김경종
25 KiB
is the stress that is work conjugate to \dot { \bar { \varepsilon } } ^ { G } . This stress measure, S, is known as the second Piola-Kirchhoff stress tensor.
For general motions including large strains S is not readily interpreted physically. But for the important case of large rotations and small strains, the second Piola-Kirchhoff stress is readily interpreted. We can perform the polar decomposition as F = R ¢ U, where R is the rotation of the principal axes of deformation and U is the right stretch matrix (the stretch written on the reference configuration). If we write the principal stretches in terms of nominal principal strains,
\lambda_ {I} = 1 + \varepsilon_ {I}, \lambda_ {I I} = 1 + \varepsilon_ {I I} \lambda_ {I I I} = 1 + \varepsilon_ {I I I},
the right stretch tensor can be written as
\mathbf {U} = (1 + \varepsilon_ {I}) \mathbf {N} _ {I} \mathbf {N} _ {I} ^ {T} + (1 + \varepsilon_ {I I}) \mathbf {N} _ {I I} \mathbf {N} _ {I I} ^ {T} + (1 + \varepsilon_ {I I I}) \mathbf {N} _ {I I I} \mathbf {N} _ {I I I} ^ {T} = \mathbf {I} + \pmb {\varepsilon}.
The deformation gradient can be written as
\mathbf {F} = \mathbf {R} \cdot (\mathbf {I} + \boldsymbol {\varepsilon}),
and the inverse deformation gradient can be approximated by
\mathbf {F} ^ {- 1} = (\mathbf {I} + \pmb {\varepsilon}) ^ {- 1} \cdot \mathbf {R} ^ {T} \approx (\mathbf {I} - \pmb {\varepsilon}) \cdot \mathbf {R} ^ {T},
since--for the small-strain case--all entries in " are very much smaller than one. In addition,
J = \det \mathbf {F} = \det (\mathbf {R}) \det (\mathbf {I} + \varepsilon) = 1 + 0 (\varepsilon).
Therefore,
\mathbf {S} \approx (1 + 0 (\pmb {\varepsilon})) (\mathbf {I} - \pmb {\varepsilon}) \cdot \mathbf {R} ^ {T} \cdot \pmb {\sigma} \cdot \mathbf {R} \cdot (\mathbf {I} - \pmb {\varepsilon}) ^ {T}.
Neglecting terms of order strain compared to unity (since this is the small-strain approximation), we obtain
Equation 1.5.2-3
\mathbf {S} = \mathbf {R} ^ {T} \cdot \pmb {\sigma} \cdot \mathbf {R}.
This result gives a very simple physical interpretation of the second Piola-Kirchhoff stress for small strains but arbitrarily large rotations: the components of S are the rotated axis components of ¾. That is, the components of S are the stress components, associated with directions in the reference configuration. Thus, if we use a rectangular Cartesian basis system, the (1; 1) component of \mathbf { S } , S ^ { 1 1 } , is the normal component of force per unit area acting on a surface that was normal to the X-axis in the reference configuration, regardless of the current orientation of that surface.
For example, consider a beam whose axis was initially parallel to the X-axis. Then, throughout the deformation, S11 will always be the axial stress in the beam, no matter how much the beam is rotated or bent (provided the strains remain small compared to unity). Thus, for this case we can think of S as a "material" or "corotational" stress: the material stress and strain are unique, to the order of the approximation, provided strains remain small.
When the small-strain approximation is no longer valid, it is essential to use appropriate measures of stress and strain. From a constitutive viewpoint we have already introduced the basic idea of the approach we will follow: we identify the natural reference for the material's elastic response and use stress and strain measures that provide a conjugate pairing so that the elastic potential can be readily expressed. Since we are often interested in the rate behavior of a material, and also because we prefer to use Cauchy stress as the most natural expression of the stress at a point, it is attractive to consider the usage of the strain measure whose rate is the rate of deformation. (We have identified this in one dimension as the logarithmic strain.) We then use the Kirchhoff stress, ¿ = J¾, with respect to the reference state for the material's elasticity, as the stress measure for our constitutive definitions; it is this stress measure that is used in forming constitutive models in ABAQUS at large strains, as will be seen in Chapter 4, "Mechanical Constitutive Theories."
