The nonlinear stress-strain behavior in solids was already described 100 years ago by Paul Ludwik in his *Elemente der Technologischen Mechanik*. In that treatise, Ludwik described the nonlinear relation between shear stress \tau and shear strain \gamma observed in torsion tests with what is nowadays called *Ludwik’s Law*:

(1)

\tau = \tau_0 + k\gamma^{1/n}

For n=1, the stress-strain curve is linear; for n=2, the curve is a parabola; and for n=\infty, the curve represents a perfectly plastic material. Ludwik just described the behavior (*Fließkurve*) of what we now call a *pseudoplastic material*.

In version 5.0 of the COMSOL Multiphysics simulation software, beside Ludwik’s power-law, the Nonlinear Structural Materials Module includes different material models within the family of nonlinear elasticity:

- Ramberg-Osgood
- Power Law
- Uniaxial Data
- Bilinear Elastic
- User Defined

In the Geomechanics Module, we have now included material models intended to represent nonlinear deformations in soils:

- Hyperbolic Law
- Hardin-Drnevich
- Duncan-Chang
- Duncan-Selig

The main difference between a nonlinear elastic material and an elastoplastic material (either in metal or soil plasticity) is the reversibility of the deformations. While a nonlinear elastic solid would return to its original shape after a load-unload cycle, an elastoplastic solid would suffer from permanent deformations, and the stress-strain curve would present hysteretic behavior and ratcheting.

Let’s open the Elastoplastic Analysis of a Plate with a Center Hole model, available in the Nonlinear Structural Materials Model Library as *elastoplastic_plate*, and modify it to solve for one load-unload cycle. Let’s also add one of the new material models included in version 5.0, the *Uniaxial data* model, and use the stress_strain_curve already defined in the model.

Here’s a screenshot of what those selections look like:

In our example, the stress_strain_curve represents the bilinear response of the axial stress as a function of axial strain, which can be recovered from Ludwik’s law when n=1.

We can compare the stress distribution after laterally loading the plate to a maximum value. The results are pretty much the same, but the main difference is observed after a full load-unload cycle.

*Top: Elastoplastic material. Bottom: Uniaxial data model.*

Let’s pick the point where we observed the highest stress and plot the *x*-direction stress component versus the corresponding strain. The green curve shows a nonlinear, yet elastic, relation between stress and strain (the stress path goes from a\rightarrow b \rightarrow a \rightarrow c \rightarrow a). The blue curve portraits a hysteresis loop observed in elastoplastic materials with isotropic hardening (the stress path goes from a\rightarrow b \rightarrow d \rightarrow e ).

With the Uniaxial data model, you can also define your own stress-strain curve obtained from experimental data, even if it is not symmetric in both tension and compression.

- P. Ludwik.
*Elemente der Technologischen Mechanik* - “Hypoelasticity“, Chapter 3.3 of
*Applied Mechanics of Solids* - Download the Elastoplastic Analysis of a Plate with a Center Hole model

There seems to be a general trend when it comes to construction. Offshore structures are constructed in deeper and deeper waters; buildings are constructed increasingly close to each other; offshore wind turbines are developed in deep waters far away from the coasts, which are likely to experience extreme loading conditions. Therefore, in recent decades, geotechnical engineers have developed numerical simulations to cope with this construction trend and ensure safe building methods.

*“Paris Metro construction 03300288-3″. Licensed under Public domain via Wikimedia Commons.*

Materials for which strains or stresses are not released upon unloading are said to behave *plastically*. Several materials behave in such a manner, including metals, soils, rocks, and concrete, for example. These give rise to an elastic behavior up to a certain level of stress, the *yield* stress, at which plastic deformation starts to occur.

The elastic-plastic behavior is path-dependent and the stress depends on the history of deformation. Therefore, the plasticity models are usually written connecting the *rates* of stress, rather than stress, and the plastic strain. The most widespread and well-known plasticity model throughout the industry is based on the von Mises yield surface for which plastic flow is not altered by pressure. Therefore, the yield condition and the plastic flow are only based on the deviatoric part of the stress tensor.

However, this model is no longer valid for soil materials since frictional and dilatation effects need to be taken into account. Let’s see how this can be worked out and briefly explain the different soil plasticity models available in the COMSOL Multiphysics® simulation software.

For materials such as soils and rocks, the frictional and dilatational effects cannot be neglected. This whole class of materials is well known to be sensitive to pressure, leading to different tensile and compressive behaviors. The von Mises model presented above is thus not suitable for these types of materials. Instead, yield functions have been worked out to take into account the behavior of frictional materials.

