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CAE Simulation of Non-Linear Analysis - Modeling of Material Model using
Isotropic Material Hardening Law
Conference Paper · January 2006
DOI: 10.13140/2.1.1202.8167
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Simulation of Non-Linear Analysis in ANSYS
L Ramanan
Principal Engineer, Advanced Mech. Engineering
GE Healthcare Technologies, GTO India.
ramanan.l@ge.com
Abstract:
Simulation of many real life problems is highly complex in nature and involves non-linear behavior to be
accurately modeled. ANSYS non-linear capabilities over the years have evolved according to the emerging
analysis needs, maturity of analysis methods and increased computing power. GE has a global
understanding with ANSYS and uses the code for some of its non-linear needs. This paper aim at explaining
the capability and settings needed in performing non-linear analysis in ANSYS through a simplified
example problem.
Introduction:
Numerical simulation plays a very important and an indispensable role in the manufacturing process [1],
reducing the cycle time in design [2] while improving the quality and performance and simulation of a
behavior like crash [3]. In the recent times analysts and designers have begun to use numerical simulation
alone as an acceptable means of validation in designing the components for six-sigma quality. In many
disciplines, virtual prototyping – employing numerical simulation tools based on finite element methodshas replaced traditional build and break prototyping. Leading world standards [American Association of
Rail Roads – AAR] for locomotive designs have standard like AAR-S-660-83 [4], AAR-S-5506 [5] for
designing some of the critical components like rail wheel, fuel tank etc. Finite element method has been an
accepted as an industry standard [6] in satisfying the regulatory needs of some of the locomotive’s critical
component.
All of the above examples quoted are highly non-linear involving all the three different type of nonlinearity namely geometric, material and boundary. For simulating the non-linear analysis in a reliable
manner, the following components are necessary in the FE analysis tool [7]
1) Element Technologies for consistent large-deformation treatment
2) Constitutive models for a variety of metals and nonmetals
3) Contact Interaction and Assembly Analysis
4) Solution of large-scale problems (where multiple nonlinearities interact in a complex manner) and
5) Infrastructure
In this paper attempt has been made to explain the generic capabilities available in ANSYS for simulating
such a large deflection, large strain problems through a very simplified problem. The actual problem in
which ANSYS has been used was to predict the effects of large deflection due to the impact of medical
equipments in the hospital environment in one of our medical devices. Simulation studies have been later
corroborated with experimental validation. Product concept developed to the medical device of MR patient
table handling system has been filed for patenting in US [8]. In explaining the non-linear problem the
material considered in this paper is steel and hence the paper does not address the capability of ANSYS in
simulating “Hyper Elastic” material model. It is impossible given the scope of the paper to address every
available analysis feature of ANSYS; rather attempt has been made to highlight the key features of interest
to most analyst and design engineers.
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Nonlinear Material Capability:
Material behavior is a very complex science. A wide variety of approaches are used to best match the
computer simulation of a material to the response actually observed. The nonlinear behavior comes into
picture when the component is loaded beyond its proportional limit. It is a known fact that the stress is
proportional to strain upto limit of proportionality [Figure 1] and is governed by Hooke’s law. For nonlinear behavior, stress is no longer proportional to strain. The various types of non-linearity and their
dependency is explained extensively in reference [9] by Ray and Guoyo. It is not within the scope of this
paper to address the entire non-linear behavior. The scope of the work is restricted with nonlinear, inelastic
and rate-independent materials, since most of the metals fall into this category at lower temperature
(temperature below half of the metal’s melting point temperature).
Plastic Behavior and Modeling in ANSYS:
When a material is subjected to an external load of such magnitude that deformation continues with no
apparent further increase in load, the material is said to have become plastic. Once in this region, the
material will not return back to its original shape when the load is removed and hence the materiel /
component has experienced permanent deformation. At the time of application of external load in plastic
region, the component experiences the total strain, which comprises of plastic strain and a small amount of
recoverable elastic strain.
Figure 1 explains the typical stress strain curve of low carbon steel (mild steel) tensile test specimen. In
general, the stress strain data, which are measured through tensile test, and the data available in the material
handbook are the “Engineering Stress and Engineering Strain”.
Figure 1 – Stress - Strain Curve of Low Carbon Steel
But most of the non-linear FE analysis tools require the data to be in the form of “True Stress to True
Strain” for FE analysis. ANSYS also from its version 5.0 required that the values of stress and strain be
inputted as true stress and true strain. The true stress to true strain curve of the above low carbon steel will
be represented as seen in Figure 2.