The work conjugacy principle implies that, for "small strains," all stress measures are indistinguishable, because in this case the strain measures are the same. One interpretation of this is that, if the stress-strain curve for a material is plotted using different stress and strain measures (for example, "true" stress versus log strain, and as nominal stress versus engineering strain) the small-strain approximation is no longer appropriate at strain levels where these two plots differ to any degree considered important to the analysis.
1.5.3 Stress invariants
Many of the constitutive models in ABAQUS are formulated in terms of stress invariants. These invariants are defined as the equivalent pressure stress,
p = - \frac {1}{3} \mathrm{trace} (\pmb {\sigma});
the Mises equivalent stress,
q = \sqrt {\frac {3}{2} (\mathbf {S} : \mathbf {S})};
and the third invariant of deviatoric stress,
r = (\frac {9}{2} \mathbf {S} \cdot \mathbf {S}: \mathbf {S}) ^ {\frac {1}{3}},
where S is the deviatoric stress, defined as
\mathbf {S} = \pmb {\sigma} + p \mathbf {I}.
1.5.4 Stress rates
Many of the materials we wish to model with ABAQUS are history dependent, and it is common for the constitutive equations to appear in rate form. In ``Stress measures,'' Section 1.5.2, it was suggested that an appropriate stress measure for stress-sensitive materials (such as yielding materials) is the Kirchhoff stress. \mathsf { W e } , therefore, need to define the rate of Kirchhoff stress for use in the constitutive equations. This definition is not simply the material time rate of Kirchhoff stress, because the Kirchhoff stress components are associated with spatial directions in the current configuration (recall that the Kirchhoff stress is J _ { \pmb { \sigma } _ { : } } , where J is the volume change from the reference configuration and ¾ is the Cauchy stress, defined by \mathbf { t } = { \pmb { \sigma } } \cdot \mathbf { n } , where t and n are vectors in the current configuration).
To illustrate the issue, consider a uniaxial tension specimen under constant axial force P , lying along the x-axis at time t _ { 1 } and rotated--with the axial force held constant--to lie along the y-axis at time t _ { 2 } (see Figure 1.5.4-1).
Figure 1.5.4-1 Rotated specimen.
text_image
y,2 P ← P x,1 time t₁ y,2 P P x,1 time t₂
Write the stress components on the global (1; 2; 3) rectangular Cartesian basis. At time t _ { 1 } , \sigma _ { 1 1 } = P / A , and all other \sigma _ { i j } = 0 . , while at time t _ { 2 } , \sigma _ { 2 2 } = P / A , and all other \sigma _ { i j } = 0 . Obviously during t _ { 1 } \to t _ { 2 } , d \sigma _ { 1 1 } \neq 0 and d \sigma _ { 2 2 } \neq 0 , but equally clearly this rate of change of stress has nothing to do with the constitutive response of the material making up the bar. (A materially based stress, such as the second Piola-Kirchhoff stress, would stay constant during the above rotation, because its components are associated with a material basis.) The problem, then, is that the components of ¾ or ¿ are associated with current directions in space and, therefore, d¿ and d¾ will be nonzero if there is pure rigid body rotation, even though from a constitutive point of view the material is unchanged. Thus, we must divide the increment of ¾ or ¿ into two parts--one attributable to rigid body motion only and a remainder that is then, presumably, associated with the rate form of the stress-strain law.
