To assign different material law s to the component and compare the result of the analysis of the different material laws

Aim: To assign different material law\'s to the component and compare the result of the analysis of the different material laws.

Problem Description :

• A circular rupture plate is impacted at its midpoint by a sphere with some velocity.

• Material of plate is aluminium and the properties are given below:

- Young’s modulus: 71000 MPa

- Poisson’s ratio: 0.3

- Density: 0.0027 g/mm3

- Yield stress: 290 MPa

- Hardening parameter: 562.3 MPa

- Hardening exponent: 0.63

• The plate is meshed with 4-node shell elements.
• The shell properties (Type 1) are:

Scope of project:

• In this project the component is circular rupture plate and it is strike by the some object or punch with some velocity.
• By applying different material laws to the plate we analyse this simulation.
• We perform radioss analysis for each material law and and observe the results like stress distribution, displacement, thickness reductionfor comapring different material laws.
• We also analyse how elements are getting deleted or crack propagation and ruputuring of elements after the collision.
• At the end we also compare the energy and mass error for each material law.

Introduction:

Before going for the comparison of the analysis result lets understand different Material Law\'s in radioss :

1) Law 1 : /MAT /ELAST

• This keyword defines an isotropic, linear elastic material using Hooke\'s law. This law represents a linear relationship between stress and strain. It is available for truss, beam (type 3 only), shell and solid elements.
• This Material law mostly used to model purely elastic amterials. The material stiffness is determined by the youngs modulus and poissons ratio.
• The shear modulus (G) can be computed using,

• Global integration approach is applied to LAW1 and shell elements when the number of integration points through the shell thickness is different from NP=1 (membranes).Failure models are not available in the case of global integration. LAW2 and LAW27 with very high yield stress may be used as a substitution to LAW1 in these cases.
• Also it is not reccomended to use elastic materials in crash analysis because for whatever amount of deformation the system is going to comeback into zero stress state when force is removed.
• Elastic materials can be used in vibration analysis or simple static analysis.

2) /MAT/LAW2 (PLAS_JOHNS) :

• This law represents an isotropic elasto-plastic material using the Johnson-Cook material model. This model expresses material stress as a function of strain, strain rate and temperature. A built-in failure criterion based on the maximum plastic strain is available
• This is an elasto-plastic material model that includes strain rate and temperature effects.
• In this model the material behaves as a linear-elastic material when the equivalent stress is lower than the plastic yield stress. For higher stress values, the material behavior is plastic and the stress is calculated as:

• \'n\' is hardening exponent and \'b\' is the hardening modulus in the above formula.
• Hardening Modulus : It is the slope of the stress strain curve after the yeild point of the materials. It tells us how well the curve is bending after the yeild point.
• Hardening exponent : As we know that strain hardening is a phenomenon where the strength of ductile metal increases when it is plastically deformed i.e increase in strenght and reduction in ductility. Hardening exponent is a measure of the ability of a metal to strain harden. Thus higher the value of n materials resistance to necking increase.

3) /MAT/PLAS_ZERIL :

• This law defines an isotropic elasto-plastic material using the Zerilli-Armstrong plasticity model.
• The Zerilli-Armstrong law is applicable only to shells and solids.

4) /MAT/LAW27 (PLAS_BRIT) :

• This law combines an isotropic elasto-plastic Johnson-Cook material model with an orthotropic brittle failure model. Material damage is accounted for prior to failure.
• Failure and damage occur only in tension. This law is applicable only for shells
• In this model, the material behaves as a linear-elastic material when the equivalent stress is lower than the plastic yield stress. For higher stress values, the material behavior is plastic and the stress is calculated as.

• Where, εp\">εp is the plastic strain and ε˙\">˙ε˙ is the strain rate. The plasticity hardening exponent, n must be less than 1.
• This law allows the modeling of material damage and brittle failure in two principal directions (1 and 2). This law is only applicable to shell elements. It is compatible with Shell Property and Sandwich Shell Property.

5) /MAT/LAW36 (PLAS_TAB) :

• This law models an isotropic elasto-plastic material using user-defined functions for the work-hardening portion of the stress-strain curve (for example, stress vs. plastic strain) for different strain rates.
• Also known as Elastic Plastic piecewise Linear Material.

• The first point of yield stress functions (plastic strain vs stress) should have a plastic strain value of zero. If the last point of the first (static) function equals 0 in stress, default value of εpmax is set to the corresponding value of εp.

