In recent years the small punch test method has become an attractive alternative ... small punch experiments simultaneously, taking advantage of neural network ...
A Unified Approach to Identify Material Properties from Small Punch Test Experiments Martin Abendroth TU Bergakademie Freiberg Institute of Mechanics and Fluid Dynamics Lampadiusstraße 4, 09599 Freiberg, Germany
Abstract In recent years the small punch test method has become an attractive alternative compared to traditional material testing procedures, especially in cases where only small amounts of material are available. It has been applied to determine the current and local material state in structural components under operating conditions. A wide range of material properties like elastic, plastic, creep, damage and fracture behavior can be obtained using this technique. But the assessment of the relevant parameters is not as simple as from standard tests, because of the non-uniform stress and deformation state. However, this can be achieved by comparing the experimental SPT results with those obtained by finite element computations of SPT using advanced material models. Then the task is to determine the parameters of the material models using special optimization techniques. This paper presents an approach to evaluate several small punch experiments simultaneously, taking advantage of neural network approximations, modern optimization strategies and data bases.
Introduction During the last three decades, the small punch test (SPT) has been established as a suitable and versatile miniaturized test method to determine the mechanical behavior of a broad range of materials. In contrast to conventional material testing techniques its great advantage is the small amount of material required. Therefore, in combination with small specimen sampling techniques the SPT becomes especially attractive, if the actual material state in structural components has to be evaluated after in service operation undergoing embrittlement, fatigue or aging. Another advantage is the opportunity to investigate local material properties as in functionally graded materials or welded joints, composite layers or coatings, where no traditional bulk specimen can be removed. Compared with other miniaturized test specimens (tensile or bending rods) the SPT has the following advantages: i) the stress state is biaxial, which meets the loading conditions in many structural components (vessels, pipelines, plates) ii) the experimental handling is comparable easy and iii) the involved material volume to be tested is relatively large compared to the specimen volume. Therefore, the potential of the SPT has been recognized by many researchers, which has been substantiated by a broad field of applications. The main drawback of the SPT consists in the non-homo-
geneous stress and strain fields within the specimen, avoiding a simple interpretation of measurement in terms of material parameters as this is possible in conventional testing procedures. For that reason, much effort has been spent in the past to find correlations between SPT results and standard material properties as yield strength, ultimate stress, Charpy energy etc. On the other hand, true material parameters in the sense of constitutive laws in continuum and damage mechanics seem to be more general and allow the transferability to larger specimens, complex stress states and even structural components. Nowadays, the numerical finite element tools are well developed to simulate complicated specimen geometries together with advanced material laws. To exploit the full information coming out from a SPT experiment, a qualified numerical analysis can be performed embedded into an optimization algorithm to identify the unknown material parameters. There exist material models which describe material behavior over a range of loading parameters as stress or temperature or which are sensitive to internal variables as damage or triaxiality. For such materials multiple tests with varying loads, geometries or changing environments are necessary to investigate the effects of interest. Then the parameter identification strategy must consider all experimental tests.
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The Small Punch Test Fig. 1 shows the principle sketch of the small punch test with the essential geometrical measures. The small punch test is used in different sizes and different types. For example the smallest specimens are standard TEM sized specimen [1], others use specimen cut from remaining pieces of Charpy specimens, which are square shaped [2]. There has been a lot of effort to standardize the SPT and its usage [3], but this is still a running process. At least, there is a common sense about the important features of the test. A disk shaped specimen with a diameter D and a thickness t is placed on a circular die with a receiving
V a crack initiates. During the steep decent in Part V the crack grows circular around the center of the specimen. The remaining force in Part VI is needed to push the punch trough the already cracked specimen. Baik et al. [5] defined the area under the LDC as the small punch fracture energy and found correlations with the values determined from CharpyV-notch specimen. Suzuki et al. [6, 7] found correlations between the maximum force Fm/t2 and the ultimate tensile strength Rm as well between the value of Fy/t2 and the yield strength Ry. Furthermore they also related the equivalent fracture strain using
Ideally would be a three-dimensional strain measuring method, which also accounts for rigid body motions of the specimen. There exist grating methods as mentioned in [12], but the accuracy of those methods is most often not sufficient, especially the distinction between rigid body motions and the overlaying small strains is problematic.
be observed together with decreasing failure times [14]. The CD-SPT is not as common as the two other types of the test. But nevertheless it could be a rather fast test to determine visco-plastic material behavior. The test starts with a predefined deflection, which is applied in a short time, followed by a longer relaxation period where the deflection is kept constant.
