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Chapter 2
Governing Equations
The purpose of this chapter is to summarize the equations that govern the response of solids to mechanical or thermal loading. The following topics will be addressed in turn: 1. The mathematical description of shape changes in a solid; 2. The mathematical description of internal forces in a solid; 3. Equations of motion for deformable solids; 4. Concepts of mechanical work and power for deformable solids; and the important principle of virtual work
2.1 Mathematical description of shape changes in solids
In this section, we list the various mathematical formulas that are used to characterize shape changes in solids (and in fluids). The formulas might look scary at first, but they are mostly just definitions. You might find it helpful to refresh your memory on vectors and matrices (Appendix A), and to read the brief discussion of Tensors (Appendix B) and Index Notation (Appendix C) before wading through this section.
As you work through the various definitions, you should bear in mind that shape changes near a point can always be characterized by six numbers. These could be could be the six independent components of the Lagrangian strain, Eulerian strain, the left or right stretch tensors, or your own favorite deformation measure. Given the complete set of six numbers for any one deformation measure, you can always calculate the components of other strain measures. The reason that so many different deformation measures exist is partly that different material models adopt different strain measures, and partly because each measure is useful for describing a particular type of shape change.
2.1.1 The Displacement and Velocity Fields
The displacement vector u(x,t) describes the motion of each point in the solid. To make this precise, visualize a solid deforming under external loads. Every point in the solid moves as the load is applied: for example, a point at position x in the undeformed solid might move to a new position y at time t. The displacement vector is defined as
We could also express this formula using index notation, which is discussed in detail in Appendix C, as
Here, the subscript i  has  values 1,2, or 3, and (for example)
The displacement field completely specifies the change in shape of the solid. The velocity field would describe its motion, as
Examples of some simple deformations
2.1.2 The Displacement gradient and Deformation gradient tensors
These quantities are defined by
where I Â is the identity tensor, with components described by the Kronekor delta symbol:
and
(for more details see Appendix B), but in practice the component
formula
Note also that
The rules of differentiation using index notation are described in more detail in Appendix C.
The concepts of displacement gradient and deformation gradient are introduced to quantify the change in shape of infinitesimal line elements in a solid body. To see this, imagine drawing a straight line on the undeformed configuration of a solid, as shown in the figure. The line would be mapped to a smooth curve on the deformed configuration. However, suppose we focus attention on a line segment dx, much shorter than the radius of curvature of this curve, as shown. The segment would be straight in the undeformed configuration, and would also be (almost) straight in the deformed configuration. Thus, no matter how complex a deformation we impose on a solid, infinitesimal line segments are merely stretched and rotated by a deformation.The infinitesimal line segments dx and dy are related by
Written out in as a matrix equation, we have
To derive this result, consider an infinitesimal line element dx in a deforming solid. When the solid is deformed, this line element is stretched and rotated to a deformed line element dy. If we know the displacement field in the solid, we can compute dy=[x+dx+u(x+dx)]-[x+u(x)] from the position vectors of its two end points
Expand
so that
We identify the term in parentheses as the deformation gradient, so
The inverse
of the deformation gradient
or alternatively
2.1.3 Deformation gradient resulting from two successive deformations
Suppose that two successive deformations are applied to a solid, as shown. Let
map infinitesimal line elements from the original configuration to the first deformed shape, and from the first deformed shape to the second, respectively, with
The deformation gradient that maps infinitesimal line elements from the original configuration directly to the second deformed shape then follows as
Thus, the cumulative deformation gradient due to two successive deformations follows by multiplying their individual deformation gradients.
To see
this, write the cumulative mapping as
2.1.4 The Jacobian of the deformation gradient
The Jacobian is defined as
It is a measure of the volume change produced by a deformation. To see this, consider the infinitessimal volume element shown with sides dx, dy, and dz in the figure above. The original volume of the element is
Here,
Recall that
so that
Recall that
so that
Hence
Observe that
Derivatives of J. When working with constitutive equations, it is occasionally necessary to evaluate derivatives of J with respect to the components of F. The following result (which can be proved e.g. by expanding the Jacobian using index notation) is extremely useful
2.1.5 The Lagrange strain tensor
The Lagrange strain tensor is defined as
The components of Lagrange strain can also be expressed in terms of the displacement gradient as
The Lagrange strain tensor quantifies the changes in length of a material fiber, and angles between pairs of fibers in a deformable solid. It is used in calculations where large shape changes are expected.
