One of the most common effects associated with an earthquake is shaking. Depending on the size and magnitude of the seismic waves, this shaking can result in various levels of destruction. In the case of buildings, such waves can produce instability or, in more extreme cases, cause structures to collapse.

Seismic control is an important consideration in the design of buildings. This is particularly relevant to taller structures, which pose a greater risk to human life during earthquakes. Such control can be achieved through various techniques, one of which includes a passive control approach where an external energy source isn’t required.

A *base isolation system* is an example of such a passive control method. As indicated by its name, a base isolation system isolates a structure’s base from its foundation with the use of a bearing. The bearing, which acts as an isolator, deflects and absorbs seismic waves, helping to protect the structure from the force of the vibrations.

*Fixed base and isolated base systems. Image by R. Sugumar, C.S. Kumar, and T.K. Datta, and taken from their presentation submission*.

Base isolation systems are valued for their simplistic construction and installation as well as their ability to be installed in existing buildings. Such systems can be used effectively in tall structures featuring multiple levels and in buildings with up to 20 stories.

Using the Nonlinear Structural Materials Module in COMSOL Multiphysics, a team of researchers from India set out to investigate the efficiency of a base isolation system as well as how to optimize its performance. They presented their findings from the study at the COMSOL Conference 2014 Bangalore.

In the analysis, a laminated rubber bearing (LRB) was used as the isolator. Consisting of steel shims between rubber layers, the LRB uses its flexibility to deflect seismic waves and, through plastic deformation, absorbs the energy from the vibrations. Additionally, its lead core assists in further dissipating the energy. Here, the steel component of the LRB was treated as an elastoplastic material, the rubber as a hyperelastic material, and the lead as an elastic perfectly plastic model.

*A schematic depicting the cross section of the LRB. Image by R. Sugumar, C.S. Kumar, and T.K. Datta, and taken from their presentation submission*.

For the structure, the research team chose a two-story, single-bay frame comprised of mild steel (i.e., low-carbon steel). In this study, two frames were analyzed: one bare frame and one frame with a base isolation system.

An eigenmode analysis was performed for the structures, and their damping properties were determined based on the first two eigenfrequencies and assuming damping ratios of 0.02. These damping properties were incorporated then into a transient analysis. The force exerted on the structures was derived from a pre-recorded earthquake time history, as illustrated in the plot below.

*Pre-recorded earthquake acceleration. Image by R. Sugumar, C.S. Kumar, and T.K. Datta, and taken from their presentation submission*.

The responses of the bare frame and the base isolation system frame to the applied earthquake acceleration were then compared. The results showed that the presence of the isolator reduced the frame’s response to the vibrations, highlighting the effectiveness of this approach in providing structures with seismic control. Furthermore, researchers noted that modifying the material properties and the dimensions of the isolator could further enhance the LRB’s ability to control vibrations.

*Comparing the response of the bare frame (uncontrolled) and the frame with the base isolation system (controlled). Image by R. Sugumar, C.S. Kumar, and T.K. Datta, and taken from their presentation submission*.

From this research, we can observe the effective nature of base isolation systems using laminated rubber bearings as a means of seismic control for structures. Simulation can help address how different parameters impact the performance of the isolator and advance its ability to stabilize vibrations within buildings. The analyses performed here can also be applied to other types of bearings, emphasizing the extensive capabilities and applications of the nonlinear material models and simulations available in the COMSOL Multiphysics FEA software.

- Access the presentation, poster, and abstract: “Seismic Control of a Structure Using Laminated Rubber Bearings“

A vehicle’s dashboard provides valuable information for a driver, from indicating the speed of the car to gauging its fuel levels. What’s as important as the instruments themselves is the manner in which they are installed. In many cases, fasteners known as snap hooks are used in the design of a car’s control panel, ensuring that the different components are securely fixed.

When inserting a snap hook into its slot, an important consideration is the force that needs to be applied to insert the hook into the slot as well as the force that is required to remove it. With COMSOL Multiphysics FEA software, you can study these forces and the resulting stresses and strains in the hook.