Figure 2: Typical results for the different types of the SPT (CDR, CF, CD) for a visco-plastic metallic material
Figure 1: left) Principal sketch of the small punch test; middle) a typical resulting load deflection curve for a ductile metal; right) axisymmetric simulation of a SPT.
hole of diameter d. This receiving die can have a round or straight chamfer of size r. The specimen can be clamped between the receiving die and a downholder. There are also cases where the specimen is not clamped, usually for testing very brittle materials [4] to avoid initial deformations during clamping. The above mentioned European guideline [3] suggests a standard geometry with values of D=8 mm, t=0.5 mm, R=1.25 mm, d=4 mm and r=0.5 mm. The middle of Figure 1 shows a typical load displacement curve (LDC) for a ductile metallic material. The LDC, the essential experimental outcome of the SPT, can be split up into several parts. Part I is mainly determined by the elastic properties of the material, Part II reflects the transition between the elastic and plastic behavior, Part III shows the hardening properties up to part IV, where geometrical softening and damage occurs. At the beginning of Part
εqf=ln(t0/tf )=β(uf/t0)n with measures of specimen thickness tf and deflection uf at fracture. Recently Garcia et al. [8] compared and discussed several alternatives how to determine yield stress and ultimate tensile stress from SPT experiments. The small punch test together with fracture mechanical simulations is used by Cardenas et al. [9] to determine fracture toughness values of ductile steel. Therefore, the specimens where notched using a micro tool creating a notch with a tip radius of 0.1 mm and 30° cut angle. The fracture toughness values can be identified using simulations and evaluating the J- integral at the point where a crack initiates. Deformation measurements of the specimen can be very useful. Egan et al. [10, 11] suggest optical methods to access the deformation profile of the small punch specimen. The measured profiles are valuable information for material parameter identification methods.
The loading can be a constant displacement rate (CDR) of the punch, a constant force (CF) applied to the punch or an initial constant displacement (CD) of the punch followed by a holding (relaxation) time. The experimental results of the test are usually the punch displacement and/or the specimen deflection and the punch force. In case of time dependent material behavior these values are stored together with the time after starting the test. The typical results for the different types of the SPT are shown in Fig. 2 considering an elastic, visco-plastic material as the most metals are. The CDR-SPT can be performed at different . punch velocities (or deflection rates) u. If the punch speed is increased the curve is shifted to higher forces due to the strain rate sensitivity and the onset of damage happens at smaller deflections [13]. The CF-SPT is used to determine the material creep behavior at different loads (stresses). As in tensile creep tests we distinguish three parts of the curve, which are related to primary, secondary and tertiary creep. The primary part of the curve is also influenced by some initial plastic deformation at rather high strain rates. The tertiary part is influenced by a localization of deformation and increasing creep damage. For increasing test forces a higher mean specimen deflection will
The main result is the decreasing part of the curve, which depends on the visco-plastic (creep) material properties. The main advantage of this test type is that very small creep rates can be achieved in a rather short time. All tests can be performed at different temperatures or other test conditions. In general one can interpret the experimental results as a set of three functions, one for each test type:
F (u ) = f CDR (u, u , pi )
(1)
u ( t ) = f CF ( t , F , pi )
(2)
F ( t ) = f CD ( t , u , pi )
(3)
where u denotes the specimen deflection or punch displacement, u the deflection rate, F the punch force, t the time and pi a set of i material parameters. The argument list of these functions could be extended by the test conditions like test temperature and geometry parameters of the SPT. In a more general mathematical framework the three functions represent boundary value problems which are solved using the finite element method (FEM).