- Element deletion:

• Once εp\">εp (plastic strain) reaches εpmax\">εpmax , in one integration point, the element is deleted.
• If ε1\">ε1 reaches εt\">εt, the stress is reduced using the following relation:

• If ε1\">ε1 (largest principal strain) reaches εm\">εm ( ε1>εm\">ε1>εmε1>εm ), the stress in element is reduced to 0 (but the element is not deleted).
• Once ε1\">ε1 (largest principal strain) reaches εf\">εf (maximum tensile failure strain), the element is deleted.

• Strain rate filtering is used to smooth strain rates.
• this material law is widely used in the industry because we can use user defined functions where we input the lab data curve, so that we can get the curve somewhat similar to the lab data which is ideal case.
• Also in lab we can test peformance at different strain rates but during actual simulation we cannot predict the exact strain rate. For this there is some interpolation happening at these curves and we use dummy curves defined for higher strain rate values and it is used to interpolate the results.

6) /MAT/LAW33 (FOAM_PLAS) :

• This law models a visco-elastic plastic foam material.
• This law is applicable only for solid elements and is typically used to model low density, closed cell polyurethane foams such as impact limiters.

7) /MAT/LAW42 (OGDEN) :

• This keyword defines a hyperelastic, viscous, and incompressible material specified using the Ogden, Mooney-Rivlin material models.
• This law is generally used to model incompressible rubbers, polymers, foams, and elastomers. This material can be used with shell and solid elements.

Case setup :

• Case 1 : Law2_epsmax_failure.

• In first case we run the model as it is and saved the analysis file as Law2_epsmax_failure.
• The value of eps_p_max is 0.151, It is failure plastic strain or we can say that elements will fail when stress causes aprroximately 15% of plastic strain.
• Run radioss simulation and get the results to check energy and mass error open the .out file with text editor.

Fig: Material of the component

Fig : Material Properties

• In first case the simulation is performed without editing the material card.

Fig: Failure model

• In Johnson failure card the parameters D1, D2, D3,D4, D5 are related to the maximum tensile failure strain. i.e εf.

Energy and mass errors :

• Since the energy error at the end of the cycle is 0.2% and the mass error is 0 also the energy error is in the acceptable range i.e for QEPH elemets the acceptable energy error is 1-2% or closer to 1% error is prefered.

Fig: Displacement contour case 1

• Max displacement is 2.758E+01 at node 3598 and minimum displacement is zero.
• when sphere hits the plate, the plate gets streched and when some elements gets deleted there is some displacement associated with them.
• So for the rest of elements which are not get deleted the displacement is less.

• The maximum velocity is 2.803E+01 at node 3598 which is associated with elements that are displace or deleted due to impact of punch.
• the velocity associated with elemets near the punch is less.

• Maximum plastic strain along the thickness direction is 1.510E-01, Here we are using Ithick = 1 for considering thickness change.

• Maximum Thickness reduction = 7.392E+00 at node 3863
• Minimum thickness reduction = -1.169E+01 at node 3896

• As we can see that maximum von mises stress is 287 Mpa.
• Initially when the rigid sphere strikes the rupture plate it expand elastically but after the yeild point the rupture plate deforms plastically or when stress reaches to 287 MPa the strain caused by this stress will be 15% of the total plastic strain.
• So the particular elements which reaches to the 15% of total plastic strain gets deleted.

• The internal energy will be increasing as the impat enegy from the sphere gets absorbed or we can say that kinetic energy will be absorbed. and since the plate is not moving kinetic enegry of plate is zero. so total enegry curve is somewhat similar to internal enegry curve.
• also there is slight increase in the kinetic energy at 4ms.

• Case 2 : Law2_epsmax_crack

• In this case first the case 1 file is imported and we need to change the values in the johnsom failure card.
• From the Failure_JOHNSON_1 card change the values of :

- Ifail_sh = 1

- Dadv = 1

- Ixfem = 1

• After editing these values Go to analysis and perform radioss analysis and get the results.
• Save the file as Law2_epsmax_crack.

Fig: Failure johnson card

Energy error:

• At the end of the cycle 49396 the energy error is more positive i.e 4% and the mass error is 0.,

• Maximum displacement is approximately 1.327E+01 mm which is less the case 1.

• As we have set Ithick as 1 i.e changing thickness need to be consider and also we have set Iplast = 1 for more accurate results.
• We can onserve from the contour that the plastic strain is the strain in the thickness direction.

• Maximum thickness reduction is 9.908E+00

• Maximum velocity is 2.311E+01 at node 3983.

• The maimum value of the von mises stress is 294 Mpa at node 4380.

