Assessment of fracture mechanics of metallic materials … Milano.pdf · 1 American Institute of Aeronautics and Astronautics Assessment of fracture mechanics of metallic materials

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American Institute of Aeronautics and Astronautics

Assessment of fracture mechanics of metallic materials for

aerospace field.

M. Russo1 and M. Anghileri2

Politecnico di Milano, 20158, Milan, Italy

C. Cofano3

CRM GROUP – AC&CS , B- 4000, Liège, Belgium

and

A. Prato4

Politecnico di Milano, 20158, Milan, Italy

Abstract

n aerospace applications, products must comply with the highest standards and quality or performance under

extreme conditions. The modern aerospace industry demands the most innovative and high quality metals able to

maintain the integrity of the structure and to be resistant to corrosion or high temperature oxidation. Because of the

high costs of tests, it is essential to limit their number preferring sophisticated numerical models necessary to predict

the behavior of the real structure. In this research work, experimental tests and numerical models on High strength

steel grades have been performed to increase the reliability of virtual testing used for the development of new

industrial solutions. High strength steels increase the cost-effectiveness of the designed solutions providing more

competitiveness when compared to other materials. In fact, High strength steels have tight controlled mechanical

properties that enable better energy absorption in the event of a crash and guarantee high performance of the

product, with a significant reduction of structure weight.

A campaign of experimental tests on specimens with different shapes has been carried out, focused on the

determination of the needed parameters to define the triaxiality stress factor. A numerical calibration and

regularization process has been required in order to provide a more precise hardening curve after necking, investigating

an engineering method to eliminate mesh size effect in solid elements.

Keyword

Aerospace applications, Ductile Materials, Fracture Mechanics, Triaxiality, Implicit Finite Element Code.

I. Introduction

The phenomenon of fracture occurs almost everywhere in daily lives. Human beings never stop to investigate the

nature and its respective issues, such as the formation of cracks and its propagation. The problem of the strength of

1 Student, Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali, via La Masa 34.

2 Thesis advisor, Associate Professor, Dipartimento di Scienze e Tecnologie Aerospaziali, via La Masa 34,

Laboratorio Last.

3 Thesis advisor, Project Leader, Centre for research in metallurgy, Allée de l’Innovation 1, B57- Quartier

Polytech 3. 4Thesis advisor, Ph. D. Candidate, Dipartimento di Scienze e Tecnologie Aerospaziali, via La Masa 34, Laboratorio

Last.

I

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materials, in fact, has been one of the most active scientific fields since Industrial Revolution. Among various materials

promoted in this period, metals have been greatly appreciated and used in new applications, thanks to their strength

and ductility.

Starting from 1950, a significant number of aspects which influence the failure behavior of ductile materials have been

contemplated, resulting in a more comprehensive understanding of damage and fracture initiation.

Between these aspects, one of the most important is the dependence on the stress state. Several authors, such as

Brigdgam, McClintock, Rice and Tracey, Mackenzie et al. and others formalized their experimental observations,

defining a scalar quantity intended to better identify the stress state in a material point, later named “triaxiality”.

Following the aforementioned works, the parameter of triaxiality has then extensively described many

phenomenological failure criteria. Gurson (1977), for instance, tried to establish constitutive equations based on

microscopic considerations according to which material degradation occurs as a result of the growth of voids in

metallic matrix, including the effect of triaxiality on the damage evolution. Wierzbicki et al. in [1], after a critical

evaluation of a number of ductile fracture models, concluded that a fracture locus formulated in the space of effective

plastic strain to fracture and stress triaxiality σ* would be able to give a good correlation with a variety of experiments.

Dubois et al. in [2] and Haufe in [3] used the triaxiality to develop accurate numerical models and failure criteria in

LsDyna [4]. Based on experimental and numerical results, Bao and Wierzbicki formulated the relation between the

equivalent strain to fracture versus the stress triaxiality, developing an empirical fracture locus not described by one

smooth monotonic function but rather consisting of three distinct branches for low, intermediate and high stress

triaxiality [5].

