EN222: Mechanics of Solids

  

 

 

Division of Engineering

    Brown University

 

 

4. Analytical techniques and fundamental principles in linear elasticity

 

Linear elasticity theory is one of the most versatile and useful branches of solid mechanics.  Cyclic plasticity cannot be tolerated in most engineering designs, so any structure or component that will be subjected to a large number of cycles of load must be designed to remain in the elastic regime.  Linear elasticity theory provides a large number of useful solutions to boundary value problems that guide design decisions – examples include solutions to contact problems, thermo-elasticity problems; and fracture mechanics.  A second major branch of the theory is concerned with modeling defects in crystalline solids.  At a short enough length scale, most solids look like ideal elastic crystals containing line and point defects.   The material between defects responds elastically, and linear elasticity theory provides a phenomenally powerful and accurate way to predict the behavior of defects such as vacancies, dislocations, interfaces and steps on surfaces.

 

4.1 Summary of governing equations of linear elasticity

 

3D elastostatics

We already listed the governing equations of linear elasticity in our discussion of linear elastic finite element methods.  A static linear elasticity problem is posed as follows.  

 

Given:

1.      The shape of the solid in its unloaded condition

2.      The initial stress field in the solid (we will take this to be zero)

3.      The elastic constants for the solid

4.      The thermal expansion coefficients for the solid, and temperature change from the initial configuration

5.      A body force distribution  acting on the solid (Note that in this section we will use b to denote force per unit volume rather than force per unit mass, to avoid having to write out the mass density all the time)

6.      Boundary conditions, specifying displacements  on a portion  or tractions on a portion  of the boundary of R

 

Calculate displacements, strains and stresses satisfying the governing equations of linear elastostatics

 

 

Elastodynamics

 

Dynamic problems are essentially identical, except that the mass density of the solid must be specified, the boundary conditions must be specified as functions of time, and the initial displacement and velocity field must be specified.  In this case the governing equations are

 

 

 

 

Plane approximations in linear elasticity

 

3D problems are hard.  One can often obtain useful information by solving an approximate 2D problem instead.  There are three common ways to obtain planar 2D approximations to the governing equations of linear elasticity, known as anti-plane shear, plane strain and plane stress, respectively.  Each approximation is outlined briefly below.

 

 

 

Anti-plane shear

 

One can induce states of anti-plane shear in a solid by loading it in a special way.  We rarely actually load solids so as to cause them to deform in anti-plane shear.  However, we will find that the governing equations and boundary conditions for anti-plane shear problems are beautifully simple: we end up solving Laplace’s equation!  Many powerful techniques are available to do this.  Whenever you are faced with solving a complex linear elastic boundary value problem (and are unwilling to resort to numerical methods), you should consider setting up a similar anti-plane shear problem first.  Your solution will have many of the features of the more general case, and may help you see how to solve the more complex problem too.

 

We will discuss anti-plane shear in the context of a three-dimensional boundary value problem.

 

Consider a cylindrical solid with arbitrary cross-section, as shown in the figure.  Assume that the length of the cylinder L greatly exceeds any cross sectional dimension.  Consider the following boundary value problem:

 

Find a solution to the field equations of linear elasticity, with body force  with

 

 

In addition, for a traction boundary value problem we must ensure that

to guarantee the existence of an static equilibrium solution.

 

The boundary conditions on  are left in weak form:

i.e. we accept any solution with zero resultant force and moment acting on the ends of the cylinder.

 

Recall the field equations

Since all forces and boundary displacements act in the  direction, it is natural to assume that the displacements are in the  direction everywhere.  Thus, try a solution of the form

The strains and stresses follow as

where Greek subscripts range from 1 to 2. The equilibrium equations reduce to

The boundary conditions may be re-written as

 

Summary:

 

To solve an anti-plane shear problem, we assume

where

Plane Stress and Strain

 

Anti-plane shear problems are nice and simple, but we rarely load a solid so as to cause anti-plane shear deformation.  The assumption of in-plane deformation is more useful. 

 

Plane stress and plane strain solutions to the governing equations of linear elasticity approximate the following three dimensional boundary value problem.

 

Consider the cylindrical solid shown above.  For this solid, we seek a solution to the governing equations of linear elasticity with boundary conditions on the lateral boundaryes

The boundary conditions on the ends of the cylindrical region  will be left unspecified for the time being.

 

Plane Strain Approximation

 

Since the loading appears to cause the solid to deform transverse to the axis of the cylinder, it is natural to attempt to approximate the solution by assuming in-plane deformations. Thus, assume that the displacement field has the form

The state of strain follows as

while the stress state (for an isotropic solid) is

The equilibrium equations reduce to  and the boundary conditions on B become

We can now examine the boundary conditions on  .  Evidently

Thus, the plane strain solution is the exact solution to a three dimensional problem involving a cylinder whose ends are constrained by rigid frictionless walls.

 

Plane Stress Approximation

 

We may find a similar approximate solution to thin plates stretched in their own plane.

Consider the cylindrical solid shown above.  Assume that the height of the cylinder is much smaller than any relevant cross-sectional dimension.

 

Find displacements, strains and stresses satisfying the field equations of linear elasticity with body force

With

To derive the plane stress field equations, we make two approximations:

 

1. Assume

 

The justification for this approximation is that  on   to satisfy the boundary conditions. In addition, the equilibrium equation

 

 

shows that    on  , since    on  .  Thus, we expect

 

2. Find field equations for the through thickness averages of the stress and displacement components.

 

Assumption (1) allows us to determine the out of plane strain component

Thus, the field equations reduce to

Now take a thickness average of these, and note that

So that

Where

The boundary conditions reduce to

The field equations for plane stress are almost identical to those for plane strain, except for the term involving Poisson’s ratio in the constitutive law. 

 

Note that, while the plane strain solution is an exact solution to a three dimensional boundary value problem, the plane stress solution is an approximate solution, and is exact only in the limit of vanishing plate thickness.  The through thickness averages of field quantities may or may not approximate the actual fields (for example, if the plate is loaded so that it bends, then there would be a significant variation in stress and strain through the plate thickness)