Chapter 10:Modeling of Stiffness,Strength, and Structure-Property Relationship in Crosslinked Silica Aerogel Samit Roy and Awlad Hossain Department of Aerospace Engineering and Mechanics,The University of Alabama,Tuscaloosa,AL 35487.USA 10.1 Introduction Mechanically stable forms of lightweight materials with porosities up to 98%were first introduced in the form of silica aerogels in the 1930s. Recently,interest in aerogels and other lightweight materials in engineering applications have increased tremendously.Native silica aerogels are chemic- ally inert,low-density,nanostructured porous materials with poor mechanical properties.They are the product of the sol-gel process whose final step involves extracting the pore-filled solvent with liquid carbon dioxide through supercritical drying.Practical applications of native aerogels are somewhat limited as they are brittle and hygroscopic,absorbing moisture from the environment which eventually leads to aerogel collapse due to capillary forces in the pores.Nevertheless,it has been recently discovered that crosslinking the nanoparticle building blocks of silica aerogels with polymeric tethers increases both modulus and strength significantly [4]. Along these lines,a novel,multifunctional,crosslinked silica aerogel,to be
Chapter 10: Modeling of Stiffness, Strength, and Structure–Property Relationship in Crosslinked Silica Aerogel Samit Roy and Awlad Hossain Department of Aerospace Engineering and Mechanics, The University of Alabama, Tuscaloosa, AL 35487, USA 10.1 Introduction Mechanically stable forms of lightweight materials with porosities up to 98% were first introduced in the form of silica aerogels in the 1930s. Recently, interest in aerogels and other lightweight materials in engineering applications have increased tremendously. Native silica aerogels are chemically inert, low-density, nanostructured porous materials with poor mechanical properties. They are the product of the sol–gel process whose final step involves extracting the pore-filled solvent with liquid carbon dioxide through supercritical drying. Practical applications of native aerogels are somewhat limited as they are brittle and hygroscopic, absorbing moisture from the environment which eventually leads to aerogel collapse due to capillary forces in the pores. Nevertheless, it has been recently discovered that crosslinking the nanoparticle building blocks of silica aerogels with polymeric tethers increases both modulus and strength significantly [4]. Along these lines, a novel, multifunctional, crosslinked silica aerogel, to be
464 S.Roy and A.Hossain Nonporous secondary particles Surface carbonate (diamete灯<1nm dense silica) Channels to (OCN-R-NM(CO)O-Si-O-Si-) micropores Channels partially Mesopores blocked 5-10nm Porous secondary particles (density~2 silica0）(间 (b) Fig.10.1.(a)Silica aerogel structure before crosslinking and (b)x-aerogel structure after crosslinking [6] referred to as x-aerogel,is derived by coating and encapsulating the skeletal framework of amine-modified silica aerogels with polyurea as depicted in Fig.10.1 Aerogels are reported to be one of the best thermal insulators.When sandwiched between two glass layers,aerogels reduce heat loss coefficient by more than a factor of 10,while preserving the capability of moderately high light transmission [5].Aerogels are being considered for different aerospace applications,such as a thermal protection system (TPS),catalyst support,or as hosts for a variety of functional materials for chemical, optical,and electronic devices. Cylindrical samples of x-aerogel manufactured in the author's labo- ratory are shown in Fig.10.2.It was observed from the mechanical char- acterization tests that x-aerogel has very good compressive,tensile,and shear properties,in addition to its low thermal conductivity [4,6,8]. Recently,manufacturing of a lightweight cryogenic propellant tank with low thermal conductivity has been proposed using novel x-aerogel material. While the use of composite sandwich panels for cryotanks is not novel,it is feasible that the delamination of facesheet from the core due to cryo- pumping,analogous to the failure that occurred in the X-33 prototype,could perhaps be prevented through the use of x-aerogel core instead of a standard honeycomb core
referred to as x-aerogel, is derived by coating and encapsulating the skeletal framework of amine-modified silica aerogels with polyurea as depicted in Fig. 10.1. Fig. 10.1. (a) Silica aerogel structure before crosslinking and (b) x-aerogel structure after crosslinking [6] Aerogels are reported to be one of the best thermal insulators. When sandwiched between two glass layers, aerogels reduce heat loss coefficient by more than a factor of 10, while preserving the capability of moderately high light transmission [5]. Aerogels are being considered for different aerospace applications, such as a thermal protection system (TPS), catalyst support, or as hosts for a variety of functional materials for chemical, optical, and electronic devices. Cylindrical samples of x-aerogel manufactured in the author’s laboratory are shown in Fig. 10.2. It was observed from the mechanical characterization tests that x-aerogel has very good compressive, tensile, and shear properties, in addition to its low thermal conductivity [4, 6, 8]. Recently, manufacturing of a lightweight cryogenic propellant tank with low thermal conductivity has been proposed using novel x-aerogel material. While the use of composite sandwich panels for cryotanks is not novel, it is feasible that the delamination of facesheet from the core due to cryopumping, analogous to the failure that occurred in the X-33 prototype, could perhaps be prevented through the use of x-aerogel core instead of a standard honeycomb core. 464 S. Roy and A. Hossain