In the Snap Hook model, we leverage the snap hook’s symmetry to analyze only half of its original geometry in an effort to decrease the size of the model. The snap hook is assumed to be composed of an elastoplastic material featuring isotropic hardening and a constant tangent hardening modulus. Meanwhile, the lock is assumed to be rigid when compared to the hook, with the space behind the lock representing the slot into which the hook should lock.

*Model geometry.*

Several boundary conditions are applied to the model, as shown in the schematic below. For boundaries in the symmetry plane, a symmetry boundary condition is used. A fixed boundary condition is implemented for the face of the lock, where it is attached to the remainder of the geometry of the locking mechanism (not modeled here). Lastly, a prescribed boundary condition is applied where the face of the hook meets the rest of the geometry.

*The applied boundary conditions.*

Before insertion into the slot, we first measure the effective stress levels in the hook. As indicated in the following plot, the maximum effective stress levels occur at parameter step 0.84. This parameter represents the point right before the hook enters the slot. Note that when passing over the edge, the hook is bent upwards. The elastic forces will tend to press the hook into the slot before it “fits.” If holding the hook, you would be pushing up until this point; however, here the hook would actually be pulled away from your hand.

*The hook’s effective stress levels prior to entering the slot.*

The next graph depicts the degree of force needed to insert and remove the fastener as a function of the parameter step. When the parameter value varies from 0 to 1, the hook is moved inwards at a constant rate to eventually sit in the slot. Then, between the parameter values 1 and 2, the hook is pulled back out of the slot.

At the parameter value 0.2, the hook first comes into contact with the fixed locking mechanism. The force rises steeply, while the hook tip is forced upwards. In reality, the hook would snap into place after reaching the peak force of around 2.5 N at the parameter value 0.23. Since we control the displacement in the simulation, we can follow the force throughout the entire process. Between the parameter values 0.7 and 0.9, the hook slides down on the back side. The change in the sign of the force indicates that the hook is actually pulled into the slot by a combination of the geometry and the elastic forces.

When trying to pull the hook back out of the slot (at parameter values greater than 1), we must apply a load that is three times higher — about 7.5 N — to remove the hook from the slot (at a parameter value of about 1.12). This is a desirable feature of a hook designed for a locking mechanism.

*Force required for the insertion and removal of the snap hook. The first positive peak in the graph can be attributed to the elastic forces pushing the hook into the slot before it “fits.” Following retraction, the hook hits a steep surface, which results in the second positive peak in the graph. After passing the corner (represented by parameter 1.2), the hook is pressed out by itself, or a negative force.*

Upon its removal from the slot, the hook is shown to have a volume in which there are plastic strains, as illustrated in the plot below. Thus, we can conclude that after inserting the hook into the slot, it is permanently deformed.

*The hook’s effective plastic strain following its removal from the slot.*

In this blog post, we have explored the role of simulation in addressing the forces behind the insertion and removal of a snap hook into a slot. By analyzing such forces, you can enhance the design of snap hooks to ensure that they provide continuous security while also being able to remove them without causing damage to the fastener. This is particularly relevant within the automotive industry in cases where certain parts of a vehicle’s control panel need to be repaired or replaced.

- Model download: Snap Hook
- Related blog post: Why All These Stresses and Strains?

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

The piezoelectric modeling interface seeks to:

- Make the modeling workflow more
- Transparent
- Flexible

- Enable you to debug the models more easily

This will allow you to successfully simulate piezoelectric devices as well as easily extend the simulation by coupling it with any other physics.

You may already be familiar with the three different modules that can be used for simulating piezoelectric materials:

Each of these modules gives you a predefined *Piezoelectric Devices* interface that you can use for modeling systems that include both piezoelectric and other structural materials. The Acoustics Module offers two predefined interfaces, namely the *Acoustic-Piezoelectric Interaction, Frequency Domain* interface and the *Acoustic-Piezoelectric Interaction, Transient* interface. These two allow you to model how piezoelectric acoustic transducers interact with the fluid media surrounding them.