We can derive a simple result for this purpose for any matrix whose components are associated with spatial directions. At some time t imagine attaching to a material point a set of base vectors, \mathbf { e } _ { \alpha } , \alpha = 1 , 2 , 3 : These vectors cannot stretch but are defined to spin with the same spin as the material. Recall that the spatial gradient of the material particle velocity at a point, \partial \mathbf { v } / \partial \mathbf { x } , was decomposed into a rate of deformation and a spin,
\mathbf {D} = \frac {1}{2} \left(\frac {\partial \mathbf {v}}{\partial \mathbf {x}} + \left[ \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \right] ^ {T}\right) \quad \mathrm{and} \quad \mathbf {\Omega} = \frac {1}{2} \left(\frac {\partial \mathbf {v}}{\partial \mathbf {x}} - \left[ \frac {\partial \mathbf {v}}{\partial \mathbf {x}} \right] ^ {T}\right).
Our concept of the motion of the base vectors \mathbf { e } _ { \alpha } in ABAQUS/Standard is that
\dot {\mathbf {e}} _ {\alpha} = \boldsymbol {\Omega} \cdot \mathbf {e} _ {\alpha}.
For shell and membrane elements a slightly different approach is used to rotate the base vectors. The differences are significant only if finite rotation of a material point is accompanied by finite shear.
Now consider any matrix T based on the current configuration: we can write it in terms of its components in the \mathbf { e } _ { \alpha } directions:
\mathbf {T} = T ^ {\alpha \beta} \mathbf {e} _ {\alpha} \mathbf {e} _ {\beta} ^ {T}.
Taking the time derivative then gives
\dot {\mathbf {T}} = \dot {T} ^ {\alpha \beta} \mathbf {e} _ {\alpha} \mathbf {e} _ {\beta} ^ {T} + T ^ {\alpha \beta} \dot {\mathbf {e}} _ {\alpha} \mathbf {e} _ {\beta} ^ {T} + T ^ {\alpha \beta} \mathbf {e} _ {\alpha} \dot {\mathbf {e}} _ {\beta} ^ {T}.
The second and third terms are the rate of T caused by the rigid body spin, so the first term is that part of T caused by other effects (in the case of stress, the rate associated with the constitutive response), called the corotational, or Jaumann, rate of T. The corotational rate of T for shells and membranes is called the Green-Naghdi rate and differs slightly from the Jaumann rate. We will write this as \mathbf { T } ^ { \nabla } , so
\dot {\mathbf {T}} = \mathbf {T} ^ {\nabla} + T ^ {\alpha \beta} \dot {\mathbf {e}} _ {\alpha} \mathbf {e} _ {\beta} ^ {T} + T ^ {\alpha \beta} \mathbf {e} _ {\alpha} \dot {\mathbf {e}} _ {\beta} ^ {T}.
From the definition of \mathbf { e } _ { \alpha } as rigid base vectors that spin with the local rigid body motion, we can rewrite this as
\dot {\mathbf {T}} = \mathbf {T} ^ {\nabla} + T ^ {\alpha \beta} \mathbf {\Omega} \cdot \mathbf {e} _ {\alpha} \mathbf {e} _ {\beta} ^ {T} + T ^ {\alpha \beta} \mathbf {e} _ {\alpha} \mathbf {e} _ {\beta} ^ {T} \cdot \mathbf {\Omega} ^ {T} = \mathbf {T} ^ {\nabla} + \mathbf {\Omega} \cdot \mathbf {T} + \mathbf {T} \cdot \mathbf {\Omega} ^ {T}.
We, thus, have the total rate of any matrix associated with spatial directions in the current configuration as the sum of the corotational rate of the matrix and a rate caused purely by the local spin. For example, the rate of change of Kirchhoff stress can be written as
\frac {d}{d t} (J \pmb {\sigma}) = \frac {d ^ {\nabla}}{d t} (J \pmb {\sigma}) + J \left(\pmb {\Omega} \cdot \pmb {\sigma} + \pmb {\sigma} \cdot \pmb {\Omega} ^ {T}\right).