In the continuum damage mechanics, the deterioration of the material capability to carry loads is described by an

internal variable of the so-called damage. In many applications, the damage can be considered isotropic and assumed

as a scalar quantity, but it differs from the ductility or the fracture strain because it is an internal quantity and often

cannot be measured directly. Thus, Xue, Bai and Wierzbicki, and others, have proposed a new plasticity model in

which the evolution of damage is considered as a non-linear process and the material deterioration has been included

by a weakening factor on the material strength [6].

Failure models have been further extended, observing the influence of the load path on the fracture behavior, especially

in case of crash events, where the strain paths are highly nonlinear.

Furthermore, mesh size and mesh geometry play an important role in calculations involving crack initiation and

propagation. Due to the softening behavior of the material, mesh size influences the fracture energy and it is often

argued that this quantity should be adjusted on experiments, involving stable crack growth in order to fit the crack

propagation energy

In addition, it is worth mentioning the dependency on strain rate, on temperature and on the so-called Lode angle, in

order to fully characterize three-dimensional stress state, but negligible at this moment.

Concerning the utility of this work, the research focuses on the development of ductile fracture model, which

introduces a precise material characterization through a calibration procedure under different loading conditions,

including non-proportional loading histories. Furthermore, it includes the damaging process which governs the

fracture initiation in ductile materials so as to be able, to all effect, to predict the behavior of real structure, especially

when instability starts. The damage accumulation and the following rupture deformation have been studied as an effect

of the level of triaxiality of the tensor of stress.

The intent is to provide a valid tool for the enterprise, increasing the reliability of virtual testing. The adopted procedure

comprises experimental study, finite element simulations and analytical solutions.

A total of 204 uniaxial tensile tests, including shear tests and tensile tests, have been carried out using standard and ad

hoc specimen shapes, properly designed and manufactured to provide clues to the dependence of fracture ductility on

the stress triaxiality in a wide range of this parameter. The loading environment is room temperature and quasi static

loading. The material is assumed to be isotropic.

Parallel numerical simulations of all the tests have been carried out using the implicit nonlinear finite element code

LS-Dyna.

Concerning the standard specimen, a calibration procedure has been developed for the complete characterization of

the material. A dedicated algorithm has been implemented in order to combine experimental testing and numerical

simulations and to define a constitutive post-necking law.

The fracture locus, instead, has been constructed in the space of the equivalent strain to fracture and the average stress

triaxiality, introduced because a path of varying triaxiality is followed during loading for most specimen types. In fact,

one of the most important observations coming from the simulation is that the critical elements of the computed

specimens almost never follow a path of constant triaxiality due to the geometrical changes of the section over

deformation. This effect has to be taken into account while creating a material card, which basically means that the

determination of failure strain is not a straightforward process.

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The evolution of an instability measure based on a critical strain has been considered, from which the coupling between

the stress tensor and the damage variable depends, validating this aspect throughout experimental tests.

In addition, spurious mesh dependence caused by strain localization has been tackled. Its effect has been compensated

in the damage accumulation through a regularization function. Five different mesh sizes have been chosen for

simulating the large tensile test of the uniaxial tensile specimen, taking into account that coarser meshes lead to

incorrect results in terms of damage and failure parameters. The procedure has included the regularization of fracture

strain and of the energy consumed during the post critical deformation, considering further regularization under shear

dominated loading and biaxial tension. In this contest, the GISSMO model, for which regularization treatment is

combined with the damage model, has been candidate for the task of reproducing properly the fracture behavior. The

numerical calibration of the GISSMO damage model has been obtained through a trial-and-error method, with the aim

to find a suitable solution able to reproduce the physical experiment.

Although in literature already exists works in which triaxiality is considered as rupture criterion for ductile materials,

the methodology here listed tries to approach the problem with more effectiveness and practicality, finding an original

database useful for future developments and also integrated in the commonly used tools by the company.

II. Specimen Preparation and Test Procedure

Ten different types of specimens have been designed and manufactured: nine different specimen geometries to be

statically tested and one specimen geometry to be dynamically tested. Standard and ad hoc specimen shapes have been

developed with the aim to create a specimen suitable for any uniaxial testing machine that achieves different stress

states without changing the type of test performed.