*The *Piezoelectric Devices* interface is available in the list of structural mechanics physics interfaces.*

*The *Acoustic-Piezoelectric Interaction, Frequency Domain *and the* Acoustic-Piezoelectric Interaction, Transient* interfaces are available in the list of acoustics physics interfaces.*

These predefined multiphysics interfaces couple the relevant physics governing equations via constitutive laws or boundary conditions. Thus, they offer a good starting point for setting up more complex multiphysics problems involving piezoelectric materials. The new piezoelectric interfaces in COMSOL Multiphysics version 5.0 provide a transparent workflow to visualize the constituent physics interfaces. There is also a separate Multiphysics node that lists how the constituent physics interfaces are connected to each other.

Let’s find out how these multiphysics interfaces are structured.

Upon selecting the *Piezoelectric Devices* multiphysics interface, you see the constituent physics: *Solid Mechanics* and *Electrostatics*. You also see the *Piezoelectric Effect* branch listed under the Multiphysics node, which controls the connection between *Solid Mechanics* and *Electrostatics*.

*Part of the model tree showing the physics interfaces and multiphysics couplings that appear upon selecting the* Piezoelectric Devices *interface.*

By default, all modeling domains are assumed to be made of piezoelectric material. If that is not the case, you can deselect the non-piezo structural domains from the branch *Solid Mechanics > Piezoelectric Material*. These domains then get automatically assigned to the *Solid Mechanics > Linear Elastic Material* branch. This process ensures that all parts of the geometry are marked as either piezoelectric or non-piezo structural materials and that nothing is accidentally left undefined.

If you are working with other material models that are available with the Nonlinear Structural Materials Module, such as hyperelasticity, you can add that as a branch under *Solid Mechanics* and assign the relevant parts of your modeling geometry to this branch. The Solid Mechanics node gives us full flexibility to set up a model that involves not only piezoelectric material but also linear and nonlinear structural materials. The best part is that if these materials are geometrically touching each other, the COMSOL software will automatically take care of displacement compatibility across them.

If some parts of the model are not solid at all, like an air gap, you can deselect them in the Solid Mechanics node.

From the Solid Mechanics node, you will also assign any sort of mechanical loads and constraints to the model.

The Electrostatics node allows you to group together all the information related to electrical inputs to the model. This would include, for example, any electrical boundary conditions such as voltage and charge sources. By default, any geometric domain that has been assigned to the *Solid Mechanics > Piezoelectric Material* branch also gets assigned to the *Electrostatics > Charge Conservation, Piezoelectric* branch. If you have any other dielectric materials in the model that are not piezoelectric, you could assign them to the *Electrostatics > Charge Conservation* branch.

The *Multiphysics > Piezoelectric Effect* branch ensures that the structural and electrostatics equations are solved in a coupled fashion within the domains that are assigned to the *Solid Mechanics > Piezoelectric Material* (and also the *Electrostatics > Charge Conservation, Piezoelectric*) branch.

The multiphysics coupling is implemented using the well-known coupled constitutive law for piezoelectric materials. Note that the *Electrostatics > Charge Conservation, Piezoelectric* branch is mainly used as a placeholder for assigning geometric domains that belong to the piezoelectric material model. This helps the *Multiphysics > Piezoelectric Effect* branch understand whether a domain assigned to the *Electrostatics* interface is piezoelectric or not.

Note: For an example of working with the

Piezoelectric Devicesinterface, check out the tutorial on modeling a Piezoelectric Shear Actuated Beam.

It is also possible to add effects of damping or other material losses in dynamic simulations. You can do so by adding one or more of the following subnodes under the *Solid Mechanics > Piezoelectric Material* branch:

*Damping and losses that can be added to a piezoelectric material.*

Subnode Name | When to Use the Subnode |
---|---|

Mechanical Damping | Allows you to add purely structural damping. Choose between using Loss Factor (in frequency domain) or Rayleigh damping (for both frequency and time domains) models. |

Coupling Loss | Allows you to add electromechanical coupling loss. Choose between using Loss Factor (for frequency domain) or Rayleigh damping (for both frequency and time domains) models. |

Dielectric Loss | Allows you to add dielectric or polarization loss. Choose between using Loss Factor (for frequency domain) and Dispersion (for both frequency and time domains) models. |

Conduction Loss (Time-Harmonic) | Allows you to add electrical energy dissipation due to electrical resistance in a harmonically vibrating piezoelectric material (for frequency domain only). |

Note: For an example of adding damping to piezoelectric models, check out the tutorial on modeling a Thin Film BAW Composite Resonator.