• Similar to case 1 the internal energy will be increasing as the strain energy which is produced from the impact of punch gets stored. and there is slight increase in the kinetic energy in between 4 to 5.

Comparison of Case 1 and Case 2 :

Fig: Case 1 Fig: Case 2

*Explanation:

• Since in case 1 is run without chnaging any values in the cards the elemts get deleted once they reach eps_max value or we can say that elements will fail when stress causes aprroximately 15% of plastic strain.
• But in case 2 we have changed values in the failure johnson card as because of that the elemets are getting cracked and we are getting more realistic failure behaviour.

*significance of Ixfem formulation :

• Ixfem is used to determine how you want to fail this material.
• It represents crack over an element
• we know that if one elemet is failing it gets deleted but if ixfem is 1 then we noticed that elemets are getting cracked and smaller elements gets form from one big elemet.
• Ixfem is only used in shell elements.

• Case 3 : Law2_epsmax_nofail :

• In this case we have deleted the johnson failure card and perform radioss simulation only with eps_max value.
• so the EPS_p_max value is 0.15 i.e we can say that the elements will be deleted when stress causes aprroximately 15% of plastic strain.
• elements will be directly deleted once the failure criteria is reached.

• Open .out document fileto get energy error and mass error. the energy error at the end of cycle 49404 is 0.8% and mass error is 0.
• Energy error is acceptable as it is closer to 1.

• The maximum displacement is 1.858E+01 at node 3844.

• The maximum velocity is 2.792E+01 at node 3592.
• The velocity is high for displaced elements.

• The maximum plastic strain along the thickness is 1.5E-01 at node 3872.

• Maximum thickness reduction is 7.410E+00 at node 3989.

• Maximum von mises stress is 312Mpa at node 4013.

• Similar to case 1 internal energy as well as total energy will go on increasing as the strain energy is absorbed and stored as internal energy.
• There will be rise in kinetic energy at 4ms.

Case 4 : Law 2

• In this case we need to delete the EPS_p_max value and also the failure johnson card is also absent.
• So we can compare case 3 and case 4 as case 3 has EPS_p_max value only.
• After editing value of EPS_p_max and deleting failure johnson car we perform the radioss analysis and obtained the result.

• The enrgy error at the end of cycle 49303 is 1.1% and mass error is zero which is in accepatable range.

• The maximum displacement is 5.011E+00 at node 3688 also there is no element which fails.
• The displcement is higher than previous cases.

• The maximum velocity is 1.134E+00 at node 3688.

• The maximum thickness reduction is 6.184E+01 at node 3688.

• The maximum plastic strain along the thickness is 1.205E+00 at node 3688.

• The maximum von mises stress is 424 Mpa at node 3984.
• In this case the EPS_p_max value is 0 and radioss will take default value of EPS_p_max which is 1E30

• Since the default value of EPS_p_max is high, the elements will not fail as such high value has not reached.
• So the Von mises stress value is also higher than all previous cases.

• The internal energy and total energy will increase and the values of both are more than all previous cases.
• This is because of the impact force produces stress and which will increase strain and strain energy gets stored as internal energy.
• Kinetic energy is zero because in this case there is only streching of plate and no elements are getting deleted.

Comparison of Case 3 and case 4:

Fig: Case 3 Fig: Case 4

*Explanation:

• In case 3 we have deleted only the failure johnson card and run the simulation but in case 4 the we have deleted both eps_max value as well as the failure johnson card.
• So in case 3 because of EPS_max value element will fail and deleted when any element reaches the stress wich will cause 15% of total plastic strain so that stress value for case 3 is 311 MPa
• But in case 4 elements will not get deleted or fail as we put the eps_max value as 0 and radioss will use the default value in this case which is 1E30 which is quite high so no elemnts will fail and plate gets streched.
• So the stress value is 424 Mpa in case 4 which is higher than all previous cases.

• Case 5: Law 1

• In this case we have converted the law into Law1 with the same material properties.
• The law 1 i.e /MAT /ELAST defines an isotropic, linear elastic material using Hooke\'s law.
• represents a linear relationship between stress and strain. It is available for truss, beam (type 3 only), shell and solid elements.
• By converting into law1we performed the radioss analysis and obtained the following results and graphs.

• The enrgy error at the end of cycle 47968 is 0.3% and the mass error is zero.

• The maximum displacement is almost equal to 5.00E+0 at node 3688 which is high as we are using law 1 which is purely elastic material.

• The maximum velocity in this case is 9.881E+01 at node 3902.

