Numerical Modeling of Non-linear
Mechanical Behavior in Solid Propellants
Maxime Coillard
Master of Science in Engineering Technology Materials Technology (EEIGM)
Luleå University of Technology
Department of Engineering Sciences and Mathematics
Research and Technology Department Calculation Laboratory
Numerical modeling of non-linear mechanical behavior in solid propellants
Master Thesis
By
Maxime COILLARD
Industrial Supervisor : M. Gilles POIREY, research engineer SAFRAN-SME University Supervisor : Dr. Roberts Joffe, Lulea University of Technology, Sweden
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Thanks
My sincere thanks go to M. Gilles POIREY for his constant help during this project, but also his clear-sightedness and kindness
I would also thank M. Pierre BRUNET, head of the Calculation Department for having allowed my participation in that project.
Last but not least:
- Dr. Roberts JOFFE, for having accepted to supervise this work from Lulea.
- The staff of the detonic/combustion laboratory
- The staff of the aerospace laboratory
- « John John » and « John » Glen
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Objectives
Nowadays, propellants are widely used in the aeronautics and armament fields where their capabilities to deliver a strong, constant and immediate energy is essential to the existence of complex systems. Indeed a large range of industrials, from the European Space Agency to the missiles manufacturers like MBDA, need those materials to propose solutions as regards the launch of satellites, the space venture or the development of missiles, sized and shaped in function of the propellants properties.
The use of solid propellants started during the Second World War where they were used as small rockets under plane wings to make the take off easier. At that time started the development of what we call today missiles, strongly dependent on the capability of the propellants to deliver an appropriate energy. The tactical interest for missiles and the operational advantages they can give, leaded, both in western countries and USSR, to dedicate a part of the military research to the development of propelling charges with better properties:
higher energy, better mechanical properties, lower weight and so on. Indeed those materials that are quite similar to polymers, are submitted to various aggression on the mechanical, thermal and pressure point of view, which can question their efficiency1,2,3.
The main goal of research is to propose solution on a chemical and physical aspect of propellants. Those solutions have to be tested before their use in applications to seen whether or not they can support different loads during their deployment. Test campaigns have to be made by the researchers. Few decades ago, they were done on real blocks of propellants which was quite expensive. The development of computation brought a solution: numerical modeling.
Today the propellant modeling is one of the main concerns in the research. Mathematical models were developed to estimate, on FEM software, the expected mechanical behavior of specific chemical compositions3,4,5. However new compositions exhibit in some aspects, unexpected evolution when they are loaded in defined conditions, that former models do not take into account3,4,5.
The idea of this project is to complete existing models with mathematical solutions representative of the unexpected effects observed experimentally. The first part of this project will point out the limitation of current models as regards the good predictions of the properties. The efficiency of the model can be corrected by the introduction of mechanical effects, that we will details the influence and the solution we propose to complete pre-existing models in a second part.
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Table of Content
Thanks ... 2
Company1 ... 6
I. Introduction to solid propellants ... 7
1) Did you say propellant? ... 7
2) Process ... 8
3) Interest for numerical modeling... 9
II. Failure prediction: elastic methodology ... 11
1) Linear elasticity ... 11
2) Non-linear elasticity... 12
3) Non-linear elastic sizing ... 12
III. Failure prediction: viscoelastic methodology with damage ... 14
1) Viscoelasticity ... 14
a) The relaxation ... 14
b) Tensor decomposition ... 16
c) Boltzmann principe3,6 ... 17
2) Damage ... 18
a) Characterization ... 18
b) Sizing with the viscoelastic and damage model... 20
c) Interpretation ... 20
IV. Triaxiality ... 22
1) Introduction ... 22
2) Function of triaxiality ... 24
3) 3D test ... 26
4) Test on a two material sample ... 28
V. Mullins effect ... 30
1) Experimental investigation ... 30
a) Introduction ... 30
b) Pre-load effect ... 31
c) Effect of the relaxation duration ... 33
2) Definition of our model ... 33
a) Sensitivity of the viscosity to load changes ... 34
b) Plasticity ... 40
Conclusions ... 44
Bibliography ... 45
Table of figures ... 47
5
Appendix ... 48 Integral of convolution ... 48
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Company1
SME, a subsidiary of the Safran group since April 2011, designs, develops and produces propelling charges and equipment for the defense and aeronautical, space and automotive industries, and strategic high-energy raw materials related to these applications. Its know-how in high-energy materials is unique in Europe and relies on its Research Center and expertise in the field of safety and environment.
