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to occur (by the "crack detection surface" of the model), the orientation of the cracks is stored, and oriented, damaged elasticity is then used to model the existing cracks. Stress components associated with an open crack are not included in the definition of the crack detection surface for detecting additional cracks at the same point, and we only allow cracks to form in orthogonal directions at a point.
Since ABAQUS/Standard is an implicit, stiffness method code and the material calculations used to define the behavior of the concrete are carried out independently at each integration point in that part of the model that is made of concrete, the solution is known at the start of the time increment. The constitutive calculations must provide values of stress and material stiffness at the end of the increment, based on the current estimate of the kinematic solution for the response at the spatial integration point during the increment that provides the (logarithmic) strain, ", at the end of the increment.
Once cracks exist at a point, the component forms of all vector and tensor valued quantities are rotated so that they lie in the local system defined by the crack orientation vectors (the normals to the crack faces). The model ensures that these crack face normal vectors will be orthogonal, so that this local system is rectangular Cartesian. This use of a local system simplifies the computation of the damaged elasticity used for the components associated with existing cracks.
The model, thus, consists of a "compressive" yield/flow surface to model the concrete response in predominantly compressive states of stress, together with damaged elasticity to represent cracks that have occurred at a material calculation point, the occurrence of cracks being defined by a "crack detection" failure surface that is considered to be part of the elasticity. The details of this model are now presented.
Elastic-plastic model for concrete
The model uses the classical concepts of plasticity theory: a strain rate decomposition into elastic and inelastic strain rates, elasticity, yield, flow, and hardening.
Strain rate decomposition
We begin with a strain rate decomposition:
Equation 4.5.1-1
d \pmb {\varepsilon} = d \pmb {\varepsilon} ^ {e l} + d \pmb {\varepsilon} _ {c} ^ {p l},
where d" is the total mechanical strain rate, d"el is the elastic strain rate (which includes crack detection strains--this elastic strain will be further decomposed when we describe the elasticity), and d \varepsilon _ { c } ^ { p l } is the plastic strain rate associated with the "compression" surface.
We assume that the elastic part of the strain is always small, so that this equation can be integrated as
Equation 4.5.1-2
\varepsilon = \varepsilon^ {e l} + \varepsilon_ {c} ^ {p l}.
Compression yield
The "compression" surface is
Equation 4.5.1-3
f _ {c} = q - \sqrt {3} a _ {0} p - \sqrt {3} \tau_ {c} = 0,
where p is the effective pressure stress, defined as
p = - \frac {1}{3} \mathrm{trace} (\pmb {\sigma}),
and q is the Mises equivalent deviatoric stress:
q = \sqrt {\frac {3}{2} \mathbf {S} : \mathbf {S}},
where \mathbf { S } = \pmb { \sigma } + p \mathbf { I } are the deviatoric stress components; a _ { 0 } is a constant, which is chosen from the ratio of the ultimate stress reached in biaxial compression to the ultimate stress reached in uniaxial compression; and \tau _ { c } ( \lambda _ { c } ) is a hardening parameter ( \tau _ { c } is the size of the yield surface on the q-axis at p = 0 _ { \mathrm { { i } } } , so that \tau _ { c } is the yield stress in a state of pure shear stress when all components of \pmb { \sigma } are zero except \sigma _ { 1 2 } = \sigma _ { 2 1 } = \tau _ { c } ) . The hardening is measured by the value of \lambda _ { c } \mathrm { : } the \tau _ { c } ( \lambda _ { c } ) relationship is defined from the user's *CONCRETE data.
This simple surface is a straight line in ( p { - } q ) space and provides a good match to experimental data over a fairly wide range of pressure stress values (up to four to five times the maximum compressive stress that can be carried by the concrete in uniaxial compression). This form of the surface means that, as the hardening ( \lambda _ { c } ) changes, the surfaces in ( p { - } q ) space are similar, so--for example--the ratio of flow stress in biaxial loading to flow stress in uniaxial loading is the same at all flow stress levels. This does not appear to be contradicted by any experimental data, and it means that only one constant ( a _ { 0 } ) is needed to define the shape of the surface.
