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increment, solely to account for the rigid body rotation in the increment:
\mathbf {a} \Rightarrow \Delta \mathbf {R} \cdot \mathbf {a}
for a vector, and
\mathbf {A} \Rightarrow \Delta \mathbf {R} \cdot \mathbf {A} \cdot \Delta \mathbf {R} ^ {T}
for a tensor.
These rotated variables are now passed to the constitutive routines, which may provide further updates to them because of constitutive effects. These constitutive effects will be associated with deformation, which must be supplied in the form of the strain increment \Delta \varepsilon . For this we proceed as follows.
Since we assume \Delta \mathbf { R } rotates the deformation basis--in the sense that it rotates the principal axes of deformation and, thus, provides a measure of average material rotation--we can define the velocity gradient L at any time during the increment, referred to the fixed basis at t + \Delta t , as
Equation 1.4.3-2
\check {\mathbf {L}} = \Delta \mathbf {R} (t + \Delta t) \cdot \left(\Delta \mathbf {R} (t + \tau) ^ {T} \cdot \mathbf {L} \cdot \Delta \mathbf {R} (t + \tau)\right) \cdot \Delta \mathbf {R} (t + \Delta t) ^ {T}, \quad 0 \leq \tau \leq \Delta t.
Then our integration of "_ is the matrix \Delta \varepsilon , , on the basis at the end of the increment, and defined by
\Delta \varepsilon = \int_ {0} ^ {\Delta t} \mathrm{sym} \big (\check {\mathbf {L}} (t + \tau) \big) d \tau .
Using Equation 1.4.3-2, this is
Equation 1.4.3-3
\Delta \pmb {\varepsilon} = \frac {1}{2} \Delta \mathbf {R} (t + \Delta t) \cdot \int_ {0} ^ {\Delta t} \Delta \mathbf {R} (t + \tau) ^ {T} \cdot \left(\mathbf {L} (t + \tau) + \mathbf {L} ^ {T} (t + \tau)\right) \cdot
\Delta \mathbf {R} (t + \tau) d \tau \cdot \Delta \mathbf {R} (t + \Delta t) ^ {T}.
Since
\mathbf {L} = \frac {d}{d t} (\Delta \mathbf {F}) \cdot \Delta \mathbf {F} ^ {- 1},
we can make use of the polar decomposition of the increment of deformation into a stretch on the axes at the start of the increment followed by rotation ( \Delta \mathbf { F } = \Delta \mathbf { R } \cdot \Delta \mathbf { U } ) to write
\mathbf {L} = \frac {d}{d t} (\Delta \mathbf {F}) \cdot \mathbf {F} ^ {- 1}
= \left(\frac {d}{d t} (\Delta \mathbf {R}) \cdot \Delta \mathbf {U} + \Delta \mathbf {R} \cdot \frac {d}{d t} \Delta \mathbf {U}\right) \cdot \left(\Delta \mathbf {U} ^ {- 1} \cdot \Delta \mathbf {R} ^ {T}\right)
= \frac {d}{d t} (\Delta \mathbf {R}) \cdot \Delta \mathbf {R} ^ {T} + \Delta \mathbf {R} \cdot \frac {d}{d t} (\Delta \mathbf {U}) \cdot \Delta \mathbf {U} ^ {- 1} \cdot \Delta \mathbf {R} ^ {T}
so that the integrand in the definition of the increment of strain is
\Delta \mathbf {R} ^ {T} \cdot (\mathbf {L} + \mathbf {L} ^ {T}) \cdot \Delta \mathbf {R} = \frac {d}{d t} (\Delta \mathbf {U}) \cdot \Delta \mathbf {U} ^ {- 1} + \Delta \mathbf {U} ^ {- 1} \cdot \frac {d}{d t} (\Delta \mathbf {U}).