- Raw materials
SME produces raw materials involved in propelling charges. It is the unique company in Europe producing the ammonium perchlorate (essential product used in the confection of propellants) for different applications: missile propulsion, automotive security systems ... The company produces also specific products linked to the field of propelling (adhesion agents, high mechanical resistance propellants and so on).
- Propelling
SME is involved for 50 years in the design, the development and the research of propellants used in strategic applications. The company manufactures the propellant used in the nuclear French missiles and carried out on the sub surface ballistic nuclear (SSBN).
It develops also a large range of products involved in systems of the nuclear dissuasion (release, ejection, ignition, separation …)
Because of their ability to give a strong impulse, high-energy materials are widely used in the aerospace field. Since the beginning of the European space venture, SME is involved in the development of specific propellants for all the family of rockets “Ariane”. It develops also the pyrotechnic equipments used in that field, like stage separators. The company will continue to be active in the development of European space programs in the next few years like Ariane 6 and Vega rockets.
Moreover with different subsidiary companies SME is present in the tactical propulsion field where she proposes solutions for missile manufacturers like MBDA.
- Gas generators
SME is present in the automotive industry through gas generators used in the deployment of car airbags or the put in tension of safety belts.
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I. Introduction to solid propellants 1) Did you say propellant?
A solid propellant is a material:
- Dense - Flammable - Gases generator
- Stable at the temperature of 20°C
Which allow the production of controlled energy, under the shape of high temperature gases able to propel a rocket engine, independently of the atmosphere (can burn without oxygen)1. On a microscopic point of view, a propellant is a composite material, where a polymeric binder contains particles. Those particles are from two types: oxidizing and reducer. The chemical reaction between these two kinds of particles will conduct to the production of gases used to create an impulse able to move a propeller. The oxidizing particles are commonly made up of ammonium perchlorate while the reducer particles are constituted with a metallic reducer like magnesium or aluminum. The volume fraction of particles in propellants can reach impressive values like 70%. Thus, the mechanical behavior will largely depends on the interface properties between the particles and the binder1,3,4,5. It is often added various components helping during the process or giving specific properties to the final product (combustion agents, binding agents and so on).
Thereby, a propellant can be seen like a charged polymer, which is relatively easy to shape.
For that reason, those materials are interesting concerning their incorporation in complexes rocket motors, like presented below1:
Figure 1: layout of solid rocket motor, with the courtesy of SAFRAN-SME
The propellant, represented in red on the draw, is incorporated in the combustion part called
“grain”. The combustion initiate the appearance of gases, canalized and ejected through the nozzle which create an impulse moving the rocket. The propellant combustion surface is proportional to the impulse created.
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Figure 2: impulse as a function of the combustion surface
Thus it is possible to give different shapes to the canal depending on which kind of impulse we want to generate:
Figure 3: star-shaped propellant
For instance a star-shaped propellant is useful is you need a constant impulse. Indeed, when it burns, the combustion surface is constant.
Those composite materials allow various combinations concerning their high-energy performances and physical properties which can be modulated to satisfy specific requirements like the stresses of the propeller.
2) Process
The process leading to the manufacturing of propellants is complex but well controlled. It is divided into 5 main steps:
- Mixing - Casting - Curing - Demolding
- Non destructive testing
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Figure 4: process, with the courtesy of SAFRAN-SME
All the process has to be well controlled. Indeed if it is not, some residual stresses or defects can be present into the material. Moreover the assembly steps and all the steps of the cycle of life (storage, ma-handling, firing) will load the propellants with different amplitudes and durations which could damage it. The material has to be well sized to avoid these mechanical properties alteration.
3) Interest for numerical modeling
Because of their composition and their specific manufacturing process, current propellants, mainly composites, have a complex mechanical behavior. Made up of crystalline oxidizing particles (ammonium perchlorate) and reducer particles (aluminum), coat into a polymeric
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binder, which can be also a source of energy, those materials exhibit a sensitivity to the strain rate. In other word, the composition has a viscoelastic behavior. This behavior is highly dependent on the interface properties involved in the macroscopic stiffness of the material, and this all the more the volume fraction of particles is important (up to 70%)1,2,3,4,5. This stiffness, when the material is loaded, can be degraded because of the presence of damage mechanisms: is it observed a progressive disentanglement between the particles and the binder. Thus, various bubbles are created which can conduct, if the stretch is too important to the initiation and the propagation of breaks.
It exists different loads responsible for this mechanical damage. The first one is the process where the propellant is directly stocken on the metallic casing of the motor body. During that step, the propellant is shaped at high temperature and then cooled. Once the cooling is started, breaks can appear into the material. Indeed the difference of the thermal coefficients between the propellant and the motor casing puts the material on load. If this load is higher than the capacity of resistance for the material, it fails.