The value of a _ { 0 } is established from the user's data as follows. In uniaxial compression \begin{array} { r } { p = \frac { 1 } { 3 } \sigma _ { c } } \end{array} and q = \sigma _ { c } , where \sigma _ { c } is the stress magnitude. Therefore, on f _ { c } = 0 ,
Equation 4.5.1-4
\frac {\tau_ {c}}{\sigma_ {c}} = \left(\frac {1}{\sqrt {3}} - \frac {a _ {0}}{3}\right).
In biaxial compression p = 2 / 3 ~ \sigma _ { b c } and q = \sigma _ { b c } , where \sigma _ { b c } is the magnitude of each nonzero principal stress. Therefore, on f _ { c } = 0 ,
Equation 4.5.1-5
\frac {\tau_ {c}}{\sigma_ {b c}} = \left(\frac {1}{\sqrt {3}} - \frac {2 a _ {0}}{3}\right).
The value of \sigma _ { b c } ^ { u } / \sigma _ { c } ^ { u } = r _ { b c } ^ { \sigma } is given on the *FAILURE RATIOS data line (typically r _ { b c } ^ { \sigma } \approx 1 . 1 6 ) . a _ { 0 } can be calculated from Equation 4.5.1-4 and Equation 4.5.1-5 as
a _ {0} = \sqrt {3} \frac {1 - r _ {b c} ^ {\sigma}}{1 - 2 r _ {b c} ^ {\sigma}}.
The "compression" surface is shown in Figure 4.5.1-2 and Figure 4.5.1-3.
Hardening
The *CONCRETE option defines the magnitude of the stress, \left| \sigma _ { 1 1 } \right| , in a uniaxial compression test as a function of the inelastic strain magnitude, \left| \varepsilon _ { 1 1 } \right| . These data are used to define the \tau _ { c } ( \lambda _ { c } ) relationship, as follows.
In uniaxial compression, \begin{array} { r } { p = \frac { 1 } { 3 } \sigma _ { c } } \end{array} and q = \sigma _ { c } , where \sigma _ { c } is the stress magnitude. During active plastic loading f _ { c } = 0 , so by using the definition of f _ { c } (Equation 4.5.1-4), we obtain \tau _ { c } immediately as
Equation 4.5.1-6
\tau_ {c} = \left(\frac {1}{\sqrt {3}} - \frac {a _ {0}}{3}\right) \sigma_ {c}.
Flow
The model uses associated flow, so if f _ { c } = 0 and d \lambda _ { c } > 0 ,
d \pmb {\varepsilon} _ {c} ^ {p l} = d \lambda_ {c} \left(1 + c _ {0} \left(\frac {p}{\sigma_ {c}}\right) ^ {2}\right) \frac {\partial f _ {c}}{\partial \pmb {\sigma}};
Equation 4.5.1-7
otherwise, d \varepsilon _ { c } ^ { p l } = 0 .
In the definition of d \varepsilon _ { c } ^ { p l } , c _ { 0 } is a constant that is chosen so that the ratio of \varepsilon _ { 1 1 } ^ { p l } in a monotonically loaded biaxial compression test to \varepsilon _ { 1 1 } ^ { p l } in a monotonically loaded uniaxial compression test is r _ { b c } ^ { \varepsilon } , a value given on the *FAILURE RATIOS option (typically r _ { b c } ^ { \varepsilon } \approx 1 . 2 8 ) . The equation defining c _ { 0 } from r _ { b c } ^ { \varepsilon } and the other constants in the compression surface is derived next.
The gradient of the flow potential for the compressive surface is
\frac {\partial f _ {c}}{\partial \pmb {\sigma}} = \frac {\partial q}{\partial \pmb {\sigma}} - \sqrt {3} a _ {0} \frac {\partial p}{\partial \pmb {\sigma}}.
Since
\frac {\partial p}{\partial \pmb {\sigma}} = - \frac {1}{3} \mathbf {I}
and
\frac {\partial q}{\partial \pmb {\sigma}} = \frac {3}{2} \frac {\mathbf {S}}{q},
then
\frac {\partial f _ {c}}{\partial \pmb {\sigma}} = \frac {3}{2} \frac {\mathbf {S}}{q} + \frac {a _ {0}}{\sqrt {3}} \mathbf {I}.