Additional damping also takes place due to the interaction between a piezoelectric device and its surroundings. This can be modeled in greater details using the Acoustic-Piezoelectric Interaction interfaces.

Upon selecting one of the Acoustic-Piezoelectric Interaction interfaces, you see the constituent physics: *Pressure Acoustics*, *Solid Mechanics* and *Electrostatics*. You also see the *Acoustic-Structure Boundary* and *Piezoelectric Effect* branches listed under the Multiphysics node.

*Part of the model tree showing the physics interfaces and multiphysics couplings that appear when selecting the *Acoustic-Piezoelectric Interaction, Frequency Domain* and the* Acoustic-Piezoelectric Interaction, Transient* interfaces.*

By default, all modeling domains are assigned to the *Pressure Acoustics* interface as well as the *Solid Mechanics > Piezoelectric Material* and* Electrostatics > Charge Conservation, Piezoelectric* branches. Note that the *Pressure Acoustics* interface is designed to simulate acoustic waves propagating in fluid media.

Since COMSOL Multiphysics cannot know a *priori* which parts of the modeling geometry belong to the fluid domain and which ones are solids, you are expected to provide that information by deselecting the solid domains from the *Pressure Acoustics, Frequency Domain* (or *Pressure Acoustics, Transient*) branch and deselecting the fluid domains from the *Solid Mechanics* and *Electrostatics* branches.

Once you do that, the boundaries at the interface between the solid and fluid domains are detected and assigned to the *Multiphysics > Acoustic-Structure Boundary* branch. This branch controls the coupling between the *Pressure Acoustics* and *Solid Mechanics* physics interfaces. It does so by considering the acoustic pressure of the fluid to be acting as a mechanical load on the solid surfaces, while the component of the acceleration vector that is normal (perpendicular) to the same surfaces acts as a sound source that produces pressure waves in the fluid.

Note: For an example of Acoustic-Piezoelectric Interaction, check out the tutorial on modeling a Tonpilz Transducer.

The transparency in the workflow as discussed above also paves the way for adding more physics and creating your own multiphysics couplings.

For example, let’s say there is some heat source within your piezoelectric device that produces nonuniform temperature distribution within the device. In order to model this, you can add another physics interface called *Heat Transfer in Solids* in the model tree and prescribe appropriate heat sources and sinks to find out the temperature profile. You could then add a *Thermal Expansion* branch under the Multiphysics node to compute additional strains in different parts of the device as a result of the temperature variation.

The *Multiphysics > Thermal Expansion* branch couples the *Heat Transfer in Solids* and the *Solid Mechanics* interfaces. It might also be possible that the piezoelectric material properties have a temperature dependency. You could represent these properties as functions of temperature and let the *Multiphysics > Temperature Coupling* branch pass on the information related to temperature distribution in the modeling geometry to the *Solid Mechanics* or even the *Electrostatics* branches, thereby producing additional multiphysics couplings.

*Part of the model tree showing the physics interfaces and multiphysics couplings that you can use to combine piezoelectric modeling with thermal expansion and temperature-dependent material properties.*

Similar to adding more physics and multiphysics couplings, it is also possible to disable one or more multiphysics couplings — or even any of the physics interfaces shown in the model tree. This could be immensely helpful for debugging large and complex models.

*The model tree on the left shows a scenario where the Piezoelectric Effect multiphysics coupling is disabled. The model tree on the right shows a scenario where the* Electrostatics* physics interface is disabled.*

For example, you can disable the *Multiphysics > Piezoelectric Effect* branch and solve for the *Solid Mechanics* and *Electrostatics* physics interfaces in an uncoupled sense. You could also solve a model by disabling either the *Solid Mechanics* or the *Electrostatics* interface.

Running such case studies could help in evaluating how the device would respond to certain inputs if there were no piezoelectric material in place. This approach could also be used to evaluate equivalent structural stiffness or equivalent capacitance of the piezoelectric material.

You could also start by adding only one of the constituent physics, say *Solid Mechanics*, and after performing some initial structural analysis, go ahead and add the *Electrostatics* physics interface to the model tree once you are ready to add the effect of a piezoelectric material.