We now assume that the incremental stretch at any time in the increment written on the basis at the beginning of the increment, ¢U, always has the same principal directions \mathbf { N } _ { I } , \mathbf { N } _ { I I } , \mathbf { N } _ { I I I } , so that
\Delta \mathbf {U} = \left(1 + (\Delta \lambda_ {I} - 1) \frac {\tau}{\Delta t}\right) \mathbf {N} _ {I} \mathbf {N} _ {I} + \left(1 + (\Delta \lambda_ {I I} - 1) \frac {\tau}{\Delta t}\right) \mathbf {N} _ {I I} \mathbf {N} _ {I I}
+ \left(1 + (\Delta \lambda_ {I I I} - 1) \frac {\tau}{\Delta t}\right) \mathbf {N} _ {I I I} \mathbf {N} _ {I I I}
and, hence,
\frac {d}{d t} (\Delta \mathbf {U}) = \frac {1}{\Delta t} \bigg ((\Delta \lambda_ {I} - 1) \mathbf {N} _ {I} \mathbf {N} _ {I} + (\Delta \lambda_ {I I} - 1) \mathbf {N} _ {I I} \mathbf {N} _ {I I} + (\Delta \lambda_ {I I I} - 1) \mathbf {N} _ {I I I} \mathbf {N} _ {I I I} \bigg)
and
\Delta \mathbf {U} ^ {- 1} = \left(\frac {1}{1 + (\Delta \lambda_ {I} - 1) (\tau / \Delta t)}\right) \mathbf {N} _ {I} \mathbf {N} _ {I} + \left(\frac {1}{1 + (\Delta \lambda_ {I I} - 1) (\tau / \Delta t)}\right) \mathbf {N} _ {I I} \mathbf {N} _ {I I}
+ \left(\frac {1}{1 + (\Delta \lambda_ {I I I} - 1) (\tau / \Delta t)}\right) \mathbf {N} _ {I I I} \mathbf {N} _ {I I I}.
We can, thus, write
\frac {1}{2} \left(\frac {d}{d t} (\Delta \mathbf {U}) \cdot \Delta \mathbf {U} ^ {- 1} + \Delta \mathbf {U} ^ {- 1} \cdot \frac {d}{d t} (\Delta \mathbf {U})\right) = \frac {(\Delta \lambda_ {I} - 1) (1 / \Delta t)}{1 + (\Delta \lambda_ {I} - 1) (\tau / \Delta t)} \mathbf {N} _ {I} \mathbf {N} _ {I}
+ \frac {(\Delta \lambda_ {I I} - 1) (1 / \Delta t)}{1 + (\Delta \lambda_ {I I} - 1) (\tau / \Delta t)} \mathbf {N} _ {I I} \mathbf {N} _ {I I} + \frac {(\Delta \lambda_ {I I I} - 1) (1 / \Delta t)}{1 + (\Delta \lambda_ {I I I} - 1) (\tau / \Delta t)} \mathbf {N} _ {I I I} \mathbf {N} _ {I I I}
and, hence,
\int_ {0} ^ {\Delta t} \frac {1}{2} \left(\frac {d}{d t} (\Delta \mathbf {U}) \cdot \Delta \mathbf {U} ^ {- 1} + \Delta \mathbf {U} ^ {- 1} \cdot \frac {d}{d t} (\Delta \mathbf {U})\right) d \tau = \ln (\Delta \lambda_ {I}) \mathbf {N} _ {I} \mathbf {N} _ {I} + \ln (\Delta \lambda_ {I I}) \mathbf {N} _ {I I} \mathbf {N} _ {I I}
+ \ln (\Delta \lambda_ {I I I}) \mathbf {N} _ {I I I} \mathbf {N} _ {I I I}
so that, finally, from Equation 1.4.3-3,
\begin{array}{l} \Delta \pmb {\varepsilon} = \ln (\Delta \lambda_ {I}) \mathbf {n} _ {I} \mathbf {n} _ {I} + \ln (\Delta \lambda_ {I I}) \mathbf {n} _ {I I} \mathbf {n} _ {I I} + \ln (\Delta \lambda_ {I I I}) \mathbf {n} _ {I I I} \mathbf {n} _ {I I I} \\ = \ln \Delta \mathbf {V}. \\ \end{array}
Thus, as long as we assume that the stretch at any time during the increment has the same principal directions as the total increment of stretch (on the fixed basis at the start of the increment), the logarithmic definition of incremental strain provides the required integral of the strain rate expressed as the rate of deformation. This assumption amounts to requiring that the components of stretch grow proportionally during the increment: that \Delta \mathbf { U } ( t + \tau ) = p \Delta \mathbf { U } ( t + \Delta t ) , where p is any scalar that we take to grow monotonically from 0 to 1 during 0 \leq \tau \leq \Delta t . This assumption might be questionable if the increments are very large, but it is consistent with the levels of approximation used in the integration of the inelastic constitutive models. We, therefore, have a simple method for calculating the strain increment for use in this type of constitutive model without any additional loss of accuracy compared to what we already accept in the constitutive integration itself.