The mechanical behavior has to be satisfying when the missile or the rocket is fired. In other words the propellant has to be enough resistant to the pressure created during its combustion.
A failure in the material could conduct to a wrong combustion regime, able to decrease the performance of the missile. If the pressure is too high for the propellant, some critic failure can be generated and some part can be removed. Those parts can obstruct the nozzle and thus the gases ejection, leading to the explosion of the missile. Finally, the internal pressure during firing can tense mechanically the metallic envelope of the motor. In the same time, this expansion loads the propellant that has to be enough strong to support this load.
Some minor loads can also be responsible for material‟s degradation. During the storage and all the manhandling steps, the material can be loaded in different ways (shock, gravity, vibration). Often active with a reduced amplitude, their accumulation can seriously damage the propellant2,7,8.
So, we have to give to the idea-mans, numerical models2,3,4,5 representatives of the mechanical behavior, to simulate the response of propellants blocs incorporated into different complex systems, submitted to a large range of loads (time, frequency, amplitude). It can be summarized in one sentence: a modeling to simulate, a simulation to predict.
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II. Failure prediction: elastic methodology
Beyong a certain level of stresses, the propellant starts to loose its mechanical integrity.
Indeed for high loads, different degradations are introduced into the material, in an irreversible way, and this all the more the material is loaded and damaged. The microscopic defects can be of two types: interface failure between the binder and the particles and/or the chains failure between two polymer chains. Their initiation and their growth is called damage and lead to the macroscopic failure of the material
Studying the history of the damage (the mechanisms of energy dissipation suppose that the thermodynamic way is not unique) is relatively complex (mathematically, numerically, and experimentally) and other approaches are possible. One of them consists in the treatment of a single point of this history: the failure.
It is the case of the elastic methodology which allows the sizing, with a certain level of security and this for the entire life of composites propellants.
The elasticity is defined by the following relation of proportionality between stress and strain:
With σij stress tensor of the material at the time t
εij strain tensor imposed to the material at the time t
E the Young‟s modulus or stiffness: characteristic constant representative of the material‟s stiffness and homogeneous to a pressure.
The strain is the report of an incremental length (the material is lengthening) over a reference length. From the definition of this reference length, different strains can be created.
1) Linear elasticity
In the case of linear elasticity, the reference length is the initial length (when the material is not under load).
This expression is the most common way to define elasticity and is relevant for strains lower than 10%. However in some cases the level of deformation reached is higher than 10%. In this situation we will use another expression of elasticity which means another definition for the reference length.
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2) Non-linear elasticity
In the case of non-linear elasticity, the reference length is the current length of the material when it is loaded:
We illustrate the modification of the length on the following chart.
Figure 5: elasticity, normalized stress as a function of strain
This expression of non-linearity is representative for materials loaded over the strain of 10%
(high strains)
3) Non-linear elastic sizing
The idea is to size with a non-linear elastic model, a numerical sample representative of a real propellant. This sample is a cylinder with a central channel concentrating stresses
We make a thermal load on the model between the polymerization temperature and a temperature of rupture (observed experimentally). The sheath of the model gets a thermal expansion coefficient much lower than the propellant thermal coefficient. During the cooling the propellant is loaded because of the contraction of the metallic sheath.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 20 40 60 80 100
Normalized Stress
Strain (%)
Normalized stress as a function of strain for different elasticity definitions
Linear elasticity Non-linear elasticiy
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Figure 6: model studied
The way to proceed is to measure for different geometries (report L/H varying), the maximal strain during the cooling between the process temperature and the temperature of rupture.
When the strain calculated is unrealistic, it seems that the model is broken. Then we check the temperature of failure which is compared to the experimental failure temperature. The experimental failure temperature was defined thanks to uni-axial tensile test submitted to a propellant block during a cooling up to the failure, which occurs at a specific temperature.
We compare a calculated difference of temperatures ΔTcalculation with an experimental difference of temperatures ΔTexp.
These calculations are made thanks to finite elements modeling software called Abaqus6,13 which include in its data, standard-behavior laws like non-linear elasticity as defined below.
For different geometries, we got the following results:
geometry 1 2 3 4
70 71.3 87.5 55.5
Table 1: elastic methodology and size calculation
This methodology is highly upper bound as regards the prediction of the material stiffness. By consequence this methodology is useless. We have to use another methodology, more representative of the material behavior and taking into account not only a point of the damage history (the break) but the story itself. We introduce in that purpose, a new methodology using a viscoelactic behavior for our material (stiffness depending on the time) and taking into account a damage law. In that way we will be sensitive to the way leading to the break.