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Numerical Simulations of the SPT This sections concentrates on aspects which come along with the finite element analysis of the SPT. A general guideline for finite element modeling states: Keep the model as simple as possible, but as detailed as necessary to capture the relevant effects or phenomena with the required accuracy. Since the geometry of the SPT is axisymmetric a two dimensional (axisymmetric) model can be used. But this implies that also the material properties used for the specimen must fulfill this symmetry condition. As long as the material under consideration is isotropic or transversal isotropic (isotropic plane perpendicular to the SPT symmetry axis) this condition is fulfilled. As soon as local damage or fracture become relevant, a three dimensional model should be used. But even then one might consider symmetries which can be used to simplify the model (half model, quarter model, or an angular section). The computation time depends mainly on the number of degrees of freedom
within the model and the bandwidth α of the α stiffness matrix: tCPU ∝ n DOF with α ≈ 2 … 3. The model size can be drastically reduced if parts of the model can be treated as rigid bodies. For the SPT everything except the specimen could be modeled as rigid bodies as the specimen deformation are large against the elastic deformations of the setup. If the elastic deformation of the setup is a substantial part of the total deformation they can be considered using a compliance correction term for the specimen deflection. Figure 3 shows the elastic deformations of a SPT setup. These deformations depend only on the applied force and not on the tested material. That’s why such a simulation needs to be performed only once to determine the setup compliance function, which is later used to correct the simulated deflection for a certain force.
Figure 4 illustrates the problem of an incorrect model geometry. It was observed that the receiving die and the punch tip contour did not have the correct geometry given in the technical drawings. Therefor, simulations where performed comparing the ideal and real geometry and significant differences were found in the resulting load displacement curves. In simulations used to identify material parameters always the real geometry should be used. One important part of the model is the contact formulation. Here, we use a node to surface algorithm with a constant friction coefficient µ . The normal contact stress σ n depends on the overclosure h between the contact surfaces. For h ≤ −c the normal contact stress σ n = 0 and for h > −c we have: σn =
σ n0
h + 1 exp h + 1 − 1 exp(1) − 1 c c (4)
This realizes a softened contact as we would have it for a surface roughness of c. σ n0 defines the normal contact stress at zero distance between the contact surfaces. In figure 5 the influence of friction on the LDC is shown. One can note that friction influences the LDC after a certain deflection. That is the point where relative motion between punch and specimen occurs. Another important step is to perform a so called mesh convergence test to check that the chosen mesh size is fine enough. The mesh is gradually refined until the changes in the re-
sults are below a predefined accuracy value. But here one have to be careful if a material model is used where localization effects can occur as it is the case for local damage models like the GTN model. Then the mesh size is part of the material model which requires a local length scale for damaged material zones. At the left of figure 5 different meshes are compared. An increasing number of elements in radial direction leads to earlier damage because a bit less energy is needed to damage smaller elements. Material Models The choice of a mechanical constitutive model depends on the material and the phenomena which are encountered in the mechanical test. Commercial finite element codes like ABAQUS or ANSYS provide a wide range of models for almost every material and loading situation. The choice of an appropriate model is the crucial task, because this choice defines which phenomena can be modeled. Here, we will concentrate on models for (ductile) metallic materials. The standard procedure to determine parameters of such materials require a number of standard tensile specimen for determining tensile and creep properties and in case of damage or fracture properties also CT- or 4PB-specimen.
Figure 3: Elastic deformations of a SPT setup (without specimen) and the corresponding compliance function.
Figure 4: Influence of geometrical details of the setup on the resulting load-deflection curves.
Figure 5: Influence of mesh size (with GTN model) and friction on the resulting load-deflection curves.
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Elastic plastic behavior
Viscoplastic behavior
Ductile metallic materials show an elastic- For a rate dependent plasticity model we deplastic behavior, where the elastic strains can fine the creep strain rate as a combination of be considered small against possible plastic strains. Prior to yield, the material response is cr cr 1 cr = ε ε p δ ij + εq nij (11) assumed to be linear elastic. The strain tensor ij 3 is split into an elastic and a plastic part. where εpcr is = the 0 pressure dependent volumetel pl σ0 n ε = ε + ε (5) ric (swelling) strain rate and εqcr the ij = εequivalent ij ij 0( A ) deviatoric creep strain rate. nij defines the Due to the axisymmetric geometry and loa- direction of the creep strain derived from the ding and of the SPT only isotropic material equivalent stress potential. behavior can be identified. Furthermore the ∂Φ SPT applies a monotonic loading thus no ki- nij = (12) ∂σ ij nematic hardening, only isotropic hardening can be identified from experimental results. The volumetric and deviatoric strain rates The stresses in terms of the elastic strains have evolution laws like: are expressed by the multiaxial Hooke’s law, ε cr h p , σ , ε cr , ε cr , θ , … (13) which for the isotropic case reads = as: p p q p q E el ν and el = ε kk δ ij σ ij (6) ε ij + 1+ν 1 − 2ν cr cr cr (14) = ε hp p , σ q , ε p , ε q , θ , … q where E denotes the Young’s modulus, ν Poisson’s ratio and δ ij the Kronecker delta. depending on the equivalent von Mises stress The plastic strain rate is derived from a flow σ q and the equivalent pressure p = − 13 σ ijδ ij . potential Φ .