1.4.4 The additive strain rate decomposition
Many useful materials, such as conventional structural metals, can carry only very small amounts of elastic strain (the elastic modulus is typically two or three orders of magnitude larger than the yield stress). We can take advantage of this behavior to simplify the description of the deformation of such a material. Since the behavior is so common, the assumption that the elastic strains are always small forms the basis of almost all of the inelastic material models provided in ABAQUS. This section discusses the description of the deformation for this case.
We begin by assuming that the material has a natural elastic reference state in the sense that, at any time in the deformation, we can imagine isolating the immediate neighborhood of a single point in the material, preventing any further inelastic deformation, removing all external forces from the isolated piece, and allowing the material to unload: the deformation associated with this unloading will then be ( \mathbf { F } ^ { e l } ) ^ { - 1 } , the reverse of the elastic deformation. The deformation between the original reference state and this elastically unloaded state is then the inelastic deformation, \mathbf { F } ^ { p l } :
\mathbf {F} ^ {p l} = (\mathbf {F} ^ {e l}) ^ {- 1} \cdot \mathbf {F}.
The total deformation can, thus, be decomposed as
Equation 1.4.4-1
\mathbf {F} = \mathbf {F} ^ {e l} \cdot \mathbf {F} ^ {p l},
from which we can obtain the velocity gradient with respect to position in the current configuration, { \bf L } = \dot { \bf F } \cdot { \bf F } ^ { - 1 } , as
\mathbf {L} = \dot {\mathbf {F}} ^ {e l} \cdot (\mathbf {F} ^ {e l}) ^ {- 1} + \mathbf {F} ^ {e l} \cdot \dot {\mathbf {F}} ^ {p l} \cdot (\mathbf {F} ^ {p l}) ^ {- 1} \cdot (\mathbf {F} ^ {e l}) ^ {- 1},
which we write as
Equation 1.4.4-2
\mathbf {L} = \mathbf {L} ^ {e l} + \mathbf {F} ^ {e l} \cdot \mathbf {L} ^ {p l} \cdot (\mathbf {F} ^ {e l}) ^ {- 1},
by defining the elastic and plastic velocity gradients, { \bf L } ^ { e l } = \dot { \bf F } ^ { e l } \cdot ( { \bf F } ^ { e l } ) ^ { - 1 } and \mathbf { L } ^ { p l } = \dot { \mathbf { F } } ^ { p l } \cdot ( \mathbf { F } ^ { p l } ) ^ { - 1 } , by analogy with the definition of the total velocity gradient.
The motion defined by F consists of rigid body motion and deformation. For the general case there is no advantage in associating rigid body motion with both the inelastic and the elastic deformation: we lose nothing by writing
\mathbf {F} = \mathbf {V} ^ {e l} \cdot \mathbf {V} ^ {p l} \cdot \mathbf {R},
where R is the rigid body rotation of the principal axes of deformation and { { \bf { V } } ^ { e l } } and \mathbf { V } ^ { p l } are each symmetric. Thus, we are writing the deforming part of the motion as an inelastic stretch along the principal axes of the total deformation and an elastic stretch along these same axes.
For the materials of concern here we now assume that, if we write { \bf V } ^ { e l } = { \bf I } + \varepsilon ^ { e l } , the principal values of the nominal elastic strain \varepsilon ^ { e l } are all very small compared to unity: \vert \varepsilon ^ { e l } { } _ { I } \vert < < 1 . Then
\begin{array}{l} \mathbf {F} ^ {e l} = \mathbf {V} ^ {e l} \\ = (1 + \varepsilon_ {I} ^ {e l}) \mathbf {n} _ {I} \mathbf {n} _ {I} + (1 + \varepsilon_ {I I} ^ {e l}) \mathbf {n} _ {I I} \mathbf {n} _ {I I} + (1 + \varepsilon_ {I I I} ^ {e l}) \mathbf {n} _ {I I I} \mathbf {n} _ {I I I}, \\ \end{array}
and so Equation 1.4.4-2 can be approximated:
\mathbf {L} = \mathbf {L} ^ {e l} + (\mathbf {I} + \pmb {\varepsilon} ^ {e l}) \cdot \mathbf {L} ^ {p l} \cdot (\mathbf {I} + (\pmb {\varepsilon} ^ {e l}) ^ {- 1})
\approx \mathbf {L} ^ {e l} + \mathbf {L} ^ {p l}.