Ø L
H
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III. Failure prediction: viscoelastic methodology with damage
The transcription at the macroscale of the microscopic organization of propellants is that those polymers exhibit a viscoelastic.behavior. The previous study demonstrated that to follow the strain in an elastic law is not sufficient to predict the failure for our materials and that why it is necessary to follow the entire mechanisms of energy dissipation: viscosity and damage.
1) Viscoelasticity
The viscoelasticity is an intermediate state between an elastic state and a vitreous state. This phenomenon is representative of the action of two effects: the elasticity thanks to its stiffness and the viscosity because of internal frictions
The viscoelasticity can be underlined through two characteristic tests:
- Creep test: a constant stress is applied and we observe the evolution of strain - Relaxation test: a constant strain is applied and we observe the evolution of stress
a) The relaxation
A relaxation test is characterized by the application of a constant strain to the material. We observe the stress evolution depending on the strain:
Figure 7: principle of viscoelasticity
An amorphous polymer is composed by an entanglement of chains. When you stretch the propellant (polymer), the chains tend to be aligned and then start sliding one each other if the load is enough strong. Because of the initial entanglement, the chains move with friction.
Thus they take a certain time to reach a position where their energy is the lowest possible.
The more the chains are entangled, the more it is necessary to provide the polymer for energy, to make the chains sliding from one each other. Thus just after the strain load, the intern
Time Strain
Stress
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stresses of the material are really high and tend to decrease when the chains disentangle. The stress relaxation being dependant of the time, we associate to each relaxation stress, a characteristic time called τ or relaxation time (higher than the time t corresponding to the strain load).
This idea of sliding with friction can be easily represented by rheological models. The model able to represent this behavior is the Maxwell model, where a spring is associated with a damper. The spring gets a stiffness E and the damper a viscosity η.
Figure 8: Maxwell rheological model
When you apply an instantaneous strain, the spring is suddenly stretched and the evolution of the system to a minimum energy state is made by the contraction of the spring, accompanied by the damper. In other words, the damper modulates the stiffness E representative of the global system stiffness.
We know that a viscoelastic material exhibits different behaviors in function of the mechanical load it sees. Indeed if you load it quickly (during a short time) or slowly (during a long time), the internal stresses will be different depending on the case. That is why, we need to restitute the material behavior and this for a large array of time duration, representative of the loads encountered by the material during its entire life.
Thus we generalize the simple model of Maxwell by the association in parallel of different simple models having different stiffness and viscosity7,8:
Figure 9: generalization of the model of Maxwell
The stiffness Einf obtained afterlong time of relaxation has to be also modeled. Indeed, when it is not under load, a material exhibits a residual stiffness. It is not because you do not load a material that is seems its stiffness is null. Thus we add to the generalized model, a stiffness through a spring with characteristic stiffness Einf. Finally we can express the total stiffness of the system through the next formula7:
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Where Einf is the material stiffness once it is completely relaxed
Ei the stiffness of the spring n°i representative of the branch n°i
τi viscosity over the stiffness for the branch n°i This report is homogeneous to a time and represents the characteristic time of the branch n°i.
NB: these parameters have to be defined thanks to an adjustment procedure made on an experimental basis.
The presence of the exponential is necessary to simulate the decrease of stiffness for our model. This decrease is function of the constants Ei and ηi, used to modulate the effect of the relaxation with the time.
Figure 10: generalized model of Mawell as a function of time
b) Tensor decomposition
It is useful to decompose the stress or strain tensor in two distinguished parts: a deviator part and a spherical part9.
The deviator part is representative of shear effects (parallel vector to the surface) while the spherical part is representative of a hydrostatic pressure (normal vector).
Courbe maîtresse de module
1.00E-01 1.00E+00 1.00E+01 1.00E+02
0.0001 0.001 0.01 0.1 1 10 100 1000 10000
temps, min
Module, MPa
Time, (min)
Modulus, (MPa)
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c) Boltzmann principe3,6
Let‟s consider a relaxation test. We applied a strain level εi at a time τi. The stress at a time t >
i can be written like:
The function E(t- i) is representative of the relaxation.
We can rewrite this function differently:
with σi the stress at the time τi
Let‟s develop this idea to a range of strains εi applied at different times τi with t > i. The principle of Boltzmann let us to show that the stress seen by the material at the time t, is only the summation of stresses that would have been seen by the material if each level of strain εi
would have been applied separately. Thus
If we consider a range of strain, not applied like a singular summation but like a time dependant function, we can change the previous formula of stress and rewrite it thanks to an integral formulation. Thus we define an incremental element of stress dσ between the times τ and τ+dτ
The strain history starts at the moment t=0, thus
This integral of convolution is complex to use. We prefer to develop new expression from this integral, much easier to calculate3,4:
with
The development of this new integral is shown in appendix.
