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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 
 
 
 
 
 Â (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 of   
 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. 
 
 
 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 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 | ||||||||||||||||||||||||||||