# Concrete and Rock Materials

In Radioss these materials can be used to represent rock or concrete materials.

These materials use a Drücker–Prager yield criterion^{1}, which is a pressure-dependent model for determining whether
a material has failed or undergone plastic yielding.

## Concrete Material (/MAT/LAW10 and /MAT/LAW21)

### Drucker-Prager Yield Criteria

- ${J}_{2}$
- Second stress invariant (von Mises stress) of the deviatoric part of the stress and $P=-\frac{{I}_{1}}{3}$.
- ${I}_{1}$
- First stress invariant (hydrostatic pressure).
- ${I}_{1}={\sigma}_{1}+{\sigma}_{2}+{\sigma}_{3}=-3P$
- ${J}_{2}=\frac{1}{6}\left[{\left({\sigma}_{1}-{\sigma}_{2}\right)}^{2}+{\left({\sigma}_{2}-{\sigma}_{3}\right)}^{2}+{\left({\sigma}_{3}-{\sigma}_{1}\right)}^{2}\right]=\frac{1}{3}{\sigma}_{VM}{}^{2}$
- In a uniaxial test.

- If $F<0$, ${J}_{2}<{A}_{0}+{A}_{1}P+{A}_{2}{P}^{2}$ the material is under yield surface and is in the elastic region.
- If $F=0$, ${J}_{2}={A}_{0}+{A}_{1}P+{A}_{2}{P}^{2}$ and the material is at the yield surface.
- If $F>0$, ${J}_{2}>{A}_{0}+{A}_{1}P+{A}_{2}{P}^{2}$ and the material is past the yield surface and has failed.
- If ${A}_{1}={A}_{2}=0$, ${\sigma}_{VM}=\sqrt{3{J}_{2}}=\sqrt{3{A}_{0}}$, which is the von Mises criterion.

### Pressure Computation

- If ${P}_{ext}=0$, the pressure is $P=\text{\Delta}P$ and the pressure limit is ${P}_{\mathrm{min}}=\text{\Delta}{P}_{\mathrm{min}}$.
- If ${P}_{ext}\ne 0$, the pressure is shifted by ${P}_{ext}$, then $P={P}_{ext}+\text{\Delta}P$ and the pressure limit is ${P}_{\mathrm{min}}={P}_{ext}+\text{\Delta}{P}_{\mathrm{min}}$.

- In traction or tension the pressure is linear and limited by $\text{\Delta}{P}_{\mathrm{min}}$.
- In compression the pressure is nonlinear also limited by $\text{\Delta}{P}_{\mathrm{min}}$.

The only difference between the material laws is that in LAW10 the material constants $${C}_{0},{C}_{1},{C}_{2},{C}_{3}$$ are used to describe the pressure versus volumetric
strain ($P-\mu $ curve). In LAW21 you can describe this curve via
function input `fct_ID`_{f}.

### Load and Unload

- In Tension ($\mu <0$)
- For LAW10, linear loading and unloading with $$P={C}_{1}\mu $$ (Figure 3).
- For LAW21,
loading is defined using the input function
`fct_ID`_{f}and linear unloading with $$P={K}_{t}\mu $$.

- In Compression ($\mu >0$), for both LAW10 and LAW21:
- If neither B and $${\mu}_{\mathrm{max}}$$ are defined, the loading and unloading path are identical.
- If either B or
$${\mu}_{\mathrm{max}}$$ is defined:
- If only B is defined, $${\mu}_{\mathrm{max}}$$ is the volumetric strain where the tangent of $$P-\mu $$ curve is equal to B with $B={\frac{dP}{d\mu}|}_{{\mu}_{\mathrm{max}}}$.
- If only
$${\mu}_{\mathrm{max}}$$ is defined, then B is the tangent
of $$P-\mu $$ curve at
$${\mu}_{\mathrm{max}}$$. The loading and unloading in
compression is:
- If $\mu >{\mu}_{\mathrm{max}}$, loading and unloading path are identical.
- If $\mu <{\mu}_{\mathrm{max}}$, loading and unloading path are different, it is linear unloading with slope B.

