A micropolar model accounting for asymmetric behavior of cold‐box sand in relation to tensile and compression tests

Cold‐box sand (CBS) belongs to the granular materials and consists of sand and a binder. The behavior of CBS is simulated with a micropolar model, whereby the additional degree of freedom of the model describes the rotation of the sand grains. The model is used to generate a shear band under pressure for three different meshes, where the force‐displacement curves of the three meshes converge so that no mesh dependence occurs. Another requirement of the model is the consideration of asymmetric behavior for compression and tension. Due to the additional degree of freedom the implicit implementation of the micropolar continuum is very time‐consuming. Therefore, an explicit implementation is considered as an alternative possibility. This paper compares the advantages and disadvantages of both methods and the results for both calculations.


EXPERIMENTS
The CBS was produced by the Chair of Foundry Engineering at RWTH Aachen, where a core box, as depicted in Figure 1A, was constructed.Into this core box a liquid sand-binder mixture was injected and then cured.In Figure 1B the finished specimen is illustrated.To investigate the material behavior of CBS, the specimen was subjected to pressure.For this purpose, a 10 kN MTS machine with an optical measuring unit consisting of two GOM systems (with a GOM system, it is possible to visualise variables such as displacements over the recorded area for different points in time) was used.The setup for the compression test is illustrated in Figure 1C.The contour plots of the compression test in the first five subfigures in Figure 2A show a branching of localized strains.In stage 6 in Figure 2A one can see a diagonal and a horizontal fracture of the specimen.The last stage is shown in more detail in Figure 2B, where the fracture concentrates on the diagonal surface.For the FE-simulation, the complex strain localization in stage 5 of localized strains in the experimental results are not axially symmetrical.Therefore, a 3D FE-simulation is indispensable to map the shear band.
In the force-displacement curve of the compression test in Figure 3A an elastic region can be seen first.It is followed by a short hardening phase and at the end by softening.In ref. [1] a tensile test was carried out for the same material.The force-displacement curve can be seen in Figure 3B.This curve also consists of an elastic, a hardening and a softening part, whereby the maximum force of approximately 1250 N is significantly lower than that of the compression test with approximately 3250 N. Accordingly, the CBS under investigation shows an asymmetrical behavior for compression and tension.
Based on the results, three requirements are made that must be fulfilled for constitutive modeling.Firstly, the model must exhibit asymmetric behavior for compression and tension.Secondly, mesh dependence should be avoided and thirdly the model should develop a shear band by softening.

General framework
For constitutive modeling the micropolar continuum is used.For this purpose, the constitutive equations are introduced in the form of a general framework in ref. [2].First, the kinematics are defined.Besides the linear strain tensor , there is another strain measure called .The tensors can be divided additively into elastic and plastic parts as Furthermore, a free energy function, which depends on the elastic parts of the two strain measures   and   as well as the equivalent plastic strain   , is introduced in the following equation as Ψ = Ψ(  ,   ,   ). ( The stress tensor  is known to be defined as the derivative of the energy function Ψ with respect to   .Similarly, the stress tensor  is the derivative of the energy function Ψ according to   and the hardening/softening function  according to   .With the density  we obtain The yield-function  as well as the plastic-potential  * are dependent on the previously introduced quantities ,  and  and become Next, the evolution equations are determined with the plastic potential as where λ describes the plastic multiplier.Then, using the known evolution equations, the dissipation equation  is established as In addition, certain conditions must be met.On the one hand, the so-called Kuhn-Tucker conditions and on the other hand the consistency condition must be fullfilled.Furthermore, initial conditions are determined for a certain starting point as

A prototype model
To fully describe the micropolar model, the equations are now specified.The asymmetric behavior for pressure and tension should be taken into account in our model.For this reason, the plasticity equations must be specified in order to activate the first invariant, as already done in ref. [4].Accordingly, we first specify the yield function as where The term   consists of the deviatoric symmetrical stress tensor    and the deviatoric skew symmetrical stress tensor    .Furthermore, the static yield point  0 , the parameters  1 and  2 as well as the first invariant  1 of the stress tensor  are introduced in term .Analogously to the yield function, the plastic potential is defined as Since in this case there is a yield function and a plastic potential, this approach is called single surface plasticity.Both functions contain the regularization parameter   , which controls mesh dependence.The parameters  1 ,  2 ,  * 1 and  * 2 are used to activate the first invariant  1 and thus to take the asymmetric behavior into account.
However, there is also a difficulty with single surface plasticity when a high equivalent plastic strain   arises.In the micropolar model, there are two dissipative mechanisms.One is the plasticity of the tensor   and the other is the rotation of the sand particles   .The problem here is that both   and   contribute to   .The idea to solve this problem is to introduce two yield-functions and two equivalent plastic strains.Then, only   contributes to   and   contributes to the additional equivalent plastic strain   .Accordingly, there are two hardening functions and two yield-functions Due to the two yield functions, this approach is called double surface plasticity.The next step is to derive the Prandtl-Reuss tensor for single surface plasticity.First, we use the consistency condition to determine the plastic multiplier λ: The evolution equations of the two stress tensors  and  are calculated according to For the determination of the Prandtl-Reuss tensor , λ is inserted into the two stress tensors and after a few rearrangements of Equation ( 16) the result is The Prandtl-Reuss tensor for double surface plasticity is derived using the same procedure.The difference here is that there are two yield-functions.From the consistency conditions it follows The stress tensors remain unchanged, while the Prandtl-Reuss tensor has certain differences compared to the single surface plasticity and is determined as