In that case, when you add the *Electrostatics* physics on top of the existing *Solid Mechanics* physics in the model tree, the COMSOL software will automatically add the Multiphysics node. From there, you can manually add the *Piezoelectric Effect* branch. Note that if you take this approach of adding the constituent physics interfaces and multiphysics effect manually, you would also have to manually add the piezoelectric modeling domains to the *Solid Mechanics > Piezoelectric Material*, the *Electrostatics > Charge Conservation, Piezoelectric*, and the *Multiphysics > Piezoelectric Effect* branches.

In a similar fashion, you can continue to add more physics interfaces and multiphysics couplings to your model based on your needs.

To learn more about modeling piezoelectric devices in the COMSOL software environment, you are encouraged to refer to these resources:

- Piezoelectric Features Overview
- Acoustics Module User’s Guide
- MEMS Module User’s Guide
- Structural Mechanics Module User’s Guide

To begin, I would like to highlight several changes in the Linear Elastic Material model of the *Membrane* interface.

First off, the previous version of the interface always assumed geometric nonlinearity. The new version listens to the “Include geometric nonlinearity” setting in the study step settings in the same way as the *Solid Mechanics* interface. The geometric linear version of the membrane can be used when it is acting as cladding on a solid surface. If the membrane is used by itself and not as a cladding, a tensile prestress is, as before, necessary in order to avoid singularity. This is because a membrane without stress or with a compressive stress has no transverse stiffness. To include the prestress effect, you must enable geometric nonlinearity for the study step.

Another update is that linear elastic materials can now also be orthotropic or anisotropic. This affects the settings of the Damping subnode as well, where non-isotropic loss factors are now allowed.

You may also notice that we have added a Hygroscopic Swelling feature as a subnode to the Linear Elastic Material node. (We described the hygroscopic swelling effect in a previous blog post. Check that out to learn more about the effect.)

All of you who use the Nonlinear Structural Materials Module may now use the *Membrane* interface to model thin hyperelastic structures by adding a Hyperelastic Material node. In order to illustrate the Hyperelastic Material model using the *Membrane* interface, we have recreated the Model Library example Inflation of a Spherical Rubber Balloon.

Tip: You can download the new version of the model in the Model Gallery by logging into your COMSOL Access account.

The *Membrane* interface works on the plane stress assumption, and it is assumed that there is no variation across the thickness of the balloon. Also, it requires a prestress to solve the model due to the absence of bending stiffness in the membrane. For this purpose, a separate study has been created before the inflation of the balloon is carried out in further studies. Results from this analysis are used as initial values for the rest of the inflation analyses. Aside from these two changes, the model is similar to the previous Solid Mechanics version.

The advantage of the Membrane version is that it is more computationally efficient. Why is that? Because the *Membrane* interface is on one geometric entity lower than the *Solid Mechanics* interface. The results obtained from the *Membrane* interface are in agreement with the analytical results. The plot below shows the inflation pressure as a function of circumferential stretch for different hyperelastic material models compared to the analytical expression for the Ogden model.

As the internal pressure increases, the balloon starts to inflate and its thickness decreases. Since the pressure is uniform over the surface, the thickness is the same along the cross section for any given inflation pressure. The next plot compares the variation of deformed thickness with applied stretch to the balloon obtained from the *Membrane* interface and the *Solid Mechanics* interface. We see that the thinning of the balloon is accurately captured by the *Membrane* interface.

We have added four new feature nodes to the *Membrane* interface.

They are as follows:

*Prescribed Velocity*— Available at the domain and boundary level*Prescribed Acceleration*— Available at the domain and boundary level*Symmetry*and*Antisymmetry*— Both available at the boundary levels

In addition to the specific improvements I just went over, we have made a few general changes to the structural mechanics interfaces that affect the *Membrane* interface. You will notice that the menus have been restructured for a number of structural mechanics interfaces.

The following interfaces now have restructured menus:

*Solid Mechanics**Shell**Plate**Membrane**Beam**Truss*

You can see a screenshot of the menu structure for the *2D Axisymmetric Membrane* interface below:

As for the Spring Foundation features, we have generalized these so that you can enter the “spring force versus displacement” and the “damping force versus velocity” relations in matrix form, rather than by component.