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Figure 2 – True Stress – True Strain Curve of Low Carbon Steel
Generally, the material data available for most of the metals from the data book are yield strength, ultimate
tensile strength (UTS) and elongation %. Typical data of ASTM-A-36 material model is shown in Table-1.
The data of engineering stress and strain of typical ASTM-A-36 material model is shown in Table 1.
Material
Young’s Modulus
(E) in psi
Poisson
’s Ratio
Yield strength in
psi
Ultimate Tensile
Strength
Elongation
percentage
ASTM-A-36
2.9 E7
0.29
36000
58000
23
Table-1 Engineering Stress Strain data of ASTM-A-36 Material
The engineering data and need to be converted to true data for use in FE analysis and can be converted
using the following equations. For low values of strain, there would not be any difference between true data
and engineering data [9].
In general, most of the non-linear FE analysis tool available in the market uses the true stress to true strain
relation ship while modeling non-linear plasticity model. Most of them uses the true plastic strain and true
stress data, hence the first data point in those codes will follow the following format for the example
material shown in Table-1
0.0, 36000 (True plastic strain at yield, True yield stress)
… (1)
0.2050, 71340 (True plastic strain at UTS, True ultimate stress)
… (2)
If the data available in between the yield and UTS, it can be converted to true format and used between the
data lines (1) & (2) above.
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Since ANSYS uses the true total strain and true stress, the first data point cannot be zero. Hence the data to
be used in ANSYS would be as follows
0.001241, 36000 (True total strain at yield, True yield stress)
… (3)
0.2070, 71340 (True total strain at UTS, True ultimate stress)
… (4)
The true total strain in the data line (3) for ANSYS has been calculated using the following relationship
ANSYS considers that there is no difference between the point A and B of Figure 1, and hence calculates the
total strain using the relationship between yield stress and Young’s modulus. The data lines (3) and (4) are
the inputs in multilinear material data table of ANSYS and is shown in Figure 3.
Figure 3. ASTM-A-36 Material Data in ANSYS for Modeling Material Plasticity-Multi Linear Option
Handling of Convergence while Modeling Plastic Behavior:
When the material enters from the elasto-plastic to perfectly plastic region [Figure 2] in the FE analysis, it
could lead to convergence issues. Generally to circumvent this, it is the practice in the Industry to have a
point above the UTS by extrapolating the data. However, care has to be taken while interpreting the results.
ANSYS has the option of modeling the plasticity data through Bi-Linear material model, which would avoid
convergence issues due to material model in plasticity analysis. The material data line for Bi-Linear
modeling in ANSYS of the ASTM-A-36 material model would be as follows.
36000 (Yield Strength of the material)
… (4)
171754.33 (Tangent Modulus)
… (5)
The tangent modulus / slope of the curve has been calculated using the true stress and true strain data of the
ASTM-A-36 material model and is as follows.
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(True stress at UTS – True Stresss at Yield)
Tangent Modulus =
(True Total Strain at UTS – True Total Strain at Yield)
(71340 – 36000)
Tangent Modulus =
(0.2070 – 0.001241)
The data lines (3) and (4) are the inputs in multilinear material data table of ANSYS and is shown in
Figure4.
Figure 4. ASTM-A-36 Material Data in ANSYS for Modeling Material Plasticity - Bi Linear Option
The true stress Vs true strain curve of the plasticity material data as entered through multi linear and Bilinear material option of ANSYS would represent the material curve as seen in Figure 5.
Figure 5. Material Curve Representation as Modeled with Multi & Bi Linear Options in ANSYS
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It was observed in one of the real life simulation, that the problem, which faced converge issues with
multilinear property could be completed with bi-linear material data as seen in Figure 6. The problem was
driven through displacement boundary condition and the reaction forces were monitored. Convergence
issues faced with severe buckling of internal members, which are like membranes. Figure 7 establishes the
fact, that there is no difference found by using bi-linear material model over multi-linear material model
Figure 6. Use of Bi-Linear Model over Multi-Linear Model in Handling Convergence
Figure 7. Bi-Linear Model or Multi-Linear Model – No Difference in Result
Nonlinear, Inelastic, rate-Independent Material Model:
Plasticity occurs after a material has yielded. For the one-dimensional case, yielding happens at a single
value of stress known as the yield stress. For more complex stress states, a method is required for
combining the various stresses to create a criterion that can be compared to the uniaxial yield stress. The
distortion energy criterion, often called the Mises criterion, is commonly used. Regardless of the method
used, the singular valued uniaxial yield strength becomes a yield curve in the two-dimensional case and a
yield surface in a three-dimensional case. Once the plastic deformation has occurred, the stress value needs
to be raised to a higher value for additional plasticity to occur [Figure 2]. This change in value where
yielding occurs is called hardening. Hardening has two types, (i) isotropic and (ii) kinematic and is
extensively covered by Ray [9]. The user can refer to “Plasticity: Theory and Application” [10] or any
other standard textbooks for a detailed understanding on the plasticity. However, for the benefit of the user,
using the FE tool, the primary differences in isotropic and kinematic hardening material model are
condensed and tabulated in Table 2.