In uniaxial compression \begin{array} { r } { p = \frac { 1 } { 3 } \sigma _ { c } , q = \sigma _ { c } } \end{array} , and S _ { 1 1 } = - { \textstyle \frac { 2 } { 3 } } \sigma _ { c } , so Equation 4.5.1-7 defines
Equation 4.5.1-8
\left(d \varepsilon_ {c} ^ {p l}\right) _ {1 1} ^ {c} = d \lambda_ {c} \left(1 + \frac {c _ {0}}{9}\right) \left(\frac {a _ {0}}{\sqrt {3}} - 1\right).
This equation can be integrated immediately to give
Equation 4.5.1-9
\left(\varepsilon_ {c} ^ {p l}\right) _ {1 1} ^ {c} = \lambda_ {c} \left(1 + \frac {c _ {0}}{9}\right) \left(\frac {a _ {0}}{\sqrt {3}} - 1\right)
so that \lambda _ { c } is known from \left( \varepsilon _ { c } ^ { p l } \right) _ { 1 1 } ^ { c } and the constants a _ { 0 } and c _ { 0 } . Equation 4.5.1-6 and Equation 4.5.1-9, therefore, define the \tau _ { c } ( \lambda _ { c } ) relationship from the *CONCRETE input data once c _ { 0 } is known.
The constant c _ { 0 } is calculated from the user's definition of r _ { b c } ^ { \varepsilon } , the ratio of \left( \varepsilon _ { c } ^ { p l } \right) _ { 1 1 } ^ { b c } \mathbf { t o } \left( \varepsilon _ { c } ^ { p l } \right) _ { 1 1 } ^ { c } , the total plastic strain components that would occur in monotonically loaded biaxial and uniaxial compression tests. In biaxial compression, when both nonzero principal stresses have the magnitude \sigma _ { b c } , \begin{array} { r } { p = \frac { 2 } { 3 } \sigma _ { b c } = \frac { 2 } { 3 } r _ { b c } ^ { \sigma } \sigma _ { c } , q = \sigma _ { b c } = r _ { b c } ^ { \sigma } \sigma _ { c } } \end{array} , and S _ { 1 1 } = - { \textstyle \frac { 1 } { 3 } } r _ { b c } ^ { \sigma } \sigma _ { c } , so the flow rule gives
\left(d \varepsilon_ {c} ^ {p l}\right) _ {1 1} ^ {b c} = d \lambda_ {c} \left(1 + \frac {4}{9} \left(r _ {b c} ^ {\sigma}\right) ^ {2} c _ {0}\right) \left(\frac {a _ {0}}{\sqrt {3}} - \frac {1}{2}\right).
Using this equation and Equation 4.5.1-8 then defines c _ { 0 } from r _ { b c } ^ { \varepsilon } and the other constants as
c _ {0} = 9 \frac {r _ {b c} ^ {\varepsilon} (\sqrt {3} - a _ {0}) + (a _ {0} - \sqrt {3} / 2)}{r _ {b c} ^ {\varepsilon} (a _ {0} - \sqrt {3}) + (r _ {b c} ^ {\sigma}) ^ {2} (2 \sqrt {3} - 4 a _ {0})}.
Crack detection and damaged elasticity
Cracking dominates the material behavior when the state of stress is predominantly tensile. The model uses a "crack detection" plasticity surface in stress space to determine when cracking takes place and the orientation of the cracking. Damaged elasticity is then used to describe the postfailure behavior of the concrete with open cracks.
Numerically we use the "crack detection" plasticity model for the increment in which cracking takes place and subsequently use damaged elasticity once the crack's presence and orientation have been
detected. As a result there is at least one increment in which we calculate crack detection "plastic" strains. Since these are really just the outcome of a numerical device to treat cracking, they are recast as elastic strains in the direction of cracking and as plastic strains in the other directions. (This means that we retain the stresses calculated for equilibrium purposes, as well as the strain decomposition of Equation 4.5.1-1.)
