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For reinforced concrete, since ABAQUS provides no direct modeling of the bond between rebar and concrete, the effect of this bond on the concrete cracks must be smeared into the plain concrete part of the model. This is generally done by increasing the value of G _ { f } ^ { I } based on comparisons with experiments on reinforced material. This increased ductility is commonly refered to as the "tension stiffening" effect.
In reinforced concrete applications the softening behavior of the concrete tends to have less influence on the overall response of the structure because of the stabilizing presence of the rebar. Therefore, it is often appropriate to define tension stiffening as a \sigma _ { t } ^ { I } { - } e _ { n n } ^ { c k } relationship directly. This option is also offered in ABAQUS.
Cracked shear models
An important feature of the cracking model is that, whereas crack initiation is based on Mode I fracture only, postcracked behavior includes Mode II as well as Mode I. The Mode II shear behavior is described next.
The Mode II model is based on the common observation that the shear behavior is dependent on the amount of crack opening. Therefore, ABAQUS offers a shear retention model in which the postcracked shear stiffness is dependent on crack opening. This model defines the total shear stress as a function of the total shear strain (shear direction nt is used as an example):
Equation 4.5.2-12
t _ {n t} = D _ {n t} ^ {I I} (e _ {n n} ^ {c k}, e _ {t t} ^ {c k}) g _ {n t} ^ {c k},
where D _ { n t } ^ { I I } ( e _ { n n } ^ { c k } , e _ { t t } ^ { c k } ) is a stiffness that depends on crack opening. D _ { n t } ^ { I I } can be expressed as
D _ {n t} ^ {I I} = \alpha (e _ {n n} ^ {c k}, e _ {t t} ^ {c k}) G,
where G is the shear modulus of the uncracked concrete and \alpha ( e _ { n n } ^ { c k } , e _ { t t } ^ { c k } ) is a user-defined dependence of the form shown in Figure 4.5.2-8.
Figure 4.5.2-8 Shear retention factor dependence on crack opening.
line
| θ | α | | ------- | ----- | | e^ck_max | 0 | | e^ck_nn | 0 |A commonly used mathematical form for this dependence when there is only one crack, associated with direction n , is the power law proposed by Rots and Blaauwendraad (1989):
Equation 4.5.2-13
\alpha (e _ {n n} ^ {c k}) = \frac {\left(1 - \frac {e _ {n n} ^ {c k}}{e _ {m a x} ^ {c k}}\right) ^ {p}}{1 - \left(1 - \frac {e _ {n n} ^ {c k}}{e _ {m a x} ^ {c k}}\right) ^ {p}},
where p and e _ { m a x } ^ { c k } are material parameters. This form satisfies the requirements that ® ! 1 as e _ { n n } ^ { c k } \to 0 (corresponding to the state before crack initiation) and \alpha \to 0 as e _ { n n } ^ { c k } \to e _ { m a x } ^ { c k } (corresponding to complete loss of aggregate interlock). Note that the bounds of \alpha , , as defined in our model using the elastic-cracking strain decomposition, are \infty and zero. This contrasts with some of the traditional shear retention models where the intact concrete and cracking strains are not separated; the shear retention in these models is defined using a shear retention factor, \rho , which can have values between one and zero. The relationship between these two shear retention parameters is
Equation 4.5.2-14
\rho = \frac {\alpha}{(\alpha + 1)}.
The shear retention power law form given in Equation 4.5.2-13 can then be written in terms of \rho as
\rho (e _ {n n} ^ {c k}) = \left(1 - \frac {e _ {n n} ^ {c k}}{e _ {m a x} ^ {c k}}\right) ^ {p}.
Since users are more accustomed to specifying shear retention factors in the traditional way (with values between one and zero), the ABAQUS input requests \rho { - } e _ { n n } ^ { c k } data. Using Equation 4.5.2-14, these data are then converted to \alpha { - } e _ { n n } ^ { c k } data for computation purposes.