Figure 1 Standard and ad-hoc specimen shapes [7].

All the shapes have been cut from the sheets and the preparation of the samples has been done according to the

reference standards.

Each of the nine shapes of specimen for the static tests covers a good range of stress triaxiality and allows determining

a point of the triaxility curve.

After the manufacturing of the samples, a total of 204 uniaxial tensile tests have been carried out using the hydraulic

uniaxial tensile machine SWICK 14186, at room temperature. The samples have been secured to the ends by means

of the clamps of the testing machine and the motion has been generated by a hydraulic piston which can be controlled

in displacement. The loading speed has been chosen sufficiently low in order to consider the test quasi-static.

Extensometers have been used to measure changes in length in the area where the fracture approximately arises, useful

for strain measurements, while the load cell measures the force values at the loaded end of the specimen.

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The experimental test ends when for each specimen fracture occurs.

Figure 2 Fracture location [7].

Concerning numerical simulations, geometry, element type, constraint mode, charge mode and measures of the models

have been chosen to reproduce the reality as faithfully as possible. A solid element discretization has been used, in

order to capture successfully the thinning and the localized deformation of the flat specimen characterized by high

ductility. All the specimens have been divided in three parts; each part has been described by a *MAT card, in order

to reproduce also the action of the machine clamps. The moving clamp has been simulated applying a displacement

in the axial direction, not constrained. A load path giving a constant velocity has been applied through a beam element

modelled with *MAT_ELASTIC, being the experimental tensile test carried out with a quasi-static load.

Thus, an implicit analysis has been performed.

The simulation data collected for calibrating the material model are:

force, as the axial force in the beam element;

displacement, as change in length between the two nodes corresponding to the measurement device position

during the test;

necking, considering a cross section plane positioned in the necked cross sectional area which captures the

thickness and width reduction of the section.

III. Calibration of Material

One of the biggest obstacles to the use of various engineering fracture criteria in industry is a poorly understood and

presented calibration procedure. The primary reason is that the effects of the damage characterization parameters are

usually coupled with each other and there is no practical way to single out individual effects. This is complicated even

more when damaging mechanism is inadequately built into the model.

Thanks to the calibration method, the damage parameters- i.e. stress triaxiality, equivalent plastic strain and critical

damage value as well as an accurate representation of the true stress-strain curve- have been obtained, combining

experimental testing and numerical simulations in terms of load-displacement curve.

The engineering tension test is widely used to provide basic design information on the strength of materials and as an

acceptance test for the specification of materials.

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The general shape of the engineering stress-strain curve is illustrated in Figure 3. Due to confidentiality reasons, all

the values hereafter have been normalized to 1.

Figure 3 Engineering stress–strain curve, standard specimen shape [7].

Since both the stress and strain have been obtained by dividing the load and the displacement by a constant factor, the

load-displacement curve has the same shape as the engineering stress-strain curve. In fact, the two curves are

frequently used interchangeably.

The limit of the engineering stress-strain curve is that it does not give a true indication of the deformation

characteristics of a metal because it is entirely based on the original dimensions of the specimen that change

continuously during the test.

In accordance with the standard EN ISO 6892-1, known the engineering stress and strain values from experimental

tests, it is possible to obtain the True stress-strain curve, which gives a true indication of deformation characteristics

because it is based on the instantaneous dimension of the specimen.

A true stress-strain curve is frequently called a flow curve because it gives the stress required to cause the metal to

flow plastically to any given strain.

Many attempts have been made to fit mathematical equations to this curve. The most common is a power expression,

𝜎𝑇 = 𝐾 휀𝑇𝑛 (1)

where n is the strain-hardening exponent and K is the strength coefficient, valid only from the beginning of plastic

flow to the maximum load at which the specimen begins to neck down due to instability [7].

An important mechanism that precipitates ductile fracture in a sheet metal is the onset of plastic instability in the form

of necking prior to ductile fracture. Necking generally begins at maximum load during the deformation of a ductile

material. It results from an instability during tensile deformation which brings the effect of a non-uniform stress

distribution on a cross section and introduces a complex triaxial state of stress in the region described as a mild notch

that under tension produces radial and transversal stresses which raise the value of longitudinal stress required to cause

plastic flow [8].