(
)
(
pl ∂Φ εij = λ ∂σ ij
(7)
The flow rule is given by
( )
Φ= σ q − σ y ε q = 0 ,
(8)
where σ q denotes the equivalent (von Mises) stress and σ y (ε qpl ) a yield function. The yield function can contain different hardening laws as the Voce law
( )
(
pl pl σ ε pl = σ 0 + σ 1ε q + σ 2 1 − exp − nε q y q
or the well known Ramberg-Osgood law. ε qpl pl σ y ε q = σ 0
( )
εq = ε0 ∑ cr
i
σ q − Bi Ai
,
(15)
) (9)
Damage behavior
(10)
(
− 1 + q1 f
In order to simulate plasticity and ductile damage the continuum damage model of Gurson [15, 16] can be used with the extensions of Tvergaard and Needleman [17, 18]. The central part of the model is the yield function
)
*
2
= 0
(16)
tion of void nuclei. For detailed information about the implementation into the FE-Code ABAQUS see [19-21].
where Σq = 32 Sij Sij denotes the macroscopic von Mises and Σ p = 13 Σii the macroscopic hydrostatic stress, expressed by the macroscopic deviatoric stresses Sij = Σij − Σ pδ ij . The mate* rial damage f f depends on the void volume fraction f . f if * * ff − fc ( f − fc ) if f = f c + f − f f c * ff if
f ≤ fc fc < f < ff f ≥ ff
(17) with f f* = q1 . Up to a critical void volume fraction f c the damage is identical with the value of the void volume fraction. Beyond f c where voids coalescence or micro crack initiation is assumed damage evolution is accelerated until a final void volume fraction f f is reached where the material fails. The evolution of the equivalent plastic strain of the matrix material is obtained from the plastic macroscopic strain rate E ijpl 1
= εq εq pl
pl 0
+
ni
which allows the modelling of multiple creep mechanisms like different diffusion and dislocation mechanisms. The McAulay brackets have the meaning as x= ( x + x ) / 2 . This ensures that a creep mechanism is only active above a respective threshold stress Bi.
1
n ε0
A simple example is the Norton creep law where εpcr = 0 and εqcr = ε0 ( σA0 ) n with the material parameters A and n. A more advanced creep model is a combination of i Norton laws
pl
)
The normal distribution of the nucleation Σq 3 Σp * strain has a mean value ε n and a standard + 2 q1 f cosh q2 pl pl deviation of sn . f n denotes the volume frac2 σ ε σ ε y ( q ) y ( q )
= Φ
pl Σ ij E ij
t
∫ (1 − f ) σ
dt.
0
(18)
q
The evolution of the void volume fraction is combined of two terms = f
fgr + fnucl
(19)
,
= f fgr describes + fnucl where the growth of voids based on the law of conversation of mass pl (20) fgr= (1 − f ) E kk
and a void nucleation part, which follows a strain controlled relationship
= fnucl
fn sn 2π
1 ε qpl − ε n pl εq (21) 2 sn
exp −
Figure 6: Top) GGG-40 CDR-SPT finite element simulation displaying the damage at failure. Bottom) P91 CF-SPT finite element model displaying the creep strain at failure.