We are assuming that the constitutive theory will define drdt (J¾), the corotational stress rate per \begin{array} { r l } { { \frac { d ^ { \nabla } } { d t } ( J { \pmb \sigma } ) } } \end{array} reference volume, in terms of the rate of deformation and past history, so this equation provides a convenient link between that material model and the overall change in "true" (Cauchy) stress (which is the stress measure defined directly from the equilibrium equations). In Chapter 4, "Mechanical Constitutive Theories," where the constitutive models in ABAQUS are discussed, "stress rate" per reference volume will mean \begin{array} { r } { \frac { d ^ { \nabla } } { d t } ( J \pmb { \sigma } ) } \end{array} , the corotational rate of Kirchhoff stress, which is the stress measure work conjugate to the rate of deformation.
1.5.5 State storage
Many of the constitutive models in ABAQUS require tensors to be stored to define the state at a material calculation point. Such "material state tensors" are stored as their components in a local, orthonormal, system at the material calculation point. The orientation of that system with respect to the global ( X , Y , Z ) spatial system is stored as a rotation from the global axis system. The purpose of this section is to define the manner in which such tensors are stored and updated.
Three types of local basis are used in ABAQUS for material calculations. For isotropic materials in continuum elements the global, spatial, (X; Y; Z) system is used--the material basis is fixed in time. For isotropic materials in structural surface elements (shells and membranes) the local system is defined by the standard ABAQUS convention described in ``Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual; and for beams and trusses it is defined with the 1-direction along the axis of the member and the 2- and 3-directions in material directions in the cross-section. Thus, with isotropic materials the material basis is always the same as the element basis, although for structural elements the material basis changes with time. For anisotropic materials the material basis is initially the local system in which the anisotropy is defined, according to the *ORIENTATION definition, and it then rotates with the average rigid body spin of the material. In this case the material basis and the element basis are not the same.
We refer to this local material basis at time t as { \mathbf { e } } _ { i } ^ { M } \mid _ { t } , where the superscript M indicates that the basis is associated with material calculations and \mid _ { t } means that the basis is taken at time t. In this section Latin subscripts (like the i above) take the range 1-3, while Greek subscripts will take the range 1-2.
Introduction and Basic Equations
Any tensor associated with the material's state, a, say (such as the stress tensor { \pmb \sigma } ) _ { } , is stored in terms of its components along the material basis:
\mathbf {a} = a _ {i j} \mathbf {e} _ {i} ^ {M} \mathbf {e} _ {j} ^ {M}.
The increment from time t to time t + \Delta t of local motion at the material calculation point is defined by the incremental deformation gradient,
\Delta \mathbf {F} \stackrel {\mathrm{def}} {=} \frac {\partial \mathbf {x} _ {t + \Delta t}}{\partial \mathbf {x} _ {t}}.
This matrix is calculated from the gradient interpolator of the finite element and the coordinates of the element's nodes at times t and t + \Delta t .
The polar decomposition of \Delta \mathbf { F } is
\Delta \mathbf {F} = \Delta \mathbf {V} \cdot \Delta \mathbf {R},
where \Delta \mathbf { R } is the average rigid body rotation at the material point and \Delta { \bf V } is a pure stretch matrix:
\Delta \mathbf {V} = \sum_ {I = 1} ^ {3} \Delta \lambda_ {I} \mathbf {n} _ {I} \mathbf {n} _ {I}
(here \Delta \lambda _ { I } is a principal stretch ratio and \mathbf { n } _ { I } is a principal stretch direction).
During an increment any material state tensor changes according to
\mathbf {a} | _ {t + \Delta t} = \Delta \mathbf {a} ^ {\nabla} + \Delta \mathbf {R} \cdot \mathbf {a} | _ {t} \cdot \Delta \mathbf {R} ^ {T},
where \Delta \mathbf { a } ^ { \nabla } is the change in a caused by constitutive behavior and \Delta \mathbf { R } is the average incremental rigid body rotation of the material. Since material tensors are written in terms of their components in the material basis system, this update is computed as
(a _ {i j}) | _ {t + \Delta t} = \Delta a _ {i j} ^ {\nabla} + \Delta R _ {i k} (a _ {k l}) | _ {t} \Delta R _ {j l}.