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2) Damage
Beyong a certain level of stress, our material starts to loose its initial integrity, and degrades due to the debonding between the binder and the particles.
Figure 11: damage, debonding binder/particles, with the courtesy of SAFRAN-SME (micrometer scale)
During the damage we generated the irreversible apparition of bubbles which affect the global properties. Thus we have to build a model taking into account this phenomenon.
a) Characterization
Two types of damage are distinguished2,4:
The first type of damage is linked to shear phenomenon, affecting the material during the debonding between the particle and the binder. When the cohesion forces are disappearing, the particles are not participating anymore to the global stiffness of the material. Thus the deviator part is affected by the loose of stiffness and the tensor is degraded thanks to a damage function γG.
The second type of damage is a damage of volume. The lack of cohesion between the particles and the binder tends to generate the appearance of voids, and thus all the more that the material is under load and degraded. The material gets new properties like becoming suddenly compressible. That why we associate to the “shear damage” a “volume damage” via the function γK,which affects the spherical part of the stress tensor2,9.
NB: during the load, we are interested in the resilient degree of integrity of our material or in other words to the level of non-damage into the material. That why the deviator and spherical parts of the tensor are affected by function whose forms are: 1-γ.
For a null damage level, the function is equal to zero. However, when the material breaks macroscopically, the damage functions are equal to 1.
It is necessary now to activate in a proper way our damage functions. For this, we calculate an average stress (like Mises), continuously compared with a characteristic capacity representative of our material behavior. Thus, if the average stress calculated is lower than the capacity of the material, it seems that the material is not damage and our functions are constant and null. However, if the equivalent stress is higher than the capacity, then the material starts to degrade itself. Thus, the function is decreased to make our model sensitive to its loose of stiffness. The damage is all the more affected that the gap between the average stress and the capacity is high.
charge
liant
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The damage functions are quite complexes, because they are built thanks to convolution integrals. Like for the viscoelasticity (Ei et τi), they use different coefficients that have to be defined. The definition of these parameters will let our model to match a certain range of experimental tests. The determination of the parameters is possible thanks to minimization tools calibrated on a large experimental base. This base is mainly composed of uni-axial tensile tests (TU) but also with some bi-axial tensile (TE) tests and tri-axial tensile tests (TT).
Figure 12: viscoelastic model with damage : definition of parameters-stress9
Figure 13: viscoelastic model with damage: definition of parameters-volume9 Extension (mm)
Extension (mm)
Normalized Stress
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Once the parameters are defined, we can show that our law (black curve) is fitting simple experimental tests (black points): in stress and volume.
We got now a viscoelastic model taking into account the history seen by the material and sensitive to the level of damage.
b) Sizing with the viscoelastic and damage model
We perform the same calculation on the numerical sample already used for the elastic methodology investigation. As previously we compare an experimental difference of temperature with a calculated difference of temperature, to check if our viscoelastic model, integrating damage effects, is more accurate than the non-linear elastic model. The results are the following:
Law geometry 1 2 3 4
Viscoelastic with damage
32.5 23.3 25.7 13
Non-linear elasticity
70 71.3 87.5 55.5
Table 2: compared results between the non-linear elastic and the viscoelastic with damage laws
We have:
Our viscoelastic law with damage is less predictive than the non-linear methodology. We cannot extrapolate our law to complexes models.
c) Interpretation
A first reason which could explain the difficulty to predict correctly the failure of our material is the way we are calculating the damage. Indeed it would be more relevant to associate to a same volume of void in the material, a same level of damage. However if we consider different tests, like TU and TE, we can clearly see that we have two different levels of damage for the same volume level in the material. Thus the damage level is different for a same volume variation (depending on the test).
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Figure 14: different level of damage (depending on the test) for the same volume variation
We can also point out the experimental basis used to define the parameters. It might be possible that the basis would not contain enough tests, mainly equi-biaxial and tri-axial tests (TE and TT). Completing the basis could let to define new parameters able to make the model fit the experiments. To develop the basis it is necessary to take into account a larger number of experiments TE and TT characterized by non-linear stress effects that our model does not take into account. In other words our model is not sensitive to those non-linear evolutions.
Thus we need first to make our law dependant on those effects and then define news parameters.
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0,0E+00 1,0E-04 2,0E-04 3,0E-04
Damage level
volume variation (mm3)
Tensile test, damage as a function of volume variation
TU TB
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IV. Triaxiality
1) Introduction
We know that the stiffness of our material is affected by energy dissipation mechanisms (viscosity and damage) that we want to define the characteristic parameters. For this we propose to increase the experimental basis on which are calibrated the parameters with experiments showing specific behavior.
