Let’s consider a symmetric two-bar structure under a compressive load, as shown in the following figure:

*Two bars under compression.*

We assume that the bars are linearly elastic so that the force in a bar, *F*, is

F=EA \frac{\Delta}{L_0}

where Δ is the elongation and *L*_{0} is the original length.

Using Pythagoras’ theorem, the vertical force can then be written as an explicit function of the vertical displacement:

P = EA \left(\sqrt{1-\kappa^2\delta(\delta-2)}-1 \right) \frac{\kappa(1-\delta)}{\sqrt{1+\kappa^2\delta(\delta-2)}} \approx EA \kappa^3\delta(\delta-2)(\delta-1)

The following quantities have been nondimensionalized:

- The normalized displacement , where is the deflection under the load. Thus, when the bars are horizontal and when the structure is upside down.
- The parameter is the aspect ratio: the initial height relative to initial bar length.

The force as a function of displacement is shown in the graph below. The example actually shows as a buckling problem with snap-through. Between points A and C, no unique solution exists. In a previous blog post, we further discuss the concept of buckling in structures.

The compressive force in the bars increases until they are horizontal (), but the vertical projection decreases even faster beyond point A. This is why the vertical force decreases.

*Force as a function of vertical displacement.*

If we build a finite element model of this structure and try to increase the load, the analysis will probably fail when we reach the first peak at point A. We can, however, easily trace the solution by prescribing the vertical displacement at the loaded point, rather than the force. The applied force can then be obtained as the reaction force. The graph above was created using that method.

The tangential stiffness for this single degree of freedom system is defined as the rate of change in force with respect to displacement:

k_{\mathrm t} = \frac{dP}{dw} = \frac{EA \kappa^3}{h}(3\delta^2-6 \delta+2)

The stiffness is thus negative between points A and B. A negative stiffness is often related to numerical and physical instabilities.

*Stiffness as a function of vertical displacement.*

There are several material models within the field of solid mechanics that contain a negative slope of the stress-strain curve, either as an intentional effect or with certain choices of parameters. For example, some models for concrete are designed like this. In the physical interpretation of this behavior, cracks form when the material model is loaded in tension. The load carried by a test specimen will then decrease. The cohesive zone models used for describing decohesion in the COMSOL Multiphysics® software also show this type of behavior.

*A strain-softening material.*

At the material level, decreasing stress with increasing strain indicates a negative stiffness:

\frac{d\sigma}{d\epsilon} < 0

Such a material can only be tested under a condition of prescribed displacement; otherwise, it will fail immediately when the peak load is reached. The negative stiffness is thus related to a material instability.

In general, the stress and strain states are multiaxial. Stress and strain are represented by second-order tensors. In the multiaxial case, we must use a more general criterion for material stability: For any small change in the strain state, the corresponding change in the stress state must be such that the sum of the products of all stress and strain components is positive. That is,

d \boldsymbol{\sigma}: d \boldsymbol{\varepsilon} >0

Or, written in component form,

\sum_{i=1}^3\sum_{j=1}^3 d\sigma_{ij} d\varepsilon_{ij} > 0

This is called *Drucker’s stability criterion* or *Hill’s stability criterion*.

The discretized form used in finite element analysis implies that the constitutive matrix relating stress increments to strain increments must be positive definite in order for the material to be stable. This is a condition that is generally computationally expensive to check for nonlinear materials. For a linear elastic material, the requirement can be directly converted into the well-known requirements and .

How can we mediate that we sometimes need to work with material models that do not fulfill the stability criterion? The important fact is that the material can be locally unstable, while the structure as such is still stable.

To understand this behavior, we can think of the material in the structure as connected springs. Some springs are purely elastic and represent the undamaged material, while a certain spring fails. Consider the three springs in the figure below.

*A three-spring system. The extension of the failing spring is denoted u _{1}.*

The spring k_{1} represents the material with the damage model, whereas the other two springs are purely elastic. The material model for the first spring is bilinear.

*Material model for the nonlinear spring. The peak force F _{m} is reached at the displacement u_{m}.*

The force in the lower branch is independent of damage:

F_l = k_3 u

Before the peak load is reached, the force in the upper branch is

F_u = \frac{k_1 k_2}{k_1 + k_2} u

since the two springs are connected in series.

The damage starts when the force in the upper branch is ; that is, when the external displacement is

u = \frac{(k_1+k_2) F_m}{k_1k_2}

and the corresponding external force is

F = F_m \left (1 +\frac{k_3(k_1+k_2)}{k_1 k_2} \right)

During the degradation, the force in the damaged spring can be written as . The same force also passes through spring 2 so that .