When the shear component under consideration is associated with only one open crack direction ( n or t), the crack opening dependence is obtained directly from Figure 4.5.2-8. However, when the shear direction is associated with two open crack directions ( n and t), then
g _ {n t} ^ {c k} = g _ {n t} ^ {c k, n} + g _ {n t} ^ {c k, t} = \frac {t _ {n t}}{D _ {n t} ^ {I I , n}} + \frac {t _ {n t}}{D _ {n t} ^ {I I , t}},
with
D _ {n t} ^ {I I, n} = \alpha (e _ {n n} ^ {c k}) G, \qquad D _ {n t} ^ {I I, t} = \alpha (e _ {t t} ^ {c k}) G,
and, therefore,
D _ {n t} ^ {I I} = \frac {t _ {n t}}{g _ {n t} ^ {c k}} = \frac {D _ {n t} ^ {I I , n} D _ {n t} ^ {I I , t}}{D _ {n t} ^ {I I , n} + D _ {n t} ^ {I I , t}}.
This total stress-strain shear retention model differs from the traditional shear retention models in which the stress-strain relations are written in incremental form (again, shear direction nt is used as an example):
Equation 4.5.2-15
\Delta t _ {n t} = D _ {n t} ^ {I I} (e _ {n n} ^ {c k}, e _ {t t} ^ {c k}) \Delta g _ {n t} ^ {c k},
where D _ { n t } ^ { I I } ( e _ { n n } ^ { c k } , e _ { t t } ^ { c k } ) is an incremental stiffness that depends on crack opening. The difference between the total model used in ABAQUS (Equation 4.5.2-12) and the traditional incremental model (Equation 4.5.2-15) is best illustrated by considering the shear response of the two models in the case when a crack is simultaneously opening and shearing. This is shown in Figure 4.5.2-9 for the total model and in Figure 4.5.2-10 for the incremental model. It is apparent that, in the total model, the shear stress tends to zero as the crack opens and shears; whereas, in the incremental model the shear stress tends to a finite value. This may explain why overly stiff responses are usually obtained with the traditional shear retention models.
Figure 4.5.2-9 ABAQUS crack opening-dependent shear retention (total) model.
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| g_nt | t_nt (t_nt) | t_nt (g_nt ck) | |------|-------------|----------------| | g_nt | 0 | 0 | | (g_nt ck)1 | 0.5 | 0.5 | | (g_nt ck)1 | 1.0 | 1.0 | | (g_nt ck)1 | 1.5 | 1.5 | | (g_nt ck)1 | 2.0 | 2.0 | | (g_nt ck)1 | 2.5 | 2.5 | | (g_nt ck)1 | 3.0 | 3.0 | | (g_nt ck)1 | 3.5 | 3.5 | | (g_nt ck)1 | 4.0 | 4.0 | | (g_nt ck)1 | 4.5 | 4.5 | | (g_nt ck)1 | 5.0 | 5.0 | | (g_nt ck)1 | 5.5 | 5.5 | | (g_nt ck)1 | 6.0 | 6.0 | | (g_nt ck)1 | 6.5 | 6.5 | | (g_nt ck)1 | 7.0 | 7.0 | | (g_nt ck)1 | 7.5 | 7.5 | | (g_nt ck)1 | 8.0 | 8.0 | | (g_nt ck)1 | 8.5 | 8.5 | | (g_nt ck)1 | 9.0 | 9.0 | | (g_nt ck)1 | 9.5 | 9.5 | | (g_nt ck)1 | 10.0 | 10.0 |Figure 4.5.2-10 Traditional crack opening-dependent shear retention (incremental) model.
line
| g_nt | t_nt (D_nt^II) | t_nt (e_nn^ck e)_tt^ck |
|---|---|---|
| 0 | 0 | 0 |
| 1 | ~0.5 | ~0.5 |
| 2 | ~1.0 | ~1.0 |
| 3 | ~1.5 | ~1.5 |
| 4 | ~2.0 | ~2.0 |
| 5 | ~2.5 | ~2.5 |
| 6 | ~3.0 | ~3.0 |
| 7 | ~3.5 | ~3.5 |
| 8 | ~4.0 | ~4.0 |
| 9 | ~4.5 | ~4.5 |
| 10 | ~5.0 | ~5.0 |
4.5.3 Constitutive model for jointed materials
The jointed material model is intended to provide a simple, continuum model for materials containing a high density of parallel joint surfaces in different orientations. The spacing of the joints of a particular orientation is assumed to be sufficiently close compared to characteristic dimensions in the domain of the model that the joints can be smeared into a continuum of slip systems. An obvious application is the modeling of geotechnical problems where the medium of interest is composed of significantly faulted rock. In this context, models similar to the one described next have been proposed in the past; see, for example, the model formulated by Zienkiewicz and Pande (1977).