## Concrete Material (/MAT/LAW24)

LAW24 uses a Drucker-Prager criteria with or without a cap in yield to model a reinforced concrete material. This material law assumes that the two failure mechanisms of the concrete material are tensile cracking and compressive crushing.

### Concrete Tensile Behavior

`H`

_{t},

`D`

_{sup}, and ${\epsilon}_{\mathrm{max}}$ can be used to describe tensile cracking and failure in tension.

In the initial very small elastic phase, the material has an elastic modulus
`E`_{c}.

`f`

_{t}is reached, the concrete starts to soften with the slope

`H`

_{t}. The maximum damage factor,

`D`

_{sup}, is significant because it enables the modeling of residual stiffness during and after a crack.

When there is crack closure, the concrete becomes elastic again, and the damage factor (for each direction) is conserved.

The bearing capacity of concrete in tensile is much lower than in compression. It is normally considered elastic when in tension.

It is recommended to choose a `D`_{sup}
value close to 1 (default is 0.99999) in order to minimize the current stiffness at
the end of the damage and consequently avoid residual stress in tension, which can
become very high if the element is highly deformed due to tension. This will happen
if the force causing the damage remains.

It is possible to adjust the `D`_{sup}
(and `H`_{t}) in order to simulate and
fit the behavior of concrete reinforced by fibers. The concrete material fails once
it reaches the total failure strain ${\epsilon}_{\mathrm{max}}$.

### Concrete Yield Surface in Compression

For concrete, the yield surface is the beginning of the plastic hardening zone which is between the failure surface, $${r}_{f}$$, and the yield surface.

- For
`I`_{cap}=0 or 1 (without a cap in yield) the yield curve is: - For
`I`_{cap}=2 (with cap in yield) the yield is:- $$r<\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)\cdot {r}_{f}$$ (green area in Figure 10)
- The material is under yield in the elastic phase.
- $r\ge {r}_{f}$ (red area in Figure 10)
- The material has failed.
- $$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)\cdot {r}_{f}<r<{r}_{f}$$ (yellow area in Figure 10)
- The material is above yield and below the failure surface which is the plastic hardening phase.

The input parameter $${\rho}_{t}$$ is the hydrostatic failure pressure in a uniaxial tension test and $${\rho}_{c}$$ is the hydrostatic pressure by failure in a uniaxial compression test.

- When $${\sigma}_{m}\ge {\rho}_{t}$$ (in tension) the scale factor $$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)=1$$. In this case, the yield surface equals the failure surface, $$r={r}_{f}$$.
- In the tension-compression region, $${\rho}_{t}>{\sigma}_{m}\ge {\rho}_{c}$$, then$$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)=1+\frac{\left(1-{k}_{0}\right)\cdot \left[{\rho}_{t}\left(2{\rho}_{c}-{\rho}_{t}\right)-2{\rho}_{c}{\sigma}_{m}+{\sigma}_{m}{}^{2}\right]}{{\left({\rho}_{c}-{\rho}_{t}\right)}^{2}}$$ with $${k}_{y}\le {k}_{0}\le 1$$
- The rest of the curve depends on the
`I`_{cap}option and the different scale factors $$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)$$ used.- For
`I`_{cap}=0 or 1 and $${\sigma}_{m}<{\rho}_{c}$$ (in compression), then $$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)={k}_{y}$$ - For
`I`_{cap}=2 (with cap in yield) and $${\rho}_{c}<{\sigma}_{m}<{f}_{k}$$ (in compression), then $$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)={k}_{y}$$ - In $${f}_{k}<{\sigma}_{m}<{f}_{0}$$ (in cap zone)$$\mathrm{k}\left({\sigma}_{m},{k}_{0}\right)={k}_{0}\left[1-{\left(\frac{{\sigma}_{m}-{f}_{k}}{{f}_{0}-{f}_{k}}\right)}^{2}\right]$$ with $$0\le {k}_{0}\le {k}_{y}$$

- For

The material constant $${k}_{y}$$ should be $$0\le {k}_{y}\le 1$$. A higher value of $${k}_{y}$$ results in a higher yield surface.