Some of those specific tests show that the stiffness can be dependent on the nature of the stress distribution.
Let‟s consider a tensile test which is explained on the next sample:
Figure 15: introduction to the aspect ratio dependence
If the length is enough important, the stress distribution in the middle of the sample is homogeneous to a 0D field. The material sees an extension without a volume variation.
However, the more we are close to the edges, the less the distribution is homogeneous. Indeed we observe 2D and 3D complex distributions when we consider areas of the sample close to the edges (clumping). In that case the material sees a high volume variation. The module of compressibility is raised. Thus, close to the edges, the stress distribution is different from the central part of the sample: it is called the edge effect.
The ability for the sample to see homogenous or heterogeneous stress distributions will depend on its geometrical lengths. For samples exhibiting a high aspect ratio (width over length), the presence of 2D or 3D stress distributions is much more important that samples characterized by a low aspect ratio. Thus the high aspect ratio samples get a stronger stiffness than low aspect ratio samples. Our numerical model, which fits the analytical calculation, is sensitive to this effect.
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Figure 16: triaxiality, gap between experiment and calculation
NB: analytical calculation was extracted from internal company documentation10.
The stiffness is calculated by dividing the stiffness calculated for each test by the lowest stiffness for the range of test.
We can see that calculations for an incompressible material show an increase of stiffness when the aspect ratio is increasing, or in other words when the sample length is reduced.
However if we compare our calculation with the experimental data, obtained for different TU that we calculate the final stiffness, we can observed that, despite we are sensitive to this effect, our predictions do not fit the experiments. The experimental over-stiffness seen are higher both for low and high aspect ratio than those calculated. The real effect seems also much more complex because of the non-homogeneity of the experimental data (every point represents a test). The hypothesis claiming that our material is incompressible seems not to be realistic. It is thus necessary to be sensitive to this stress effect.
We have previously used the idea of aspect ratio to show the stiffness stress dependence. We concluded that for high aspect ratio we had high stiffness (higher for the experiment compared with the calculation). This conclusion is not always true. Indeed if we consider a shear test, this test has a very high aspect ratio but we do not observe any over-stiffness.
Indeed this test is not affected by a volume variation. The compressibility modulus is not changed.
Thus, we propose to work with a new field of investigation. We want to follow an average stress, accessible in every element of the sample. We use the Mises average stress:
1
2 3 4 5 6
0,01 0,10 1,00
Normalized Stiffness
Aspect ratio
Modulus evolution as a function of the aspect ratio
FE Calculation
Analytical calculation Experimental tests
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We want to work with a non-dimensional criterion. For this, we need to normalize our Mises stress. We choose to normalize it by the pressure. The pressure being negative during a tensile test we choose finally the negative value of the pressure:
We obtain a new magnitude named the triaxaility rate. τtriaxiality, defined by :
Then we apply this criterion to different simple characteristic tests that we know they have or not over-stiffness: uni-axial tensile test (TU, one main component of the stress tensor is loaded), bi-axial tensile test (TE, two main components of the stress tensor are loaded) and equi-triaxial tensile test (TT, all the main component of the stress tensor are loaded). We have the following results:
Test Triaxiality test Over-stiffness
Shear nul
TU 1 nul
TB 0.5 low
TT 0 high
Table 3: triaxiality rate as a function of test stiffness
NB : « nul »,« low », et « high », are only illustrative notions. They help us for reasoning.
2) Function of triaxiality
We know that the equi-biaxial and equi-triaxial tests are those for whom the gap between the calculation and the experiments is the most important. They are associated to a high proportion of non-homogeneous stress distribution or a quite low rate of triaxiality. However, the TU tests do not exhibit any over-stiffness. Thus we have to create a function which associate high stiffness to a low rate of triaxiality (<0.5) and no over-stiffness for tensile test (TU) which have a high triaxiality rate (≈1). We propose a gaussian type function, depending on the rate of triaxiality and representative of the experimental behavior. This function is affected to the stress tensor, thus it has to get a value equal to one when there are no over- stiffness.
The slope, between the low and the high rate of triaxiality can be modulated by the values of the parameters a and b.
The next chart illustrates the main behavior of the function:
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Figure 17: influence of parameters on the function of triaxiality"
Then we have to define the way we affect the stress tensor. If we express the Young‟s modulus as a function of the shear modulus and the modulus of compressibility2,3,4,9 we understand that it is necessary to add the function to the deviator and spherical part.
Finally we have the equation:
If we affect only the deviator part, it would restrain the effect of the function because of the presence of the compressibility modulus K.
