18.1.1 Tension Softening Relations

The relation between the crack stress $\sigma_{{nn}}^{{\mathrm{cr}}}$ and the crack strain $\varepsilon_{{nn}}^{{\mathrm{cr}}}$ in the normal direction can be written as a multiplicative relation

$$\sigma_{{nn}}^{{\mathrm{cr}}} \varepsilon_{{nn}}^{{\mathrm{cr}}} \left(\vphantom{ \frac{{\varepsilon _{nn}^{\mathrm{cr}}}}{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}} }\right. {\frac{{{\varepsilon _{nn}^{\mathrm{cr}}}}}{{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}}}} \left.\vphantom{ \frac{{\varepsilon _{nn}^{\mathrm{cr}}}}{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}} }\right)$$ (18.20)

in which f_t is the tensile strength and $\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$ the ultimate crack strain. The general function y(...) represents the actual softening diagram. In DIANA both the tensile strength and ultimate strain may be a function of temperature, (moisture) concentration or maturity. Therefore the development of tensile strength and fracture energy in time can be simulated. If the softening behavior on the constitutive level is related to the Mode-I fracture energy G_f^I through an equivalent length or crack bandwidth denoted as h , the following relation can be derived

$$\int_{{ {\varepsilon _{nn}^{\mathrm{cr}}}= 0 }}^{{ {\varepsilon _{nn}^{\mathrm{cr}}}= \infty }} \sigma_{{nn}}^{{\mathrm{cr}}} \varepsilon_{{nn}}^{{\mathrm{cr}}} \varepsilon_{{nn}}^{{\mathrm{cr}}}$$ (18.21)

Substitution of (18.20) into (18.21) results in

$$\int_{{ {\varepsilon _{nn}^{\mathrm{cr}}}=0 }}^{{ {\varepsilon _{nn}^{\mathrm{cr}}}=\infty }} \left(\vphantom{ \frac{{\varepsilon _{nn}^{\mathrm{cr}}}}{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}} }\right. {\frac{{{\varepsilon _{nn}^{\mathrm{cr}}}}}{{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}}}} \left.\vphantom{ \frac{{\varepsilon _{nn}^{\mathrm{cr}}}}{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}} }\right) \varepsilon_{{nn}}^{{\mathrm{cr}}}$$ (18.22)

with the assumption that f_t is a constant. Change from the variable $\varepsilon_{{nn}}^{{\mathrm{cr}}}$ to

$${\frac{{{\varepsilon _{nn}^{\mathrm{cr}}}}}{{{\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}}}}$$ (18.23)

and consequently  d$\varepsilon_{{nn}}^{{\mathrm{cr}}}$ = $\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$ dx results in the relation

$$\biggl( \int_{{ x=0 }}^{{ x=\infty }} \biggr) \varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$$ (18.24)

where it is tacitly assumed that the ultimate crack strain $\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$ is finite. The final expression for the ultimate crack strain is now given by

$$\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}} {\frac{{1}}{{\alpha}}} {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} }}{{ h f_{\mathrm{t}} }}}$$ (18.25)

with the factor $\alpha$ determined by the integral

$$\alpha \int_{{ x = 0 }}^{{ x = \infty }}$$ (18.26)

The factor $\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$ is assumed to be constant during the analysis and is considered to be an element-related material property, which can be calculated from the material properties, the tensile strength f_t , the fracture energy G_f^I and the element area represented by the equivalent length h .

Mesh objectivity.

The Mode-I fracture energy will be released in an element if the tensile strength is violated and the deformations localize in the element. With this approach the results which are obtained with the analysis are objective with regard to mesh refinement. Unfortunately, it is possible that the elements of the discretization are so large that the equivalent length of an element results in a snap-back in the constitutive model and the concept of objective fracture energy which has been assumed is no longer satisfied.

A snap-back in the constitutive model is possible if the absolute value of the initial slope of the softening diagram is greater than the Young's modulus of the material, if it is assumed that the initial tangent of the tension softening diagram results in the greatest value of the tangent stiffness. The condition which has to be fulfilled then reads

$${\frac{{ \,\mathrm{d}{\sigma _{nn}^{\mathrm{cr}}}}}{{ \,\mathrm{d}{\varepsilon _{nn}^{\mathrm{cr}}}}}} \Bigg\vert_{{ {\varepsilon _{nn}^{\mathrm{cr}}}= 0 }}^{} \geq$$ (18.27)

This can be expressed as

$${\frac{{ f_{\mathrm{t}} }}{{ {\varepsilon _{nn.\mathrm{ult}}^{\mathrm{cr}}}}}} {\frac{{\,\mathrm{d}y}}{{\,\mathrm{d}x}}} \Bigg\vert_{{ x=0 }}^{} \geq$$ (18.28)

which results in an expression for the ultimate crack strain which reads

$$\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}} \geq {\frac{{ f_{\mathrm{t}} }}{{ E }}} {\frac{{\,\mathrm{d}y}}{{\,\mathrm{d}x}}} \Bigg\vert_{{ x=0 }}^{} \varepsilon_{{nn}}^{{\mathrm{cr}}}_{{{\mathrm{.ult.min}}}}^{{}}$$ (18.29)

with $\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$ determined by (18.25). If the condition given in (18.29) is violated, there are various possibilities to solve this problem. Firstly, it is possible to decrease the equivalent length h , but this property is an element property and consequently a fixed value. Secondly, it is possible to increase the fracture energy G_f^I since this will result in an increase the ductility of the material. The final possibility is to decrease the tensile strength f_t which results implicitly in an increase of the ductility since the fracture energy remains constant in this case.

