21.5.1 Two-dimensional Interface Model

A plane stress interface model was formulated by Lourenço & Rots [63], and enhanced by Van Zijl [106]. It is based on multi-surface plasticity, comprising a Coulomb friction model combined with a tension cut-off and an elliptical compression cap [Fig.21.16].

Figure 21.16: Two-dimensional interface model

Softening acts in all three modes and is preceded by hardening in the case of the cap mode. The interface model is derived in terms of the generalized stress and strain vectors:

$$\boldsymbol\sigma \left\{\vphantom{ \negthickspace \begin{array}{c} \sigma \\ \tau \end{array} \negthickspace }\right. \begin{array}{c} \sigma \\ \tau \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \sigma \\ \tau \end{array} \negthickspace }\right\} \boldsymbol\varepsilon \left\{\vphantom{ \negthickspace \begin{array}{c} u \\ v \end{array} \negthickspace }\right. \begin{array}{c} u \\ v \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} u \\ v \end{array} \negthickspace }\right\}$$ (21.34)

with $\sigma$ and u the stress and relative displacement respectively in the interface normal direction and $\tau$ and v the shear stress and relative displacement respectively. In the elastic regime the constitutive behavior is described by

$$\boldsymbol\sigma \boldsymbol\varepsilon$$ (21.35)

with the stiffness matrix

$$\left[\vphantom{ \, k_{n} \: k_{s} \, }\right. \left.\vphantom{ \, k_{n} \: k_{s} \, }\right]$$ (21.36)

Shear slipping.

A Coulomb friction yield/crack initiation criterion

$$\tau \sigma \Phi$$ (21.37)

describes the shear-slipping, with $\Phi$ the friction coefficient equal to tan$\phi$ , the friction angle and c the adhesion. Both adhesion softening and friction softening are captured. The adhesion softening is described by

$$\sigma \kappa {\dfrac{{ c_{\mathrm{o}} }}{{ G_{\mathrm{f}}^{\mathrm{II}} }}} \kappa$$ (21.38)

where c_o is the initial adhesion of the brick-mortar interface and G_f^II the shear-slip fracture energy. The friction softening is coupled to the adhesion softening via

$$\Phi \sigma \kappa \Phi_{{\mathrm{o}}}^{} \left(\vphantom{ \Phi_{\mathrm{r}} - \Phi_{\mathrm{o}} }\right. \Phi_{{\mathrm{r}}}^{} \Phi_{{\mathrm{o}}}^{} \left.\vphantom{ \Phi_{\mathrm{r}} - \Phi_{\mathrm{o}} }\right) {\frac{{ c_{\mathrm{o}} - c }}{{ c_{\mathrm{o}} }}}$$ (21.39)

with $\Phi_{{\mathrm{o}}}^{}$ the initial and $\Phi_{{\mathrm{r}}}^{}$ the residual friction coefficient. The adhesion and friction parameters are found by linear regression of the micro-shear experimental data, while the fracture energy is determined by the appropriate integration of the stress-crack width response. Note that this integration produces the total energy dissipated by both the adhesion and the friction softening, which amounts to

$$\left(\vphantom{ 1 + \frac{ \sigma }{ c_{\mathrm{o}} } \left( \Phi_{\mathrm{r}} - \Phi_{\mathrm{o}} \right) }\right. {\frac{{ \sigma }}{{ c_{\mathrm{o}} }}} \left(\vphantom{ \Phi_{\mathrm{r}} - \Phi_{\mathrm{o}} }\right. \Phi_{{\mathrm{r}}}^{} \Phi_{{\mathrm{o}}}^{} \left.\vphantom{ \Phi_{\mathrm{r}} - \Phi_{\mathrm{o}} }\right) \left.\vphantom{ 1 + \frac{ \sigma }{ c_{\mathrm{o}} } \left( \Phi_{\mathrm{r}} - \Phi_{\mathrm{o}} \right) }\right)$$ (21.40)

The experimentally observed linear relation between the fracture energy and the normal confining stress is captured by letting

$$\begin{cases} a \, \sigma + b \quad & \text{if $\sigma < 0$} \\ b & \text{if $\sigma \geq 0$} \end{cases}$$ (21.41)

with a and b constants to be determined by linear regression of the experimental data. If the contribution of the friction softening energy is significant, which is revealed upon evaluation of the second term between the large parentheses of (21.40), the regressed coefficients a and b should be adjusted to avoid a too high energy dissipation at high compressive stresses.

Dilatancy.