Then we apply the function to different models characterized by different rate of triaxiality:
0 1 2 3 4 5 6 7 8 9
0 0,2 0,4 0,6 0,8 1
Triaxiality function
Rate of triaxiality
Function of triaxiality as a function of the triaxiality rate
a=5 b=10 a=5 b=20 a=8 b=10
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Figure 18:triaxiality and correction due to the function of triaxiality
NB: we represent the data in a chart taking in abscissa the aspect ratio. Indeed event if thinking with the aspect ratio is not pertinent, it stay useful as regards the presentation of the results, all the more that the rate of triaxiality is inversely proportional to the aspect ratio.
However we do not know the strict equivalence between these two fields.
The function of triaxiality lets to generate higher modulus than the previous models used for the prediction. The previous chart show experimental data which are only illustrative. Indeed they present the tendency of evolution but there are not representative of all the loads that our material can be submitted to. That why we do not fit the data. We can add that, as the viscosity and the damage we have to define properly the parameters a and b to make then the law representative of a large range of tests.
Nevertheless we possess now a model able to take into account the effect of triaxiality.
3) 3D test
Despite we do not fit, for the reasons we explained before, a large range of experiments, we can see the influence of our function on a 3D test that we follow during the evolution of time.
We will see if it is possible to defined parameters able to fit an experiment, those parameters being only representative of this test and not of a large range a test. We do a tensile test on a propellant with a low rate of triaxiality.
1 2 3 4 5 6
0,01 0,10 1,00 10,00
Normalized Stiffness
aspect ratio
Modulus evolution as a function of the aspect ratio
FE Calculation Analytical calculation Experimental tests FE calculation + triaxiality
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Figure 19: triaxiality, tensile test9
NB: we can notice on the picture that the propellant is devastated by internal bubbles, showing a high volume load and consequently a pronounced effect of triaxiality.
The results are presented on the next chart:
Figure 20: triaxiality and 3D test
The use of the correction thanks to the function of triaxiality lets to recover the experimental test. There is not a perfect superimposition between our law and the test because the parameters were found by “manual” incrementation without using any tool of minimization (that has to be used to define properly the coefficients on a large range of tests).
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0 0,2 0,4 0,6 0,8 1
Normalized Stress
Normalized Strain
Triaxiality, stress as a function of strain Viscoelastic law
Viscoelastic law + triaxiality Experimental test
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4) Test on a two material sample
With the previous models used to size the propelling charges, it was difficult to represent over-stiffness for 3D samples made with two materials (propellant itself and the liner). We have to check the effect of the new correction of triaxiality on those samples. On the next picture we can see that the rate of triaxiality is suitable. Indeed it is equal or close to 1 in the center part of the sample and equal to zero close to the edges.
Figure 21: rate of triaxiality for a two material sample (liner+propellant)
If we consider two tensile tests, one with a duration of 1s and the second of 100s, we observe a rate of triaxiality varying with the load speed.
Figure 22: modulation of the triaxiality by the material interface
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We do not pretend to have the same level of stiffness but the same distribution between the two tests. This variation is certainly due to the properties of the liner used during the calculation. It was defined, in the FEM software, as an elastic material instead of a viscoelastic material as it is supposed to be.
We can reasonably claim that the triaxiality is directed by the interface stiffness. We can also blame the way the campaign of experimentation is led. To have more accurate results it would be better to work on pure propellant samples.
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V. Mullins effect
The complexity of modern composite propellants is responsible for specific mechanical behavior. Among those unusual characteristics, we find evolution of stiffness grouped and named “Mullins effect”. Their importance was demonstrated on experimental tests made in the mechanical laboratory. Our model has to take into account this effect in order to be able to open the experimental basis required in the definition of parameters involved in the viscoelastic law and the damage.
This effect seen experimentally is only partially integrated in models representative of polymers behavior. In the literature or in standard laws of the software Abaqus11 this effect is present but only in some aspect. Thus we have to:
- List the specificities of the Mullins effect that we want to be sensitive to - Propose a mathematical model of the Mullins effect using the software Fortran
The first step will be done thanks to experimental tests. The second through the UMAT formalism used for the whole models developed in that project.
1) Experimental investigation a) Introduction
The Mullins effect is a term traducing the presence of over-stiffness and lower-stiffness when appear changes during strain or stress loads for a material. This effect can be seen like the dependence for the stress (or strain) to the strain rate.