To visualize the physical significance of E, suppose we mark out an imaginary
tensile specimen with (very short) length
Note that this definition of strain is similar to
the definition
We proceed to derive this result. Note that
is an infinitesimal vector with length and orientation of our undeformed specimen. From the preceding section, this vector is stretched and rotated to
The length of the deformed specimen is equal to the length of dy, so we see that
Hence, the strain for our line element is
giving the results stated.
2.1.6 The Eulerian strain tensor
The Eulerian strain tensor is defined as
Its physical significance is similar to the Lagrange strain tensor, except that it enables you to compute the strain of an infinitesimal line element from its orientation after deformation.
Specifically, suppose that n denotes a unit vector parallel to the deformed material fiber, as shown in the picture. Then
The proof is left as an exercise.
2.1.7 The Infinitesimal strain tensor
The infinitesimal strain tensor is defined as
where u is the displacement vector. Written out in full
The infinitesimal strain tensor is an approximate deformation measure, which is only valid for small shape changes. It is more convenient than the Lagrange or Eulerian strain, because it is linear.
Specifically, suppose the deformation gradients are
small, so that all
so the infinitesimal strain approximates the Lagrange strain. You can show that it also approximates the Eulerian strain with the same accuracy.
Properties of the infinitesimal strain tensor
 (see below for more details)
is illustrated in the figure.
To relate this to the infintesimal strain tensor, let
2.1.8 Engineering shear strains
For a general strain tensor (which could be any of
The shear strains are sometimes reported as ‘Engineering Shear Strains’ which are related to the formal definition by a factor of 2 i.e.
This factor of 2 is an endless source of confusion. Whenever someone reports shear strain to you, be sure to check which definition they are using. In particular, many commercial finite element codes output engineering shear strains.
2.1.9 Decomposition of infinitesimal strain into volumetric and deviatoric parts
The volumetric infinitesimal
strain is defined as The deviatoric infinitesimal
strain is defined as
The volumetric strain is a measure of volume changes, and for small
strains is related to the Jacobian of the deformation gradient by
The deviatioric strain is a measure of shear deformation (shear deformation involves no volume change).
2.1.10 The Infinitesimal rotation tensor
The infinitesimal rotation tensor is defined as
Written out as a matrix, the components of
Observe that
A skew tensor represents a rotation through a small
angle. Specifically, the operation
To visualize the significance ofÂ
The rotation of the specimen depends on its original
orientation, represented by the unit vector m. One can show (although
one would rather not do all the algebra) that
As a final remark, we note that a general deformation can always be decomposed into an infinitesimal strain and rotation
Physically, this sum of
first stretches the infinitesimal line element, then rotates it.
2.1.11 Principal values and directions of the infinitesimal strain tensor
The three principal values
Clearly,
Their significance can be visualized as follows. 1. Note that the decomposition 2. The formula 3. Thus, if you draw a small cube with its faces
perpendicular to 4. Finally, w rotates the small cube through a small angle onto its configuration in the deformed solid.
2.1.12 Cauchy-Green Deformation Tensors
There are two Cauchy-Green deformation tensors
They are called `left’ and `right’ tensors because
of their relation to the `left’ and ‘right’ stretch tensors defined
below. They can be regarded as
quantifying the squared length of infinitesimal fibers in the deformed
configuration, by noting that if a material fiber
2.1.13 Rotation tensor, and Left and Right Stretch Tensors
The definitions of these quantities are
To calculate these quantities you need to remember how to calculate the square root of a matrix. For example, to calculate the square root of C, you must 1. Calculate the eigenvalues of C 2. Calculate the eigenvectors of C and normalize them so
they have unit magnitude. We will
denote the eigenvectors by 3. Finally, calculate 4. As an additional bonus, you can quickly compute the inverse square root (which is needed to find R) as
To see the physical significance of these tensors, observe that 1. The definition of the rotation tensor shows that
2. The multiplicative decomposition of a constant
tensor 3. R is proper orthogonal (it satisfies
and det(R)=det(F)det(U-1)=1
where
The decompositions
are known as the right and left polar decomposition of F. (The right and left refer to the positions of U and V). They show that every homogeneous deformation can be decomposed into a stretch followed by a rigid rotation, or equivalently into a rigid rotation followed by a stretch. The decomposition is discussed in more detail in the next section.