Fig. 6 shows results from finite element simulations of the SPT and Fig. 7 the corresponding specimens after testing. The GGG-40 is a ductile cast iron containing spherical graphite inclusions which act as voids. Therefore the GTN damage model is used to simulate the onset of failure. On the right hand side of Figs. 6 and 7 results for a P91 specimen are shown. P91 is a material used for high temperature components under internal pressure, whereas the creep behavior is of great interest. The simulations predict the correct locations of failure (GGG-40) and necking (P91) very well. 4
The error for the CD-SPT is defined as the sum of the integrals of the normalized difference of the punch force from all CD-SPT simulations and experiments in a predefined time interval timin … timax . e
CD
=
n
∑t i =1
Figure 7:
Left) GGG-40 CDR-SPT specimen just after failure. Right) P91 CF-SPT specimen after creep testing [14]
The above described model belongs to the quality or exactness of the simulations for class of local damage models. It is well known each test. Each pair of experiment and correthat the results of these models are mesh de- sponding simulation gets its own error value, pendent. A damage zone usually localizes which is multiplied by a certain weight wi. within an one element thick band or plane. Those weights represent the importance of To avoid this one can use non-local damage an experiment or the confidence the user has models, see [22–24] for details. Such models for this single experiment. For the CDR-SPT usually introduce a characteristic length as an the error is defined as the sum of integrals of additional parameter which can be related to the normalized difference of the punch force the spacing of voids or the width of localized from all CDR-SPT simulations and experidamage bands. Using these models element ments in a predefined interval uimin … uimax . sizes smaller than the characteristic length 2 CDR n are required, which can lead to larger mou wi Fi sim ( ui ) − Fi exp (ui ) CDR = dels than those using a local damage model e max min exp du i u u F − i =1 ui i i and solving additional field equations might 2 be necessary, which requires the use of nonCDR n u w Fi sim ( ui ) − Fi exp (ui ) CDR standard solvers and/or elements ewithin = the max i min (22) exp du i u Fi − ui i =1 ui finite element codes. CDR
∑
CDR
∑
Parameter identification As shown in the previous section the material model contains a set of parameters pi which are to be determined from experimental results. Especially for the creep parameters more than one CF-SPT experiment at different loads is necessary. The general way to find parameter sets is to fit the model to the experimental results, which could be a set of different tests (CDR, CF and CD). Mathematically it is an optimization process where the difference between experiments and simulations needs to be minimized. The value to be minimized is an error, which measures the
∫
CD
wi
CD
max i
min
− ti
∫
max
ti
min
ti
2
Fi sim ( ti ) − Fi exp ( ti ) exp dti Fi
(24) As normalizing values the observed mean values of the punch force Fi exp or the punch displacement uiexp are used. Finally, the three errors are summed up and divided by the sum of the weights for all experiments. e=
e n
CDR
CDR
∑w
CDR
i
CF
+e +e n
CF
CDR n
CD
+ ∑ wi + ∑ wi CF
CD
(25)
=i 1 =i 1 =i 1
The minimization of this error is done within an optimization loop as shown in figure 8. One may notice that the finite element computations are not a direct part of the optimization loop.
Instead finite element computations are done in advance using parameters which are varied in reasonable bounds. These computed results are used to train neural networks, which represent an approximation of the finite element simulation. The quality of the neural network approximations can be measured by comparing predictions of the networks with simulation results that had not been part of the training. Each test type (CDR, CF or CD) requires separate neural networks. For creep tests two networks are used, one predicting the failure time and a second predicting the deflection over time. All the networks used here have the structure of feed forward neural networks as shown in figure 9. They consist of at least three layers of neurons. The two first (left) layers have neurons with sigmoid functions, the last (right) layer has neurons with linear functions. All the layers are fully forward connected, which means that each neuron of one layer is connected with each neuron of the subsequent layer. The number of neurons for the first (input) layer is similar to the number of arguments of the function which the network should approximate.
max
∫
i
min
i
max
i
min
i
The error for the CF-SPT is the sum of the integrals of the normalized difference of thepunch displacement from all CF-SPT simulations and experiments in a predefined time interval timin … timax plus an error which expresses the normalized diffrence beween the times of failure for all the simulations and experiments.
1 e = ∑ wi max min i =1 ti − ti CF
n
CF
CF
∫
max
ti
min
ti
2
uisim (ti ) − uiexp (ti ) exp dti ui
2 t sim − t exp + fi exp fi t fi
(23)
Figure 8:
Scheme for the identification process of material parameters using the SPT and neural networks as an approximation for finite element computations.