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Sl.
No.
Isotropic Hardening
Kinematic Hardening
1.
Yield Surface grows equally in all direction
and remains centered about its initial location
as the material hardens.
Yield surface remains constant in size and shifts
the location
2.
Absolute value of compressive yield surface is
always equal to tensile yield stress
If a specimen is first loaded and deformed in
uniform tension, the load is then removed and
the specimen is loaded in compression, the
compressive yield stress will be less than the
initial tensile yield stress.
3.
Recommended for large strain (True strains
greater than 5 to 10% in metals)
Recommended for small strain (True strain less
than 5 to 10% in metals)
4
Recommended for proportional / non-cyclic
plastic loading
Recommended
plastic loading
for
non-proportional
cyclic
Table-1 Difference Between Isotropic and Kinematic Hardening
However, for a monotonic loading behavior, the plasticity material modeled either through the isotropic /
kinematic hardening is expected to produce same results. This has been tested in a real life problem having
all the three different type of non-linearity namely geometric, material and boundary. The problem has been
simulated through displacement boundary conditions and the reaction forces are monitored. Figure 8,
displays the Load Vs Deflection curve for a real life problem with large strain effects having same
boundary conditions except for material hardening law. It could be noticed from Figure 8, for the load upto
23000 lbs, the both the curve overlaps with each other. After this point the structure was observed to enter
into elsto-plastic region.
Figure 8. Load Vs Deflection Curve - Modeled with Isotropic & Kinematic Hardening Options in ANSYS
Experiments with A Simplified Problem:
The real problem for which ANSYS is used in GE Healthcare Technologies is highly complex. The
problem involves all the different type of nonlinearity, namely material, geometric and boundary. The
structure also had some internal members, which were severely buckling leading to convergence problems.
ANSYS has large varieties and options for performing a non-linear analysis. The analysis has been
performed on a simplified problem to arrive at the options that were to be used for the complex model and
analysis. ANSYS gives the user, the option of using either “Shell43” or “Shell181” for large strain
problem. Both the elements are capable of handling plasticity. For the monotonic loading, the plasticity
material model can be modeled either with kinematic hardening / isotropic hardening. The problem
involves modeling a rigid body. The effect of including material properties and null material properties to
the rigid body has been studied. With these factors, a Design of Experiments [DoE] has been performed.
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FE analysis has been performed through displacement approach, by pressing the rigid surface against the
deformable plate.. The experimental model along with the boundary condition is shown in Figure 9.
Reaction forces, Plastic strain and the solution time are the responses for this DoE. The DoE data are
shown in the Figure 10. It could be concluded from the DoE, that Shell 181 element predicts the plastic
strain in a shorter time of the analysis when compared with Shell 43 element and has been opted for the
complex analysis. Multilinear Isotropic hardening material model behaves better for the large strain
analysis, however its effect on solution time has no effect. In the absence of friction, material property with
actual data or with null data for the rigid body, there is no significant difference in prediction of plastic
strain or reaction force. It has no effect on the solution time. By arriving at these details and keeping the
mesh density constant, again a DoE was conducted by varying the plate thickness, displacement and
including another commercial non-linear FE solver for validation and we did not find a difference. The
predicted results of ANSYS had a very closer matching with a non-linear FE tool in the market. The
predictions from ANSYS in another non-linear simulation correlated with the data from the field.
Figure 9. Simplified Model Plasticity FE Analysis Experiment in ANSYS
Figure 10. DoE Data and Effect of the Factors over the Response
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Conclusion
All the experiments performed were useful in choosing the best choice of element, method of addressing
convergence issues and other best options in ANSYS for plasticity analysis. We have observed usage of
Shell 181 in large strain problems gives us a faster turn-around time, when compared with Shell 43.
Reference:
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the 15th ISPE/IEE International Conference on CAD/CAM, Robotics and Factories of the Future,
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11) ANSYS Users Manual
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