Once the necking region is described, the curve for the FE model has been prepared.

In particular, the curve to give in the LS-Dyna code has been described by the true stress and the effective plastic

strain, where the effective plastic strain represents the residual true strain after unloading elastically [7].

The aim of the calibration procedure is to find the same load-displacement curve from tensile tests and simulations.

For this reason, an iterative method has been required to correct the curve given as input to LS-Dyna step by step.

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Remembering that the stress state is uniform up to the point of maximum load, two different behaviors have been

analyzed, respectively before and after diffuse necking occurs. The iterative procedure starts after computing the initial

True stress-strain curve for the simulation from experimental tensile data. Referring to the first iteration, in the range

before diffuse necking the difference between the values of force referred to the test and to the first simulation has

been evaluated.

∆ 𝐹 = 𝐹𝑡𝑒𝑠𝑡(𝛿) − 𝐹𝑠𝑖𝑚(𝛿). (2)

Thanks to this measure it is possible to build a new vector of force, simply by adding the value of ΔF to the vector of

experimental force values

𝐹𝑛𝑒𝑤 = 𝐹𝑡𝑒𝑠𝑡 (𝛿) + 𝛥 𝐹. (3)

The engineering stress and strain values have been obtained by considering the new vector of force calculated above

and the corresponding vector of displacements returned from the simulation. The True stress-strain curve has been

constructed by applying the correction between the engineering and true measures. After the peak load, necking

deformation brings the effect of a non-uniform stress distribution on a cross section and the True stress-strain curve

can only be obtained by the trial-and-error method. To estimate the true stress after necking, it is necessary to observe

the local area, taking into account not only the reduction of area proportional to the reduction in thickness but also the

change in shape [9]. Therefore, the behavior after diffuse necking has been described by updating the power law, in

terms of strain, strain-hardening exponent and strength coefficient. The hardening exponent n of the power law has

been determined by

𝑛𝑖 = ln (1 + 𝛿𝑖

𝐿0) (4)

where δi is the magnitude of displacement corresponding to the maximum load in tensile load-displacement curve, for

the i-th iteration [10]. The equivalent plastic strain has been conveniently obtained considering all the aforesaid

treatment as

휀𝑇𝑖= ln (

𝐴0

𝐴𝑠𝑖𝑚𝑖

) (5)

with 𝐴𝑠𝑖𝑚𝑖 which represents the necked cross sectional area until fracture, for the i-th iteration.

Also the strength coefficient Ki has been computed starting from the definition, considering as numerator the value of

stress at which necking experimentally occurs and replacing the value of hardening exponent calculated by the

equation (4) at the current step. Finally, the new power law which expresses the behavior after necking has been

computed. In this way, it is possible to build the new input for LS-Dyna and continue the iterative process. From the

second iteration forward, the shape of the curve before necking calculated at the first iteration has been kept unchanged

while the last steps which describe the behavior post-necking have been repeated until 𝐹𝑡𝑒𝑠𝑡(𝛿) ≈ 𝐹𝑠𝑖𝑚(𝛿).

In fact, only when the correlation is almost perfect, it is possible to obtain from the simulation the parameters useful

for fracture criteria [7].

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The comparison between the load-displacement curves of experimental tests and simulations has been summarized in

Figure 4.

Figure 4 Comparison between numerical and experimental data [7].

It is possible to notice that already for the first iteration the numerical values match well experimental data up necking,

while the trend of the numerical curve seems to follow the experimental data after the peak load as desired. It has

been necessary to perform in total three iterations to obtain a good correlation between the simulation and the actual

stress-strain curve.

IV. Treatment of Spurious Mesh Dependency

Generally, failure in metals is the result of a damage process. However, prior to failure a process of plastic instability

usually takes place, which lead to a loss of the homogeneous state of deformation, to plastic strain localization and

consequently mesh dependency of the numerical results. Spurious mesh dependence represents a concern when

simulating material behavior up to fracture, as in case of high speed car crash scenarios. In addition, the mesh

dependence issue poses challenges when mapping results from forming to crash simulation.