The basis of the postcracked behavior is the brittle fracture concept of Hilleborg (1976). We assume that the fracture energy required to form a unit area of crack surface, G _ { f } , , is a material property. This value can be calculated from measuring the tensile stress as a function of the crack opening displacement (Figure 4.5.1-4), as
G _ {f} = \int \sigma_ {t} d u.
Figure 4.5.1-4 Cracking behavior based on fracture energy.
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| displacement | σ_t | | ------------ | ------- | | u^σ | σ^u_t | | u^el | σ^u_t | | u | 0 |Typical values of G _ { f } range from 40 N/m (0.22 lb/in) for a typical construction concrete (with \sigma _ { c } ^ { u } \approx 2 0 MPa, 2850 lb/in2 ) to 120 N/m (0.67 lb/in) for a high strength concrete (with \sigma _ { c } ^ { u } \approx 4 0 MPa, 5700 lb/in2).
The implication of assuming that G _ { f } is a material property is that, when the elastic part of the displacement, u ^ { e l } ; is eliminated, the relationship between the stress and the remaining part of the displacement, \boldsymbol { u } ^ { c r } = \boldsymbol { u } - \boldsymbol { u } ^ { e l } ; is fixed, regardless of the specimen size. We may think of a specimen developing a single crack across its section as tensile displacement is applied to it: u ^ { c r } is the displacement across the crack and is not changed by using a longer or shorter specimen in the test (so long as the specimen is significantly longer than the width of the crack band, which will typically be of the order of the aggregate size). Thus, one important part of the cracked concrete's tensile behavior is
defined in terms of a stress/displacement relationship. In the finite element implementation of this model we must, therefore, compute the relative displacement at an integration point to provide u ^ { c r } . We do this in ABAQUS by multiplying the strain by a characteristic length associated with the integration point. The characteristic crack length is based on the element geometry: for beams and trusses we use the integration point length; for shell and planar elements we use the square root of the integration point area; for solid elements we use the cube root of the integration point volume. This definition of the characteristic length is used because we do not necessarily know in which direction the concrete will crack and so cannot choose the length measure in any particular direction. Thus, if there are elements in the model that have large aspect ratios, the model will likely provide different results if it is loaded in different directions and cracking occurs in such elements. This effect should be considered by the user in defining values for the material properties.
In reinforced concrete the \boldsymbol { \sigma } { - } \boldsymbol { u } ^ { c r } relationship must also represent the action of the bond between the concrete and the rebar as the concrete cracks. We assume this is accommodated by increasing the value of G _ { f } based on comparisons with experiments on reinforced material.
We first describe the crack detection plasticity model and then discuss the damaged elasticity.
Strain rate decomposition
We decompose the elastic strain rate of Equation 4.5.1-1 as
Equation 4.5.1-10
d \pmb {\varepsilon} ^ {e l} = d \pmb {\varepsilon} _ {d} ^ {e l} + d \pmb {\varepsilon} _ {t} ^ {p l},
where d \pmb { \varepsilon } ^ { e l } is the total mechanical strain rate for the crack detection problem, d \pmb { \varepsilon } _ { d } ^ { e l } is the elastic strain rate, and d \pmb { \varepsilon } _ { t } ^ { p l } is the plastic strain rate associated with the crack detection surface.
Yield
The crack detection surface is the Coulomb line
Equation 4.5.1-11
f _ {t} = \hat {q} - \left(3 - b _ {0} \frac {\sigma_ {t}}{\sigma_ {t} ^ {u}}\right) \hat {p} - \left(2 - \frac {b _ {0}}{3} \frac {\sigma_ {t}}{\sigma_ {t} ^ {u}}\right) \sigma_ {t} = 0,
where \sigma _ { t } ^ { u } is the failure stress in uniaxial tension and b _ { 0 } is a constant that is defined from the value of the tensile failure stress, \sigma _ { I } , in a state of biaxial stress when the other nonzero principal stress, \sigma _ { I I } , is at the uniaxial compression ultimate stress value, \sigma _ { c } ^ { u } . \sigma _ { t } ( \lambda _ { t } ) is a hardening parameter ( \sigma _ { t } is the equivalent uniaxial tensile stress). The hardening is measured by \lambda _ { t } . , with the \sigma _ { t } ( \lambda _ { t } ) relationship defined from the user's *TENSION STIFFENING data (see Figure 4.5.1-5).
