MAT_MM_CONCRETE
Material properties
Beta command
This command is in the beta stage and the format may change over time.
"Optional title"
mid, $\rho$, $G$
$K_0$, $K_L$, $cid_{cmp}$, $f_t$, $f_c$, $\xi$, $\lambda$, $\gamma$
$\xi_y$, $\xi_r$, $\varepsilon_{p,u0}$, $\varepsilon_{p,r0}$, $\psi_p$, $\psi_r$, $\varepsilon_{p,u}^{min}$, $\varepsilon_{p,r}^{min}$
$m$, $bulk$, $bulk_{cap}$, $cid_{src}$, $cid_{srt}$, $c$, $\sigma_{y,min}$, $\sigma_{y,max}$
$\mu$, $G_{r0}$, $L_{ref}$, $nsplit$
Parameter definition
Description
Elastic and viscous stresses
The total stress, $\boldsymbol{\sigma}$, is the sum of an elastic component $\boldsymbol{\sigma^e}$ and a viscous component $\boldsymbol{\sigma^v}$.
$\boldsymbol{\sigma} = \boldsymbol{\sigma^e} + \boldsymbol{\sigma^v}$
Elastic component:
$\boldsymbol{\sigma^e} = 2\cdot G\cdot \boldsymbol{\varepsilon_{dev}^e} + K\cdot \boldsymbol{\varepsilon_{vol}^e}\cdot \mathbf{I}$
$G$ is the shear modulus, $\boldsymbol{\varepsilon_{dev}^e}$ is the elastic deviatoric strain, $K$ is the bulk modulus (defined below) and $\boldsymbol{\varepsilon_{vol}^e}$ is the elastic volumetric strain.
Viscous component:
$\boldsymbol{\sigma^v} = c\cdot \boldsymbol{\dot\varepsilon}$
$c$ is an input parameter and $\boldsymbol{\dot\varepsilon}$ is the total strain rate.
Inelastic compaction
Compaction is described by a curve of compaction pressure vs. densification (inelastic volumetric strain), defined by the user. From the defined curve, parameters $p_0$, $p_L$ and $\varepsilon_L$ are extracted by the solver. An example curve and the parameters extracted from it is presented below. The compaction pressure, $p_c\left(\varepsilon_{vol}^p\right)$, is increased from $p_0$ to $p_L$ during inelastic compaction, which occurs in region 3 (defined below).
![](/documents/manual/images/figs/mat_mm_concrete_compaction_curve.png)
Functions $f_u$ and $f_r$
Function $f_u\left(p,p_c,g\left(\theta\right)\right)$ is divided into three regions. Pressure, $p$, determines active region:
$\begin{array}{ccc} \mbox{Region} & \mbox{Pressure} & f_u \\ 1 & p \leq \frac{f_c}{3} & max\left(0, \eta\cdot \left(p-p_s\right)\right) \\ 2 & \frac{f_c}{3} \lt p \leq \xi\cdot p_c & f_u\left(\frac{f_c}{3},p_c,g\left(\theta\right)\right) + \lambda\cdot \eta\cdot \left(\xi\cdot p_c - f_c/3\right)\cdot \left(1 - \left(1 - \alpha\right)^{1/\lambda}\right) \\ 3 & p \gt \xi\cdot p_c & f_u\left(\xi\cdot p_c, p_c, g\left(\theta\right)\right) \end{array}$
where:
$\eta = \frac{g\left(\theta\right)\cdot f_c}{f_c/3-p_s}$
$p_s = \frac{\left(1+1/(2-\gamma)\right)\cdot f_t\cdot f_c}{3\cdot\left(f_t - f_c/(2-\gamma)\right)}$
$\alpha = \frac{p-f_c/3}{\xi\cdot p_c - f_c/3}$
$f_c$, $f_t$, $\xi$, $\lambda$ and $\gamma$ are input parameters and $g\left(\theta\right)$ is a function of Lode angle.
Function $f_r\left(p,p_c,g\left(\theta\right)\right)$ is defined as $f_r = max(0, f_u + \eta\cdot p_s)$. Functions $f_u$ and $f_r$ are used to describe the flow surface and the transition strains.
![](/documents/manual/images/figs/mat_mm_concrete_functions_f.png)
Ultimate surface
The ultimate surface, $\sigma_u(p,p_c,g\left(\theta\right)$, is defined as function $f_u$ but with a cap in region 3:
$\displaystyle{\sigma_u = \left\{ \begin{array}{ccc} f_u & : & \mbox{Region 1 and 2} \\ f_u\left(\xi\cdot p_c,p_c,g\left(\theta\right)\right)\cdot \sqrt{1-min\left(1,\left(\frac{p-\xi\cdot p_c}{p_c\cdot (1-\xi)}\right)^2\right)} & : & \mbox{Region 3} \end{array} \right. }$
![](/documents/manual/images/figs/mat_mm_concrete_f_sigma_u.png)
Parameters $f_c$, $f_t$, $\xi$, $\lambda$, $\gamma$ and $p_0$ control the shape of the ultimate surface:
![](/documents/manual/images/figs/mat_mm_concrete_xi.png)
![](/documents/manual/images/figs/mat_mm_concrete_lambda.png)
![](/documents/manual/images/figs/mat_mm_concrete_gamma.png)
![](/documents/manual/images/figs/mat_mm_concrete_gamma_pi_plane.png)
![](/documents/manual/images/figs/mat_mm_concrete_p_0.png)
In addition to demonstrating the influence of $p_0$, the figure above also demonstrates the evolution of the ultimate surface during inelastic compaction (from $p_c = 0.5\cdot p_{0,ref}$ to $p_c = p_{0,ref}$).