The model implemented in ABAQUS/Standard provides for opening of the joints, or frictional sliding of the joints, in each of these systems (a "system" in this context is a joint orientation in a particular direction at a material calculation point). In addition to the joint systems, the model includes a bulk material failure mechanism. This is based on the Drucker-Prager failure criterion.
Joint system definitions
We consider a particular joint a oriented by the normal to the joint surface \mathbf { n } _ { a } . We define \mathbf { t } _ { a \alpha } , \alpha = 1 ; 2 as two unit, orthogonal vectors in the joint surface. The local stress components are the pressure stress across the joint
p _ {a} \stackrel {\mathrm{def}} {=} \mathbf {n} _ {a} \cdot \pmb {\sigma} \cdot \mathbf {n} _ {a},
and the shear stresses across the joint
\boldsymbol {\tau} _ {a \alpha} = \mathbf {n} _ {a} \cdot \boldsymbol {\sigma} \cdot \mathbf {t} _ {a \alpha},
where ¾ is the stress tensor. We define the shear stress magnitude as
\tau_ {a} = \sqrt {\tau_ {a \alpha} \tau_ {a \alpha}}.
The local strain components are the normal strain across the joint
\varepsilon_ {a n} = \mathbf {n} _ {a} \cdot \pmb {\varepsilon} \cdot \mathbf {n} _ {a},
and the engineering shear strain in the ®-direction in the joint surface
\gamma_ {a \alpha} = \mathbf {n} _ {a} \cdot \pmb {\varepsilon} \cdot \mathbf {t} _ {a \alpha} + \mathbf {t} _ {a \alpha} \cdot \pmb {\varepsilon} \cdot \mathbf {n} _ {a},
where " is the strain tensor.
Strain rate decomposition
A linear strain rate decomposition is assumed, so that
Equation 4.5.3-1
d \varepsilon = d \varepsilon^ {e l} + d \varepsilon^ {p l},
where d" is the total strain rate, d \pmb { \varepsilon } ^ { e l } is the elastic strain rate, and d \varepsilon ^ { p l } is the inelastic (plastic) strain rate. Supposing that several systems are active (we designate an active system by i, where i = b indicates the bulk material system and i = a is a joint system a), we write
Equation 4.5.3-2
d \pmb {\varepsilon} ^ {p l} = \sum_ {i} d \pmb {\varepsilon} _ {i} ^ {p l}.
Elasticity and joint opening/closing
When all joints at a point are closed, the elastic behavior of the material is assumed to be isotropic and linear and, thus, must be defined with *ELASTIC, TYPE=ISOTROPIC. The material cannot be elastically incompressible (Poisson's ratio must be less than 0.5).
We use a stress-based joint opening criterion whereas joint closing is monitored based on strain. Joint system a opens when the estimated pressure stress across the joint (normal to the joint surface) is no longer positive:
p _ {a} \leq 0.
In this case the material is assumed to have no elastic stiffness with respect to direct strain across the joint system. Open joints, thus, create anisotropic elastic response at a point. The joint system remains open as long as
\varepsilon_ {a n (p s)} ^ {e l} \leq \varepsilon_ {a n} ^ {e l},
where \varepsilon _ { a n } ^ { e l } is the component of direct elastic strain across the joint and \varepsilon _ { a n ( p s ) } ^ { e l } is the component of direct elastic strain across the joint calculated in plane stress as
\varepsilon_ {a n (p s)} ^ {e l} = - \frac {\nu}{E} (\sigma_ {a 1} + \sigma_ {a 2}),
where E is the Young's modulus of the material, º is the Poisson's ratio, and
\sigma_ {a \alpha} = \mathbf {t} _ {a \alpha} \cdot \boldsymbol {\sigma} \cdot \mathbf {t} _ {a \alpha},
are the direct stresses in the plane of the joint.