For 2D Axisymmetric cases, there is a new load type called “Point Load (on Axis)”. With this option, it is now possible to apply loads on a point on a symmetry axis.

For 2D Axisymmetric cases, a Point Load is actually a line load (N/m) since a point represents a ring in axisymmetry. To follow better naming conventions, such a load is now called “Ring Load” in both the *Solid Mechanics* interface and the *Membrane* interface.

Models that were made with COMSOL Multiphysics version 4.4 or earlier still use the old *Membrane* interface and new functionality is not available. To utilize the new functionality for old models, we suggest that you add a new *Membrane* interface and copy all the nodes from the previous interface to the new one.

As always, do not hesitate to contact us if you have any questions.

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Residual stresses are self-equilibrating stresses that remain after performing the unloading of an elastic-plastic structure. During the manufacturing process of a mechanical part, residual stresses will be introduced. These will influence the part’s fatigue, failure, and even corrosion behaviors.

Indeed, uncontrolled residual stresses may cause a structure to fail prematurely. Although residual stresses may alter the performance, or even lead to the failure of manufactured products, some applications actually rely on them. For instance, brittle materials, such as glass in smartphone screens, are often manufactured so that compressive residual stresses are induced on the surface to avoid crack-tip propagation.

For these reasons, residual stresses play an important role in mechanical projects as a whole. Only through qualitative and quantitative analysis of these stresses is it possible to determine the most suitable machining processes for a given application. These types of analyses also help you discover the optimal amount of material to be used for their reliability or the most suitable shape that needs to be designed, in order to avoid malfunctions and failures.

Let’s consider the following slender beam with a rectangular cross section, depth a, and width b. The beam is fixed at the left-hand side and a bending moment is applied on the free end.

Based on the beam theory, it turns out that the bending moment is constant in this case and the stress can be written as:

(1)

\sigma_x=\frac{M_\mathrm{b}}{I_z}y

where I_z is the moment of inertia about the *z*-axis.

As M_\mathrm{b} increases, the beam first behaves in an elastic manner, but after reaching its yield moment, M_y, it begins to take on plastic behavior. This leads to an elastic-plastic cross section. Once the plastic zone has propagated through the entire cross section, the ultimate bending moment, M_\mathrm{ult}, that the beam can carry is determined. Here, it is assumed that the beam will collapse at such a moment and that it has a perfectly plastic behavior.

The outer fibers of the beam will reach the yield point first, while the core fibers remain elastic. Thus, the previous equation applied to the outer fibers of the beam provides the first yielding moment:

(2)

M_y=\frac{\sigma_\mathrm{yield} I_z}{b/2}=\frac{\sigma_\mathrm{yield} ab^3/12}{b/2}=\frac{ab^2\sigma_\mathrm{yield}}{6}

where \sigma_\mathrm{yield} is the yield stress.

Under an elastic-plastic moment, M_\mathrm{ep} < M_\mathrm{ult}, the plastic zone propagates through the thickness by a distance of h_\mathrm{p} at each side of the beam, as shown below.

*Plastic zone penetration in a rectangular cross-section beam.*

The total moment can be divided into an elastic part, M_e, and a plastic part, M_p, such that:

(3)

M_\mathrm{ep}=M_\mathrm{e}+M_\mathrm{p}=\frac{2\sigma_\mathrm{yield}I_\mathrm{e}}{b-2h_\mathrm{p}}+\sigma_\mathrm{yield}ah_\mathrm{p}(b-h_\mathrm{p})

where I_\mathrm{e}=\frac{a(b-2h_\mathrm{p})^3}{12} is the elastic core moment of inertia along the *z*-axis.