The maxium strength is increased with increased crushing pressure as:
$\sigma_u\left(\xi\cdot p_c, p_c, g\left(\theta\right)\right) = \sigma_u\left(\xi\cdot p_0, p_0, g\left(\theta\right)\right)\cdot \frac{\xi\cdot p_c - f_c/3}{\xi\cdot p_0 -f_c/3}$
Initial yield surface
The initial yield surface, $\sigma_{y0}\left(p,p_c,g\left(\theta\right)\right)$, is defined as:
$\sigma_{y0} = \xi_y\cdot \sigma_u$
$\xi_y$ is an input parameter.
![](/documents/manual/images/figs/mat_mm_concrete_xi_y.png)
Residual surface
The residual surface, $\sigma_r\left(p,p_c,g\left(\theta\right)\right)$, is defined as:
$\displaystyle{\sigma_r = \left\{ \begin{array}{ccc} \beta\cdot f_r & : & \mbox{Region 1 and 2} \\ \beta\cdot f_r\left(\xi\cdot p_c,p_c,g\left(\theta\right)\right)\cdot \sqrt{1-min\left(1,\left(\frac{p-\xi\cdot p_c}{p_c\cdot (1-\xi)}\right)^2\right)} & : & \mbox{Region 3} \end{array} \right. }$
$\beta = min\left(1, \xi_r\cdot \eta\cdot \frac{p}{f_r}\right)$
$\xi_r$ is an input parameter.
![](/documents/manual/images/figs/mat_mm_concrete_xi_r.png)
Yield criterion and plastic flow
$\begin{array}{ccc} \sigma_{eff} = \sigma_y \rightarrow \mbox{Yielding} & : & \mbox{Region 1 and 2} \\ \left(\frac{\sigma_{eff}}{\sigma_y}\right)^2 + \left(\frac{p/p_c-\xi}{1-\xi}\right)^2 \rightarrow \mbox{Yielding} & : & \mbox{Region 3} \end{array}$
Parameter $bulk$ controls the type of plastic flow in region 1 and 2. With $bulk = 0$, the plastic flow is purely deviatoric. With $bulk = 1$, associated flow is used, meaning that the plastic flow is both deviatoric and volumetric. The volumetric strain caused by bulking can be capped with parameter $bulk_{cap}$. Radial return is used in region 3.
![](/documents/manual/images/figs/mat_mm_concrete_bulking.png)
Damage
Damage is divided into tensile damage, $D_t$ and crushing damage, $D_c$. Crushing damage is divided into deviatoric crushing damage, $D_{c,dev}$, and volumetric crushing damage, $D_{c,vol}$.
Pressure and region determines which type of damage that accumlates. Damage grows with plastic flow and is initiated once the ultimate surface is reached. Node splitting, $nsplit$, is optional and is only used for tensile damage.
Tensile damage developes during negative pressures and does not affect the materials behavior in compression $\left(p \ge 0 \right)$.
$D_t = min\left(1,\sum{\frac{d\varepsilon_{eff}^p}{\varepsilon_r}}\right)$
Crushing damage develops during positive pressures and affects the materials behavior in tension $\left(p \lt 0 \right)$.
$D_c = \sqrt{D_{c,dev}^2 + D_{c,vol}^2}$
$D_{c,dev} = min\left(1,\sum{\frac{d\varepsilon_{eff}^p}{\varepsilon_r}}\right)$
$D_{c,vol} = min\left(1,\sum{\frac{d\varepsilon_{vol}^p}{\varepsilon_{L}}}\right)$
A damage factor, $D_{fac}$ is defined as:
$\displaystyle{D_{fac} = \left\{ \begin{array}{ccc} \left(1-D_c\right)^m & : & p \ge 0 \\ \left(1-D_c\right)^m\cdot \left(1-D_t\right)^m & : & p \lt 0 \end{array} \right. }$
$m$ is an input parameter.
Contour plot attribute "Damage" in the GUI displays maximum of tensile and crushing damage.