In this contribution, the strategy adopted has been to adjust the fracture curve to the corresponding element size, in

order to obtain a local value of deformation to fracture equal to experimental one. In particular, it has been necessary

to introduce the local value of deformation at fracture to have the correct behavior in the model [11]. However, an

inherent mesh-dependency of the local rupture strain value occurs after localization, understood in terms of diffuse

necking. Concerning the true local strain value, it can be approximated as a function of the element characteristic

length. In this way, it has been possible to predict the strain at fracture by a curve of regularization which allows to

generalize the fracture behavior of the model, taking into account the variability of the characteristic length of the

elements of various meshes [7]. Typically, the curve is a hyperbolic function of the mesh length and its shape is

monotonic. To ensure accurate results for a wide range of mesh sizes, simulations of a tensile test on a standard

uniaxial specimen have been carried out using the hardening curve of material calibrated above.

The GISSMO model allows for regularizing not only fracture strains but also the energy consumed during the post

critical deformation. This has been done by activating the two flags which describe the numerical material card,

respectively named LCREGD and FADEXP. In addition, further regularization under shear dominated loading and

biaxial tension has been considered through the parameters REGSHR and RGBIAX [12].

Concerning the calibration of the hardening curve, a constant mesh size Lc= 0.55mm has been used in the simulation

of the specimens, in order to capture the fracturing process with moderate resolution.

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However, crash simulations require larger element sizes in practical applications. For this reason, five different mesh

sizes have been chosen for simulating the large tensile test of the uniaxial tensile specimen.

Table 1 Average element size for different configurations.

M1 M2 M3 M4 M5

0.5 mm 1.25 mm 3.55 mm 6.55 mm 10 mm

Especially due to the pronounced mesh dependence under strain localization prior to failure, the element size has to

be kept as constant as possible in the damaging and fracturing zones during the numerical procedure. From

experiments, global force-displacement curves have been obtained. Parallel numerical simulations have been carried

out in order to obtain the same global force-displacement responses. The purpose is to modify the regularization factors

so that for the considered mesh size the value of strain at fracture is equal to that experimental. In this contest, both

REGSHR and RGBIAX have been set to 1.0, assuming that no strain localization, and therefore no spurious mesh

dependence, takes place under shear and biaxial stress state. Both regularization curves have been constructed point

by point: to each abscissa value corresponding to the size of the considered mesh, the scale factor for the equivalent

plastic strain at fracture and the fading exponent have been respectively associated in ordinate.

A smoothing spline interpolation has been used to fit the points obtained from the simulations.

Figure 5 Regularization factor and fading exponent [7].

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Finally, once the dependency on the mesh size has been compensated, adapting to mesh size the true material

characteristic in the post-necking region, a good correlation between the force-displacement curve from test and from

simulations has been achieved.

Figure 6 Results of the large tensile specimen with different mesh sizes [7].

The regularization procedure acts on the displacement corresponding to the rupture, not on the entire load path. To

confirm the accuracy of the results, fracture has to occur for the same displacement, therefore at the same instant of

time, in all meshes considered.

V. Calibration of Fracture Locus

Experimental evidence has shown the dependence of metallic fracture on the stress state. Under the assumption of

plane stress state and isotropic material, fracture has been defined to be only dependent on the stress triaxiality ratio

σ*, leading to the definition of a fracture envelope in 휀�� – σ* space.

It ought to be interpreted as the fracture curve for proportional strain paths, but it is not the case of real life experiments.

In fact, generally the stress state changes during the loading history and fracture does not necessarily occur on the

fracture curve. One way proposed to consider the strain path nonlinearity is, from a numerical point of view, to include

the damage evolution as an incremental accumulation of a failure variable.

The material degradation, expressed through a variable D, has been interpreted as a phenomenological scalar failure

variable which indicates the imminence of fracture when reaching unity [13].

In a similar manner to the damage variable, an additional variable F to represent an instability measure has been

defined. It can be interpreted as an instability criterion which designates the point from which the material experiences

an accelerated, localized straining behavior up to fracture. The nonlinearity of strain path has been assumed to affect

the onset of the critical behavior without considering geometrical effects.