• The maximum plastic strain in this case is zero since we are using the Law 1 which represents purely plastic materials and so the deformation is only elastic.
• there is no plastic deformation in this case as we are using law 1 and hence plastic strain along the thickess is zero.

• The maximum thickness reduction is 4.544E+01 at node 3989

• In this case the value of von mises case is comparitively higher than previous cases as we are using elastic material law the plate will strech completely when we apply force through punch.
• So there is no failing of elements and so the the elements will not be deleted.

• As we can see that the internal energy and total energy will increse when we apply the force throuh punch and it will decrease as we remove the force as we are using law 1 which represents law 1.
• It means when sphere will impact the plate the kintic energy will be absorbed and so the internal energy is increasing and it will decrease when the force is removed.

• Case 6 : Law 36

• In this case Law 36 material card is used.
• The law 36 material card represents the /MAT/LAW36 (PLAS_TAB) i.e isotropic elasto-plastic material
• this is widely used in industry as we can use user defined functions where we input the lab data curve so that we can get the curve somewhat similar to the lab data which is ideal.

Fig: Material card

Where,

EPS_t = Tensile failure strain at which stress starts to reduce

EPS_m = Maximum tensile failure strain at which the stress in element is set to zero

EPS_p_max = Failure plastic strain ( 10^30 default)

Fig : Creating Test curve

• Sometimes during the analysis to avoid poor extrapolation of the curve during strain calculation, we we use dummy curves defined for high strain rate value and it is used to interpolate the results.
• The curves need to be defined in such a way that they doont intersect themselves in strain range.

• The energy error for this case is -1.5% and the mass error is zero at the end of cycle 52202.

• The maximum displacement is 3.368E+01 at node 3820.

• The maximum velocity is 3.128E+01 at node 3829.

• The maximum plastic strain along the thickness is 1.363E-01 at node 3688.

• The maximum thickness reduction is 1,321E+01 at node 3688.

• The maximum von mises stress value for this case is 753MPa at node 3688.
• We get this value based on the input data we have provided in the curve.

• Internal energy will increase upto some value and then there is some fluctuation of value as elemts will be deleted after 3msso it will increases upto some value and then it will remain constant.
• The samething is observed for the total energy.
• for kinetic energy there is some rise after 3ms where the element deletion will takes place.

• Case 7 : Law 27

• In this case we need to use law 27 material card i.e /MAT/LAW27 (PLAS_BRIT)
• This law combines an isotropic elasto-plastic Johnson-Cook material model with an orthotropic brittle failure model.
• In this model, the material behaves as a linear-elastic material when the equivalent stress is lower than the plastic yield stress.
• This law is only applicable for shells.

Fig: Material card

• Damage and rupture parameters definedin material card are,

- EPS_t : It represents the strain at the begining of the tensile failure.

- EPS_m : It is the maximum tensile strain at which the stress in element is set to value dependent on d_max1.

- d_max : It is the maximum damage factor.It is the factor that is used to accelerate the element removal or it describe how quikly the failure will happen.

- EPS_f : It is the maximum tensile strain for element deletion.

Fig : Shell Property values changed

• The energy error at the end of the cycle 49355 is 1.1% and the mass error is zero.

• The mximum displacement is 6.013E+00 at the node 3990.

• The maximum velocity is 6.906E+00 at node 3990.

• The maximum plastic strain along the thickness is 1.766E-01

• The maximum thickness reduction is 6.669E+00.

• The maximum von mises stress is 277Mpa at the node 3820.

• The internal enrgy and total energy will increases linearly as the punch hiths the plate as it will resist the force and generate the internal energy after some value the both energies will decrease as elements will be deleted after 4ms.
• due to elements will fail and gets deleted from the system we can see some rise in kinetic energy after 4ms.

Result and Conclusion :

• In this project we apply different material cards to the component to study each material law and to study how material behaves to applied force.
• We compare the 7 different cases on various parameters like energy error and mass error, von mises stress, displacement, number of cycles etc and obtained below results :

• Mass error is zero for all the cases as there is no mass adition to increase the time step also we have provide the same Tstop value for all the cases so the run time is same for each case.
• Also it is observed that hourglass energy and contact energy is zero as we have used all the recommended shell properties in each cases so there is physical stabilization of the hourglass.
• The selection of material depends upon lot of factors and our application. If we want to study elastic behaviour of then we need to assign Law 1 to the component.
• But in crash analysis it is not recommended to use elastic material because for whatever level of deformation we create the system is going to comeback into zero stress state.so elastic materials are not recommended for the parts which are undergoing large deformation.
• also the case 2 where we have used Ixfem for crack propagation and it represents the crack over the element.