Let’s illustrate the frictional behavior and plastic flow for these materials by considering the block shown here:

The block is loaded as shown by a normal load N and a tangential load Q. Assuming that the block rests on a surface with a coefficient of static friction \mu, according to Coulomb’s law, the maximum force that the block can withstand before sliding is given by F=\mu N. Therefore, the onset of sliding occurs when the following condition is reached:

(1)

f=Q-\mu N=0

The direction of sliding is horizontal. For tangential loads such as f<0, the block will not slide, but as soon as f=0, the block will slide in the direction of the applied load Q. The *Mohr-Coulomb* criterion — the first soil plasticity model ever developed — is a generalization of this approach to continuous materials and a multiaxial state of stress. It is defined such that yielding and even rupture occur when a critical condition that combines the shear stress and the mean normal stress is reached on any plane. This condition is stated as below:

(2)

\tau=c-\mu\sigma

Here, \tau is the shear stress, \sigma is the normal stress, c is the cohesion representing the shear strength under zero normal stress, and \mu=\tan\phi is the coefficient of internal friction coming from the well-known Coulomb model of friction. This equation represents two straight lines in the Mohr plane. A state of stress is safe if all three Mohr’s circles lie between those lines, while it is a critical state (onset of yielding) if one of the three circles is tangent to the lines.

*Mohr-Coulomb yield behavior. The Mohr circles are based on the principal stresses \sigma_1, \sigma_2, and \sigma_3. As you can see, one of the circles is tangential to the yield surface, and so the onset of yielding is occurring.*

According to the figure above, the stress state is given by \tau=\frac{1}{2}(\sigma_1-\sigma_2)\cos\phi and \sigma=\frac{1}{2}(\sigma_1+\sigma_3)+\frac{1}{2}(\sigma_1-\sigma_3)\sin \phi. The yield criterion and Equation 2 can therefore be re-written in a generalized form as follows:

(3)

f_y({\bf\sigma})=\sigma_1-\sigma_3+(\sigma_1+\sigma_3)\sin\phi-2c\cos\phi

It can even be seen as a particular case of a more general family of criteria based on Coulomb friction and written by equations based on invariants of the stress tensor:

(4)

f_y({\bf\sigma})=F(J_2,J_3)+\lambda I_1-\beta

*Representation of the Mohr-Coulomb yield function. *

The *Mohr-Coulomb* criterion defines a hexagonal pyramid in the space of principal stresses, which makes it straightforward for this criterion to be treated analytically. But, the constitutive equations are difficult to handle from a numerical point of view because of the sharp corners (for instance, the normal of this yield surface is undefined at the corners).

In order to avoid the issue associated with the sharp corners, another yield criterion of this family, the *Drucker-Prager* yield criterion, has been developed by modifying the von Mises yield criterion to take into account the Coulomb friction, i.e., incorporate a hydrostatic pressure dependency:

(5)

f_y({\bf\sigma})=\sqrt{J_2}+\alpha I_1-k

This represents a smooth circular cone in the plane of principal stress, rather than a hexagonal pyramid. If the coefficients \alpha and k are chosen such that they match the coefficients in the *Mohr-Coulomb* criterion, as follows:

(6)

\alpha=\frac{2}{\sqrt{3}}\frac{\sin\phi}{3\pm\sin\phi},\quad k=\frac{2\sqrt{3}c\cos\phi}{3\pm\sin\phi}

the *Drucker-Prager* yield surface passes through the inner or outer apexes of the Mohr-Coulomb pyramid, depending on whether the symbol \pm is positive or negative. The plastic flow direction is taken from the so-called “plastic potential”, which can be either the same, associative plasticity, or different, non-associative plasticity, than the onset of yielding (the yield function). Many different non-associative flow rules can be developed.

Using an associative law for the Drucker-Prager model leads the volumetric plastic flow to be nonzero. Therefore, there is a change in volume under compression. However, this is contradictory to the behavior of many soil materials, particularly granular materials. Instead, a non-associative flow rule can be used such that the plastic behavior is isochoric (volume preserving) — a much better reflection of the plastic behavior of granular materials.

*Representation of the Drucker-Prager yield function.*

Next, I will show you how to use a non-associative law for soil plasticity in COMSOL Multiphysics. Non-associative plastic laws can be used regardless of the plasticity model used in the software.

If you’re using the Mohr-Coulomb model, there are basically two different approaches to handling non-associative plasticity. The plastic potential can either be taken from the Drucker-Prager model or be the same as the *Mohr-Coulomb* yield function but with a different slope with respect to the hydrostatic axis, i.e., the angle of friction is replaced by the *dilatation angle* (see screenshot below).