The shear response of open joints is governed by the shear retention parameter, f _ { s r } , which represents the fraction of the elastic shear modulus retained when the joints are open ( \ f _ { s r } { = } 0 means no shear stiffness associated with open joints, while f _ { s r } { = } 1 corresponds to elastic shear stiffness in open joints; any value between these two extremes can be used).
Plastic behavior of joint systems
The failure surface for sliding on joint system a is defined by
Equation 4.5.3-3
f _ {a} = \tau_ {a} - p _ {a} \tan \beta_ {a} - d _ {a} = 0,
where \beta _ { a } is the friction angle for system a, and d _ { a } is the cohesion for system a (see Figure 4.5.3-1).
Figure 4.5.3-1 Joint system material model.
text_image
dεₐ^p₁ τₐ ψₐ βₐ fₐ dₐ pₐ
As long as f _ { a } < 0 , joint system a does not slip. When f _ { a } = 0 joint system a slips. The inelastic ( " \mathrm { { p l a s t i c } " } ) strain on the system is then given by
d \gamma_ {a \alpha} ^ {p l} = d \overline {{\varepsilon}} _ {a} ^ {p l} \frac {\tau_ {a \alpha}}{\tau_ {a}} \cos \psi_ {a}
d \varepsilon_ {a n} ^ {p l} = d \overline {{\varepsilon}} _ {a} ^ {p l} \sin \psi_ {a},
where d \gamma _ { a \alpha } ^ { p l } is the rate of inelastic shear strain in direction ® on the joint surface, d \overline { { \varepsilon } } _ { a } ^ { p l } is the magnitude of the inelastic strain rate, \psi _ { a } is the dilation angle for this joint system (choosing \psi _ { a } = 0 provides pure
shear flow on the joint, while \psi _ { a } > 0 causes dilation of the joint as it slips), and d \varepsilon _ { a n } ^ { p l } is the inelastic strain normal to the joint surface. In order to add the plastic flow contributions from different systems we write the tensorial plastic strain rate for joint a as
Equation 4.5.3-4
d \pmb {\varepsilon} _ {a} ^ {p l} = d \varepsilon_ {a n} ^ {p l} \mathbf {n} _ {a} \mathbf {n} _ {a} + d \gamma_ {a \alpha} ^ {p l} (\mathbf {n} _ {a} \mathbf {t} _ {a \alpha} + \mathbf {t} _ {a \alpha} \mathbf {n} _ {a}).
The sliding of the different joint systems at a point is independent, in the sense that sliding on one system does not change the failure criterion or the dilation angle for any other joint system at the same point. The model provides for up to three joint systems at a point.
Plastic behavior of bulk material
In addition to the joint systems, the model includes a bulk material failure mechanism. This is based on the Drucker-Prager failure criterion,
Equation 4.5.3-5
q - p \tan \beta_ {b} - d _ {b} = 0,
where q \ { \stackrel { \mathrm { d e f } } { = } } \ { \sqrt { \frac { 3 } { 2 } \mathbf { S } : \mathbf { S } } } is the Mises equivalent deviatoric stress (here S is the deviatoric stress \mathbf { S } \ { \stackrel { \mathrm { d e f } } { = } } \ { \pmb { \sigma } } + p \mathbf { I } ) , p \ { \stackrel { \mathrm { d e f } } { = } } \ - { \frac { 1 } { 3 } } \mathbf { I } : { \pmb { \sigma } } is the equivalent pressure stress, \beta _ { b } is the friction angle for the bulk material, and d _ { b } is the cohesion for the bulk material (see Figure 4.5.3-2).
Figure 4.5.3-2 Bulk material model.
text_image
q dε_b^pl ψ_b f_b β_b d_b p
If this failure criterion is reached, the bulk inelastic flow is defined by
d \pmb {\varepsilon} _ {b} ^ {p l} = d \overline {{\varepsilon}} _ {b} ^ {p l} \frac {1}{1 - \frac {1}{3} \tan \psi_ {b}} \frac {\partial g _ {b}}{\partial \pmb {\sigma}},
Equation 4.5.3-6
where
Equation 4.5.3-7
g _ {b} = q - p \tan \psi_ {b}
is the flow potential. Here, d \overline { { \mathcal { E } } } _ { b } ^ { p l } is the magnitude of the inelastic flow rate (chosen so that d \overline { { \varepsilon } } _ { b } ^ { p l } = | ( d \varepsilon _ { b } ^ { p l } ) _ { 1 1 } | in uniaxial compression in the 1-direction) and \psi _ { b } is the dilation angle for the bulk material. This bulk failure model is a simplified version of the extended Drucker-Prager model described in ``Models for granular or polymer behavior,'' Section 4.4.2. As with the joint systems, this bulk failure system is independent of the joint systems, in that bulk inelastic flow does not change the behavior of any joint system.