The most obvious choice is to reduce the tensile strength because this has some physical meaning. The probability of a reduced strength is larger if the sampling area is larger. This implies that the tensile strength should be reduced in larger elements since stress concentrations are not captured with these elements. So, if the condition of (18.29) is violated, the tensile strength should be reduced to

$${\frac{{ G_{\mathrm{f}}^{\mathrm{I}} E }}{{ \alpha h \dfrac{\,\mathrm{d}y}{\,\mathrm{d}x} \Bigg\vert_{x=0} }}}$$ (18.30)

Alternatively, the element size could be reduced such that the crack bandwidth h , is equal to a maximum of

$${\frac{{ G_{\mathrm{f}}^{\mathrm{I}} E }}{{ \alpha f_{\mathrm{t}}^{2} \dfrac{\,\mathrm{d}y}{\,\mathrm{d}x} \Bigg\vert_{x=0} }}}$$ (18.31)

18.1.1.1 Brittle Cracking

Brittle behavior is characterized by the full reduction of the strength after the strength criterion has been violated [Fig.18.4].

Figure 18.4: Brittle cracking behavior

This model involves a discontinuity. Before the peak, there is only elastic strain. Beyond the peak, the stress drops to zero immediately; the elastic strain vanishes and we have only crack strain. The sudden stress drop, indicated by the dashed line in Figure 18.4, in fact involves an energy dissipation which is related to the peak strain $\varepsilon_{{nn}}^{{\mathrm{peak}}}$ and the crack band width:

$${\tfrac{{1}}{{2}}} \varepsilon_{{nn}}^{{\mathrm{peak}}}$$ (18.32)

with $\varepsilon_{{nn}}^{{\mathrm{peak}}}$ being a fixed value equal to f_t/E .

Consequently, a change of the element size, i.e., a change of the crack band width h , involves a different energy consumption in case of brittle cracking. In other words, a brittle cracking model is not objective with regard to mesh refinement. With tension-softening models, the ultimate strain is adapted to h , but with the brittle cracking model the ultimate strain is fixed and always equal to f_t/E so that a change in h leads to a different energy G_f being consumed. For more information see Bazant & Cedolin [5].

The issue is relevant especially for large scale unreinforced structures. Then, the element dimensions and the crack band width may be large, so that the softening diagram becomes very steep, brittle or even of the snap-back type. A solution may be to refine the mesh and make sure that the ultimate strain of the softening diagram (see next section) is larger than f_t/E . For reinforced structures, the issue is less relevant as the post-peak input is based on tension-stiffening considerations for distributed cracking rather than fracture energy considerations for a single localized crack.

18.1.1.2 Linear Tension Softening

In case of linear tension softening [Fig.18.5]

Figure 18.5: Linear tension softening

the relation of the crack stress is given by

$${\frac{{{\sigma _{nn}^{\mathrm{cr}}}({\varepsilon _{nn}^{\mathrm{cr}}})}}{{f_{\mathrm{t}}}}} \begin{cases} 1 - \dfrac{{\varepsilon _{nn}^{\mathrm{cr}}}}{... ...{cr}}}< {\varepsilon _{nn}^{\mathrm{cr}}}< \infty$} \end{cases}$$ (18.33)

The factor $\alpha$ for the ultimate crack strain is given by

$$\begin{split}\alpha = \int_{0}^{\infty} y ( x ) \,\mathrm{d}x... ... \int_{0}^{1} ( 1 - x ) \,\mathrm{d}x = \frac{1}{2} \end{split}$$ (18.34)

which results in an ultimate crack strain

$$\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}} {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} }}{{ h f_{\mathrm{t}} }}}$$ (18.35)

It is easily verified that

$${\frac{{\,\mathrm{d}y}}{{\,\mathrm{d}x}}} \Bigg\vert_{{x=0}}^{}$$ (18.36)

The minimum value of the ultimate crack strain is then given by

$$\varepsilon_{{nn}}^{{\mathrm{cr}}}_{{{\mathrm{.ult.min}}}}^{{}} {\frac{{ f_{\mathrm{t}} }}{{ E }}}$$ (18.37)

and the reduced tensile strength reads

$$\sqrt{{ 2 \frac{ G_{\mathrm{f}}^{\mathrm{I}} E } { h } }}$$ (18.38)