`I`

_{cap}=2 (yield with cap), the difference of yield surface between $${k}_{y}=0.8$$ and $${k}_{y}=0.6$$ (Figure 16). The default value of $${k}_{y}$$ in LAW24 is 0.5.

### Concrete Plastic Flow Rule in Compression

- $$\alpha $$
- Plastic dilatancy.
- $$\alpha =\frac{\partial g}{\partial {I}_{1}}$$
- Governs the volumetric plastic flow.
- $${I}_{1}$$
- First stress invariant (hydrostatic pressure).

- If $${k}_{0}={K}_{y}$$
- then, $$\alpha ={\alpha}_{y}$$ which means the material is in yield.
- If $${k}_{0}<{K}_{y}$$
- then, $$\alpha $$ becomes negative is the cap region.
- If $${k}_{0}=1$$
- then, $$\alpha ={\alpha}_{f}$$ which means the material has failed.

The values of $${\alpha}_{y},\text{}{\alpha}_{f}$$ are used to describe the material beyond yield, but before failure. It is recommended to use -0.2 and -0.1 for $${\alpha}_{y},\text{}{\alpha}_{f}$$ in LAW24. If very small values of $${\alpha}_{y},\text{}{\alpha}_{f}$$ are used, there is no volumetric plasticity (no cap region).

### Concrete Crushing Failure in Compression

- $${f}_{t}$$
- Uniaxial tension (triaxiality is 1/3)
- $${f}_{c}$$
- Uniaxial compression (triaxiality is -1/3)
- $${f}_{b}$$
- Biaxial compression (triaxiality is -2/3)
- $${f}_{2}$$
- Confined compression strength (tri-axial test)
- $${s}_{0}$$
- Under confined pressure

Load Type | Surface Point | Default Input | Criteria $$r$$ | Pressure $${\sigma}_{m}$$ | Lode Angle $\theta $ |
---|---|---|---|---|---|

Compression | $\left({f}_{c},0,0\right)$ | Mandatory | $$r=\sqrt{2/3}$$ | $${\sigma}_{m}=-1/3$$ | $$\mathrm{cos}\theta =-1$$ |

Direct Tensile | $\left({f}_{t},0,0\right)$ | ${f}_{t}\text{=0}\text{.1}{f}_{c}$ | $$r=\sqrt{2/3}\left(\frac{{f}_{t}}{{f}_{c}}\right)$$ | $${\sigma}_{m}=1/3\left(\frac{{f}_{t}}{{f}_{c}}\right)$$ | $$\mathrm{cos}\theta =1$$ |

Biaxial Compression | $\left({f}_{2},{s}_{0},{s}_{0}\right)$ | If
I_{cap} =
1 ${f}_{2}\text{=4}\text{.0}{f}_{c}$
If ${s}_{0}\text{=1}\text{.25}{f}_{c}$ |
$$r=\sqrt{2/3}\left(\frac{{f}_{t}}{{f}_{c}}\right){\sigma}_{m}$$ | $${\sigma}_{m}=2/3\left(\frac{{f}_{t}}{{f}_{c}}\right)$$ | $$\mathrm{cos}\theta =1$$ |

Compression Strength under Confinement Pressure | $\left({f}_{b},{f}_{b},0\right)$ | ${f}_{b}\text{=1}\text{.2}{f}_{c}$ | $$r=\sqrt{2/3}\frac{{f}_{2}-{s}_{0}}{{f}_{c}}$$ | $${\sigma}_{m}=\frac{{f}_{2}+2{s}_{0}}{3{f}_{c}}$$ | $$\mathrm{cos}\theta =-1$$ |

Where the failure curve is defined using $$r=\sqrt{2{J}_{2}}=\sqrt{\frac{2}{3}}{\sigma}_{VM}$$ and $${\sigma}_{m}=\frac{{I}_{1}}{3}$$ is the mean stress (pressure), then $${I}_{1}$$ and $${J}_{2}$$ are the first and second stress invariants.

