Question

Plane Strain In the case of plane strain, \(\varepsilon_{3}=0,\) for example, and \(\varepsilon_{1}\) and \(\varepsilon_{2}\) are nonzero. Figure \(3-7\) illustrates a plane strain situation. A long bar is rigidly confined between supports so that it cannot expand or contract parallel to its length. In addition, the stresses \(\sigma_{1}\) and \(\sigma_{2}\) are applied uniformly along the length of the bar. Equations \((3-1)\) to \((3-3)\) reduce to $$ \begin{aligned} \sigma_{1} &=(\lambda+2 G) \varepsilon_{1}+\lambda \varepsilon_{2} \\ \sigma_{2} &=\lambda \varepsilon_{1}+(\lambda+2 G) \varepsilon_{2} \\ \sigma_{3} &=\lambda\left(\varepsilon_{1}+\varepsilon_{2}\right) \end{aligned} $$. From Equation \((3-6)\) it is obvious that $$ \sigma_{3}=v\left(\sigma_{1}+\sigma_{2}\right) $$ This can be used together with Equations \((3-4)\) and (3-5) to find $$ \begin{array}{l} \varepsilon_{1}=\frac{(1+v)}{E}\left\\{\sigma_{1}(1-v)-v \sigma_{2}\right\\} \\\ \varepsilon_{2}=\frac{(1+v)}{E}\left\\{\sigma_{2}(1-v)-v \sigma_{1}\right\\} \end{array} $$

Short answer

Strains \(\varepsilon_1\) and \(\varepsilon_2\) are related to stresses \(\sigma_1\) and \(\sigma_2\) using given equations for plane strain conditions.

Step-by-step solution

Understand given equations and conditions

We are dealing with a plane strain condition where \(\varepsilon_3=0\). The normal strains \(\varepsilon_1\) and \(\varepsilon_2\) are nonzero, and two normal stresses \(\sigma_1\) and \(\sigma_2\) are applied. The relevant equations for stress-strain relationship are provided: \[ \sigma_{1} =(\lambda + 2G) \varepsilon_{1} + \lambda \varepsilon_{2} \\sigma_{2} =\lambda \varepsilon_{1} + (\lambda + 2G) \varepsilon_{2} \\sigma_{3} = \lambda (\varepsilon_{1} + \varepsilon_{2})\] Where \(\lambda\) and \(G\) are Lame's parameters. Also given, \(\sigma_3=v(\sigma_1+\sigma_2)\) from Equation (3-6).

Use compatibility conditions

Introduce Poisson's ratio \(v\) and Young's modulus \(E\) into the equations. Recognize that \(\sigma_3 = v(\sigma_1 + \sigma_2)\) implies there is a relation between applied stresses and resulting strains. We will use this relation to express strains in terms of applied stresses.

Express strains in terms of stresses

We are given the final forms for \(\varepsilon_1\) and \(\varepsilon_2\) based on the existing material model: \[\begin{align*}\varepsilon_1 &= \frac{(1+v)}{E} \left[ \sigma_1(1-v) - v \sigma_2 \right] \\varepsilon_2 &= \frac{(1+v)}{E} \left[ \sigma_2(1-v) - v \sigma_1 \right]\end{align*}\] These are derived using the fact that, in the plane strain state, \(\sigma_3\) links the out-of-plane stress to in-plane stresses via Poisson's ratio, thereby allowing the calculation of \(\varepsilon_1\) and \(\varepsilon_2\) directly from \(\sigma_1\) and \(\sigma_2\).

Key concepts

Stress-Strain Relationship

When studying the deformation of materials, the stress-strain relationship is a fundamental concept. It describes how the stress on a material relates to the strain it undergoes. In a plane strain scenario, like the one described in the exercise, this relationship becomes particularly clear. Here, the material is constrained such that strain occurs primarily in two dimensions. The standard form is given by the equations:

• \( \sigma_1 = (\lambda + 2G) \varepsilon_1 + \lambda \varepsilon_2 \)
• \( \sigma_2 = \lambda \varepsilon_1 + (\lambda + 2G) \varepsilon_2 \)
• \( \sigma_3 = \lambda (\varepsilon_1 + \varepsilon_2) \)

These equations show that stress components \(\sigma_1\) and \(\sigma_2\) are related to strains \(\varepsilon_1\) and \(\varepsilon_2\) through Lame's parameters, \(\lambda\) and \(G\). The equations describe how material deformation under a set of applied stresses is influenced by both the material's inherent properties and the constraints of the strain state.

Poisson's Ratio

Poisson's Ratio, denoted as \(v\), is a key parameter in understanding material behavior under stress. It defines the ratio of transverse strain to axial strain when a material is subjected to uniaxial stress. In the context of plane strain, where the third strain component \(\varepsilon_3\) is zero, Poisson's Ratio ties the in-plane strains to the out-of-plane stress through the equation:
\[ \sigma_3 = v(\sigma_1 + \sigma_2) \]
This implies that if stresses \(\sigma_1\) and \(\sigma_2\) are applied, the material will also experience some out-of-plane stress \(\sigma_3\) proportional to Poisson's Ratio. Additionally, Poisson's Ratio is used to derive the expressions for in-plane strains in terms of the applied stresses, as seen in the equations for \(\varepsilon_1\) and \(\varepsilon_2\). This interconnectedness highlights how stress in one direction affects strain in another, key to predicting material behavior in engineering applications.

Lame's Parameters

Lame's Parameters, \(\lambda\) and \(G\), are essential in the stress-strain equations for understanding material elasticity. These parameters characterize how a material responds elastically to stress. In the given exercise, they appear in the relationships:

• \( \sigma_1 = (\lambda + 2G) \varepsilon_1 + \lambda \varepsilon_2 \)
• \( \sigma_2 = \lambda \varepsilon_1 + (\lambda + 2G) \varepsilon_2 \)

\(\lambda\), called the first Lame parameter, primarily accounts for volumetric response, while \(G\), also known as the shear modulus, relates to the material's ability to withstand shearing. Together, they define the stiffness of a material in response to different types of deformation. Understanding these parameters helps predict how materials will perform under complex loading conditions, such as those seen in various structural applications.