibrium about the x₃-axis leads to $σ_{12} - σ_{21} = 0$ $math-4597.png$ . The equations of moment equilibrium at point P are

$\begin{align} σ_{12} = σ_{21} σ_{13} = σ_{31} σ_{23} = σ_{32} . & (A.85) \end{align}$ $math-4598.png$

Hence, the stress matrix (A.71) is symmetric.

A.2.2 Transformation of stresses between two Cartesian coordinate systems

At point P coordinates $(x_{1}^{'}, x_{2}^{'}, x_{3}^{'})$ $math-4599.png$ are linearly related to coordinates $(x_{1}, x_{2}, x_{3})$ $math-4600.png$ by eq. (A.41). The stress components σ_ij are functions in the variables $(x_{1}, x_{2}, x_{3})$ $math-4601.png$ , and the stresses $σ_{i j}^{'}$ $math-4602.png$ are functions in the variables $(x_{1}^{'}, x_{2}^{'}, x_{3}^{'})$ $math-4603.png$ . The stress vectors acting on the x_i-faces are denoted by $\overset{⇀}{T}$ _i, and those acting on the $x_{i}^{'}$ $math-4604.png$ -faces are denoted by ${\overset{⇀}{T}}_{i}^{'}$ $math-4605.png$ . These stress vectors are written in their respective coordinate systems by

$\begin{align} [\begin{matrix} {\overset{⇀}{T}}_{1} \\ {\overset{⇀}{T}}_{2} \\ {\overset{⇀}{T}}_{3} \end{matrix}] = [\begin{matrix} σ_{11} & σ_{12} & σ_{13} \\ σ_{21} & σ_{22} & σ_{23} \\ σ_{31} & σ_{32} & σ_{33} \end{matrix}] [\begin{matrix} î_{1} \\ î_{2} \\ î_{3} \end{matrix}], and [\begin{matrix} {\overset{⇀}{T}}_{1}^{'} \\ {\overset{⇀}{T}}_{2}^{'} \\ {\overset{⇀}{T}}_{3}^{'} \end{matrix}] = [\begin{matrix} σ_{11}^{'} & σ_{12}^{'} & σ_{13}^{'} \\ σ_{21}^{'} & σ_{22}^{'} & σ_{23}^{'} \\ σ_{31}^{'} & σ_{32}^{'} & σ_{33}^{'} \end{matrix}] [\begin{matrix} î_{1}^{'} \\ î_{2}^{'} \\ î_{3}^{'} \end{matrix}] . & (A.86) \end{align}$ $math-4606.png$

In eq. (A.86) the stress matrices are

$\begin{align} [σ] = [\begin{matrix} σ_{11} & σ_{12} & σ_{13} \\ σ_{21} & σ_{22} & σ_{23} \\ σ_{31} & σ_{32} & σ_{33} \end{matrix}], and [σ^{'}] = [\begin{matrix} σ_{11}^{'} & σ_{12}^{'} & σ_{13}^{'} \\ σ_{21}^{'} & σ_{22}^{'} & σ_{23}^{'} \\ σ_{31}^{'} & σ_{32}^{'} & σ_{33}^{'} \end{matrix}] . & (A.87) \end{align}$ $math-4607.png$

Note that the stress matrix [σ] is symmetric by eq. (A.85). The stress transformation equations between the Cartesian coordinate systems $(x_{1}, x_{2}, x_{3})$ $math-4608.png$ and $(x_{1}^{'}, x_{2}^{'}, x_{3}^{'})$ $math-4609.png$ are determined by selecting the unit normal in eq. (A.77) to be either $î_{1}^{'}$ $math-4610.png$ , $î_{2}^{'}$ $math-4611.png$ , or $î_{3}^{'}$ $math-4612.png$ . First let $\hat{n} = î_{1}^{'}$ $math-4613.png$ such that ${\overset{⇀}{T}}^{(n)} = {\overset{⇀}{T}}_{1}^{'}$ $math-4614.png$ in eq. (A.77). From eq. (A.44) we have $\hat{n} = λ_{11} î_{1} + λ_{12} î_{2} + λ_{13} î_{3}$ $math-4615.png$ . Hence, eq. (A.77) becomes

$\begin{align} {\overset{⇀}{T}}_{1}^{'} = λ_{11} {\overset{⇀}{T}}_{1} + λ_{12} {\overset{⇀}{T}}_{2} + λ_{13} {\overset{⇀}{T}}_{3} . & (A.88) \end{align}$ $math-4616.png$

Second, let $\hat{n} = î_{2}^{'}$ $math-4617.png$ such that ${\overset{⇀}{T}}^{(n)} = {\overset{⇀}{T}}_{2}^{'}$ $math-4618.png$ . From eq. (A.44) we have $\hat{n} = λ_{21} î_{1} + λ_{22} î_{2} + λ_{23} î_{3}$ $math-4619.png$ . Hence, eq. (A.77) becomes