Figure 4.5.1-5 "Tension stiffening" model.
line
| Strain, ε | Stress, σ |
|---|---|
| 0 | 0 |
| ε_t^u = σ_t^u / E | σ_t^u |
| ε_t^u = σ_t^u / E | σ_t^u |
| ε_t^u = σ_t^u / E | σ_t^u |
| ε_t^u = σ_t^u / E | σ_t^u |
| ε_t^u = σ_t^u / E | σ_t^u |
| ε_t^u = σ_t^u / E | σ_t^u |
| ε_t = 0 | 0 |
| ε_t = 0.5 | 0.5 |
| ε_t = 1.0 | 1.0 |
| ε_t = 1.5 | 1.5 |
| ε_t = 2.0 | 2.0 |
| ε_t = 2.5 | 2.5 |
| ε_t = 3.0 | 3.0 |
| ε_t = 3.5 | 3.5 |
| ε_t = 4.0 | 4.0 |
| ε_t = 4.5 | 4.5 |
| ε_t = 5.0 | 5.0 |
| ε_t = 5.5 | 5.5 |
| ε_t = 6.0 | 6.0 |
| ε_t = 6.5 | 6.5 |
| ε_t = 7.0 | 7.0 |
| ε_t = 7.5 | 7.5 |
| ε_t = 8.0 | 8.0 |
| ε_t = 8.5 | 8.5 |
| ε_t = 9.0 | 9.0 |
| ε_t = 9.5 | 9.5 |
| ε_t = 10.0 | 10.0 |
The stress measures \hat { p } and \hat { q } are defined in the same way as p and q , except that all stress components \sigma _ { \alpha \beta } associated with open cracks (that is, if ® or \beta is a crack direction in which the direct strain \varepsilon { \frac { e l } { \alpha \alpha } } > 0 \mathrm { ~ o r ~ } \varepsilon { \frac { e l } { \beta \beta } } > 0 ) are not included in these measures: they are invariants in subspaces of the stress space.
This surface has a simple mathematical form but matches plane stress data quite well. The hardening is introduced in the particular form shown in Equation 4.5.1-11 so that, as \sigma _ { t } \to 0 , the surface becomes q - 3 p = 0 , which in ( p { - } q ) space is the cone containing the principal axes of stress. This means that, as the tension stiffening is exhausted in a plane stress test, the stress point will drop back onto the nearest principal stress axis.
The value of b _ { 0 } is obtained as follows. The user's *FAILURE RATIOS option includes a definition of f , a ratio that states that, in a plane stress test cracking would occur when one principal stress has the value \sigma _ { I } = - \sigma _ { c } ^ { u } \left( \sigma _ { c } ^ { u } \right. is the magnitude of the ultimate stress in uniaxial compression) and the other nonzero principal stress has the value \sigma _ { I I } = f \sigma _ { t } ^ { u } . Another value provided on the *FAILURE RATIOS option is r _ { t } ^ { \sigma } , which defines
\sigma_ {t} ^ {u} = r _ {t} ^ {\sigma} \sigma_ {c} ^ {u}.
Cracking would, therefore, occur at the point with principal stresses - \sigma _ { c } ^ { u } , f r _ { t } ^ { \sigma } \sigma _ { c } ^ { u } , and 0. For these values
p = \frac {1}{3} \left(1 - f r _ {t} ^ {\sigma}\right) \sigma_ {c} ^ {u}
and
q = \sigma_ {c} ^ {u} \sqrt {1 + \left(f r _ {t} ^ {\sigma}\right) ^ {2} + f r _ {t} ^ {\sigma}}.