These two relations determine *u*_{1} as a function of the external displacement:

u_1 = \frac{k_2u-F_m-k_fu_m}{k_2-k_f}

In order to give a reasonable solution, *u _{1}* must increase when the external displacement is increased. Thus, it is necessary that . This is actually a clue to a very general result. A quick decrease in the force (or stress) is more susceptible to instability than a slower decrease.

Finally, we can derive the relation between the total external force and displacement during the degradation phase:

F = F_l + F_u = k_3 u-k_2\left( \frac{k_fu-F_m -k_fu_m}{k_2-k_f} \right) =\left ( k_3-\frac{k_2k_f}{k_2-k_f} \right ) u +\left( \frac{F_m+k_fu_m}{k_2-k_f} \right)

Thus, the external force can either increase or decrease when the external displacement increases, depending on the relative stiffness in the two branches. This simple model can thereby predict two types of instability:

- The upper branch can become unstable if
- Even if , the total system will be unstable if

In either case, a slower decrease of the force in the damage model is beneficial. In other words, the stiffer the surroundings, the more plausible it is that the whole system will be stable.

*A globally stable system (left) and a system where the stiffness in the lower branch is too small to maintain stability (right).*

In reality, we are not free to make arbitrary choices about force and stiffness. The area under the triangular force-displacement curve in the material model represents the energy dissipated by the process. The energy dissipation and the displacement (or strain) at final failure have a physical meaning.

The damaged part of a structure elongates while its force decreases. If the external displacement remains fixed, then the elastic parts of the structure must contract to compensate. This means that elastic energy is released. The only way the energy can be absorbed is by doing work on the damaged part. If, for a certain incremental displacement , the energy released by the elastic parts is larger than the work needed to produce the same displacement in the cracking part, the state is unstable.

Years ago, a friend of mine at the Department of Solid Mechanics at KTH Royal Institute of Technology in Stockholm performed some interesting experiments where he studied the stability of cracks in a ductile steel using extremely long three-point bend test specimens. The tests highlighted that crack stability is not only a function of the local stress state, but also of the capacity that the stored energy in the test specimen has to drive crack propagation. The longest test specimen in the experiments was 26 meters and occupied a large portion of the lab! The experiment was reported in the article “The stability of very long bend specimens” in the *International Journal of Pressure Vessels and Piping*.

With softening material models, it is extremely difficult to achieve convergence in a finite element model if the stress state is homogeneous.

In a physical material, the strength does not have a perfectly uniform distribution. When increasing the load, a crack will form at the location with the lowest strength, even if the stress state is homogeneous. When this happens, the surrounding material is unloaded.

Consider this example of three elastic blocks joined by two glue layers:

In real life, one glue layer will fail before the other. The slightly stronger layer will then be unloaded as the force through the part decreases. We cannot predict which layer will fail, since that is controlled by manufacturing inaccuracies. In the mathematical model, however, both layers fail simultaneously. Numerically, the iterations may not converge because the failure jumps back and forth between the two layers.

In a finite element model, the stresses are evaluated at each integration point within each element. When the load is increased above the maximum value, the failure may even jump between the elements or individual integration points within the same element (if the stress is the same everywhere).

This behavior implies that if we implement our own material model containing strain softening, we should test it using a single first-order element and under prescribed displacements. In this way, we ensure a homogeneous prescribed strain field and the stress is the same everywhere in the element. One example is Mazar’s damage model, which we described in a previous blog post. If we were to change the element shape functions to quadratic in that model, the analysis would no longer converge.

Does this mean that damage models are meaningless? Not at all. However, we must be careful to avoid indeterminate states. If a structure and its boundary conditions are symmetric, that symmetry must be employed in order to avoid indeterminacy. We can often solve problems with axial symmetry by using an axially symmetric model, while this may be impossible using a model of a 3D solid sector. Another approach is to allow a slight random spatial disturbance of the material data. This approach actually mimics nature, where strength values are randomly distributed. Also, it is important to increase the loading slowly in order to avoid large portions of the structure switching to a failed state at the same time.

In some material models, for example, within soil plasticity, strongly mesh-dependent thin layers with high shear strains can occur. These layers are called *shear bands*. When yielding is first initiated, the surrounding elements or even integration points are unloaded. The first elements to yield continue to accumulate plastic strains. It is interesting that this type of instability can actually be seen in real soil and is not only an artifact in the numerical model. Just as in nature, we cannot predict the exact location and distribution of the shear bands in the model.