Taking the symmetric part,
Equation 1.4.4-3
\dot {\pmb {\varepsilon}} \approx \dot {\pmb {\varepsilon}} ^ {e l} + \dot {\pmb {\varepsilon}} ^ {p l},
where \dot { { \boldsymbol \varepsilon } } = \mathrm { s y m } ( { \bf L } ) , \dot { { \boldsymbol \varepsilon } } ^ { e l } = \mathrm { s y m } ( { \bf L } ^ { e l } ) , and \dot { \varepsilon } ^ { p l } = \mathrm { s y m } ( \mathbf { L } ^ { p l } ) are the total, elastic, and inelastic strain rates. Equation 1.4.4-3 is the classical "strain rate decomposition" of plasticity theory. We see that it derives from the general decomposition Equation 1.4.4-1 when we use the symmetric part of the velocity gradient with respect to current position (the rate of deformation, "_) as the measure of strain rate and when the total elastic strain is always small compared to one. The strain rate decomposition is used in this form in almost all of the inelastic constitutive models in ABAQUS: whenever we refer to an elastic or inelastic strain rate, we imply \dot { \pmb { \varepsilon } } ^ { e l } or \dot { \varepsilon } ^ { p l } .
1.5 Equilibrium, stress, and state storage
1.5.1 Equilibrium and virtual work
Many of the problems to which ABAQUS is applied involve finding an approximate (finite element) solution for the displacements, deformations, stresses, forces, and--possibly--other state variables such as temperature in a solid body that is subjected to some history of "loading," where "loading" implies some series of events to which the body's response is sought. The exact solution of such a problem requires that both force and moment equilibrium be maintained at all times over any arbitrary volume of the body. The displacement finite element method is based on approximating this equilibrium requirement by replacing it with a weaker requirement, that equilibrium must be maintained in an average sense over a finite number of divisions of the volume of the body. In this section we develop the exact equilibrium statement and write it in the form of the virtual work statement for later reduction to the approximate form of equilibrium used in a finite element model.
Let V denote a volume occupied by a part of the body in the current configuration, and let S be the surface bounding this volume. (Again, we should emphasize that we are adopting a Lagrangian viewpoint: the volume being considered is a volume of material in the body--specifically, V is the volume of space occupied by this material at the "current" point in time, which is distinct from the Eulerian approach, where we are examining a volume in space and watch material flowing through that volume.) Let the surface traction at any point on S be the force t per unit of current area, and let the body force at any point within the volume of material under consideration be f per unit of current volume. Force equilibrium for the volume is then
Equation 1.5.1-1
\int_ {S} \mathbf {t} d S + \int_ {V} \mathbf {f} d V = 0.
The "true" or Cauchy stress matrix ¾ at a point of S is defined by
Equation 1.5.1-2
\mathbf {t} = \mathbf {n} \cdot \boldsymbol {\sigma},
where n is the unit outward normal to S at the point. Using this definition, Equation 1.5.1-2 is
\int_ {S} \mathbf {n} \cdot \pmb {\sigma} d S + \int_ {V} \mathbf {f} d V = 0.
Gauss's theorem allows us to rewrite a surface integral as a volume integral according to
\int_ {S} \mathbf {n} \cdot (\mathbf {\theta}) d S = \int_ {V} \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot (\mathbf {\theta}), d V,
where ( ) is any continuous function--scalar, vector or tensor.
Applying the Gauss theorem to the surface integral in the equilibrium equation gives
\int_ {S} \mathbf {n} \cdot \pmb {\sigma} d S = \int_ {V} \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} d V.
Since the volume is arbitrary, this equation must apply pointwise in the body, thus providing the differential equation of translational equilibrium:
Equation 1.5.1-3
\left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} + \mathbf {f} = 0.