The flow rule

$$\dot{{\boldsymbol{\varepsilon }}}_{{\mathrm{p}}}^{} \left\{\vphantom{ \negthickspace \begin{array}{c} \dot{u}_{\mathrm{p}} \\ \dot{v}_{\mathrm{p}} \end{array} \negthickspace }\right. \begin{array}{c} \dot{u}_{\mathrm{p}} \\ \dot{v}_{\mathrm{p}} \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \dot{u}_{\mathrm{p}} \\ \dot{v}_{\mathrm{p}} \end{array} \negthickspace }\right\} \dot{{\lambda}} {\frac{{ \partial g }}{{ \partial \boldsymbol{\sigma } }}}$$ (21.42)

provides a way of describing the dilatancy, by choice of a suitable potential function

$${\frac{{ \partial g }}{{ \partial \boldsymbol{\sigma} }}} \left\{\vphantom{ \negthickspace \begin{array}{c} \Psi \\ \mathrm{sign}( \tau ) \end{array} \negthickspace }\right. \begin{array}{c} \Psi \\ \mathrm{sign}( \tau ) \end{array} \left.\vphantom{ \negthickspace \begin{array}{c} \Psi \\ \mathrm{sign}( \tau ) \end{array} \negthickspace }\right\}$$ (21.43)

$\Psi$ = tan$\psi$ being the mobilized dilatancy coefficient. Following directly from the flow rule

$$\Psi {\frac{{ \dot{u}_{\mathrm{p}} }}{{ \dot{v}_{\mathrm{p}} }}} \tau$$ (21.44)

By integration the shear-slip induced normal uplift is found to be

$$\int \Psi \left\vert\vphantom{ \Delta v_{\mathrm{p}} }\right. \Delta \left.\vphantom{ \Delta v_{\mathrm{p}} }\right\vert$$ (21.45)

There is experimental evidence that dilatancy depends on the confining stress and the shear-slip. A dilatancy formulation of separate variables, i.e.,

$$\Psi \Psi_{{1}}^{} \sigma \Psi_{{2}}^{}$$ (21.46)

simplifies curve fitting and ensures convexity of the potential function g

$$\int \left(\vphantom{ \frac{ \partial g } { \partial \boldsymbol{\sigma } } }\right. {\frac{{ \partial g }}{{ \partial \boldsymbol{\sigma } }}} \left.\vphantom{ \frac{ \partial g } { \partial \boldsymbol{\sigma } } }\right)^{{\mathrm{\scriptscriptstyle{T}}}}_{} \boldsymbol\sigma \tau \Psi_{{2}}^{} \int \Psi_{{1}}^{} \sigma \sigma$$ (21.47)

Therefore, a description of the normal uplift upon shear-slipping is chosen as

$$\begin{cases} 0 &\text{if $\sigma < \sigma_{\mathrm{u}}$} \\... ...v_{\mathrm{p}}}} \right) &\text{if $\sigma \geq 0$} \end{cases}$$ (21.48)

which yields after differentiation

$$\Psi \begin{cases} 0 &\text{if $\sigma < \sigma _{\mathrm{u}}$} \... ...ta \, v_{\mathrm{p}} } } &\text{if $\sigma \geq 0$} \end{cases}$$ (21.49)

The dilatancy $\Psi_{{\mathrm{o}}}^{}$ at zero normal confining stress and shear slip, the confining (compressive) stress $\sigma_{{\mathrm{u}}}^{}$ at which the dilatancy becomes zero, and the dilatancy shear slip degradation coefficient $\delta$ are material parameters to be obtained by, for instance, a least squares fit of (21.48) to experimental test data. Note that for tensile stress a stress-independent dilatancy coefficient is assumed.

Softening.

A strain softening hypothesis is employed, where the softening is governed by shear-slipping, yielding

$$\Delta \kappa \Delta \Delta \lambda$$ (21.50)

upon substitution of (21.42) and (21.43). The stress-update can be cast in the standard plasticity predictor-corrector fashion and the corrected stresses, together with the plastic strain increment $\Delta$$\kappa$ , or $\Delta$$\lambda$ can be solved by a Newton-Raphson iterative scheme. A consistent tangent modulus is employed for the global convergence iterations, which ensures quadratic convergence (see Van Zijl [106]).

Tension cut-off.