![](/documents/manual/images/figs/mat_mm_concrete_damage.png)
Strain rate dependency
Strain rate dependency is defined by curves containing stress vs. plastic strain rate. Parameters $cid_{src}$ and $cid_{srt}$ can refer to the same curve. In the case of different curves, the strain rate term, $\sigma_y^{rate}\left(sr,p\right)$, is pressure dependent and the following holds:
$\begin{array}{ccc} \mbox{Pressure} & & \sigma_y^{rate} \mbox{ defined by}\\ p \leq -\frac{f_t}{3} & : & \mbox{curve with id. $cid_{srt}$} \\ -\frac{f_t}{3} \lt p \leq \frac{f_c}{3} & : & \mbox{interpolation} \\ p \gt \frac{f_c}{3} & : & \mbox{curve with id. $cid_{src}$} \end{array}$
Parameter $sr$ is the plastic strain rate, defined as:
$\displaystyle{sr = \left\{ \begin{array}{ccc} \dot{\varepsilon}_{eff}^p & : & \mbox{Region 1 and 2} \\ \sqrt{\left(\dot{\varepsilon}_{eff}^p\right)^2 + \left(\dot{\varepsilon}_{vol}^p\right)^2} & : & \mbox{Region 3} \end{array} \right. }$
The strain rate term is added to the quasi-static flow stress and crushing pressure:
$\sigma_y^{dyn} = \sigma_y^{qs} + \sigma_y^{rate}$
$p_c^{dyn} = p_c^{qs} + \sigma_y^{rate}$
Transition strains
Transition from initial yield surface to ultimate surface and from ultimate surface to residual surface is controlled by parameters $\varepsilon_{p,u}\left(p,g\left(\theta\right),sr\right)$ and $\varepsilon_{p,r}\left(p,g\left(\theta\right),sr\right)$, respectively.
$\varepsilon_{p,u} = \varepsilon_{p,u0}\cdot\left(1 + \psi_p\cdot \left(\frac{f_u^*+\psi_r\cdot\sigma_y^{rate}}{f_c}-1\right)\right)$
$\varepsilon_{p,r} = \varepsilon_{p,r0}\cdot\left(1 + \psi_p\cdot \left(\frac{f_u^*+\psi_r\cdot\sigma_y^{rate}}{f_c}-1\right)\right)$
$\varepsilon_{p,r0}$, $\varepsilon_{p,u0}$, $\psi_r$ and $\psi_p$ are input parameters. Note that strain rate dependency requires $\psi_p\gt 0$ and $\psi_r\gt 0.$
Function $f_u^*$ is defined as function $f_u$ but independent of inelastic compaction, i.e. $f_u^*\left(p,g\left(\theta\right)\right) = f_u\left(p,p_0,g\left(\theta\right)\right)$.
Lower caps on the transition strains can be defined with input parameters $\varepsilon_{p,u}^{min}$ and $\varepsilon_{p,r}^{min}$.
An energy-based transition from ultimate to residual surface can be defined with parameter $G_{r0}$. The transition strain for negative pressures is then defined as:
$\varepsilon_{p,r}\left(p,g\left(\theta\right),sr,V_e\right) = \frac{G_{r0}}{V_e^{1/3}\cdot f_c}\cdot \left(1 + \psi_p\cdot \left(\frac{f_u^*+\psi_r\cdot\sigma_y^{rate}}{f_c}-1\right)\right)$
$V_e$ is the element volume, automatically set for each element at initialization. Note that $\varepsilon_{p,r0}$ still must be defined, since $G_{r0}$ only operates at negative pressures.
If parameter $L_{ref}$ is defined, parameter $\varepsilon_{p,r}^{min}$ is scaled with element size by a factor $L_{ref}/V_e^{1/3}$.
![](/documents/manual/images/figs/mat_mm_concrete_transition_strains2.png)
Flow stress
The flow stress, $\sigma_y\left(p,p_c,g\left(\theta\right),\varepsilon_{eff}^p,sr,D_{fac}\right)$, is defined as:
$\displaystyle{\sigma_y = \left\{ \begin{array}{ccc} \sigma_{y0} & : & \varepsilon_p^{eff} = 0 \mbox{ and } D_{fac} = 1 \\ \sigma_u\cdot h & : & \varepsilon_p^{eff} \gt 0 \mbox{ and } D_{fac} = 1 \\ \sigma_u\cdot D_{fac} + \sigma_r\left(1-D_{fac}\right) & : & \varepsilon_p^{eff} \gt 0 \mbox{ and } D_{fac} \lt 1 \end{array} \right. }$
The plastic hardening, $h$, is defined as:
$h = min\left(1,\sum{\frac{d\varepsilon_{eff}^p}{\varepsilon_u}}\right), h(0) = \xi_y$
![](/documents/manual/images/figs/mat_mm_concrete_transitions.png)
Caps on the flow stress can be defined with input parameters $\sigma_{y,max}$ and $\sigma_{y,min}$.
Modeling of material inhomogeneity
Deviations to surfaces and transition strains can be introduced with input parameter $\mu$.
$rnd_f^1 = 1 + \mu\cdot\left(2\cdot rnd - 1\right)$
$rnd_f^2 = 1 - \mu\cdot\left(2\cdot rnd - 1\right)$
$rnd$ is a random number between 0 and 1. Parameters $p_0$, $p_L$, $f_t$, $f_c$ are scaled with $rnd_f^1$ and parameter $\varepsilon_L$ with $rnd_f^2$. The scaling is done on integration point level at initialization.
![](/documents/manual/images/figs/mat_mm_concrete_deviations.png)