Components of Strain In terms of a given coordinate system, the

state of strain at a point

p of a material is defined by the

components of strain :

$$ \left[\begin{array}{ccc}{\varepsilon}_x& {\gamma}_{xy}& {\gamma}_{xz}\\ {}{\gamma}_{yx}& {\varepsilon}_y& {\gamma}_{yz}\\ {}{\gamma}_{zx}& {\gamma}_{zy}& {\varepsilon}_z\end{array}\right]. $$

(8.1)

The components ε _{x} , ε _{y} , and ε _{z} are the normal strains in the x , y , and z directions. The component γ _{xy} = γ _{yx} is the shear strain referred to the positive directions of the x and y axes, and the components γ _{yz} = γ _{zy} and γ _{xz} = γ _{zx} are defined similarly.

Consider a volume

dV _{0} of material in a reference state. Its volume in the deformed state is

$$ dV=\left(1+{\varepsilon}_x+{\varepsilon}_y+{\varepsilon}_z\right)\;{dV}_0. $$

(8.2)

The

dilatation is the change in volume of the material per unit volume:

$$ e=\frac{dV-{dV}_0}{dV_0}={\varepsilon}_x+{\varepsilon}_y+{\varepsilon}_z. $$

(8.3)

Transformations of Plane Strain The strain at a point

p is said to be a state of

plane strain if it is of the form

$$ \left[\begin{array}{ccc}{\varepsilon}_x& {\gamma}_{xy}& 0\\ {}{\gamma}_{yx}& {\varepsilon}_y& 0\\ {}0& 0& 0\end{array}\right]. $$

(8.4)

In terms of a coordinate system

x ^{′} y ^{′} z ^{′} oriented as shown in Fig. (a), the components of strain are

$$ {\varepsilon}_{x^{\prime }}=\frac{\varepsilon_x+{\varepsilon}_y}{2}+\frac{\varepsilon_x-{\varepsilon}_y}{2}\cos\;2\theta +\frac{\gamma_{xy}}{2}\sin\;2\theta, $$

(8.7)

$$ {\varepsilon}_{y^{\prime }}=\frac{\varepsilon_x+{\varepsilon}_y}{2}-\frac{\varepsilon_x-{\varepsilon}_y}{2}\cos\;2\theta -\frac{\gamma_{xy}}{2}\sin\;2\theta, $$

(8.8)

$$ \frac{\gamma_{x^{\prime }{y}^{\prime }}}{2}=-\frac{\varepsilon_x-{\varepsilon}_y}{2}\sin\;2\theta +\frac{\gamma_{xy}}{2}\cos\;2\theta . $$

(8.9)

For an isotropic linear elastic material, Eqs. (8.7 ), (8.8 ), and (8.9 ) also apply to the components of strain resulting from a state of plane stress.

Strain Gauge Rosette Suppose that a material is subject to an unknown state of plane strain relative to the

x -

y coordinate system in Fig. (a). A

strain gauge rosette measures the normal strain

\( {\varepsilon}_{x^{\prime }} \) in three different directions:

θ _{a} ,

θ _{b} , and

θ _{c} . Let the measured strains be

ε _{a} ,

ε _{b} , and

ε _{c} . Using Eq. (

8.6 ) to express these strains in terms of the strain components

ε _{x} ,

ε _{y} , and

γ _{xy} gives

$$ {\displaystyle \begin{array}{c}{\varepsilon}_a={\varepsilon}_x\;{\cos}^2{\theta}_a+{\varepsilon}_y\;{\sin}^2{\theta}_a+{\gamma}_{xy}\;\sin\;{\theta}_a\;\cos\;{\theta}_a,\\ {}{\varepsilon}_b={\varepsilon}_x\;{\cos}^2{\theta}_b+{\varepsilon}_y\;{\sin}^2{\theta}_b+{\gamma}_{xy}\;\sin\;{\theta}_b\;\cos\;{\theta}_b,\\ {}{\varepsilon}_c={\varepsilon}_x\;{\cos}^2{\theta}_c+{\varepsilon}_y\;{\sin}^2{\theta}_c+{\gamma}_{xy}\;\sin\;{\theta}_c\;\cos\;{\theta}_c.\end{array}} $$

(8.10)

This system of equations can be solved for the strain components ε _{x} , ε _{y} , and γ _{xy} .

Maximum and Minimum Strains in Plane Strain A value of

θ for which the normal strain is a maximum or minimum is determined from the equation

$$ \tan\;2{\theta}_{\mathrm{p}}=\frac{\gamma_{xy}}{\varepsilon_x-{\varepsilon}_y}. $$

(8.11)

The values of the principal strains can be obtained by substituting

θ _{p} into Eqs. (

8.7 ) and (

8.8 ). Their values can also be determined from the equation

$$ {\varepsilon}_1,{\varepsilon}_2=\frac{\varepsilon_x+{\varepsilon}_y}{2}\pm \sqrt{{\left(\frac{\varepsilon_x-{\varepsilon}_y}{2}\right)}^2+{\left(\frac{\gamma_{xy}}{2}\right)}^2}, $$

(8.14)

although this equation does not indicate their directions. An infinitesimal square element oriented as shown in Fig. (b) is subjected to the principal strains in the

x ^{′} and

y ^{′} directions and undergoes no shear strain.