\operatorname { I f } \psi \neq \beta in any system the flow in that system is "nonassociated." This implies that the material stiffness matrix is not symmetric, so that the unsymmetric matrix solution scheme should be used (by setting UNSYMM=YES on the *STEP option). If the difference between \psi and \beta is not large, a symmetric approximation to the matrix can provide an acceptable rate of convergence of the equilibrium equations, and hence a lower overall solution cost. For this reason the UNSYMM parameter is not automatically invoked by this option. However, it is recommended for all cases where \psi and \beta are very different on any joint system.
Integration of the model
The constitutive equations described above are integrated using the backward Euler method generally used with the plasticity models in ABAQUS. A material Jacobian consistent with this integration operator is used for the overall equilibrium iterations.
4.6 Large-strain elasticity
4.6.1 Hyperelastic material behavior
The constitutive behavior of a hyperelastic material is defined as a total stress-total strain relationship, rather than as the rate formulation that has been discussed in the context of history-dependent materials in previous sections of this chapter. Therefore, the basic development of the formulation for hyperelasticity is somewhat different. Furthermore, hyperelastic materials are often incompressible or very nearly { \bf s o } ; hence, mixed ("hybrid") formulations can be used effectively. In this section the hyperelastic model provided in ABAQUS is defined, and the mixed variational principles used in ABAQUS/Standard to treat the fully incompressible case and the almost incompressible case are introduced.
Definitions and basic kinematic results
We first introduce some definitions and basic kinematic results that will be used in this section. Some of these items have already been discussed in Chapter 1, "Introduction and Basic Equations": they are repeated here for convenience.
Writing the current position of a material point as x and the reference position of the same point as \mathbf { X } _ { : } ,
the deformation gradient is
\mathbf {F} \stackrel {\mathrm{def}} {=} \frac {\partial \mathbf {x}}{\partial \mathbf {X}}.
Then J , the total volume change at the point, is
J \stackrel {\mathrm{def}} {=} \det (\mathbf {F}).
For simplicity, we define
\overline {{\mathbf {F}}} \stackrel {\mathrm{def}} {=} J ^ {- \frac {1}{3}} \mathbf {F}
as the deformation gradient with the volume change eliminated.
We then introduce the deviatoric stretch matrix (the left Cauchy-Green strain tensor) of \overline { { \mathbf { F } } } as
\overline {{\mathbf {B}}} \stackrel {\mathrm{def}} {=} \overline {{\mathbf {F}}} \cdot \overline {{\mathbf {F}}} ^ {T}
so that we can define the first strain invariant as
Equation 4.6.1-1
\overline {{I}} _ {1} \stackrel {\mathrm{def}} {=} \operatorname{trace} \overline {{\mathbf {B}}} = \mathbf {I}: \overline {{\mathbf {B}}},
where I is a unit matrix, and the second strain invariant as
Equation 4.6.1-2
\overline {{I}} _ {2} \stackrel {\mathrm{def}} {=} \frac {1}{2} \left(\overline {{I}} _ {1} ^ {2} - \mathrm{trace} \left(\overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}}\right)\right) = \frac {1}{2} \left(\overline {{I}} _ {1} ^ {2} - \mathbf {I}: \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}}\right).
The variations of \overline { { \mathbf { B } } } , \overline { { \mathbf { B } } } \cdot \overline { { \mathbf { B } } } , \overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } , and J will be required during the remainder of the development. We first define some variations of basic kinematic quantities that will be needed to write these results.
The gradient of the displacement variation with respect to current position is written as
\delta \mathbf {L} \stackrel {\mathrm{def}} {=} \frac {\partial \delta \mathbf {u}}{\partial \mathbf {x}}.