Combining the last two expressions, we get the following:

(4)

M_\mathrm{ep}=\frac{ab^2\sigma_\mathrm{yield}}{6}\left[1+\frac{2h_p}{b}\left(1-\frac{h_p}{b}\right)\right]

When an elastic-perfectly plastic beam is unloading from M_\mathrm{ep}, a state of residual stress, \sigma_r, remains in the beam cross section. The beam attempts to recover its initial shape following recovery of elastic bending stress, \sigma_\mathrm{e}. Here, it is assumed that purely elastic unloading occurs after being loaded at M_\mathrm{ep}, corresponding to a state of elastic-plastic stress, \sigma. The residual stresses can be computed from the difference between the elastic-plastic stress and the purely elastic stress — i.e., the stress you would have if plastic behavior was not involved.

(5)

The elastic bending theory gives the recovered elastic stress as:

(6)

\sigma_\mathrm{e}=\frac{M_\mathrm{tot}y}{I_z}=\frac{2\sigma_\mathrm{yield}}{b}\left[1+\frac{2h_\mathrm{p}}{b}\left(1-\frac{h_\mathrm{p}}{b}\right)\right]y

Assuming a perfectly plastic behavior, the stress \sigma in the plastic zone (in other terms, \frac{b}{2}-h_\mathrm{p} \le |y| \le \frac{b}{2}) remains constant and equal to \sigma_\mathrm{yield}. Therefore, according to the Equation (5), the residual stresses can be written as:

(7)

\sigma_\mathrm{r}=\sigma_\mathrm{yield}-\frac{2\sigma_\mathrm{yield}}{b}\left[1+\frac{2h_\mathrm{p}}{b}\left(1-\frac{h_\mathrm{p}}{b}\right)\right]y

In the elastic zone (in other terms, 0 \le |y| \le \frac{b}{2}-h_\mathrm{p}), the beam theory provides the applied stress as:

(8)

\sigma_\mathrm{e}=\frac{M_\mathrm{e}y}{I_\mathrm{e}}=\frac{2y\sigma_\mathrm{yield}}{b-2h_\mathrm{p}}

Therefore, the residual stress is then deduced as:

(9)

\sigma_\mathrm{r}=\sigma_\mathrm{yield}\left[\frac{2}{b-2h_\mathrm{p}}-\frac{2}{b}\left[1+\frac{2h_\mathrm{p}}{b}\left(1-\frac{h_\mathrm{p}}{b}\right)\right]\right]y

Note that after the external moment has been removed, the beam will still have some permanent displacement due to plastic deformation, but it will also have recovered some of the displacement that was present at the peak load. This *springback* effect is important when you want to achieve a controlled plastic deformation.

When modeling the beam in 2D, we could choose a *plane stress* assumption taking Poisson’s ratio, \nu=0, to match with the 1D beam theory, which does not account for the Poisson effect. In COMSOL Multiphysics, you can model 2D plane stress by selecting a 2D space dimension and choosing the *Solid Mechanics* interface.

Here, we will show how to use the *Solid Mechanics* interface in 2D to compute the residual stresses in the beam cross section.

*A snapshot of the 2D beam model using the* Solid Mechanics *interface.*

According to the snapshot above, we define variables to evaluate the theoretical residual stresses we worked out in the section above. Those values will be used to compare the computed results with the theoretical ones.

The applied bending moment is ramped progressively. A Plasticity node is added to account for the uniaxial plastic behavior that may occur through the beam thickness. Plastic flow begins once \sigma_x reaches the critical value \sigma_\mathrm{yield}. Any fiber that has reached this value will remain at a constant state of stress during loading.

In the graph below, you can see the stress distribution along the *Y*-axis of the cross section. The applied bending moment has been computed from Equation (4) for a plastic zone with depth h_\mathrm{p}=\frac{b}{4}=0.01 \ \mathrm{m}. According to the blue curve, COMSOL Multiphysics results match perfectly with this value. The red curve represents the residual stresses after one loading-unloading cycle. It is worth noting that the residual stresses obtained may also be found by subtracting the elastic curve (green) from the elastic-perfectly plastic curve (blue).

*Stress value after elastic-plastic loading, elastic loading, and unloading.*

Equations (7) and (9) have been defined as variables and compared to the solution computed in COMSOL Multiphysics. As shown in the previous screenshot, you can create a “switch” using the if() operator, so that the two expressions representing the analytical residual stresses are gathered together in one expression. The next graph shows both analytical and computed residual stresses after two loading-unloading cycles.

