Modeling Hyperelastic Material

Hi, welcome. Welcome to everyone. I am Alessandro Massafini and I work in the Numerics group. And today I'm going to talk about hyper elasticity and how to model hyper elastic material. Hyper elasticity represents a wet clasp of material which can undergo to extremely large deformation and still remain fully fully elastic. Everything that I'm going to show you today is taken from the Hyper elasticity and solid mechanics monograph that you can find in the in the documentation. Today I'm going to briefly introduce the hyper elasticity and how to model hyper elastic material. I want to model some mechanical problem and also the new features that we introduced for version 14 of Mathematica. Then I want also to show you some simple example that show you how to implement this solid mechanical problem. And then I want to show how we can use hyper elasticity to model some multiphysical problem. First of all, in the find of solid mechanics and then of the hyper elasticity we need to describe the displacement and the deformation of material. So the determinant variable that we need is the displacement. We can describe displacement in both one in one, two or three dimension. To remain as general as possible, we can express this in three dimension with three different variable U, V and W, each of one depend on the independent variable X, Y and Z. Of course we can have time dependent problem, but today we are going only to talk about stationary problem. And of course we need some material parameters to represent different material. We can now see two different configurations. The reference one expressed is Omega 0 with the local coordinate big X and the deformation configuration of our current configuration is omega and is function of the displacement map. And to map the surrounding of the big X to the small X in the current configuration we need the deformation gradient that is mathematically expressed as the identity matrix plus the gradient of the displacement. To represent the physical phenomenal behind the deformation of the body we need some equation, of course the Mechanical equilibrium equation, I.e. a partial differential equation. And now we can express this in both the reference of the or in the current configuration. They are two different equations but representing the same physical behavior and the same phenomenon. But we need to map the coordinate between one configuration and another depending on the deformation of the body. And for the same reason we can now have different stress measure like the period Kirchhoff 1 or the Cauchy stress, but they are both the stress inside the material. But we can see different expression of the stress in the reference configuration or in the current configuration and they are linked from a coordinate mapping. Then we also need some boundary condition in order to Close the problem and be able to solve. And of course we need a material behavior which represents the relation between the deformation of the material and the stress in the material. And this is where the epialasticity comes out. In the epriel assisted framework we are able to describe the material as depending on a potential which represents the strain energy of the material, so the energy that the material needs to deform. And we can express the stress as the derivative of the strain energy with respect to a strain measure like the deformation gradient. For instance, in this chart we can now see a very nonlinear relation that link the uniaxial stress and uniaxial stretch. This is a uniaxial tensile test example. And this sample is stretch like three times the original size. And we can see this nonlinear relation that show different behavior by increasing the stretch. And this is an example taken from the EO model. So, in the version 14 there are some news about hyper elasticity. We introduce different introduce several isotropic material models with the possibility to model learning compressible behavior. And also for the bidimensional cases, the plane stress and the plane strain model form. Then we also introduce the transverse isotropic materials with the fiber reinforced example and the possibility to model the different material load to experimental data of tensile or compression test. Then we also introduced the possibility to use the multiplicative decomposition of the deformation gradient to represent different typical phenomena. And we described this new application example that show you how to start and how to implement also complex deformation. Everything is inside the documentation, mostly in the hyperelasticity monograph and in the linked page. Now I'm going to talk a little better about some of this point. So we introduce different material model, some phenological model like the yeo, the moonlight rearing to five or nine parameters. So mechanic is mechanical, air mechanistic model like the Newcan or the Aruda Boys and some hybrid model like the gan. They are based on different strain energy, of course, with different material parameters which correspond to very different behavior. For instance, this chart is extracted from the hyper elastic model comparison example that show you how to calibrate your different material law on experimental data. And we can also see how these different models show very different nonlinear behaviors outside the fitting range. We also introduce some reinforced material and these materials are described by splitting the strain energy into an isotropic part which represents the bulk or the isotropic matrix of the materials and is described by some of the strain energy that I introduced before. And then we have this contribution which represents the fiber and depends on the fiber orientation and on the fiber stiffness. For instance, we can see this plot by fixing the fiber stiffness and varying the fiber direction. We can observe very different behavior of this unit square that is fixed to the left pull and to the right. By varying the orientation, we can see very different deformation. Then we introduce the kinematic decomposition that allow us to split the deformation gradient into an elastic contribution which represents the hyperelastic material model deformation. And then we can also include some inelastic strain within the inelastic deformation gradient that we can use to model several phenomena like thermal expansions or plasticity or some growth of soft tissues. I'm going to talk a little better about that in the in the following slide, but I want first to introduce an example. We can start with a very simple example of this rectangle of size one by two. And this is a fixed to the to the left and pull it to the right. Of course, to describe this problem in the solving mechanic world, we need the displacement variable U along X and V along there Y direction, both depending on the special coordinates X and Y. Then we want to model a new kernel material that is based on this law. And to model this material we only need to specify inside the parameters variable the solid mechanic material model association within the new kind isotropic name. And of course the material parameter that for the new current model is the Hessian models. Then we can define the domain Omega. It is basically a rectangle with size two by one. And then we can include some boundary conditions. We need to express the load that is a normal boundary condition. And to do that we can use this the built in function solid boundary load value. Let me apply this function where on the right side, when where x is equal to 2 and we specify this is a like a function as a function that take F as an input the force and uses this