These are the three familiar differential equations of force equilibrium. In deriving them we have made no approximation with respect to the magnitude of the deformation or rotation--the equations are an exact statement of equilibrium so long as we are precise about our definitions of surface tractions, body forces, stress (Cauchy stress, defined by Equation 1.5.1-2), volume, and area.
Moment equilibrium is most simply written in the general case by taking moments about the origin:
\int_ {S} (\mathbf {x} \times \mathbf {t}) d S + \int_ {V} (\mathbf {x} \times \mathbf {f}) d V = 0.
Use of the Gauss theorem with this equation then leads to the result that the ``true'' (Cauchy) stress matrix must be symmetric:
Equation 1.5.1-4
\pmb {\sigma} = \pmb {\sigma} ^ {T},
so that at each point there are only six independent components of stress. Conversely, by taking the stress matrix to be symmetric, we automatically satisfy moment equilibrium and, therefore, need only consider translational equilibrium when explicitly writing the equilibrium equations. (The moment equilibrium equation written above assumes that there are no point couples acting on the volume. If there are, the stress matrix does not have the symmetry property of Equation 1.5.1-4. Continuum mechanics models that allow for such point couples have been developed, but they are not relevant to any of the models provided in ABAQUS.)
The basis for the development of a displacement-interpolation finite element model is the introduction of some locally based spatial approximation to parts of the solution. To develop such an approximation, we begin by replacing the three equilibrium equations represented by Equation 1.5.1-3 by an equivalent "weak form"--a single scalar equation over the entire body, which is obtained by multiplying the pointwise differential equations by an arbitrary, vector-valued "test function," defined, with suitable continuity, over the entire volume, and integrating. As the test function is quite arbitrary, the differential equilibrium statement in any particular direction at any particular point can always be recovered by choosing the test function to be nonzero only in that direction at that point. For this case of equilibrium with a general stress matrix, this equivalent "weak form" is the virtual work principle. The test function can be imagined to be a "virtual" velocity field, ±v, which is completely arbitrary except that it must obey any prescribed kinematic constraints and have sufficient continuity: the dot product of this test function with the equilibrium force field then represents the "virtual" work rate.
Taking the dot product of Equation 1.5.1-3 with ±v results in a single scalar equation at each material point that is then integrated over the entire body to give
\int_ {V} \left[ \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} + \mathbf {f} \right] \cdot \delta \mathbf {v} d V = 0.
Equation 1.5.1-5
The chain rule allows us to write
\left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot (\pmb {\sigma} \cdot \delta \mathbf {v}) = \left[ \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} \right] \cdot \delta \mathbf {v} + \pmb {\sigma}: \left(\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}}\right),
so that
\int_ {V} \left[ \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} \right] \cdot \delta \mathbf {v} d V = \int_ {V} \left[ \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot (\pmb {\sigma} \cdot \delta \mathbf {v}) - \pmb {\sigma}: \left(\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}}\right) \right] d V
= \int_ {S} \mathbf {n} \cdot \pmb {\sigma} \cdot \delta \mathbf {v} d S - \int_ {V} \pmb {\sigma}: \left(\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}}\right) d V,
(using the Gauss theorem with the ¯rst term)
= \int_ {S} \mathbf {t} \cdot \delta \mathbf {v} d S - \int_ {V} \pmb {\sigma}: \left(\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}}\right) d V
(using the de¯nition of Cauchy stress with the ¯rst term).
Thus, the virtual work statement, Equation 1.5.1-5, can be written
\int_ {S} \mathbf {t} \cdot \delta \mathbf {v} d S + \int_ {V} \mathbf {f} \cdot \delta \mathbf {v} d V = \int \pmb {\sigma}: \left(\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}}\right) d V.