2.1.14 Principal stretches
The principal stretches can be calculated from any one of the following (they all give the same answer)
The principal stretches are also related to the eigenvalues of the Lagrange and Eulerian strains. The details are left as an exercise.
There are two sets of principal stretch directions, associated with the undeformed and deformed solids.
To
visualize the physical significance of principal stretches and their
directions, note that a deformation can be decomposed as
Note also that
The
decomposition
We could compare the undeformed and deformed cubes
by placing them side by side, with the vectors
2.1.15 Generalized strain measures
The polar decompositions
The correspoinding Eulerian strain measures are
Another strain measure can be defined as
This can be computed directly from the deformation gradient as
and is very similar to the Lagrangean strain tensor, except that its principal directions are rotated through the rigid rotation R.
We now list several measures of the rate of deformation. The velocity gradient is the basic measure of deformation rate, and is defined as
It quantifies the relative velocities of two material particles at positions y and y+dy in the deformed solid, in the sense that
The velocity gradient can be expressed in terms of the deformation gradient and its time derivative as
To see this, note that
and recall that
2.1.17 Stretch rate and spin tensors
The stretch rate tensor is defined as The spin tensor is defined as
A general velocity gradient can be decomposed into the sum of stretch rate and spin, as
The stretch rate quantifies the rate of stretching of material fibers in the deformed solid, in the sense that
is
the rate of stretching of a material fiber with length l and orientation n in
the deformed solid. To see this, let
By definition,
Hence
Finally,
take the dot product of both sides with n,
note that since n is a unit vector
The spin tensor W can be shown to provide a measure of the average angular velocity of all material fibers passing through a material point.Â
2.1.18 Infinitesimal strain rate and rotation rate
For small strains the rate of deformation tensor can be approximated by the infinitesimal strain rate, while the spin can be approximated by the time derivative of the infinitesimal rotation tensor
Similarly, you can show that                                                   Â
2.1.19 Other deformation rate measures
The rate of deformation tensor can be related to time derivatives of other strain measures. For example the time derivative of the Lagrange strain tensor can be shown to be
Other useful results are
For small strains the rate of change of Lagrangian strain E  is approximately equal
to the rate of change of infinitesimal strain
2.1.20Â Strain Equations of Compatibility for infinitesimal strains
It
is sometimes necessary to invert
the relations between strain and displacement
For infinitesimal motions the relation between strain and displacement is
Given
the six strain componets
Second, we need to be sure that the strain-displacement relations can be integrated at all. The strain is a symmetric second order tensor field, but not all symmetric second order tensor fields can be strain fields. The strain-displacement relations amount to a system of six scalar differential equations for the three displacement components ui.
To be integrable, the strains must satisfy the compatibility conditions, which may be expressed as Or, equivalently
Or, once more equivalently
It is easy to show that all strain fields must satisfy these conditions - you simply need to substitute for the strains in terms of displacements and show that the appropriate equation is satisfied. For example,
and similarly for the other expressions. Not that for planar
problems for which
It can be shown that  (i) If the strains do not satisfy the equations of compatibility, then a displacement vector can not be integrated from the strains. (ii) If the strains satisfy the compatibility equations, and the solid simply connected (i.e. it contains no holes that go all the way through its thickness), then a displacement vector can be integrated from the strains. (iii) If the solid is not simply connected, a
displacement vector can be calculated, but it may not be single valued
Now, let us return to the question posed at the beginning of this section. Given the strains, how do we compute the displacements?
2D strain fields
For 2D (plane stress or plane strain) the procedure is quite simple and is best illustrated by working through a specific case
As
a representative example, we will use the strain field in a 2D (plane stress)
cantilever beam with Young’s modulus E
and Poisson’s ratio
We first check that the strain is compatible. For 2D problems this requires
which is clearly satisfied in this case.
For
a 2D problem we only need to determine
The first two of these give
We
can integrate the first equation with respect to
where
We can re-write this as
The
two terms in parentheses are functions of
where
where c and d are two more arbitrary constants. Finally, the displacement field follows as
The
three arbitrary constants
3D strain fields
For a general, three dimensional field a more formal procedure is required. Since the strains are the derivatives of the displacement field, so you might guess that we compute the displacements by integrating the strains. This is more or less correct. The general procedure is outlined below.
We
first pick a point
Actually,
you don’t exactly integrate the strains
where
Here,
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(c) A.F. Bower, 2008 |