input to express F Newton as a force along the first coordinate, which correspond to the X to the horizontal direction. Then we with the with solid fixed condition, we can express the fixed left side, where X is equal to zero. Then we can express the model. To express the model we need to write the equilibrium equation. And we can use the built in function solid mechanics PD components that allow us to express them the equilibrium equation. And it's very useful to use this built in function, because manually they equilibrium equation could be very very long and complex. Because now we are representing very nonlinear behavior. So we can use this built in function to express in a very compact form this model which take F as the input and use this input to express the boundary load. Then we have now defined the model and we can use this to to solve the problem. We only need to describe the force and uses a force to express the model. Now, and we can use this ndisolve value to solve the problem. And then we can observe this displacement field that now it's very large displacement, which is more or less three times the original size of the rectangle. And of course, we can also increase the external load from 12 kilonewton to 20 kn, and solve with the same procedure the problem. And we can see a displacement, a larger displacement. Now that is around six times the original size. And we can also try to increase understand the load from 20 to 40kn. But this time we can observe. We observe an error with the solution, because now we are trying to apply a very, very large deformation, which leads to the model far from the convergence. So, to solve this problem, we can use the parametric solve and express. Now we can express the force with respect to a parameter P that represent a scaling factor for the load. And we can iteratively solve this problem by gradually increasing the load from 0, the scaling factor from 0 to 1, so the force from 0 to the maximum force. And for each solving step, we can use the solution of the previous step as an initial seeding for the actual solution. And with that we can solve also for this very, very Large displacement of 13 times the original size. Starting from this simple model, we can introduce several other behaviors. For instance, we can introduce some fibers to reinforce the materials. And to do that we only need to specify the fibers direction. We start from the same new kinesotropic description for the bulk contribution, for the stropic contribution. And then we add some fiber. And to add the fibers, we only have to specify the reinforcement stiffen factor, which link to a fiber stiffness to K times the stiffness of the isotropic materials. And then we need to specify the direction of the fiber. With this specification inside the parameters variable, we can follow the same identical procedure to specify the model in function of the boundary force, and then specify the force and solve the model. And now we can see a very different displacement. This is the same example fixed to the left. Pull it to the right. And then now we can see a smaller displacement and a very different and non isotropic displacement. Starting from the same example, we can now try also to use and couple this problem with the different physical phenomena. We can use the kinematic decomposition to split the deformation gradient into the elastic contribution, that is the same deformation that I described in the previous slides. And then add some an elastic contribution that we can use to describe several inelastic deformation like the thermal expansion, the hygroscopic swelling, the plasticity problem or some growth of biological tissue. For instance, thermoelasticity. Link the hyper elastic material with the deformation induced by temperature, which that we can represent inside the thermal inelastic deformation gradient defined as the identity matrix plus the thermal strain. Thermal strain is defined as the difference between the local temperature and the reference temperature scaled with the thermal expansion coefficient alpha. We can also use this thermal PDE models thermal strains built in function to express this thermal strain which represent the deformation induced by the temperature. We know that material expand when temperature is higher than the referential values and with the opposite side situation when temperature is below the reference temperature. So we can set up the same rectangular example with the new kinestropic model. And we only need to specify inside the inelastic deformation gradient parameters and link these parameters to the determining inelastic deformation gradient that I defined in the previous slide. So we need also to specify other parameters to use the built in function for the thermal strains. In particular, we need to specify the temperature that is a t function of X and Y, the reference temperature 25 degrees and the thermal expressions. That basically is a number, but it could be also a function. We can, and it is done in the monograph, link this problem with the heat equation and solve a full couplet example. But for now, in this presentation I want to manually define the temperature with like a simple a simple file which vary only on the vertical direction from 10 degrees in the bottom side to 100 degrees on the top side. This is the temperature file. And by apply this temperature we can set up the same procedure this model. Now the applied force is zero, but we still need the parametric solver to gradually apply the thermal load. So we can solve this problem with this parametric solver. And in the end we can observe a very different displacement field. Now, beside the force being at zero. Now we can see this deformation and then expansions to the top side and this contraction on the bottom side, which lead to a general bending of this rectangular circle. We can of course use this inelastic contribution within different physical phenomena. And it is described in the monograph with several other examples. As I mentioned before, everything that I show you today is is inside the hybrid elasticity monograph. I really suggest you to have a look to this to both the heel elasticity and the solid mechanic reference page to have a better idea on how to model this solid mechanical problem. I want also to thanks Oliver which drive our numerics group and Martin. He isn't here to present, but he had done a lot of work on the fiber and forced and the transverse isotopic materials. My presentation is ending now and feel free to ask everything you want to ask about hyper elasticity and solving mechanical problem. Hi. Yes, Harald. So the current limitations are mostly about the model size, number of number of elements and it's not a real limitation. But of course for very complex problem due to the ergonominearity you need to iteratively solve the problem with a very small step. It will of course requires a very long, very long solving time. Other limitation could be to the multi physical coupling because we need to a different load to couple with different phenomena. And now we can mostly model the physical interaction that I mentioned before. But of course this is in development for include different other phenomena and to include different field of physics within this solid mechanical deformation. To search for the hyper elasticity monograph you can search in the documentation for the PD PDE models monograph and then this is linked to the epiderasys in the monograph. Or you can search into the Wolf from mathematical documentation for directory for ever mechanical monograph you will of course find the documentation page that I mentioned.