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)  represents the three Cartesian components of the vector y.


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


 Volume preserving uniaxial extension



 Simple shear



 Rigid rotation through angle  about  axis


 General rigid rotation about the origin


where R must satisfy , det(R)>0. (i.e. R is proper orthogonal). I  is the identity tensor with components  


Alternatively, a rigid rotation through angle  (with right hand screw convention) about an axis through the origin that is parallel to a unit vector n can be written as


The components of R are thus


 where  is the permutation symbol, satisfying


 General homogeneous deformation




where  are constants.


The physical significance of a homogeneous deformation is that all straight lines in the solid remain straight under the deformation.  Thus, every point in the solid experiences the same shape change.  All the deformations listed above are examples of homogeneous deformations.







2.1.2 The Displacement gradient and Deformation gradient tensors


These quantities are defined by

 Displacement Gradient Tensor:    is a tensor with components  

 Deformation Gradient Tensor:  

where I  is the identity tensor, with components described by the Kronekor delta symbol:


and  represents the gradient operator. Formally, the gradient of a vector field u(x) is defined so that


(for more details see Appendix B), but in practice the component formula  is more useful.


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  as a Taylor series


so that


We identify the term in parentheses as the deformation gradient, so



The inverse of the deformation gradient  arises in many calculations.  It is defined through


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  and apply the chain rule





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,  is the permutation symbol. The element is mapped to a paralellepiped with sides dr, dv, and dw with volume given by


Recall that


so that


Recall that


so that




Observe that

 For any physically admissible deformation, the volume of the deformed element must be positive (no matter how much you deform a solid, you can’t make material disappear).  Therefore, all physically admissible displacement fields must satisfy J>0

 If a material is incompressible, its volume remains constant.  This requires J=1.

 If the mass density of the material at a point in the undeformed solid is , its mass density in the deformed solid is  


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  on our deforming solid, as shown in the picture. The orientation of the specimen is arbitrary, and is specified by a unit vector m, with components .  Upon deformation, the specimen increases in length to . Define the strain of the specimen as


Note that this definition of strain is similar to the definition  you are familiar with, but contains an additional term.  The additional term is negligible for small . Given the Lagrange strain components , the strain of the specimen may be computed from


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 . Then the Lagrange strain tensor is


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


 For small strains, the engineering strain of an infinitesimal fiber aligned with a unit vector m can be estimated as


 Note that


 (see below for more details)

 The infinitesimal strain tensor is closely related to the strain matrix introduced in elementary strength of materials courses.  For example, the physical significance of the (2 dimensional) strain matrix


is illustrated in the figure.


To relate this to the infintesimal strain tensor, let  be a Cartesian basis, with  parallel to x and  parallel to y as shown.  Let  denote the components of the infinitesimal strain tensor in this basis.  Then





2.1.8 Engineering shear strains


For a general strain tensor (which could be any of ,  or , among others), the diagonal strain components  are known as `direct’ strains, while the off diagonal terms   are known as ‘shear strains’


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 .  To see this, recall that


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  are


Observe that  is skew symmetric:


A skew tensor represents a rotation through a small angle.  Specifically, the operation  rotates the infinitesimal line element  through a small angle  about an axis parallel to the unit vector .  (A skew tensor also sometimes represents an angular velocity). 


To visualize the significance of  , consider the behavior of an imaginary, infinitesimal, tensile specimen embedded in a deforming solid.  The specimen is stretched, and then rotated through an angle  about some axis q.  If the displacement gradients are small, then .


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  represents the average rotation, over all possible orientations of m, of material fibers passing through a point.


As a final remark, we note that a general deformation can always be decomposed into an infinitesimal strain and rotation


Physically, this sum of  and  can be regarded as representing two successive deformations  a small strain, followed by a rotation, in the sense that


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  and directions  of the infinitesimal strain tensor satisfy


Clearly,  and  are the eigenvalues and eigenvectors of .  There are three principal strains and three principal directions, which are always mutually perpendicular. 


Their significance can be visualized as follows.

1.      Note that the decomposition  can be visualized as a small strain, followed by a small rigid rotation, as shown in the picture.

2.      The formula  indicates that a vector n is mapped to another, parallel vector by the strain.

3.      Thus, if you draw a small cube with its faces perpendicular to  on the undeformed solid, this cube will be stretched perpendicular to each face, with a fractional increase in length .  The faces remain perpendicular to  after deformation.

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  defined through

 The Right Cauchy Green Deformation Tensor    

 The Left Cauchy Green Deformation Tensor      

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  in the undeformed solid is stretched and rotated to  in the deformed solid, then





2.1.13 Rotation tensor, and Left and Right Stretch Tensors


The definitions of these quantities are

 The Right Stretch Tensor  

 The Left Stretch Tensor    

 The Rotation Tensor          


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  we will call these , with n=1,2,3.  Since C and B are both symmetric and positive definite, the eigenvalues  are all positive real numbers, and therefore their square roots  are also positive real numbers.

2.       Calculate the eigenvectors of C and normalize them so they have unit magnitude.  We will denote the eigenvectors by .  They must be normalized to satisfy  

3.       Finally, calculate , where  denotes a dyadic product (See Appendix B). In components, this can be written  

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  can be regarded as a sequence of two homogeneous deformations  U, followed by R.  Similarly,  is R followed by V.