Similar to material damaging, also material instability is a rather complex physical phenomenon as well.

The GISSMO damage model combines proven features of damage and failure description available in crashworthiness

calculations with the possibility of mapping various history data from sheet metal forming to final crash loading [14].

It is intended to provide a maximum in variability for the description of damage for a variety of metallic materials.

The input of parameters is based on tabulated data, allowing to directly convert test data to numerical input [12].

For instance, a failure curve as a function of triaxiality is used for the nonlinear accumulation of damage, assuming

that the strain path changes. Furthermore, an instability curve can also be included in GISSMO: if instability achieves

a critical value, the stresses are assumed to be coupled with damage, leading to a ductile dissipation of energy upon

fracture [15]. For both strategies a numerical calibration based on experimental data has been required, taking into

account the strain path changes.

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Considering an incremental scheme,

𝛥𝐷 = 𝐷𝑀𝐺𝐸𝑋𝑃

𝑓(𝜎∗) 𝐷(1−

1

𝐷𝑀𝐺𝐸𝑋𝑃)𝛥휀𝑃, (6)

𝛥𝐹 = 𝐷𝑀𝐺𝐸𝑋𝑃

𝑐𝑟𝑖𝑡(𝜎∗)𝐹(1−

1

𝐷𝑀𝐺𝐸𝑋𝑃)𝛥휀𝑃. (7)

Both equations have been evaluated at every time-step of the simulation and for every integration point. Focusing on

the equation (6), D is the damage value which varies between 0 and 1, 𝛥휀𝑃 is the equivalent plastic strain increment

while 휀��(𝜎∗) is the equivalent plastic strain to failure, determined from LCSDG as a function of the current triaxiality

value [12]. The input of this failure strain is realized as a tabulated curve definition of failure strain values vs.

triaxiality, which allows for an arbitrary definition of triaxiality-dependent failure strains. Using the same relation as

for the accumulation of ductile damage to failure, the possibility of accumulating a measure of instability is expressed

through the equation (7). The difference between the two equations above is limited to the use of a different weighting

function, which for F, instead, is defined as a curve of critical equivalent plastic strain vs. triaxiality.

The equivalent plastic strain to instability is determined from ECRIT. This flag allows for a definition of triaxiality-

dependent material instability, which takes into account that instability and localization occur depending on the actual

load case.

Thus, a nonlinear damage accumulation with respect to equivalent plastic strain has been considered in the present

research. The generated fracture surface depends on the damage exponent. From the micromechanical point of view,

the void formation and evolution are followed by a progressive void coalescence, described as nonlinear damage

accumulation, as shown by some research [16]. In this research, the exponent for nonlinear damage accumulation has

been set DMGEXP=2, assuming a quadratic accumulation of damage along the loading path.

The experimental results of eight specimens with different shape have been used.

The material parameters have been calibrated using a trial-and-error strategy until the force-displacement curves of

the different specimens could be reproduced. In order to calibrate the fracture and the critical strain curves, few points

have been firstly defined in the triaxiality-plastic strain space. Each test gives one data-point, meaning that all

simulations have been performed until the failure displacement is the same as that measured during the experimental

tests.

The LCSDG and ECRIT curves have been interpolated using a cubic spline, connecting the points obtained after

comparing experiments and numerical results. In this way, both smooth curves have been generated, then put in input

of the damage/failure model [7].

The equivalent plastic strain to fracture has been evaluated as the equivalent strain which corresponds to the

displacement to fracture. Generally, the stress triaxiality is not constant during the entire deformation. In order to

construct both fracture and instability locus, an average stress triaxiality concept has been introduced. In fact, since

the stress triaxiality in all tests varies in a rather narrow range, the definition of an average value would not introduce

large errors [7].

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Plotting the results for each shape, the trend of the desired curves has been obtained interpolating all the values.

Figure 7 Calibrated fracture and critical plastic strain curves for the GISSMO model [7].

VI. Conclusion

Prediction of ductile fracture of metals in engineering structures is the topic of this research work. Concerning its

utility, the research focuses on the importance of sophisticate numerical models, which introduce a precise material

characterization and take into account fracture criteria, which are necessary to predict the behavior of the real structure,

especially when instability starts.