It is, therefore, necessary to project \Delta \mathbf { R } onto the material basis systems at the start and end of the increment to define the update of material tensor components:
\Delta R _ {i j} = \mathbf {e} _ {i} ^ {M} | _ {t + \Delta t} \cdot \Delta \mathbf {R} \cdot \mathbf {e} _ {j} ^ {M} | _ {t}.
For isotropic materials the { \mathbf { e } } _ { i } ^ { M } have been chosen for geometric convenience only, so the \Delta R _ { i j } are quite general.
For anisotropic materials the material basis system, { \mathbf { e } } _ { i } ^ { M } , rotates with the average rigid body rotation of
the material, ¢R, and so is updated by
\mathbf {e} _ {i} ^ {M} | _ {t + \Delta t} = \Delta \mathbf {R} \cdot \mathbf {e} _ {i} ^ {M} | _ {t}.
In this case we see that
\Delta R _ {i j} = \delta_ {i j},
and so the update of a material tensor's components simplifies to
(a _ {i j}) | _ {t + \Delta t} = \Delta a _ {i j} ^ {\nabla} + (a _ {i j}) | _ {t}.
However, since in this case the material basis system is not the same as the element basis system, \mathbf { e } _ { i } ^ { E } (which is chosen for geometric convenience for element calculations), transformations must be done to change basis system. Specifically, at the start of the material calculation routines, the strain increments are rotated from the element basis into the material basis:
\Delta \varepsilon_ {i j} ^ {M} = \Delta \varepsilon_ {k l} ^ {E} [ \mathbf {e} _ {i} ^ {M} \cdot \mathbf {e} _ {k} ^ {E} ] [ \mathbf {e} _ {j} ^ {M} \cdot \mathbf {e} _ {l} ^ {E} ].
Likewise, at the end of the material calculation routines, the stress increments are rotated back to the element basis for integration into the discretized approximation to the equilibrium equations:
\Delta \sigma_ {i j} ^ {E} = \Delta \sigma_ {k l} ^ {M} [ \mathbf {e} _ {i} ^ {E} \cdot \mathbf {e} _ {k} ^ {M} ] [ \mathbf {e} _ {j} ^ {E} \cdot \mathbf {e} _ {l} ^ {M} ].
In addition, the material stiffness matrix,
D _ {i j k l} \stackrel {\mathrm{def}} {=} \frac {\partial \sigma_ {i j}}{\partial \varepsilon_ {k l}}
must also be rotated from the material basis to the element basis:
D _ {i j k l} ^ {E} = D _ {m n p q} ^ {M} [ \mathbf {e} _ {i} ^ {E} \cdot \mathbf {e} _ {m} ^ {M} ] [ \mathbf {e} _ {j} ^ {E} \cdot \mathbf {e} _ {n} ^ {M} ] [ \mathbf {e} _ {k} ^ {E} \cdot \mathbf {e} _ {p} ^ {M} ] [ \mathbf {e} _ {l} ^ {E} \cdot \mathbf {e} _ {q} ^ {M} ].
For a shell or membrane only two-dimensional rotations are required--for example,
\Delta \varepsilon_ {\alpha \beta} ^ {M} = \Delta \varepsilon_ {\gamma \delta} ^ {E} [ \mathbf {e} _ {\alpha} ^ {M} \cdot \mathbf {e} _ {\gamma} ^ {E} ] [ \mathbf {e} _ {\beta} ^ {M} \cdot \mathbf {e} _ {\delta} ^ {E} ],
since { \bf e } _ { 3 } ^ { M } \equiv { \bf e } _ { 3 } ^ { E } because both are unit vectors along the normal to the surface.
1.5.6 Energy balance
The conservation of energy implied by the first law of thermodynamics states that the time rate of change of kinetic energy and internal energy for a fixed body of material is equal to the sum of the rate of work done by the surface and body forces. This can be expressed as
Equation 1.5.6-1
\frac {d}{d t} \int_ {V} \left(\frac {1}{2} \rho \mathbf {v} \cdot \mathbf {v} + \rho U\right) d V = \int_ {S} \mathbf {v} \cdot \mathbf {t} d S + \int_ {V} \mathbf {f} \cdot \mathbf {v} d V,
where
½
is the current density,
v
is the velocity field vector,
U
is the internal energy per unit mass,
t
is the surface traction vector,
f
is the body force vector, and
n
is the normal direction vector on boundary S.