Therefore, with \sigma _ { t } = \sigma _ { t } ^ { u }
f _ {t} = \sigma_ {c} ^ {u} \sqrt {1 + \left(f r _ {t} ^ {\sigma}\right) ^ {2} + f r _ {t} ^ {\sigma}} - \frac {1}{3} \left(3 - b _ {0}\right) \left(1 - f r _ {t} ^ {\sigma}\right) \sigma_ {c} ^ {u} - \left(2 - \frac {b _ {0}}{3}\right) r _ {t} ^ {\sigma} \sigma_ {c} ^ {u} = 0,
so
b _ {0} = 3 \frac {1 + (2 - f) r _ {t} ^ {\sigma} - \sqrt {1 + (f r _ {t} ^ {\sigma}) ^ {2} + f r _ {t} ^ {\sigma}}}{1 + r _ {t} ^ {\sigma} (1 - f)}.
The crack detection surface is shown in Figure 4.5.1-2 and Figure 4.5.1-3.
Flow
The crack detection model uses the assumption of associated flow, so if f _ { t } = 0 and d \lambda _ { t } > 0 ,
Equation 4.5.1-12
d \pmb {\varepsilon} _ {t} ^ {p l} = d \lambda_ {t} \frac {\partial f _ {t}}{\partial \pmb {\sigma}};
otherwise, d \pmb { \varepsilon } _ { t } ^ { p l } = 0
Hardening
The *TENSION STIFFENING option defines the magnitude of the stress, \sigma _ { t } . , in a uniaxial tension test as a function of the inelastic strain. (When the fracture energy concept is used to define the postfailure behavior, "strain" is now defined as u ^ { c r } / c , where c is the characteristic length associated with the integration point.) The \sigma _ { t } ( \lambda _ { t } ) relationship is defined as follows.
Using the definition of f _ { t } , Equation 4.5.1-11, in the flow rule above we write
d \pmb {\varepsilon} _ {t} ^ {p l} = d \lambda_ {t} \left(\frac {3}{2} \frac {\mathbf {S}}{q} + \left(1 - \frac {b _ {0}}{3} \frac {\sigma_ {t}}{\sigma_ {t} ^ {u}}\right) \mathbf {I}\right).
In uniaxial tension \begin{array} { r } { S _ { 1 1 } = \frac { 2 } { 3 } \sigma _ { t } } \end{array} and q = \sigma _ { t } . Therefore, in uniaxial tension,
\left(d \varepsilon_ {t} ^ {p l}\right) _ {1 1} = d \lambda_ {t} \left(2 - \frac {b _ {0}}{3} \frac {\sigma_ {t}}{\sigma_ {t} ^ {u}}\right);
and, hence,
Equation 4.5.1-13
d \lambda_ {t} = \left(d \varepsilon_ {t} ^ {p l}\right) _ {1 1} \bigg / \left(2 - \frac {b _ {0}}{3} \frac {\sigma_ {t}}{\sigma_ {t} ^ {u}}\right) .
Upon integration Equation 4.5.1-13 gives \lambda _ { t } from \left( d \varepsilon _ { t } ^ { p l } \right) _ { 1 1 } ; and, therefore, the \sigma _ { t } ( \lambda _ { t } ) relationship is obtained from the *TENSION STIFFENING input data.
Damaged elasticity
Following crack detection we use damaged elasticity to model the failed material. The elasticity is written in the form
Equation 4.5.1-14
\pmb {\sigma} = \mathbf {D}: \pmb {\varepsilon} ^ {e l},
where D is the elastic stiffness matrix for the concrete.
Let ® represent a cracked direction, with corresponding direct stress \sigma _ { \overline { { \alpha \alpha } } } and direct elastic strain \varepsilon { \frac { e l } { \alpha \alpha } } . In these expressions and in the remainder of this section, no summation is implied by repeated indices with a bar over them. If the fracture energy concept is used, the strains are related to the stress/displacement definition in the *TENSION STIFFENING option by \varepsilon = u / c , , where c is the characteristic length associated with the integration point.