18.1.1.3 Multilinear Tension Softening

Multilinear behavior is completely defined by the user. If you define the behavior as shown in Figure 18.6

Figure 18.6: Multilinear tension softening

then the initial slope should comply with (18.29), so

$${\frac{{ f_{\mathrm{t}.1} - f_{\mathrm{t}.0} }}{{ {\varepsilon _{nn}^{\mathrm{cr}}}_{.1} }}} \geq$$ (18.39)

18.1.1.4 Nonlinear Tension Softening (Moelands and Reinhardt)

The softening diagram proposed by Moelands & Reinhardt [83] is a modification of the linear tension softening diagram according to

$${\frac{{{\sigma _{nn}^{\mathrm{cr}}}({\varepsilon _{nn}^{\mathrm{cr}}})}}{{f_{\mathrm{t}}}}} \begin{cases} 1 - \left( \dfrac{{\varepsilon _{nn}^{\mathrm{... ...{cr}}}< {\varepsilon _{nn}^{\mathrm{cr}}}< \infty$} \end{cases}$$ (18.40)

with c₁ = 0.31 [Fig.18.7].

Figure 18.7: Nonlinear tension softening (Moelands and Reinhardt)

The factor $\alpha$ for the ultimate crack strain is now given by

$$\begin{split}\alpha = \int_{0}^{\infty} y(x)\,\mathrm{d}x & =... ...{1}} ) \,\mathrm{d}x = \frac{ c_{1} } { 1 + c_{1} } \end{split}$$ (18.41)

which results for a parameter c₁ = 0.31 in $\alpha$ = 0.23664122 and in an ultimate crack strain for the tension softening diagram of Moelands and Reinhardt of

$$\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}} {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} }}{{ h f_{\mathrm{t}} }}}$$ (18.42)

The linear tension softening diagram is recovered if c₁ = 1 which results in the factor $\alpha$ = ${\tfrac{{1}}{{2}}}$ . The initial slope of the diagram is infinite which can be observed from Figure 18.7 and the initial tangent given by

$${\frac{{\,\mathrm{d}y}}{{\,\mathrm{d}x}}} \Bigg\vert_{{x=0}}^{} \Bigg\vert_{{x=0}}^{} \infty$$ (18.43)

which results in a value equal to - $\infty$ if the parameter c₁ is less or equal to one. For this tension softening diagram ( c₁ = 0.31 ), the initial stiffness  dy/ dx = - $\infty$ which implies that the condition of (18.29) is always violated.

18.1.1.5 Nonlinear Tension Softening (Hordijk et al.)

Hordijk, Cornelissen & Reinhardt [17,43] proposed an expression for the softening behavior of concrete which also results in a crack stress equal to zero at a crack strain $\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}}$ [Fig.18.8]. The function is defined by

$${\frac{{{\sigma _{nn}^{\mathrm{cr}}}({\varepsilon _{nn}^{\mathrm{cr}}})}}{{f_{\mathrm{t}}}}} \begin{cases} \left( 1 + \left( c_{1} \dfrac{{\varepsilon _{... ...{cr}}}< {\varepsilon _{nn}^{\mathrm{cr}}}< \infty$} \end{cases}$$ (18.44)

with the parameters c₁ = 3 and c₂ = 6.93 .

Figure 18.8: Nonlinear tension softening (Hordijk et al.)

The parameter $\alpha$ for the ultimate crack strain is given by

$$\begin{split}\alpha &= \int_{0}^{\infty} y(x)\,\mathrm{d}x = ... ...ots \quad 2 c_{2}^{4} \exp( c_{2} ) \hspace{20ex} } \end{split}$$ (18.45)

which results in $\alpha$ = 0.195 for the parameters c₁ = 3 and c₂ = 6.93 . The ultimate crack strain then reads

$$\varepsilon_{{nn.\mathrm{ult}}}^{{\mathrm{cr}}} {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} }}{{ h f_{\mathrm{t}} }}}$$ (18.46)

For the softening diagram of Hordijk et al., the following relation can be derived

$$\begin{split}\frac{ \,\mathrm{d}y } { \,\mathrm{d}x } \Bigg\v... ... -c_{2} - \left( 1 + c_{1}^{3} \right) \exp(-c_{2}) \end{split}$$ (18.47)

The minimum value of the ultimate crack strain is then given by

$$\varepsilon_{{nn}}^{{\mathrm{cr}}}_{{{\mathrm{.ult.min}}}}^{{}} {\frac{{ f_{\mathrm{t}} }}{{ E }}}$$ (18.48)

and the reduced tensile strength reads

$$\left(\vphantom{ 0.739 \frac{ G_{\mathrm{f}}^{\mathrm{I}} E }{ h } }\right. {\frac{{ G_{\mathrm{f}}^{\mathrm{I}} E }}{{ h }}} \left.\vphantom{ 0.739 \frac{ G_{\mathrm{f}}^{\mathrm{I}} E }{ h } }\right)^{{\tfrac{1}{2}}}_{}$$ (18.49)