On an experimental point of view, we observe over and lower stiffness when the strain rate is discontinuous, like it is the case in certain steps of the cycle of life (cooling and firing for instance). This observation can be illustrated by conventional tests made by different cycles like: traction-relaxation-traction (TRT) and traction-release-traction (TDT) like explained on the next charts9:
Figure 23: Tests TRT and TDT15
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If we consider a TDT test, between the steps V1 and V2, it appears a high decrease of stresses: the real material is much supple that can predict a standard viscoelastic model of Maxwell. Moreover between the step V2 and V3, the stiffness of the material is stronger.
If we consider a TRT test, between the first load and the relaxation step, it appears a high decrease of stress and between the relaxation and the step V2, the material exhibits a stronger stiffness compared to the standard model of Maxwell.
Thus irregularities appear when the strain rate changes suddenly: the material gets a softer stiffness when the rate of load is decreasing. On the other hand the stiffness of the material becomes stronger when the rate is increasing.
b) Pre-load effect
The Mullins effect is also dependent on the strain level when appears the discontinuity. For TDT and TRT repeated cycles with relaxation and creep present at different level of strain, we observe non-linear effects as regards stresses and stiffness.
i. TRT tests
In that type of load we are interested in the stress decrease and the modulus when the material is newly stretched after the relaxation3,4,13,14,15,16,17
.
Figure 24: TRT tests, increasing of strain rate15
Indeed during the relaxation, stresses in the material are strongly decreased compared with a standard viscoelastic material and this all the more that the strain rate is important.
During the step following the relaxation, it occurs a kind of recovery for the stiffness. Indeed the material seems to express the will to recover the monotonic tensile test curve. The material wants to respect a finite break energy (e d ) whatever the way it is loaded.
The recovery is all the more strong that the energy lost during the relaxation is important.
strain Normalized Recharge modulus
0 0.09
0.025 0.61
0.05 0.78
0.1 1
0.15 1
0.2 0.83
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Thus, during that step, the stresses generated are stronger than stress generated with a standard law of Maxwell and this all the more that the strain rate is important (see table above). This over-stiffness stop increasing around 15% because of the damage effect.
ii. TDT tests
The TDT tests are really interesting tests giving a lot of information. On those tests we can work with different stiffness: the release modulus, the recharge modulus, and the inflexion modulus (modulus when the stress-strain curve has an inflexion point)3,4,13,14,15,16,17
.
During the release of TDT tests, the material sees its stiffness decreasing strongly, and thus all the more that the strain (when the release occurs) is high. When it is submitted to a new traction after release, the material stiffness is lower than the initial stiffness (first traction or monotonic load). To put it differently, the more we tense the material, the more the stiffness during the release is important and the less the stiffness during the tension phase after release is high3,4,13,14,15
. The modulus is all the more low that the strain in the material is high.
We could claim that the modulus during recharging (after release) is a degraded modulus due to damage. However when the strain is close to the maximal strain obtained during the first load, the modulus or stiffness increases to get values around the stiffness of the monotonic load. Thereby this phenomenon cannot be explained like damage. It comes the same idea as exposed before, which tends to claim that the material wants to accumulate the same energy whatever the way it is loaded.
NB: release and inflexion modulus get the same evolutions that the recharge modulus for the TRT cycle tests
strain Normalized
recharge modulus
0 1
0.025 0.6
0.05 0.48
0.1 0.36
0.15 0.3
0.2 0.2
Figure 25: TDT tests, increasing of strain rate15
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c) Effect of the relaxation duration
The Mullins effect or the over/lower-stiffness is also dependant of the duration between the changes of strain rate, like the duration of the traction. Let‟s consider a TRT test.
In a stress-strain chart, we make TRT tests where the duration for the relaxation phase is varying. This time can be seen like the time of storage for our material. We see that for 3 increasing durations t1, t2, t3, the interval on which the over-stiffness is active during the recharging is increased. In other words, the activation of the Mullins effect is all the more present that the time of relaxation is important. Thus:
2) Definition of our model
From experimental observations, we can think to a phenomenology of the Mullins effect. Our material is quite dissipative on an energy point of view: both in relaxation and release, the stiffness decrease much more that it can be predicted by a standard viscoelastic law. And this whatever the strain rate reached. There no doubt that the internal reorganization of the material is responsible for this irreversible consumption of energy.
We know also that the material is « damaged » quicker than we can imagine. The recharge modulus is all the more low that the previous maximal strain reached was important (TDT test).
However, the material wants to reach the same energy leading to the failure. For this, it imposes high stiffness during the reload even if the initial stiffness of reload would be
“damaged”. This is quite similar to the behavior of elastoplastic materials submitted to TDT cycle test.
Strain
∆ε 3
Stress
t1
t2
t3
∆ε 2
∆ε1
Figure 26: influence of the relaxation duration on the Mullins effect3