A value of

θ for which the in-plane shear strain is a maximum or minimum is determined from the equation

$$ \tan\;2{\theta}_{\mathrm{s}}=-\frac{\varepsilon_x-{\varepsilon}_y}{\gamma_{xy}}. $$

(8.15)

The corresponding shear strain can be obtained by substituting

θ _{s} into Eq. (

8.9 ). The magnitude of the maximum in-plane shear strain can be determined from the equation

$$ {\gamma}_{\mathrm{max}}=\sqrt{{\left({\varepsilon}_x-{\varepsilon}_y\right)}^2+{\gamma}_{xy}^2}. $$

(8.18)

The

absolute maximum shear strain is

$$ {\gamma}_{\mathrm{abs}}=\operatorname{Max}\left(\left|{\varepsilon}_1\right|,\left|{\varepsilon}_2\right|,\left|{\varepsilon}_1-{\varepsilon}_2\right|\right). $$

(8.22)

Mohr’s Circle for Plane Strain Given a state of plane strain ε _{x} , ε _{y} , and γ _{xy} , establish a set of horizontal and vertical axes with normal strain measured to the right along the horizontal axis and shear strain measured downward along the vertical axis. Plot two points, point P with coordinates (ε _{x} , γ _{xy} /2) and point Q with coordinates (ε _{x} , −γ _{xy} /2). Draw a straight line connecting points P and Q . Using the intersection of the straight line with the horizontal axis as the center, draw a circle that passes through the two points (Fig. (c)). Draw a straight line through the center of the circle at an angle 2θ measured counterclockwise from point P . Point P ^{′} at which this line intersects the circle has coordinates \( \left({\varepsilon}_{x^{\prime }},{\gamma}_{x^{\prime }{y}^{\prime }}/2\right), \) and point Q ^{′} has coordinates \( \left({\varepsilon}_{y^{\prime }},-{\gamma}_{x^{\prime }{y}^{\prime }}/2\right) \) (Fig. (d)).

Stress-Strain Relations A material for which the state of stress at a point is a single-valued function of the current state of strain at that point is said to be

elastic . If the function is invariant under changes of orientation of the coordinate system relative to the material, the material is

isotropic . If the components of strain are linear functions of the components of stress, the material is

linear elastic . The stress-strain relations for an isotropic linear elastic material are

$$ {\varepsilon}_x=\frac{1}{E}{\sigma}_x-\frac{v}{E}\left({\sigma}_y+{\sigma}_z\right), $$

(8.43)

$$ {\varepsilon}_y=\frac{1}{E}{\sigma}_y-\frac{v}{E}\left({\sigma}_x+{\sigma}_z\right), $$

(8.44)

$$ {\varepsilon}_z=\frac{1}{E}{\sigma}_z-\frac{v}{E}\left({\sigma}_x+{\sigma}_y\right), $$

(8.45)

$$ {\gamma}_{xy}=\frac{1}{G}{\tau}_{xy}, $$

(8.46)

$$ {\gamma}_{yz}=\frac{1}{G}{\tau}_{yz}, $$

(8.47)

$$ {\gamma}_{xz}=\frac{1}{G}{\tau}_{xz}. $$

(8.48)

The shear modulus is related to the modulus of elasticity and Poisson’s ratio by

$$ G=\frac{E}{2\left(1+v\right)}. $$

(8.50)

The stress-strain relations can also be expressed as

$$ {\sigma}_x=\left(\lambda +2\mu \right){\varepsilon}_x+\lambda \left({\varepsilon}_y+{\varepsilon}_z\right), $$

(8.51)

$$ {\sigma}_y=\left(\lambda +2\mu \right){\varepsilon}_y+\lambda \left({\varepsilon}_x+{\varepsilon}_z\right), $$

(8.52)

$$ {\sigma}_z=\left(\lambda +2\mu \right){\varepsilon}_z+\lambda \left({\varepsilon}_x+{\varepsilon}_y\right), $$

(8.53)

$$ {\tau}_{xy}={\mu \gamma}_{xy}, $$

(8.54)

$$ {\tau}_{yz}={\mu \gamma}_{yz}, $$

(8.55)

$$ {\tau}_{xz}={\mu \gamma}_{xz}, $$

(8.56)

where

λ and

μ are the

Lamé constants . The constant

μ =

G , and the constant

λ is given in terms of the modulus of elasticity and Poisson’s ratio by

$$ \lambda =\frac{vE}{\left(1+v\right)\left(1-2v\right)}. $$

(8.57)

Let an isotropic linear elastic material be subjected to a pressure

P so that

σ _{x} = −

P ,

σ _{y} = −

P , and

σ _{z} = −

P . The

bulk modulus K of the material is the ratio of −

P to the dilatation

$$ K=\frac{-P}{e}. $$

(8.58)

In terms of the modulus of elasticity and Poisson’s ratio, the bulk modulus is

$$ K=\frac{E}{3\left(1-2v\right)}. $$

(8.59)