Then, in D, D _ { \overline { { \alpha \alpha } } \beta \gamma } is the usual elasticity of the concrete if \begin{array} { r } { \varepsilon _ { \overline { { \alpha \alpha } } } \leq 0 . \mathrm { I f } \varepsilon _ { \overline { { \alpha \alpha } } } ^ { \mathrm { o p e n } } > \varepsilon _ { \overline { { \alpha \alpha } } } > 0 . } \end{array} ,
D _ {\overline {{\alpha \alpha \alpha \alpha}}} = \sigma_ {\overline {{\alpha \alpha}}} ^ {\mathrm{open}} \Bigg / \varepsilon_ {\overline {{\alpha \alpha}}} ^ {\mathrm{open}},
where \sigma _ { \overline { { \alpha \alpha } } } ^ { \mathrm { o p e n } } is the stress corresponding to \varepsilon { \frac { \mathrm { o p e n } } { \alpha \alpha } } (as defined in the *TENSION STIFFENING option), and
\varepsilon_ {\overline {{\alpha \alpha}}} ^ {\mathrm{open}} = \max _ {\mathrm{overhistory}} \left(\varepsilon_ {\overline {{\alpha \alpha}}} ^ {e l}\right).
\mathrm{If} \varepsilon_ {\overline {{\alpha \alpha}}} = \varepsilon_ {\overline {{\alpha \alpha}}} ^ {\mathrm{open}},
D _ {\overline {{\alpha \alpha \alpha \alpha}}} = d \sigma_ {\overline {{\alpha \alpha}}} ^ {\mathrm{open}} \Bigg / d \varepsilon_ {\overline {{\alpha \alpha}}} ^ {\mathrm{open}}
defined from the *TENSION STIFFENING option.
We also assume no Poisson effect for open cracks: for \begin{array} { r } { \varepsilon _ { \alpha \alpha } ^ { e l } > 0 , D _ { \overline { { \alpha \alpha } } \beta \gamma } = 0 } \end{array} for \beta \neq \alpha , \gamma \neq \alpha
The shear terms in the elasticity associated with existing crack directions are
D _ {\overline {{\alpha \beta \alpha \beta}}} = \hat {G}, \quad \beta \neq \alpha ,
where \hat { G } = \varrho ^ { \mathrm { c l o s e } } G for \varepsilon { \overline { { \alpha \alpha } } } \leq 0 ; and \hat { G } = \varrho ^ { \mathrm { o p e n } } G for \varepsilon { \frac { } { \alpha \alpha } } > 0 : In these expressions G is the elastic shear modulus, \varrho ^ { \mathrm { c l o s e } } is a constant defined in the *SHEAR RETENTION input option (see Figure 4.5.1-6),
Figure 4.5.1-6 Shear retention.
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| ε | ρ |
|---|---|
| ε_max | 0 |
| close | 1.0 |
and \varrho ^ { \mathrm { o p e n } } is a linear function of \bar { \varepsilon } ^ { e l } , \varrho ^ { \mathrm { o p e n } } = ( 1 - \bar { \varepsilon } _ { \alpha \alpha } ^ { e l } / \varepsilon ^ { \mathrm { m a x } } ) and is also defined in the *SHEAR RETENTION option. Here \begin{array} { r } { \bar { \varepsilon } ^ { e l } = \langle \varepsilon _ { \alpha \alpha } ^ { e l } \rangle + \langle \varepsilon _ { \beta \beta } ^ { e l } \rangle } \end{array} , where h and i are Macauley brackets, defining
\langle f \rangle = \left\{ \begin{array}{l l} f & \text {if} f \geq 0 \\ 0 & \text {otherwise} \end{array} \right.
for any function f . .
Cracking
As soon as the crack detection surface ( f _ { t } ) has been activated, we assume that cracking has occurred. The crack direction, \mathbf { n } _ { \alpha } , is taken to be the direction of that part of the maximum principal plastic strain increment conjugate to the crack detection surface, \Delta \varepsilon _ { t } ^ { p l } , that is orthogonal to the directions of any existing cracks at the same point. This crack orientation is stored for subsequent calculations, which are done for convenience in a local coordinate system oriented so that one of the coordinate directions is the crack direction, \mathbf { n } _ { \alpha } . Cracking is irrecoverable in the sense that, once a crack has occurred at a