*Analytical vs. computed residual stresses.*

COMSOL Multiphysics enables you to model the hysteresis cycle of a given material. In the case of perfectly plastic behavior, as depicted below, the second load cycle already provides a stable stress-strain response that is representative of each consecutive load cycle. For instance, you can use these load cycles to carry out a fatigue analysis.

*Hysteresis behavior after three loading-unloading cycles.*

Last but not least, let’s find out how strain-hardening behavior influences residual stresses and loading-unloading cycles. So far, we have been dealing with a perfectly plastic material. The yield stress remains constant, no matter the number of cycles or whether a tensile or a compressive is applied. Equation (5) is only valid as long as reverse yielding does not occur. Since reverse plastic deformation during unloading has a negative effect on the performance, it is quite important to figure out under which condition reverse yielding is likely to occur.

A ductile material that is subjected to an increasing stress in one direction (in tension, for instance) and then unloaded, will behave differently when loaded in the reverse direction. It is found that the *compressive* yield stress is now lower than that measured in *tension*. This is called the *Bauschinger effect*. Similarly, an initial compression provides a lowered tensile yield stress. The figure below displays this effect over two stress cycles:

*Hysteresis behavior with kinematic strain hardening.*

Now, let’s move on to a more sophisticated mechanical process in which residual stresses are of great importance: the sheet metal forming process.

Die forming is a widespread sheet metal forming manufacturing process. The workpiece, usually a metal sheet, is permanently reshaped around a die through plastic deformation by forming and drawing processes. A blankholder applies pressure to the blank, leading the metal sheet to flow against the die.

In order to avoid cracks, tears, wrinkles, and too much thinning and stretching, you can turn to simulations. They can also be useful to estimate and overcome the *springback phenomenon*. This refers to how the workpiece will attempt to recover its initial shape once the forming process is done and the forming tools are removed. Springback can lead the formed blank to reach an unexpected state of warping. To cope with this effect, the sheet can be over-bent. Thus, the die, punch, and blankholder must be manufactured not only to match the actual shape of the object, but also to allow for springback.

In this study, the sheet is made of aluminum. A Hill’s orthotropic elastoplastic material model with isotropic hardening is used to characterize the plastic deformation. It has been observed that metal sheets in deep drawing process no longer behave isotropically. There tends to be less plastic deformation through the thickness. Therefore, in die forming and deep drawing of sheets, we need a kind of anisotropy where the sheet is isotropic in-plane and has an increased strength in the perpendicular direction, called *transverse isotropy*.

Below, we have illustrated the forming tools that are used in the process.

*Forming tools: The die is shown in red, punch in blue, blankholder in pink, and the blank in gray.*

As mentioned above, simulations can allow for handling several tasks that need to be taken into account whenever such a mechanical process is worked out. For instance, optimization of the corner radius of both the die and the punch can be carried out properly to prevent tearing of the metal sheet. It may also be useful to carry out simulations in order to get the clearance that is needed between the punch and the die, to avoid shearing or cutting of the metal blank.

One of the most challenging aspects is to figure out how much of the metal sheet should be over-bent. When the sheet has been formed, the residual stresses cause the material to spring back towards its initial position, so the sheet must be over-bent to achieve the desired bend angle. Therefore, you have to properly model residual stresses as not to over- or underestimate the springback phenomenon.

The two animations below show the sheet metal forming as well as the springback of the metal blank.

*Representation in the* RZ*-plane of the spingback phenomena.*

*Simulation of sheet metal forming.*

When subjecting the structure to other mechanical loads, the superposition of the residual stresses can reduce the reliability of the structure or even cause irreversible damages. Therefore, the residual stresses must be released as much as possible or be managed so that the structure can withstand the external loads that may be applied. The plot below shows the Hill effective residual stresses that remain around the bend regions after the deep-drawn cup process.

Today, we studied residual stresses in structural mechanics. We introduced a conventional definition, which was first applied to a bending beam example. We simulated this bending example using COMSOL Multiphysics and compared our results to the analytical solution from the beam theory. Then, we explored the importance of the residual stresses in a sheet metal forming example. We saw that any mechanical process induces residual stresses and particular care must be given to release them properly or, at least, be certain that they will not cause any damage.

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