$\begin{align} {\overset{⇀}{T}}_{2}^{'} = λ_{21} {\overset{⇀}{T}}_{1} + λ_{22} {\overset{⇀}{T}}_{2} + λ_{23} {\overset{⇀}{T}}_{3} . & (A.89) \end{align}$ $math-4620.png$

Third, let $\hat{n} = î_{3}^{'}$ $math-4621.png$ that ${\overset{⇀}{T}}^{(n)} = {\overset{⇀}{T}}_{3}^{'}$ $math-4622.png$ and $\hat{n} = λ_{31} î_{1} + λ_{32} î_{2} + λ_{33} î_{3}$ $math-4623.png$ . Hence,

$\begin{align} {\overset{⇀}{T}}_{3}^{'} = λ_{31} {\overset{⇀}{T}}_{1} + λ_{32} {\overset{⇀}{T}}_{2} + λ_{33} {\overset{⇀}{T}}_{3} . & (A.90) \end{align}$ $math-4624.png$

The three selections for the unit normal in eq. (A.77) relate the tractions acting on the $x_{i}^{'}$ $math-4625.png$ coordinate faces to the tractions acting on the x_i-faces by

$\begin{align} [\begin{matrix} {\overset{⇀}{T}}_{1}^{'} \\ {\overset{⇀}{T}}_{2}^{'} \\ {\overset{⇀}{T}}_{3}^{'} \end{matrix}] = [\begin{matrix} λ_{11} & λ_{12} & λ_{13} \\ λ_{21} & λ_{22} & λ_{23} \\ λ_{31} & λ_{32} & λ_{33} \end{matrix}] [\begin{matrix} {\overset{⇀}{T}}_{1} \\ {\overset{⇀}{T}}_{2} \\ {\overset{⇀}{T}}_{3} \end{matrix}] . & (A.91) \end{align}$ $math-4626.png$

Substitute the expressions for the stress vectors from eq. (A.86) into eq. (A.91) to get

$\begin{align} [\begin{matrix} σ_{11}^{'} & σ_{12}^{'} & σ_{13} \\ σ_{21}^{'} & σ_{22}^{'} & σ_{23} \\ σ_{31}^{'} & σ_{32}^{'} & σ_{33} \end{matrix}] [\begin{matrix} î_{1}^{'} \\ î_{2}^{'} \\ î_{3}^{'} \end{matrix}] = [\begin{matrix} λ_{11} & λ_{12} & λ_{13} \\ λ_{21} & λ_{22} & λ_{23} \\ λ_{31} & λ_{32} & λ_{33} \end{matrix}] [\begin{matrix} σ_{11} & σ_{12} & σ_{13} \\ σ_{21} & σ_{22} & σ_{23} \\ σ_{31} & σ_{32} & σ_{33} \end{matrix}] [\begin{matrix} î_{1} \\ î_{2} \\ î_{3} \end{matrix}] . & (A.92) \end{align}$ $math-4627.png$

The inverse eq. (A.44) is

$\begin{align} [\begin{matrix} î_{1} \\ î_{2} \\ î_{3} \end{matrix}] = [\begin{matrix} λ_{11} & λ_{21} & λ_{31} \\ λ_{12} & λ_{22} & λ_{32} \\ λ_{13} & λ_{23} & λ_{33} \end{matrix}] [\begin{matrix} î_{1}^{'} \\ î_{2}^{'} \\ î_{3}^{'} \end{matrix}] . & (A.93) \end{align}$ $math-4628.png$

Substitute eq. (A.93) into the right-hand side of eq. (A.92) and rearrange the result to find

$\begin{align} ([\begin{matrix} σ_{11}^{'} & σ_{12}^{'} & σ_{13}^{'} \\ σ_{21}^{'} & σ_{22}^{'} & σ_{23}^{'} \\ σ_{31}^{'} & σ_{32}^{'} & σ_{33}^{'} \end{matrix}] - [\begin{matrix} λ_{11} & λ_{12} & λ_{13} \\ λ_{21} & λ_{22} & λ_{23} \\ λ_{31} & λ_{32} & λ_{33} \end{matrix}] [\begin{matrix} σ_{11} & σ_{12} & σ_{13} \\ σ_{21} & σ_{22} & σ_{23} \\ σ_{31} & σ_{32} & σ_{33} \end{matrix}] [\begin{matrix} λ_{11} & λ_{21} & λ_{31} \\ λ_{12} & λ_{22} & λ_{32} \\ λ_{13} & λ_{23} & λ_{33} \end{matrix}]) [\begin{matrix} î_{1}^{'} \\ î_{2}^{'} \\ î_{3}^{'} \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}] . & (A.94) \end{align}$ $math-4629.png$