Moreover, when using the Drucker-Prager matched to a *Mohr-Coulomb* criterion, it is easy to adapt the dilatation angle to match with the non-associative law that you want to use. For instance, the non-associative law presented above can be worked out by taking the dilatation angle null.

Last but not least, a useful feature called *elliptic cap* has been developed to avoid unphysical behavior of the material beyond a certain level of pressure. Indeed, real-life material cannot withstand infinite pressure and still deform elastically. Therefore, to cope with this, we can use the elliptic cap feature available in COMSOL Multiphysics.

*Soil Plasticity feature settings window.*

Let’s try to put into practice everything we’ve learned so far by analyzing the example of a tunnel excavation. This will also be an opportunity to figure out what the effects of the different features we mentioned above are.

The simulation of a tunnel excavation process is especially important in predicting the necessary reinforcements that the workers need to use to avoid the collapse of the construction.

The following model aims to simulate the soil behavior during a tunnel excavation. The surface settlement (i.e., the vertical displacement along the free ground surface) and the plastic region are computed and compared between the different soil models used to carry out this simulation. The geometry we’ll use is presented in the figure below. To make our model realistic, infinite elements have been used to enlarge the soil domain, while keeping the computational domain small enough to get the solution in a relatively short time.

*The geometry consists of a soil layer that is 100 meters deep and 100 meters wide plus 20 meters of infinite elements. A tunnel 10 meters in diameter is placed 10 meters away from the symmetry axis and 20 meters below the surface.*

First of all, we need to add the in-situ stresses in the soil before the excavation of the tunnel. Then, we can compute the elastoplastic behavior once the soil corresponding to the tunnel is removed. The in-situ stresses must be incorporated in this second step. This is fairly straightforward to set up in COMSOL Multiphysics.

We can begin by adding a stationary step where the in-situ stresses will be computed. Then, in a second step but still within the same study, we add a soil plasticity feature. Finally, we compute the solution. In order to get the pre-stresses incorporated into the second step, we should add an Initial Stress and Strain feature under the *Solid Mechanics* interface, as shown below.

*Initial Stress and Strain feature used to incorporate the in-situ stresses from the first step as initial stresses for the second step, during which excavation occurs. The variables solid.sx, solid.sxy, etc. are the *x*-components of the stress tensor, the *xy*-components of the stress tensor, etc.*

The first plot shows the in-situ stresses computed from the first step. These stresses result from the gravity load.

*The von Mises stress in the soil before the excavation of the tunnel.*

The second plot shows the stress distribution after excavating the tunnel. In-situ stresses are taken from the first step. Note, as expected, the increase in the von Mises stress around the tunnel as well as the deformation of the tunnel shape.

*The von Mises stress in the soil after excavation of the tunnel.*

As mentioned previously, while removing the tunnel domain, a plasticity feature is added and the soil experiences a plastic behavior. This is depicted in the figure below of a Drucker-Prager model with associative plastic flow. The plastic region is concentrated around the near surroundings of the tunnel. The analysis of this region is quite important in gaining insight into how the soil is more likely to deform. Therefore, it allows us to handle the necessary reinforcements in order to avoid collapse and get the desired tunnel shape.

*Plastic region after excavating the tunnel.*

This tunnel excavation simulation has been carried out in four different cases in order to compare the different soil models presented in the previous section as well as understand the influence of the cohesion on the soil’s behavior. The results are shown by taking the surface settlement as the criterion.

Below, we have a 1D plot from which we can observe the following relationship: The lower the cohesion, the greater the deformation. We can also note that the Mohr-Coulomb model tends to, somehow, make the soil stiffer than the Drucker-Prager model. The non-associative law with a null dilatation angle prevents the soil from dilating under compression and so the surface settlement becomes greater.

*Surface settlement comparison between different plasticity models and material properties.*

There are also a couple of other plasticity models for soil, rocks, and concrete available in COMSOL Multiphysics. Please check out the links below to get further information about geotechnical simulations and the Geomechanics Module of COMSOL Multiphysics.

Also, be sure to watch the video on how to build a model of an excavation:

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There are two ways to model an excavation in COMSOL Multiphysics, both of which include a parametric sweep. One option involves a sweep of the geometry, removing the geometry (excavation) one step at a time. As the soil is removed, the support it supplies is removed as well, subjecting the retaining wall to soil stresses from the non-excavated side. The other option is to start with the already excavated geometry, and simulate the excavation using a boundary load. The boundary load applies a force on the excavation side of the retaining wall, equal to (and therefore negating) the in-situ stresses, for any part of the wall that is below the virtual excavation depth.