Using Gauss' theorem and the identity that t = ¾ ¢ n on the boundary S, the first term of the right-hand side of Equation 1.5.6-1 can be rewritten as
Equation 1.5.6-2
\int_ {S} \mathbf {v} \cdot \mathbf {t} d S = \int_ {V} \left(\frac {\partial}{\partial \mathbf {x}}\right) \cdot (\mathbf {v} \cdot \pmb {\sigma}) d V
= \int_ {V} [ (\frac {\partial}{\partial \mathbf {x}} \cdot \pmb {\sigma}) \cdot \mathbf {v} + \frac {\partial \mathbf {v}}{\partial \mathbf {x}}: \pmb {\sigma} ] d V
= \int_ {V} [ (\frac {\partial}{\partial \mathbf {x}} \cdot \pmb {\sigma}) \cdot \mathbf {v} + \dot {\pmb {\varepsilon}}: \pmb {\sigma} ] d V,
since we also know (see ``Equilibrium and virtual work,'' Section 1.5.1) that
\frac {\partial \mathbf {v}}{\partial \mathbf {x}}: \boldsymbol {\sigma} = \dot {\boldsymbol {\varepsilon}}: \boldsymbol {\sigma},
where "_ is the symmetric part of the velocity gradient tensor (see ``Rate of deformation and strain increment,'' Section 1.4.3). Substituting Equation 1.5.6-2 into Equation 1.5.6-1 yields
Equation 1.5.6-3
\frac {d}{d t} \int_ {V} \bigl (\frac {1}{2} \rho \mathbf {v} \cdot \mathbf {v} + \rho U \bigr) d V = \int_ {V} \bigl [ \bigl (\frac {\partial}{\partial \mathbf {x}} \cdot \pmb {\sigma} + \mathbf {f} \bigr) \cdot \mathbf {v} + \pmb {\sigma}: \dot {\pmb {\varepsilon}} \bigr ] d V.
From Cauchy's equation of motion we have
\frac {\partial}{\partial \mathbf {x}} \cdot \pmb {\sigma} + \mathbf {f} = \rho \frac {d \mathbf {v}}{d t}.
Substituting this into Equation 1.5.6-3 gives
\begin{array}{l} \frac {d}{d t} \int_ {V} (\frac {1}{2} \rho \mathbf {v} \cdot \mathbf {v} + \rho U) d V = \int_ {V} (\rho \frac {d \mathbf {v}}{d t} \cdot \mathbf {v} + \pmb {\sigma}: \dot {\pmb {\varepsilon}}) d V \\ = \int_ {V} [ \frac {d}{d t} (\frac {1}{2} \rho \mathbf {v} \cdot \mathbf {v}) + \pmb {\sigma}: \dot {\pmb {\varepsilon}} ] d V. \\ \end{array}
From this we get the energy equation
\rho \frac {d U}{d t} = \pmb {\sigma}: \dot {\pmb {\varepsilon}}.
The internal energy, E _ { U } , is then defined as
E _ {U} = \int_ {V} \rho U d V = \int_ {0} ^ {t} \left(\int_ {V} \pmb {\sigma}: \dot {\pmb {\varepsilon}} d V\right) d t.
To make the energy balance (Equation 1.5.6-1) more convenient to use, we integrate it in time:
\int_ {V} \frac {1}{2} \rho \mathbf {v} \cdot \mathbf {v} d V + \int_ {V} \rho U d V = \int_ {0} ^ {t} \dot {E} _ {W F} d t + \mathrm{constant},
where \dot { E } _ { W F } is the rate of work done to the body by external forces and contact friction forces between the contact surfaces, defined as
\dot {E} _ {W F} = \int_ {S} \mathbf {v} \cdot \mathbf {t} d S + \int_ {V} \mathbf {f} \cdot \mathbf {v} d V.