From the previous section we recognize
\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}} = \delta \mathbf {L}
as the virtual velocity gradient in the current configuration. We can decompose the gradient into a symmetric and an antisymmetric part:
\delta \mathbf {L} = \delta \mathbf {D} + \delta \boldsymbol {\Omega},
where
\delta \mathbf {D} = \mathrm{sym} (\delta \mathbf {L}) = \frac {1}{2} (\delta \mathbf {L} + \delta \mathbf {L} ^ {T})
is the virtual strain rate (the virtual rate of deformation) and
\delta \pmb {\Omega} = \mathrm{asym} (\delta \mathbf {L}) = \frac {1}{2} (\delta \mathbf {L} - \delta \mathbf {L} ^ {T})
is the virtual rate of spin. With these definitions
\boldsymbol {\sigma}: \delta \mathbf {L} = \boldsymbol {\sigma}: \delta \mathbf {D} + \boldsymbol {\sigma}: \delta \boldsymbol {\Omega}.
Since ¾ is symmetric,
\pmb {\sigma}: \delta \pmb {\Omega} = \frac {1}{2} \pmb {\sigma}: \delta \mathbf {L} - \frac {1}{2} \pmb {\sigma}: \delta \mathbf {L} ^ {T} = \frac {1}{2} \pmb {\sigma}: \delta \mathbf {L} - \frac {1}{2} \pmb {\sigma}: \delta \mathbf {L} = 0.
Finally, we obtain the virtual work equation in the classical form
Equation 1.5.1-6
\int_ {V} \pmb {\sigma}: \delta \mathbf {D} d V = \int_ {S} \delta \mathbf {v} \cdot \mathbf {t} d S + \int_ {V} \delta \mathbf {v} \cdot \mathbf {f} d V.
Recall that t, f , and ¾ are an equilibrium set,
\mathbf {t} = \mathbf {n} \cdot \pmb {\sigma}, \quad \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot \pmb {\sigma} + \mathbf {f} = 0, \quad \pmb {\sigma} = \pmb {\sigma} ^ {T};
±D and ±v are compatible,
\delta \mathbf {D} = \frac {1}{2} \left(\frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}} + \left[ \frac {\partial \delta \mathbf {v}}{\partial \mathbf {x}} \right] ^ {T}\right);
and ±v is compatible with all kinematic constraints. We can show that any two of these three statements (virtual work, equilibrium, and compatibility of the test function ±v) imply the other: we can thus use the virtual work principle, with a suitable test function, as a statement of equilibrium.
The virtual work statement has a simple physical interpretation: the rate of work done by the external forces subjected to any virtual velocity field is equal to the rate of work done by the equilibrating stresses on the rate of deformation of the same virtual velocity field. The principle of virtual work is the "weak form" of the equilibrium equations and is used as the basic equilibrium statement for the finite element formulation that will be introduced in ``Procedures: overview and basic equations, '' Section 2.1.1. Its advantage in this regard is that it is a statement of equilibrium cast in the form of an integral over the volume of the body: we can introduce approximations by choosing test functions for the virtual velocity field that are not entirely arbitrary, but whose variation is restricted to a finite number of nodal values. This approach provides a stronger mathematical basis for studying the approximation than the alternative of direct discretization of the derivative in the differential equation of equilibrium at a point, which is the typical starting point for a finite difference approach to the same problem.
1.5.2 Stress measures
The virtual work statement (Equation 1.5.1-6) expresses equilibrium in terms of Cauchy ("true") stress
and the conjugate virtual strain rate, the rate of deformation. (Here "conjugate" means work conjugate, in the sense that the product of the stress and the strain rate defines work per current volume.) It is natural to think of stress and strain as conjugate quantities, but so far we only have "true" stress and a wide range of possible strain measures. By defining the concept of conjugacy more precisely, we can define a stress matrix conjugate to any strain matrix that we might choose to use. This exercise has some value, although--as we develop the argument--it is worth remembering that the Cauchy ("true") stress is--from the engineer's viewpoint--probably the only measure of stress of practical interest as an output value from a computer code like ABAQUS, because it is a direct measure of the traction being carried per unit area by any internal surface in the body under study. For this reason ABAQUS always reports the stress as the Cauchy stress. One of the alternative stress definitions developed in this section (Kirchhoff stress) is relevant to the constitutive development in Chapter 4, "Mechanical Constitutive Theories." The other (second Piola-Kirchhoff stress) is discussed because it is frequently mentioned in standard texts.