The virtual rate of deformation is the symmetric part of ±L:
\delta \mathbf {D} \stackrel {\mathrm{def}} {=} \operatorname{sym} (\delta \mathbf {L}) = \frac {1}{2} \left(\delta \mathbf {L} + \delta \mathbf {L} ^ {T}\right),
which we decompose into the virtual rate of change of volume per current volume (the "virtual volumetric strain rate"),
\delta \varepsilon^ {\mathrm{vol}} \stackrel {\mathrm{def}} {=} \mathbf {I}: \delta \mathbf {D},
and the virtual deviatoric strain rate,
\delta \mathbf {e} \stackrel {\mathrm{def}} {=} \delta \mathbf {D} - \frac {1}{3} \delta \varepsilon^ {\mathrm{vol}} \mathbf {I}.
The virtual rate of spin of the material is the antisymmetric part of \delta \mathbf { L } :
\delta \pmb {\Omega} \stackrel {\mathrm{def}} {=} \mathrm{asym} (\delta \mathbf {L}) = \frac {1}{2} \left(\delta \mathbf {L} - \delta \mathbf {L} ^ {T}\right).
The variations of \overline { { \mathbf { B } } } , \overline { { \mathbf { B } } } \cdot \overline { { \mathbf { B } } } , \overline { { I } } _ { 1 } , \overline { { I } } _ { 2 } , and J are obtained directly from their definitions above in terms of these quantities as
\delta \overline {{\mathbf {B}}} = \delta \mathbf {e} \cdot \overline {{\mathbf {B}}} + \overline {{\mathbf {B}}} \cdot \delta \mathbf {e} + \delta \boldsymbol {\Omega} \cdot \overline {{\mathbf {B}}} - \overline {{\mathbf {B}}} \cdot \delta \boldsymbol {\Omega} = \mathbf {H} _ {1}: \delta \mathbf {e} + \delta \boldsymbol {\Omega} \cdot \overline {{\mathbf {B}}} - \overline {{\mathbf {B}}} \cdot \delta \boldsymbol {\Omega},
where
(H _ {1}) _ {i j k l} \stackrel {\mathrm{def}} {=} \frac {1}{2} \left(\delta_ {i k} \overline {{B}} _ {j l} + \overline {{B}} _ {i k} \delta_ {j l} + \delta_ {i l} \overline {{B}} _ {j k} + \overline {{B}} _ {i l} \delta_ {j k}\right);
\delta (\overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}}) = \delta \mathbf {e} \cdot \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} + \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \cdot \delta \mathbf {e} + 2 \overline {{\mathbf {B}}} \cdot \delta \mathbf {e} \cdot \overline {{\mathbf {B}}} + \delta \boldsymbol {\Omega} \cdot \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} - \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \cdot \delta \boldsymbol {\Omega},
= \mathbf {H} _ {2}: \delta \mathbf {e} + \delta \boldsymbol {\Omega} \cdot \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} - \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}} \cdot \delta \boldsymbol {\Omega},
where
(H _ {2}) _ {i j k l} \stackrel {\mathrm{def}} {=} \frac {1}{2} \left(\delta_ {i k} \overline {{B}} _ {j p} \overline {{B}} _ {p l} + \overline {{B}} _ {i p} \overline {{B}} _ {p k} \delta_ {j l} + \delta_ {i l} \overline {{B}} _ {j p} \overline {{B}} _ {p k} + \overline {{B}} _ {i p} \overline {{B}} _ {p l} \delta_ {j k}\right) + \overline {{B}} _ {i k} \overline {{B}} _ {j l} + \overline {{B}} _ {i l} \overline {{B}} _ {j k};
Equation 4.6.1-3
\delta \overline {{I}} _ {1} = 2 \overline {{\mathbf {B}}}: \delta \mathbf {e};
Equation 4.6.1-4
\delta \overline {{I}} _ {2} = 2 (\overline {{I}} _ {1} \overline {{\mathbf {B}}} - \overline {{\mathbf {B}}} \cdot \overline {{\mathbf {B}}}): \delta \mathbf {e};
and
Equation 4.6.1-5
\delta J = J \delta \varepsilon^ {\mathrm{vol}}.