Triaxial characterization is also an import tool that contributes to a more complete definition of the rupture modelling

in ductile metals.

The failure criterion based on the stress triaxiality, in fact, unlike that based on the effective plastic strain, allows not

to have deletion of the elements only upon reaching of a local deformation value imposed by the user, but it is able to

discern the type of stress to which each element is subjected, correctly evaluating the possible deletion of the same.

The numerical calibration of the GISSMO damage model has been obtained through a trial-and-error method, with

the aim to find a suitable solution able to reproduce the physical experiment. However, it is not a simple curve fitting:

since the strain paths of the different specimens are nonlinear, not only the force-displacement curves, then the

engineering stress-strain curves, have to match using a single set of damage and failure parameters, but ideally also

the deformation and the strain fields of the specimens itself [7]. For this reason, pictures of the fractured specimens

have been used to visually compare the deformations, confirming the numerical results.

Although the model has shown a good ability to reproduce the fracture behavior of the material, there is still ample

room for further improvements which should be addressed in the future. The introduction of upsetting tests on

cylindrical specimens could be a useful continuation of the work, being able to fully describe the triaxiality curve, also

including the localized shear band for negative values of σ*. The dependence on the Lode angle together with the

introduction of dynamic phenomena, already seen in the course of this research, which also take into account of the

changes in temperature, would be a further interesting extension of the fracture locus.

Concerning the numerical simulations, the behavior of same parameters of the GISSMO damage model could be even

better understood by considering their variation, being already treated previously but not extensively investigated, for

example the regularization factors for shear and biaxial conditions and the role of the damage exponent.

Finally, a suitable criterion for the crack propagation, in combination with numerical implementation, mesh-

independent for the biggest range of values, together with the transferability of fracture data from solid to shell

elements, remains an important challenge for the future.

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Appendix

In Structural Engineering, a material is said isotropic when it manifests the same elastic properties in all directions.

More precisely, this means that in carrying out a mechanical experiment on a specimen of this material, which

induces in it any stress state, the elastic deformation that is measured is indifferent respect to an arbitrary relative

rotation between the specimen and the testing machine. Or in other words, subjecting the sample always at the same

stress but changing its direction of application, if the specimen always gives the same answer in terms of elastic

deformation, the material can be defined isotropic [7].

On the contrary, if the response is different, the material is considered anistropic.

Concerning this research work, a total of thirteen standard specimens have been cut using a water-jet machining:

five in the longitudinal direction;

five in the transversal direction;

three in the 45° direction.

In order to verify the validity of the assumption of isotropic material, the results of the experimental tensile tests have

been plotted, normalized to 1 for confidentiality reasons, from which some considerations have been made.

Figure II.1 Comparison between Longitudinal, Transversal and 45° Directions [7].

Taking as reference the longitudinal direction and reasoning in terms of mean values, it has been possible to estimate

the deviation in percentage between the curves considered in the other directions.

In particular, referring to the Young’s modulus, the transversal direction deviates from the reference by 10 percent

while the 45° direction differs by 0.04 percent. Same identical results have been achieved reasoning in terms of Yield

Strength.

For completeness, referring to the Tensile Strength values, the transversal and the 45° direction differ from the

longitudinal direction by 0.03 and 0.04 percent respectively.

This confirms that the material can be considered isotropic, without committing any error.

The entire work has been performed using the following numerical software:

LS-Dyna Version 7.1.1, SMP double precision.

LS-PrePost.

SolidWorks.

Altair HyperMesh.

Excel.

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Word.

Powerpoint.

MS. Paint.

Teamviewer.

Acknowledgments

Many thanks to Professor Paul Du Bois and to Ing. André Haufe, because their correspondence, although only by

mail, has been of fundamental importance.

References

1. Wierbicki –T., Bao Y., Lee Y. W., Bai Y. ‘Calibration and evaluation of seven fracture models’, International

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Documents

Assessment of fracture mechanics of metallic materials … Milano.pdf · 1 American Institute of Aeronautics and Astronautics Assessment of fracture mechanics of metallic materials