As mentioned in the initial example, using prescribed displacements rather than prescribed forces is a good way to stabilize the numerical problem. However, this approach is essentially limited to the following cases:

- A single point load can be replaced with a prescribed displacement.
- The strain state is homogeneous, as when performing single-element tests of a material model, so that the displacement is the same over an entire boundary.
- The instability can be controlled by the external boundary conditions. In the spring device discussed above, a global instability caused by too small of a value of
*k*_{3}could be cured by using a prescribed displacement, but not an instability within the upper branch.

There is a more general method, which we can use to continue solving past a point of instability. In this method, we first prescribe an arbitrary quantity that is known to monotonically increase and then add an extra equation that solves for the corresponding value of the prescribed load or displacement.

To display this technique, let’s augment the initial example with a spring, so that the load is applied by prescribing the deformation of the end of the spring. If the spring is very stiff, this is essentially the same as prescribing the displacement directly.

*Bar system loaded through a spring.*

If the spring is softer, the system may become unstable, since too much energy can be released by the spring. The critical value is

k>\frac{EA \kappa^3}{h}

This is the most “negative stiffness” of the bar assembly, which occurs when the bars are horizontal. The relation between force and displacement at point 1 when varying the spring stiffness is shown below. The spring stiffness is given as

k = \beta \frac{EA \kappa^3}{h}

where the coefficient *β* is varied from an essentially stiff spring to values below the critical value.

*Force as a function of the displacement at point 1 when varying the spring stiffness.*

For values of *β* smaller than one, the solution fails when the spring stiffness equals the “negative” stiffness of the bar assembly.

If a prescribed force is used instead, all solutions will fail at the first peak load. By using prescribed displacement, it is possible to continue the analysis further. For lower spring stiffness values, we are still limited by the state when the internal instability causes failure.

The solution that we want to track has a monotonous vertical displacement at point 2, but prescribing it directly is not possible, since this would change the problem fundamentally. Instead, we add an equation stating: “Set the spring end displacement at point 1 so that the monitored displacement at point 2 has the prescribed value.” To do this, we add a *Global Equation* node in which a new unknown variable `disp_at_P1`

is added.

*The Global Variable definition.*

The equation determining the value of `disp_at_P1`

states that `disp_at_P2-delta = 0`

. The variable delta is the monotonous parameter incremented in the Stationary study step and ` disp_at_P2`

is a variable that contains the current value of the displacement at point 2.

*Settings for the study step, where delta is used as the auxiliary sweep parameter.*

The displacement at point 2 is then prescribed to have the value that satisfies the global equation.

*Settings for the prescribed displacement at point 1.*

With this modification, it is possible to trace the solution through the instability. As seen in the following graph, even strong instabilities can be bypassed using this method.

*Force as a function of the displacement at point 1 when varying the spring stiffness after stabilization with a* Global Equation *node.*

- Browse these Application Gallery examples to learn more about modeling negative stiffness and instabilities in structures:
- S. Kaiser, “The stability of very long bend specimens”,
*Int. J. Pres. Ves, & Piping.*17 (1984) 1–17.

When you enter an old building, such as an ancient church or castle, you may feel as if you are surrounded by history. The building around you has survived through battering storms and rain, stayed standing through wars and conflict, and remained the same as the world around it has changed. Although the initial design of these cultural heritage sites contributes to their longstanding existence, additional protection is often needed to avoid deterioration. As an example, let’s take a look at a chapel in Lemiers, the Netherlands called Saint Catherine’s Chapel, which was registered as a Dutch state monument in 1967.

*St. Catherine’s Chapel. Image by Sigibert — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

St. Catherine’s Chapel was built in the 11^{th} or 12^{th} century, although the remains of an older building exist within its foundation, possibly from the 8^{th} century. In its entirety, the chapel is the only complete surviving example of the Netherlands’ first type of stone church; however, this is not the chapel’s only claim to fame. It is also home to famous paintings by the artist Hans Truijen, which are displayed on the internal walls.

Today, these paintings are decaying at an increasing rate, especially in areas closer to the floor, where entire sections have disappeared. But what is causing such deterioration? We can look at the natural environment surrounding the chapel to answer this question.

*Painting deterioration inside St. Catherine’s Chapel. Images by Henk Schellen et al. and taken from their COMSOL Conference 2015 Grenoble presentation.*

High groundwater levels and a nearby creek with a propensity to flood create problems for the structure. These combined factors cause the crypt underneath the chapel to flood, resulting in water traveling up the chapel walls. As the moisture rises, the salt that it carries is entrained within the walls. Then, when the salt dries, it forms crystals that have as much as 80 times the volume of the salt in a solute state. This expands the stucco and overworked paint on the walls, damaging the precious artwork.