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In our case the last (output) layer has only one neuron, representing the return value of the function. The number of neurons in the middle (hidden) layers depends on the complexity of the function which the network should approximate. For the networks, which are used to predict creep failure time and SPT creep curves around 10 hidden neurons are sufficient. More complex predictions like for the CDR-SPT under consideration of damage models can need up to 50 neurons in the hidden layer(s). The training of a neural network is also an optimization process. Details about the training of neural networks can be found in [13, 25].
Figure 9:
WEB based parameter identification In the sections above all necessary parts for a successful identification of material parameters have been explained. What is missing is an application bringing all the tools together. This application should be accessible to the experimentalists, some of them might not have access to or experience with finite element analysis and optimization tools running on high performance computers (HPC). The most common tool to connect everything with the users (experimentalists) is of course the Internet. This section will explain a structure for a WEB based parameter identification tool.
It is build of four blocks (see Fig. 10), which could run on different computers. The arrows represent the flow of information (data, commands) through the network. At least one data base is used where experimental and simulated results can be stored. This data base also holds the information about the users who want to use the application and the information about the setups for simulations. An important part is the HPC-Cluster. This is the machine which does all the heavy computational work, i.e. are the simulations of the experiments, the training of the neural networks and the optimization processes. The HPCcluster gets the experimental results from the data base to compare it with its computations. When one thinks of a simulation as an artificial experiment producing similar results it makes sense to store those results in the same data base and just mark them as simulations. This creates the opportunity to compare new incoming experiments with simulations already done or with already processed expe-
riments and to make predictions or at least suggestions what a good parameter set for a reasonable material model could be. Those two main tools (Data Base and HPC-cluster) are controlled by a central WEB server, which hosts interactive WEB-pages, where external users can login and upload experimental results, running simulations or starting optimization (parameter identification) processes. The whole structure is very flexible. It’s not necessary to have a single date base, instead it is possible to distribute especially the experimental data on different machines, so that certain users can have full control over their data. The software which controls the simulations, the construction and training of the neural networks and the interaction between HPC-cluster and the data base(s) is programmed in PYTHON, which is a platform independent high-level programming language. Figure 11 shows a typical graphical user interface for managing all simulations, network training and parameter identification.
Structure of neural networks used to approximate the finite element simulations.
Figure 10: Scheme of the WEB based parameter identification.
Figure 11: Graphical user interface for the parameter identification procedure
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Application and Results This section shows a collection of different applications of the SPT and results, which are obtained using the above explained parameter identification techniques. Hardening Parameters for Nickel-Base Super-Alloy A thermal sprayed nickel-base super-alloy (modified IN625) is investigated using the high temperature small punch test (HTSPT). This alloy is used as a corrosion protection coating for components used in heat exchanging devices which can be exposed to a corrosion aggressive environment as in waste incineration plants. The specimens are cut from the substrate using a core drill. The separation from the substrate is done using a diamond blade saw (Strüers Accutom 50). All specimens are finally grinded to a final thickness of 300 m. The test conditions for the HT-CDR-SPT are u=0:5 mm/min at three different test temperatures (773, 973, 1173 K). At the highest temperature the test environ-
ment was either air or argon. Fig. 13 shows four specimen tested at different temperatures and environments. It is obvious that the damage mechanism changes from brittle failure at 773K indicated by the star shaped crack pattern over a brittle-ductile transition failure at 973K to ductile failure at 1173 K, where the final crack has a circular shape. The load deflection curves in Fig. 12 also reflects this behavior. At 773K the specimen already fails within the elastic region. The load drops at the end of the curve correspond to crack initiation events. Finally the specimen breaks into five almost identical pieces. At 973K the star shaped cracks occur first, followed by plastic bending and further circular crack growth close to the region where the specimen is in contact with the chamfer of the lower die. At 1173K the load deflection curve is fully developed indicating a pure ductile failure. To identify the hardening paramters of the alloy for the different temperatures CDR-SPT simulations where done using a Voce hardening law (see Eq. 9). The identified parameters are listed in table 1.
Figure 13: Nickel-base super alloy SPT specimen testet at a) 773 K, b) 973 K, c) 1173 K, d)1173K at argon.