To satisfy eq. (A.94) we find that the stress components $σ_{i j}^{'}$ $math-4630.png$ in the $x_{i}^{'}$ $math-4631.png$ -system are related to the stress components σ_ij in the x_i-system by

$\begin{align} [\begin{matrix} σ_{11}^{'} & σ_{12}^{'} & σ_{13} \\ σ_{21}^{'} & σ_{22}^{'} & σ_{23}^{'} \\ σ_{31}^{'} & σ_{32}^{'} & σ_{33}^{'} \end{matrix}] = [\begin{matrix} λ_{11} & λ_{12} & λ_{13} \\ λ_{21} & λ_{22} & λ_{23} \\ λ_{31} & λ_{32} & λ_{33} \end{matrix}] [\begin{matrix} σ_{11} & σ_{12} & σ_{13} \\ σ_{21} & σ_{22} & σ_{23} \\ σ_{31} & σ_{32} & σ_{33} \end{matrix}] [\begin{matrix} λ_{11} & λ_{21} & λ_{31} \\ λ_{12} & λ_{22} & λ_{32} \\ λ_{13} & λ_{23} & λ_{33} \end{matrix}] . & (A.95) \end{align}$ $math-4632.png$

Equation (A.95) in compact form is

$\begin{align} [σ^{'}] = [λ] [σ] {[λ]}^{T} . & (A.96) \end{align}$ $math-4633.png$

Pre-multiply eq. (A.96) by [λ]^T, post-multiply it by [λ], and note that ${[λ]}^{T} [λ] = [λ] {[λ]}^{T} = [I]$ $math-4634.png$ to find the inverse transformation

$\begin{align} [σ] = {[λ]}^{T} [σ^{'}] [λ] . & (A.97) \end{align}$ $math-4635.png$

The transpose of eq. (A.96) is [λ][σ]^T[λ]^T, but [σ]^T = [σ], so the stress matrix [σ^′] is also symmetric. Comparing the strain transformation eq. (A.63) to the stress transformation eq. (A.96), it is clear that the transformation of strains ε_ij is the same form as the transformation of the stresses σ_ij.

A.2.3 Cartesian tensors

A tensor is a system of numbers or functions, whose components obey a certain law of transformation when the independent variables undergo a linear transformation. If the independent variables are the rectangular cartesian systems x_i and $x_{i}^{'}$ $math-4636.png$ transforming by the linear relations given by {x^′} = [λ]{x} at point P, then the systems obeying certain laws of transformation are called Cartesian tensors.

Definition. A system of order two may be defined to have nine components ε_ij in x_i and nine components $ε_{i j}^{'}$ $math-4637.png$ in $x_{i}^{'}$ $math-4638.png$ . If

$\begin{align} [ε_{i j}^{'}] = [λ] [ε_{i j}] {[λ]}^{T} & (A.98) \end{align}$ $math-4639.png$

then the functions ε_ij and $ε_{i j}^{'}$ $math-4640.png$ are the components in their respective variables of a second order Cartesian tensor. Similarly, functions σ_ij and $σ_{i j}^{'}$ $math-4641.png$ are the components in their respective variables of a second order Cartesian tensor.

A.3 Linear elastic material law

To this point in the study of the mechanics of a solid body we have eighteen unknown functions of the Cartesian coordinates x₁, x₂, and x₃. These are the three displacements u₁, u₂, and u₃, the six strains ε₁₁, ε₂₂, ε₃₃, γ₂₃, γ₃₁, and γ₁₂, and nine stresses σ₁₁, σ₁₂, σ₁₃, σ₂₁, σ₂₂, σ₂₃, σ₃₁, σ₃₂, and σ₃₃. There are twelve equations relating these unknowns; the six strain-displacement equations (A.35) and (A.36), and the six equilibrium equations (A.81) and (A.85). Therefore we need six more equations to get the number of unknowns equal to number of equations. The additional six equations come from the relations between the strains and the stresses, which express the material law. This relation between strains and stresses for different materials is established by material characterization tests on standard test specimens.

Solid bodies that can instantly recover their original size and shape when the forces producing the deformation are removed are called perfectly elastic. The elastic limit is defined as the greatest stress that can be applied without resulting in permanent strain on release of the stress. Elasticity is applicable to any body provided the stresses do not exceed the elastic limit. For many solid bodies there is a region where the stress is very nearly proportional to strain. The proportional limit is defined as the greatest stress for which the stress is still proportional to the strain. Both the elastic limit and proportional limit cannot be precisely determined from test data since they are defined by the limiting cases of no permanent deformation and no deviation from linearity. In practice the definition of the yield strength of a material is used to determine the limit of elastic behavior. See article 4.2 on page 77 for a discussion on yield criteria.

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A.2.1 Equilibrium differential equations

A.2.2 Transformation of stresses between two Cartesian coordinate systems

A.2.3 Cartesian tensors

A.3 Linear elastic material law