To help prevent this, a research team from the Eindhoven University of Technology used physical experiments and the simulation capabilities of COMSOL Multiphysics to better understand the damage to cultural heritage sites that is caused by rising moisture levels. They also looked into solutions to the issue, including drying strategies and drainage.

During a period of two consecutive years, the researchers collected a wide range of data from St. Catherine’s Chapel. These measurements include the indoor and outdoor air temperature and relative humidity, heat flux on the external walls, CO_{2} concentration, solar radiation, microclimate conditions near the walls, and moisture content at different wall points, as well as indoor surface temperature and relative humidity.

One such measurement, moisture content, required the researchers to measure the chapel crypt twice, once when it was flooded up to the entrance hatch and once when it was dry. From these measurements, they discovered that the high moisture levels were located in areas where painting loss is most severe: close to the floor and in the wall above the crypt. Further, their measurements indicated that high groundwater levels caused the rising moisture in the chapel.

*Measurements showing the averaged moisture content in the chapel walls. The blue line shows the groundwater level. Images by Henk Schellen et al. and taken from their COMSOL Conference 2015 Grenoble paper.*

With this, the research team had verified the dampness problem at the chapel, but not how the indoor and outdoor climate conditions relate to the deterioration of the chapel’s paintings. Their next goal was to use the collected measurements to provide data for their simulations and validate their model.

To begin, the research team built a multizone hygrothermal building, or HAMBase (heat, air, and moisture), model of the chapel. The model divides the chapel into three zones: chapel (nave and choir), attic, and crypt. This model enabled them to calculate the indoor temperature and relative humidity that occurs as a result of building properties and changes in the outdoor climate, climate control system, and building use.

In an effort to minimize simulation errors, the team also utilized inverse modeling techniques. Here, they created a thermal and hygric model for each of the three zones of the chapel. The comparison showed that the chapel and attic simulation results were similar to measured results, with the attic showing a higher temperature in the summer when compared to measured results.

*Comparing measured and simulated results for the chapel and attic. Images by Henk Schellen et al. and taken from their COMSOL Conference 2015 Grenoble paper.*

The crypt, on the other hand, showed the most deviation of any zone. Although the absolute humidity for the crypt compared well with measured data, there were deviations in relative humidity. With that in mind, the team used inverse modeling to take a closer look at the relative humidity and discovered that slight deviations in air temperature can result in large deviations in relative humidity. This effect, as well as the inaccurate measuring equipment and lack of additional moisture sources, may have caused the deviation between simulated and measured results.

*Left: Comparing measured and simulated results for the crypt. Right: Inverse modeling of the relative humidity. Images by Henk Schellen et al. and taken from their COMSOL Conference 2015 Grenoble paper and presentation.*

While the results thus far were useful, the indoor climate condition models didn’t enable the researchers to verify why the chapel walls were damaged. So, they decided to model the moisture transport in the chapel walls using a user-defined moisture transport equation. Their goal here was to create a model that accurately described the wetting and drying of a porous medium — in this case, the chapel walls.

*Top: Schematic illustrating water uptake. Bottom: Moisture content in the wall after one year. Images by Henk Schellen et al. and taken from their COMSOL Conference 2015 Grenoble paper submission.*

Although the exact material properties of the chapel are unknown, the simulation results successfully illustrated the effect of water uptake on the wall. By looking at the transportation of salts and evaporation of water, the researchers were able to confirm that rising water is the primary cause of the deteriorating wall paintings at St. Catherine’s Chapel.

Understanding the cause of deterioration at St. Catherine’s Chapel is the first step toward finding a way to protect this cultural heritage site. Now, we can look at methods for preventing rising moisture before the damage is beyond repair. Even though solutions can be costly and rigorous, the research team did find one potential method for protecting the art at the chapel: By injecting the foundation with a water-repellent chemical fluid, it is possible to prevent moisture from rising up the walls. With this approach, the paintings can be preserved for future generations. Hopefully, the results from this research can be used to protect other cultural heritage sites in the future.

- Check out the full paper: “The Use of COMSOL Multiphysics® Software to Explore Flooding and Rising Dampness Problems Related to Cultural Heritage“
- Watch an archived webinar about modeling the built environment in COMSOL Multiphysics: “Model the Physics Behind the Built Environment with COMSOL“
- Catch up on some past blog posts about avoiding damage in buildings with simulation:

Rye Brook, New York is home to the William E. Ward House, the oldest reinforced structure in the United States. When the house — made up entirely of concrete, aside from wooden paneling, doors, and window frames — was first built in the 1870s, neighbors were skeptical of the new building material and nicknamed the house “Ward’s Folly”. But, after recognizing the strength and durability of the reinforced concrete, the house’s nickname changed to “Ward’s Castle”. The house, shown below, is an early testament to the strength of reinforced concrete and its effectiveness as a building material.