We can further split the traction, t, into the surface distributed load, \mathbf { t } ^ { l } , , and the frictional traction, \mathbf { t } ^ { f } . \dot { E } _ { W F } can be written as
\dot {E} _ {W F} = \left(\int_ {S} \mathbf {v} \cdot \mathbf {t} ^ {l} d S + \int_ {V} \mathbf {f} \cdot \mathbf {v} d V\right) - \left(- \int_ {S} \mathbf {v} \cdot \mathbf {t} ^ {f} d S\right) = \dot {E} _ {W} - \dot {E} _ {F},
where \dot { E } _ { W } is the rate of work done to the body by external forces and \dot { E } _ { F } is the rate of energy dissipated by contact friction forces between the contact surfaces. An energy balance for the entire model can be written as
E _ {U} + E _ {K} + E _ {F} - E _ {W} = \text { constant },
Equation 1.5.6-4
where E _ { K } , the kinetic energy, is given by
E _ {K} = \int_ {V} \frac {1}{2} \rho \mathbf {v} \cdot \mathbf {v} d V.
For convenience, the internal energy is split into two contributions:
\begin{array}{l} E _ {U} = \int_ {0} ^ {t} \left(\int_ {V} \boldsymbol {\sigma}: \dot {\boldsymbol {\varepsilon}} d V\right) d t = \int_ {0} ^ {t} \left[ \int_ {V} \left(\boldsymbol {\sigma} ^ {c} + \boldsymbol {\sigma} ^ {d}\right): \dot {\boldsymbol {\varepsilon}} d V \right] d t \\ = \int_ {0} ^ {t} \left(\int_ {V} \boldsymbol {\sigma} ^ {c}: \dot {\boldsymbol {\varepsilon}} d V\right) d t + \int_ {0} ^ {t} \left(\int_ {V} \boldsymbol {\sigma} ^ {d}: \dot {\boldsymbol {\varepsilon}} d V\right) d t \\ = E _ {I} + E _ {V}, \\ \end{array}
where { \pmb { \sigma } } ^ { c } is the stress derived from the user-specified constitutive equation and \pmb { \sigma } ^ { d } is the viscous stress (defined for bulk viscosity, material damping, and dashpots), E _ { V } is the energy dissipated by viscous effects, and E _ { I } is the remaining energy, which we continue to call the internal energy. If we introduce the strain decomposition, \dot { \pmb { \varepsilon } } = \dot { \pmb { \varepsilon } } ^ { e l } + \dot { \pmb { \varepsilon } } ^ { p l } + \dot { \pmb { \varepsilon } } ^ { c r } (where \dot { \varepsilon } ^ { e l } , \dot { \varepsilon } ^ { p l } , and \dot { \varepsilon } ^ { c r } are elastic, plastic, and creep strain rates, respectively), the internal energy, E _ { I } , can be expressed as
Equation 1.5.6-5
\begin{array}{l} E _ {I} = \int_ {0} ^ {t} \left(\int_ {V} \pmb {\sigma} ^ {c}: \dot {\pmb {\varepsilon}} d V\right) d t \\ = \int_ {0} ^ {t} \left(\int_ {V} \pmb {\sigma} ^ {c}: \dot {\pmb {\varepsilon}} ^ {e l} d V\right) d t + \int_ {0} ^ {t} \left(\int_ {V} \pmb {\sigma} ^ {c}: \dot {\pmb {\varepsilon}} ^ {p l} d V\right) d t + \int_ {0} ^ {t} \left(\int_ {V} \pmb {\sigma} ^ {c}: \dot {\pmb {\varepsilon}} ^ {c r} d V\right) d t \\ = E _ {S} + E _ {P} + E _ {C}, \\ \end{array}
where E _ { S } is the recoverable elastic strain energy, E _ { P } is the energy dissipated by plasticity, and E _ { C } is the energy dissipated by time-dependent deformation (creep, swelling, and viscoelasticity).