The video shown below uses the latter of the two strategies to model a 26-meter excavation. As the excavation deepens, three struts are activated using a ramp function, and boolean expressions. As the excavation reaches their depths, the struts are activated as long as the horizontal wall deflection is greater than what we allow it to be. There are four sets of results showing the deformation of the soil and retaining wall, the plastic deformation, wall deflection, and the surface settlement.

Check out our Deep Excavation Model for more information. Logging into your COMSOL Access account enables you to download the documentation for this model as well.

*In this example, a 26-meter excavation is modeled by means of a parametric sweep, with a step size of 2 meters. Struts are activated once the excavation reaches their depths by using a boolean expression. A symmetry is used because the modeled excavation is only the right half of the full excavation.*

For this example, we will be modeling in 2D plane strain, using the solid mechanics interface and a stationary study.

These are the parameters we will be using later on, which represent in-situ stresses, properties of the metal struts, excavation steps, along with a couple other related parameters.

Create the ramp function for activating the struts. The ramp location is set to be -U_max. Once a strut is activated, it will generate a force proportional to the struts stiffness and the horizontal displacement.

The geometry has been previously created for this model, but all the steps are outlined in the model file to build the upper and lower layers of the soil, as well as the retaining wall and three embedded struts. Forming an assembly creates an identity pair between the wall soil and wall diaphragm boundaries.

Two boundary selections have been created and renamed wall_diaphragm and wall_soil. As you can see, they are in the same location but are different boundaries. The extrusion operators constrain the normal displacement between the retaining wall and soil, forcing them to stay in contact.

Add a soil plasticity node and you can see that the yield criterion is Drucker-Prager, but we still want to match it to the Mohr-Coulomb criterion. An initial stress is added to the model as well to simulate the in-situ stresses in the x, y, and z directions.

Now we can add the boundary constraints, including a symmetry on the left, a fixed constraint for the bottom boundary, and a roller for the right boundaries.

We can choose a prescribed displacement to make sure the soil at boundary 4 only moves in the y direction. Enter the expression for the general extrusion operator from earlier.

A prescribed displacement is also needed for the wall_soil boundary created earlier. This time in the x direction and using the second general extrusion operator in the u0 field.

For the final part of the physics set up, we are going to add five boundary loads to the model.

The first boundary load is added for the horizontal soil boundaries. Therefore we want stress in the y-direction.

The second boundary load is added for the vertical retaining wall boundaries. We want stress in the negative x-direction here.

The rest of the boundary loads will describe the three struts. Select the top strut and then total force as the load type. Enter in the equation shown which is the ramp function of the wall deflection, with an added term limiting the expression to occur only when the depth is below the strut. Right click the Boundary Load 3 node to rename it Strut_1. Right click again to duplicate it twice, since we want similar settings for the second and third struts. Clear the selection and add the middle strut, then change the expression from stage 1 to stage 2 to activate it only when the depth moves below the second strut. Rename this one Strut_2. Similar to the first two, for the third strut, add the third strut boundary and change stage 2 to stage 3. Rename the third strut to finish off the physics set up.

The materials have already been created and set up for this model. The upper layer soil, the lower layer soil, and the retaining wall.

For meshing the model, sometimes an automatic mesh is sufficient, but we want to make our own for this model to improve convergence on the soil-wall boundary. First a mapped mesh for the retaining wall domain. Add a distribution for the wall diaphragm and enter 60 for the number of elements. Add a second distribution for the bottom boundary, and enter 2 for the number of elements.

Now, add a free triangular for the remaining geometry. First add a size node to make sure the mesh is finer. Then we add three distributions, one corresponding to each of the mapped mesh distributions. 2 for the wall diaphragm totaling 60 elements, and one for the bottom boundary, with 3 elements.

Now we can build the mesh and zoom in on the soil-wall boundary, to see the improved mesh.

The last step before computing the model is defining a range of depth parameters for the parametric sweep. Add depth as the continuation parameter, and click the range button. We want a depth ranging from 0 to -26 meters with a step size of two meters.

Right-click study 1 to compute the model.

Once the model has finished computing we can add some post processing to better view the results.

The default plot shows the von Mises stress. Click replace expression and go to solid mechanics, displacement, and choose total displacement. Click plot to view the displacement at the different excavation depths. The player button feature allows you to see all the excavation depth results in one animation.

Create a second plot group and surface plot, then enter in the expression solid.epe>0 to view the plastic deformation in the different soil layers. Click the player button again to view all the parameter values in succession.

Now we will create a 1D plot and a line graph, with the wall diaphragm as the selection. For the y-axis data, the expression is y, and for the x axis, the expression is u, with millimeters as units. This graph shows the wall deflection or horizontal displacement as a function of depth for different excavation steps.