*The William E. Ward House. Image by Daniel Case — Own work. Licensed under CC BY-SA 3.0, via Wikimedia Commons.*

In modern construction, reinforced concrete is often used in the development of foundations, frames, walls, and beams. Typically, the material consists of rebars (steel bars or steel meshes) that are embedded in the concrete. The rebars are meant to avoid stress within the concrete that could potentially lead to cracking or failure of the structure. Because the concrete shows weakness in tension, it is often prestressed in compression by pretensioning the rebars. Ensuring the functionality of each reinforced concrete design is, of course, an important step in the building process.

*A construction worker creating a rebar for reinforced concrete.*

Physical prototypes are one way to test reinforced concrete, but such an approach can be both expensive and time consuming. Think back to the Ward House, for instance. In that situation, deflection tests were performed over multiple years and the strength of the floor had to be tested *after* the construction of the house.

Today, you can use simulation to quickly analyze multiple reinforced concrete designs without the need for physical prototypes. Apps take this power to the next level, allowing you to share simulation capabilities with your colleagues and customers through an intuitive and easy-to-use interface.

As an example, let’s take a look at the Parameterized Concrete Beam demo app, which is available in our Application Gallery.

The Parameterized Concrete Beam demo app is based on a model of a concrete beam that is reinforced by steel bars. The focus of the app is to compute the deflection and axial stress of the beam, with a body load as the self weight and a surface load on the beam’s top face.

Within the app, users have the option to modify a range of parameters. Such parameters include the geometry of the beam, steel and concrete material properties, the distribution of the reinforcement bars, and boundary conditions at the beam’s end. In regards to the app’s layout, the user inputs are primarily placed on the left side, with the results and plots on the right side.

When designing your own app, you have control over the parameters that are available for modification as well as how they are presented in the app’s user interface. This not only makes it easier for users to navigate the app, but it also helps to ensure accuracy in the simulation results.

Going back to our demo app, let’s take a look at the input sections. These sections are tabbed under three panels: *Geometry*, *Material Properties*, and *Loads and Constraints*.

Within the *Geometry* panel, you can alter the beam geometry as well as the distribution of the reinforcement bars. Once the inputs are modified, a new cross section is plotted in the results panel (shown below), enabling users to visualize the updated beam design in a simplified format. Additionally, the modified geometry is checked to make sure that the rebars are consistently located within the concrete beam geometry and don’t interfere with one another’s geometry. If such a situation does occur, a warning message will pop up in the geometry check area.

*Schematic of the beam cross section, showing the main geometrical parameters.*

The *Materials Properties* panel features the material properties that can be modified for the steel and concrete used within the reinforcement bars.

In the *Loads and Constraints* panel, there are four sets of boundary conditions to choose from for the ends of the beam. You may also set the applied loads, pretension of the rebars, and choose whether gravity should be taken into account in the simulation.

In the Results and Plots section, you can choose at which step the results are displayed: after the pretension of the rebars, after gravity is applied, or after loads are applied. The default plot is the axial stress in concrete, with tensile stress plotted in red and compressive stress in blue. You can also plot the stress in the rebars, the plastic regions, and the deflection of the beam. The numerical values of those quantities are displayed for the three steps in the table below the plots.

Apps create a smoother design process by empowering people of varying levels of expertise to run their own simulation tests. In the case of a reinforced concrete beam, building an app provides an easier route for testing and analyzing different design schemes. This helps to ensure the optimal safety and performance of the beams and thus the structure in which they are used.

The app presented here is just one example of what you can design with the Application Builder. We hope that it both inspires and guides you in your own app-building process.

- Download the demo app: Parameterized Concrete Beam
- Eager to see how others are using apps to improve their design workflows? Take a look at these blog posts:

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 and shear strain observed in torsion tests with what is nowadays called *Ludwik’s Law*:

(1)

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

For , the stress-strain curve is linear; for , the curve is a parabola; and for , 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 .

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 ). The blue curve portraits a hysteresis loop observed in elastoplastic materials with isotropic hardening (the stress path goes from ).

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 and a tangential load . Assuming that the block rests on a surface with a coefficient of static friction , according to Coulomb’s law, the maximum force that the block can withstand before sliding is given by . 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 , the block will not slide, but as soon as , the block will slide in the direction of the applied load . 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, is the shear stress, is the normal stress, is the cohesion representing the shear strength under zero normal stress, and 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 , , and . 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 and . 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 and 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 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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