NUMERICAL IMPLEMENTATION
There are several possibilities for the numerical implementation.A distinction is made between the two approaches single surface and double surface plasticity.Within these two approaches we can choose between an implicit and an explicit implementation.The advantage of the implicit radial return method is that large time steps are possible.But there are also some challenges that have to be overcome.These include the consideration of the first invariant, the determination of the consistent tangent and convergence problems in softening regions, which leads to the need to calculate with the smallest possible time steps.For the double surface plasticity method there is another challenge.Due to the existence of two flow functions, it must be found out which of the two functions is active or whether both or neither are active.The active-set method is used for this.The advantages of the explicit Prandtl-Reuss method are obvious.On the one hand, no iteration is necessary and on the other hand, the Prandtl-Reuss tensor is equal to the consistent tangent, which saves a lot of work.The challenge of all explicit methods is to achieve stability.To achieve this, small time steps are necessary.

NUMERICAL EXAMPLES
In this section, two examples are calculated using the single surface plasticity approach.The first example is the comparison of the explicit and implicit algorithm for one element.The element is a cube with an edge length of 1 mm and this cube is compressed uniaxially by a displacement  = 0.15 mm.In both simulations, 500 time steps are used.In Figure 4 the stress-displacement curves of the two algorithms are illustrated.It can be seen that no stability problems occur for the explicit method.When one compares the two curves, one can see that there are no serious differences between implicit and explicit implementation.
The second example is an FE-simulation of a cylinder under compression and tension.In Figure 5A and Figure 5B we see the geometry and boundary conditions, where  = 17.5 mm and  = 1.5 mm.In this example, three different meshes (128, 896 and 2880 elements) with hexaeder elements are simulated.The medium mesh with 896 elements is illustrated in Figure 5C.
At the beginning of this paper, three requirements were defined for the model.One of them is the consideration of the first invariant in order to be able to represent the asymmetric behavior under compression and tension.In Figure 6 the force-displacement curves for compression and tension are shown using the coarse mesh.It is obvious that by activating the first invariant, the asymmetric behavior is taken into account.
The second requirement for constitutive modeling is the avoidance of mesh dependence.For this purpose, the cylinder is simulated under compressive load for three different meshes.First, the calculation is carried out without regularization, as shown in Figure 7A.Due to convergence problems in the softening area, the two finer meshes could not be calculated completely.Nevertheless, it can be seen at the beginning of the softening that the three curves have different courses for the different meshes, whereby one can conclude that there is a mesh dependence.When the algorithm is simulated with regularization, that is the regularization parameter   (here   = 1.6 mm) is activated, the three curves at the beginning of the softening run almost identically, see Figure 7B.This means that there is no longer a mesh dependence.The last requirement is the development of a shear band.In Figure 8 the contourplots for the cylinder under compression are shown for all three meshes (with regularization), and a diagonal shear band can be seen in each case.The shear band geometry as well as the equivalent strain intensity are in accordance with the experimental results that are shown in Figure 2A.It is also obvious that there is no axial symmetry.Thus, a 2D simulation is not sufficient, which makes a 3D simulation inevitable.

CONCLUSION AND OUTLOOK
In summary, CBS shows asymmetric behavior for compression and tension.In addition, a shear band is formed under pressure.For these reasons, the micropolar continuum is ideal as a material model.For the numerical implementation, a choice is made between single surface and double surface plasticity.For both approaches, there are the possibilities of implicit and explicit implementation.For the generation of simulation results, the single surface plasticity approach was implemented in this work.It was found that the implicit and explicit approaches produce almost the same results.With the implicit radial return method, the asymmetric behavior could then be shown, mesh dependence could be avoided and a diagonal shear band could be generated.Topics to be addressed in the future are parameter identification, an algorithm for large strains and an eigenvalue analysis for the explicit Prandtl-Reuss implementation.

F I G U R E 1
Production and setup: (A) Production of CBS, (B) CBS specimen, (C) setup for compression test.CBS, cold-box sand.F I G U R E 2 Results of compression test: (A) Equivalent strain, (B) fracture of the specimen.
Figure 2A is simplified by placing the focus on the diagonal shear band at an angle of approximately 45 • .Fracture will not be an issue of modeling.It is obvious that the F I G U R E 3 Comparison of compression and tensile test: (A) Compression, (B) tension.

F I G U R E 4
Example 1: Comparison of implicit and explicit algorithm (1 element).F I G U R E 5 Example 2: Input for FE-simulation (A) Boundary conditions (y-z-plane), (B) boundary conditions (x-y-plane), (C) medium mesh.

F I G U R E 6
Example 2: Comparison of compression and tension for cylinder (coarse mesh).F I G U R E 7 Example 2: Reaction force-displacement curve for cylinder under compression -(A) No regularization, (B) regularization.F I G U R E 8 Example 2: Contourplots for cylinder under compression (  ) -(A) Coarse mesh, (B) medium mesh, (C) fine mesh.