Creep Behavior of P91
Figure 12: Load deflection curves for Nickel-base super alloy SPT. left) Experiments, right) Comparison with FEM simulations using identified hardening parameters. Table 1:
Identified hardening parameter for the nickel-base super alloy
T [K]
E [MPa]
σ1 [MPa]
σ2 [MPa]
σ3 [MPa]
n
973
33786
139
50.1
53.0
28.2
1173
15422
6.8
26.1
75.2
29.3
In Fig. 14 CF-SPT experiments (exp) taken from [14] are shown done at a temperature of 873K at different loads under argon atmosphere together with simulated results (res) and corresponding neural network predictions (nna). The finite element simulations where done using a user creep law considering two Norton laws (see Eq. 15). For the creep parameter identification two different neural networks where used. One just predicting the time of failure depending on the material parameters and a second one just approximates the specimen deflection over a normalized
time, which is taken from the first network. The optimization routine is a SQP algorithm as explained in detail in [13]. At the right side of Fig. 14 a single CF-SPT simulation together with the corresponding neural network approximation is shown to demonstrate the accuracy of the neural networks. All seven experiments where evaluated together using the approach in section 5. The identified parameters are listed in table 2.
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Figure 14: Left) CF-SPT experiments for P91 at 823K and different loads together with the simulated tests and the corresponding neural network approximations. right) A single simulation with its neural network approximation.
Table 2:
Identified parameters for P91
Figure 16: Simulation of a CT-25 specimen using the identified parameters for 22NiMoCr37, left) force-load line opening, right) crack resistance curve.
Table 3:
Identified hardening and damage parameters for 22NiMoCr37
A1 [MPa]
B1 [MPa]
n1 [-]
A2 [MPa]
B2 [MPa]
n2 [-]
µ [-]
E [GPa]
σ0 [MPa]
ε
n
f0
fc
ff
fN
ε
sN
q1
q2
550
0.445
11.01
456.4
144.3
12.11
0.5
199
430
0.00492
6.44
0.002
0.117
0.2
0.05
0.3
0.1
0.846
1.03
Damage and Fracture Properties of 22NiMoCr37 Fig. 15 shows the comparison between experimental and simulated CDR-SPT results for reactor pressure vessel steel 22NiMoCr37. This material (similar to ASTM A508 cl. 2) is used for pressure vessels and pipings in nuclear power plants. Such components are exposed to neutron radiation, which can cause embrittlement of the material, whereas the current fracture toughness has to be supervised during the lifetime of the respective component.
Here, the GTN model was used to simulate hardening and ductile damage due to void nucleation, void growth and coalescence. The identified parameters from the SPT were used to simulate a standard tensile (Fig. 15 right) and compact tension tests (CT-25, see Fig. 16 right). It was found that these models can predict other tests quite well, only using material parameters from SPT evaluations. More detailed information about the training of the neural networks and the modeling of the fracture toughness specimen can be found in [12].
Conclusion This paper presents a general approach to identify material parameters obtained from small punch experiments by means of finite element simulations. A fully parametric model is used to simulate SPT experiments. The model allows geometry variations as well as the choice of an appropriate material model and specifying material parameters. All SPT types can be simulated with regard to different loading conditions. The parameter identification process can take several experiments into account, making it possible to characterize even complex material behavior. The graphical user interface which has been developed makes this approach accessible to experimentalists which don’t have own access or experience in finite element analysis. Finite element analysis using advanced constitutive material models allow us to transfer the identified parameters (from SPT) to simulations of standard specimens and the prediction of material behavior under more general situations.
Acknowledgement The financial support for this project by the EFRE fund of the European Union is gratefully acknowledged. Furthermore the author thanks Dr. Petr Dymácek from IPM Brno for providing the P91 creep data. Special thanks goes also to the students and co-workers Christin Heinig, Carolin Ranft, Tobias Kaden, Stefan Soltysiak and Wolfgang Kilian who helped to develop the universal SPT-model and some of the optimization tools. Without the staff from the labs Dagmar Schmidt and Kurt Fredersdorf who were preparing countless specimens and running all the experiments this work could not be done. And last but not least the author thanks Prof. Meinhard Kuna for many fruitful scientific discussions.
Figure 15: left) CDR-SPT results for 22NiMoCr37 at RT. right) Simulation of a tensile test using the identified parameters for 22NiMoCr37.
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