It is convenient to think of a solid material as having a natural, elastic, reference state to which it will return upon unloading. For a fully elastic material like rubber, this state will always be the original, unstressed, state. For a material that yields, such as a metal, this reference state will be modified by the inelastic deformation to which the material is subjected. Further, we expect the elasticity of the material to be derivable from a thermodynamic potential written about this reference state so that, for isothermal deformations, there will be a potential function for the elastic strain energy per unit of the natural reference volume. On this basis we formalize the concept of conjugacy by writing the work rate per unit of volume in this elastic reference state as
Equation 1.5.2-1
d W ^ {0} = \pmb {\tau}: d \pmb {\varepsilon},
where " is a particular choice of strain matrix, derived on the basis of the discussion in ``Strain measures,'' Section 1.4.2, and ¿ is now the stress matrix that is work conjugate to d". Equation 1.5.2-1 defines a conjugate stress measure for any chosen strain measure.
The internal virtual work rate was expressed in Equation 1.5.1-6 directly in terms of Cauchy stress, and the virtual velocity gradient. This internal virtual work rate may be rewritten as an integral over the natural reference volume:
\int_ {V} \pmb {\sigma}: \mathbf {D} d V = \int_ {V ^ {0}} J \pmb {\sigma}: \mathbf {D} d V ^ {0},
where J = d V / d V ^ { 0 } is the Jacobian of the elastic deformation between the natural reference and the current volume--the ratio of the material's volume in the current and natural configurations. According to the work conjugacy concept just defined, the stress measure defined by
Equation 1.5.2-2
\pmb {\tau} = J \pmb {\sigma}
is work conjugate to the strain measure whose rate is the rate of deformation
\mathbf {D} = \mathrm{sym} \left(\frac {\partial \mathbf {v}}{\partial \mathbf {x}}\right).
This measure of stress is called Kirchhoff stress. It is useful in the development of constitutive models at large strain because it is the most directly available stress measure when we wish to think of the strain rate measured by the rate of deformation and are considering a material with an elastic reference state.
The discussion of strain suggested that Green's strain is convenient for the description of problems involving small strains but rotations that are not small, because Green's strain matrix, \pmb { \varepsilon } ^ { G } , can be computed directly from the deformation gradient F. We now develop the stress measure work conjugate to Green's strain. From ``Strain measures,'' Section 1.4.2, the standard Green's strain matrix was defined with respect to the reference configuration as
\pmb {\varepsilon} ^ {G} = \frac {1}{2} (\mathbf {F} ^ {T} \cdot \mathbf {F} - \mathbf {I}),
so the rate of Green's strain is
\dot {\pmb {\varepsilon}} ^ {G} = \frac {1}{2} (\dot {\mathbf {F}} ^ {T} \cdot \mathbf {F} + \mathbf {F} ^ {T} \cdot \dot {\mathbf {F}}).
From the discussion of the rate of deformation (``Rate of deformation and strain increment,'' Section 1.4.3) we have { \dot { \mathbf { F } } } = \mathbf { L } \cdot \mathbf { F } so that
\begin{array}{l} \dot {\pmb {\varepsilon}} ^ {G} = \frac {1}{2} \mathbf {F} ^ {T} \cdot \left(\mathbf {L} + \mathbf {L} ^ {T}\right) \cdot \mathbf {F} \\ = \mathbf {F} ^ {T} \cdot \mathbf {D} \cdot \mathbf {F} \\ \end{array}
and, thus,
\mathbf {D} = \mathbf {F} ^ {- T} \cdot \dot {\pmb {\varepsilon}} ^ {G} \cdot \mathbf {F} ^ {- 1},
where \mathbf { F } ^ { - T } means the inverse of the transpose of F.
Since the work rate per unit reference volume is
d W ^ {0} = J \boldsymbol {\sigma}: \mathbf {D},
it follows that
d W ^ {0} = J \pmb {\sigma}: (\mathbf {F} ^ {- T} \cdot \dot {\pmb {\varepsilon}} ^ {G} \cdot \mathbf {F} ^ {- 1}) = J (\mathbf {F} ^ {- 1} \cdot \pmb {\sigma} \cdot \mathbf {F} ^ {- T}): \dot {\pmb {\varepsilon}} ^ {G}.
(This last manipulation is most readily seen by looking at the equation in component form.) Thus,
\mathbf {S} = J \mathbf {F} ^ {- 1} \cdot \pmb {\sigma} \cdot \mathbf {F} ^ {- T}