The yield function for the tension cut-off (criterion number 2 of the interface model) is:

$$\sigma \sigma_{{\mathrm{t}}}^{}$$ (21.51)

with $\sigma_{{\mathrm{t}}}^{}$ the tensile, or brick-mortar bond strength. The strength is assumed to soften exponentially

$$\sigma_{{\mathrm{t}}}^{} {\dfrac{{ f_{\mathrm{t}} }}{{ G_{\mathrm{f}}^{\mathrm{I}} }}} \kappa_{{2}}$$ (21.52)

with f_t the bond strength and G_f^I the Mode-I fracture energy. The softening is governed by a strain softening hypothesis:

$$\Delta \kappa_{{2}}^{} \left\vert\vphantom{ \Delta u_{\mathrm{p}} }\right. \Delta \left.\vphantom{ \Delta u_{\mathrm{p}} }\right\vert$$ (21.53)

which, upon consideration of an associated flow rule

$$\Delta \boldsymbol\varepsilon \Delta \lambda_{{2}}^{} {\frac{{ \partial f_{2} }}{{ \partial \boldsymbol{\sigma } }}}$$ (21.54)

reduces to

$$\Delta \kappa_{{2}}^{} \Delta \lambda_{{2}}^{}$$ (21.55)

Compression cap.

The yield function for the compression cap, here referred to as criterion number 3 (with 1 being the shear mode), is

$$\sigma^{{2}}_{} \tau^{{2}}_{} \sigma_{{\mathrm{c}}}^{{2}}$$ (21.56)

with C_s a parameter controlling the shear stress contribution to failure and $\sigma_{{\mathrm{c}}}^{}$ the compressive strength. The latter is assumed to evolve according to the strain hardening hypothesis:

$$\Delta \kappa_{{3}}^{} \sqrt{{ \Delta \boldsymbol{\varepsilon }_{\mathrm{p}}^{\mathrm{\scriptscriptstyle{T}}} \Delta \boldsymbol{\varepsilon }_{\mathrm{p}}}}$$ (21.57)

which, upon consideration of an associated flow rule

$$\Delta \boldsymbol\varepsilon \Delta \lambda_{{3}}^{} {\frac{{ \partial f_{3} }}{{ \partial \boldsymbol{\sigma } }}}$$ (21.58)

becomes

$$\Delta \kappa_{{3}}^{} \Delta \lambda_{{3}}^{} \sqrt{{ \sigma ^{2} + \left( C_{\mathrm{s}} \tau \right)^{2} }}$$ (21.59)

The yield surface hardens, as described by a parabolic hardening rule, followed by parabolic/exponential softening [Fig.21.17]. The peak strength f_c.x is reached at the plastic strain $\kappa_{{\mathrm{p}}}^{}$ . Subsequently, the softening branch is entered, governed by the fracture energy G_fc .

Figure 21.17: Hardening-softening law for interface compression cap

For practical reasons, all stress values in Figure 21.17 are related to the peak strengths f_c as follows: $\bar{{\sigma }}_{{\mathrm{i}}}^{}$ = ${\tfrac{{1}}{{3}}}$f_c , $\bar{{\sigma }}_{{\mathrm{m}}}^{}$ = ${\tfrac{{1}}{{2}}}$f_c , and $\bar{{\sigma }}_{{\mathrm{r}}}^{}$ = ${\frac{{1}}{{7}}}$f_c . The three regions of this hardening-softening rule are given by

$$\begin{split}\bar{\sigma }_{1} ( \kappa_{3} ) &= \bar{\sigma ... ...{m}} - \bar{\sigma }_{\mathrm{r}} } \right) \right) \end{split}$$ (21.60)

Corners.

At each of the intersections of the Coulomb friction criterion with the tension cut-off and the compression cap the plastic strain increment is given by

$$\Delta \boldsymbol\varepsilon \Delta \lambda_{{1}}^{} {\frac{{ \partial g_{1} }}{{ \partial \boldsymbol{\sigma } }}} \Delta \lambda_{{i}}^{} {\frac{{ \partial g_{i} }}{{ \partial \boldsymbol{\sigma } }}}$$ (21.61)

where the subscript 1 refers to the shear criterion and i refers to tension cut-off (i = 2 ) and to the compression cap (i = 3 ). Lourenço [61] describes this procedure in detail. The corners are treated consistently. In both the shear/tension corner and the shear/compression corner the stress corrections can be written in standard predictor-corrector fashion and solved for, together with the two plastic strain increments $\Delta$$\lambda_{{1}}^{}$ or $\Delta$$\lambda_{{i}}^{}$ , by a Newton-Raphson iterative scheme. Also here consistent tangent moduli are employed for the global convergence iterations to ensure quadratic convergence.