*Create a second line graph, showing the surface settlement, or vertical displacement, as a function of the distance from the wall. Add boundary 8 and change the y-axis data expression to v and use millimeters as the unit. And when you check the legends box, the plot will automatically be generated.*

*Schematic showing orthogonal fracture propagation. From the paper “Investigation of Hydraulic Fracture Re-Orientation Effects in Tight Gas Reservoirs“. Image courtesy of B. Hagemann ^{1} J. Wegner^{1} L. Ganzer^{1}*

^{1}Clausthal University of Technology, Clausthal-Zellerfeld, Germany.

Fracturing is simple to describe; it’s just cracks in the earth. The mechanics involved, however, are complicated, especially when you consider fluid flow through formations within the earth. Moreover, it’s not easy to observe what’s happening hundreds or thousands of meters beneath the surface of the earth. These facts have made hydraulic fracturing a hot topic of active research, and since the underlying phenomena are relatively unknown, it is ideal for multiphysics simulation.

That fracturing is ideal for multiphysics modeling is evidenced by the number of papers and posters related to the topic presented at the COMSOL Conference earlier this year. Consider the following:

3D Modeling of Fracture Flow in Core Samples Using μ-CT Data*, S. Hoyer ^{1} U. Exner^{2} M. Voorn^{1} A. Rath^{3}*

^{1}Department of Geodynamics and Sedimentology, University of Vienna, Austria

^{2}Museum of Natural History, Vienna, Austria

^{3}OMV ESG-D Production Geology, Vienna, Austria

Here the authors used CT scans of fractured samples of rock to perform simulations measuring permeability without destroying the samples — a difficult task using physical experiments.

3D Simulations of an Injection Test Done Into an Unsaturated Porous and Fractured Limestone*, A. Thoraval ^{1} Y. Guglielmi^{2} F. Cappa^{3}*

^{1}INERIS, Nancy, France

^{2}CEREGE, Aix-en-Provence, France

^{3}GEOAZUR, Valbonne, France

The researchers behind this paper developed an advanced COMSOL model including stress-strain constitutive law, two-phase flows, and hydro-mechanical coupling, then applied it to an actual slope location in France.

A Coulomb Stress Model to Simulate Induced Seismicity Due to Fluid Injection and Withdrawal in Deep Boreholes*, G. Perillo ^{1} G. De Natale^{2} C. Troise^{2} A. Troiano^{2} M.G. Di Giuseppe^{2} A. Tramelli^{2}*

^{1}University of Naples Parthenope, Naples, Italy

^{2}INGV, Osservatorio Vesuviano, Naples, Italy

This paper presents a stress model to account for cases of fluid injection in deep boreholes, with particular application to a geothermal site in northeastern France.

Finite Element Solution of Nonlinear Transient Rock Damage with Application in Geomechanics of Oil and Gas Reservoirs*, S. Enayatpour ^{1} T. Patzek^{1}*

^{1}The University of Texas at Austin, Austin, TX, USA

A novel script was written for a first-principles PDE model of rock damage until fracture.

Fracture-Matrix Flow Partitioning and Cross Flow: Numerical Modeling of Laboratory Fractured Core Flood*, R. Sanaee ^{1} G.F. Oluyemi^{1} M. Hossain^{1} B.M. Oyeneyin^{1}*

^{1}Robert Gordon University, Aberdeen, United Kingdom

This paper is part of an effort to better understand subsurface flow that includes both fractures and matrix flow. Under certain stress conditions (fracture closure and overburden), the flow was studied and further physical experiments were suggested.

Investigation of Hydraulic Fracture Re-Orientation Effects in Tight Gas Reservoirs*, B. Hagemann ^{1} J. Wegner^{1} L. Ganzer^{1}*

^{1}Clausthal University of Technology, Clausthal-Zellerfeld, Germany

Not just fracturing, but re-fracturing! This is an important consideration for tight gas formations where the critical question is: when should you perform re-fracturing for optimal productivity?

Poroelastic Models of Stress Diffusion and Fault Re-Activation in Underground Injection*, R. Nopper ^{1} J. Clark^{2} C. Miller^{1}*

^{1}DuPont Company, Wilmington, DE, USA

^{2}DuPont Company, Beaumont, TX, USA

Flow plus poroelastic deformation models developed in COMSOL point toward promising criteria for rock failure.

No matter where you might stand on the issue, it’s great to see that COMSOL is enabling those who are at the forefront of fracture modeling to simulate, understand, and advance the technology. Knowledge, I think we can agree, is the key to understanding the risks and rewards of fracking.

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