3.       R is proper orthogonal (it satisfies  and det(R)=1), and therefore represents a rotation.  To see this, note that U is symmetric, and therefore satisfies , so that


and det(R)=det(F)det(U-1)=1

  1. U can be expressed in the form


where  are the three (mutually perpendicular) eigenvectors of U. (By construction, these are identical to the eigenvectors of C).  If we interpret  as basis vectors, we see that U is diagonal in this basis, and so corresponds to stretching parallel to each basis vector, as shown in the figure below.


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)

  1. The eigenvalues of the right stretch tensor U
  2. The eigenvalues of the left stretch tensor V
  3. The square root of the eigenvalues of the right Cauchy-Green tensor C
  4. The square root of the eigenvalues of the left Cauchy-Green tensor B

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.

  1. The principal stretch directions in the undeformed solid are the (normalized) eigenvectors of U or C.  Denote these by .
  2. The principal stretch directions in the deformed solid are the (normalized) eigenvectors of V or B. Denote these by .


To visualize the physical significance of principal stretches and their directions, note that a deformation can be decomposed as  into a sequence of a stretch followed by a rotation. 


Note also that

  1. The principal directions  are mutually perpendicular.  You could draw a little cube on the undeformed solid with faces perpendicular to these directions, as shown above.
  2. The stretch U will stretch the cube by an amount  parallel to each .  The faces of the stretched cube remain perpendicular to .
  3. The rotation R will rotate the stretched cube so that the directions  rotate to line up with .
  4. The faces of the deformed cube are perpendicular to  


The decomposition  can be visualized in much the same way.  In this case, the directions  are first rotated to coincide with .  The cube is then stretched parallel to each  to produce the same shape change.


We could compare the undeformed and deformed cubes by placing them side by side, with the vectors  and  parallel, as shown in the figure. 





2.1.15 Generalized strain measures


The polar decompositions  and  provide a way to define additional strain measures.  Let  denote the principal stretches, and let  and  denote the normalized eigenvectors of U and V.  Then one could define strain tensors through


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.





2.1.16 The velocity gradient


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 , so 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 , so that


By definition,




Finally, take the dot product of both sides with n, note that since n is a unit vector  must be perpendicular to n and therefore .  Note also that , since W is skew-symmetric.  It is easiest to show this using index notation: .  Therefore


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. 




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





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 a pure rotation , or equivalently .  To see this, recall that  and evaluate the time derivative. 

 If the deformation gradient is decomposed into a stretch followed by a rotation as  then  and  

 .  The trace of D is therefore a measure of rate of change of volume.


For small strains the rate of change of  Lagrangian strain E  is approximately equal to the rate of change of infinitesimal strain :





2.1.20  Strain Equations of Compatibility for infinitesimal strains


It is sometimes necessary to invert the relations between strain and displacement  that is to say, given the strain field, to compute the displacements. In this section, we outline how this is done, for the special case of infinitesimal deformations.


For infinitesimal motions the relation between strain and displacement is


Given the six strain componets  (six, since  ) we wish to determine the three displacement components . First, note that you can never completely recover the displacement field that gives rise to a particular strain field.  Any rigid motion produces no strain, so the displacements can only be completely determined if there is some additional information (besides the strain) that will tell you how much the solid has rotated and translated.  However, integrating the strain field can tell you the displacement field to within an arbitrary rigid motion.


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  all of these compatibilty equations are satisfied trivially, with the exception of the first:  


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  i.e. you may get different solutions depending on how the path of integration encircles the holes.


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  loaded at one end by a force P. The beam has a rectangular cross-section with height 2a and out-of-plane width b.  We will show later (Sect 5.2.4) that the strain field in the beam is



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  and  such that


The first two of these give


We can integrate the first equation with respect to  and the second equation with respect to  to get


where  and  are two functions of  and , respectively, which are yet to be determined.  We can find these functions by substituting the formulas for  and  into the expression for shear strain


We can re-write this as


The two terms in parentheses are functions of  and , respectively.  Since the left hand side must vanish for all values of   and , this means that


where  is an arbitrary constant.  We can now integrate these expressions to see that


where c and d are two more arbitrary constants. Finally, the displacement field follows as


The three arbitrary constants , c and d can be seen to represent a small rigid rotation through angle  about the  axis, together with a displacement (c,d) parallel to  axes, respectively.



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  in the solid, and arbitrarily say that the displacement at  is zero, and also take the rotation of the solid at   to be zero. Then, we can compute the displacements at any other point x in the solid, by integrating the strains along any convenient path.  In a simply connected solid, it doesn’t matter what path you pick.


Actually, you don’t exactly integrate the strains  instead, you must evaluate the following integral




Here,  are the components of the position vector at the point where we are computing the displacements, and  are the components of the position vector  of a point somewhere along the path of integration. The fact that the integral is path-independent (in a simply connected solid) is guaranteed by the compatibility condition.  Evaluating this integral in practice can be quite painful, but fortunately almost all cases where we need to integrate strains to get displacement turn out to be two-dimensional.


(c) A.F. Bower, 2008
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