The material fails once it reaches the failure curve $${r}_{f}$$.

### Concrete Reinforcement

- One way is to use beam or truss elements and connect them to the concrete with kinematic conditions.
- Another way is to use the parameters in LAW24 along with the orthotropic
solid property /PROP/TYPE6 to define the reinforced
direction. Parameters $${\alpha}_{1},\text{}{\alpha}_{2},\text{}{\alpha}_{3}$$ in LAW24 are used to define the
reinforcement cross-section area ratio to the whole concrete section area in
direction 1, 2, 3.
(13) $${\alpha}_{i}=\frac{Are{a}_{steel}}{Are{a}_{concrete}}$$

## Concrete Material (/MAT/LAW81)

LAW81 can be used to model rock or concrete materials.

### Drucker-Prager Yield Criteria

- $$q$$
- von Mises stress with $$q={\sigma}_{VM}=\sqrt{3{J}_{2}}$$
- $$p$$
- Pressure is defined as $$p=\frac{1}{3}{I}_{1}$$

- The linear part ($$p\le {p}_{a}$$), where the scale function is $${\mathrm{r}}_{c}\left(p\right)=1$$ which leads to the von Mises stress being
linearly proportional to pressure:
(15) $$q=p\mathrm{tan}\varphi +c$$ Where,- $$c$$
- Cohesive and is the intercept of yield envelope with the shear strength.
- $$\varphi $$
- Angle of internal friction, which defines the slope of the yield envelope.

$$c$$ and $$\varphi $$ are also used to define the Mohr-Coulomb yield surface. The Drucker-Prage yield surface is a smooth version of the Mohr-Coulomb yield surface.

- The second part ($${p}_{a}<p<{p}_{b}$$) of the yield surface simulates a cap limit.
An increase of pressure in a rock or concrete material will increase the
yield of the material; but, if pressure increases enough, then the rock or
concrete material will be crushed. The Drucker-Prager model with the cap
limit can be used to model this behavior. The cap limit defined in part
and uses the scale
function:
(16) $${\mathrm{r}}_{c}\left(p\right)=\sqrt{1-{\left(\frac{p-{p}_{a}}{{p}_{b}-{p}_{a}}\right)}^{2}}$$ The von Mises stress is:(17) $$q=\sqrt{1-{\left(\frac{p-{p}_{a}}{{p}_{b}-{p}_{a}}\right)}^{2}}\cdot \left(p\mathrm{tan}\varphi +c\right)$$ Where,- $${p}_{b}$$
- Curve is defined using the
`fct_ID`_{Pb}input - $${p}_{a}$$
- Computed by Radioss using the input $$\alpha $$ ratio value.

$${p}_{a}=\alpha \cdot {p}_{b}$$ with $$0<\alpha <1$$.

Where, $${p}_{0}$$ is the maximum point of yield curve, where $$\frac{\partial F}{\partial p}\left({p}_{0}\right)=0$$

If $$p={p}_{b}$$, then $${\mathrm{r}}_{c}\left({p}_{b}\right)=0$$ and the yield function is then,

$$q=0\cdot \left(p\mathrm{tan}\varphi +c\right)=0$$ which means the material is crushed.

- $$G=q-p\cdot \mathrm{tan}\psi =0$$ if $$p\le {p}_{a}$$
- $$G=q-\mathrm{tan}\psi \left(p-\frac{{\left(p-{p}_{a}\right)}^{2}}{2\left({p}_{0}-{p}_{a}\right)}\right)=0$$ if $${p}_{a}<p\le {p}_{0}$$
- $$G=F$$ if $$p>{p}_{0}$$

^{1}Han, D. J., and Wai-Fah Chen. "A nonuniform hardening plasticity model for concrete materials." Mechanics of materials 4, no. 3-4 (1985): 283-302