Chapter 10:Modeling of Stiffness,Strength,and Structure 465 江 可 Fig.10.2.Cylindrical samples of x-aerogel It is envisioned that aerogel material can be used as the central core bonded between two facesheets of a sandwich plate.As aerogels are highly porous,facesheets will be necessary to make the sandwich composite panels impermeable for storing cryogenic fuels.Facesheets can be made of carbon fiber-reinforced polymer(CFRP)having high tensile load-bearing capability.Schematics of a typical sandwich plate with a standard honey- comb core and an x-aerogel core at the center are shown in Fig.10.3.As another example of a practical application of x-aerogel,a conceptual design of a prototype cryogenic propellant tank using x-aerogel material is shown in Fig.10.4. Honeycomb core Facesheet Facesheet x-aerogel (a) (b) Fig.10.3.(a)Traditional sandwich panel with honeycomb core and (b)novel sandwich panel with x-aerogel as central core material
Fig. 10.2. Cylindrical samples of x-aerogel It is envisioned that aerogel material can be used as the central core bonded between two facesheets of a sandwich plate. As aerogels are highly porous, facesheets will be necessary to make the sandwich composite panels impermeable for storing cryogenic fuels. Facesheets can be made of carbon fiber-reinforced polymer (CFRP) having high tensile load-bearing design of a prototype cryogenic propellant tank using x-aerogel material is shown in Fig. 10.4. Fig. 10.3. (a) Traditional sandwich panel with honeycomb core and (b) novel sandwich panel with x-aerogel as central core material Chapter 10: Modeling of Stiffness, Strength, and Structure capability. Schematics of a typical sandwich plate with a standard honeycomb core and an x-aerogel core at the center are shown in Fig. 10.3. As another example of a practical application of x-aerogel, a conceptual 465
466 S.Roy and A.Hossain Inlet Port Sensor Port Vacuum Port -Outlet Port Aluminum Liner Aluminized Mylar Storage Tank X-Aerogel Maintained at Vacuum Outlet Jacket- Fig.10.4.Prototype cryotank design concept with x-aerogel material 10.2 Nanostructural Features of Silica Aerogel Aerogel is a class of monolithic material that possesses porous structure Scanning electron microscopy (SEM)[1]and transmission electron micro- scopy (TEM)[9]are widely used to produce direct images of mesoporous structures.For the crosslinked silica aerogel,SEM imaging was conducted for different loading stages,as shown in Fig.10.5.In this figure,the clusters of secondary nanoparticles are clearly visible along with the mesopores The mechanical,thermal,electrical,and optical properties exhibited by x-aerogels are related to their mesoporous cluster assemblies,as depicted earlier in Fig.10.1.The sol-gel manufacturing process can control the geo- metry,porosity,and physical properties of mesoporous silica aerogels by manipulating its chemistry and processing parameters.The stiffness and strength of x-aerogels strongly depend on their microstructural features, such as particle connectivity.However,there is no experimental technique currently available to measure this connectivity directly.As an alternative metric,the self-similar characteristics of aerogel structures can be investi- gated by evaluating their fractal dimension from geometric correlations
Fig. 10.4. Prototype cryotank design concept with x-aerogel material 10.2 Nanostructural Features of Silica Aerogel Aerogel is a class of monolithic material that possesses porous structure. Scanning electron microscopy (SEM) [1] and transmission electron microscopy (TEM) [9] are widely used to produce direct images of mesoporous structures. For the crosslinked silica aerogel, SEM imaging was conducted for different loading stages, as shown in Fig. 10.5. In this figure, the clusters of secondary nanoparticles are clearly visible along with the mesopores. S. Roy and A. Hossain The mechanical, thermal, electrical, and optical properties exhibited by x-aerogels are related to their mesoporous cluster assemblies, as depicted earlier in Fig. 10.1. The sol–gel manufacturing process can control the geometry, porosity, and physical properties of mesoporous silica aerogels by manipulating its chemistry and processing parameters. The stiffness and strength of x-aerogels strongly depend on their microstructural features, such as particle connectivity. However, there is no experimental technique currently available to measure this connectivity directly. As an alternative metric, the self-similar characteristics of aerogel structures can be investigated by evaluating their fractal dimension from geometric correlations. 466
Chapter 10:Modeling of Stiffness,Strength,and Structure 467 Fig.10.5.SEM images of crosslinked silica aerogels.The clusters of secondary nanoparticles (round particles)and the mesopores (dark spots)are clearly visible: (a)30%strain-no appreciable change in mesoporous structure,(b)45%strain- gradual decrease in the mesoporosity,few dark spots,and (c)77%strain appreciable loss of porosity,particles are squeezed closer to one another The shape of clusters or cluster configuration,the existence of voids of all sizes,and the gradual loss of connectivity among mesoporous particles suggest that a fractal dimension can be attributed to the x-aerogel structures as a useful descriptive parameter [5].The fractal dimension of a mesoporous structure can be determined from its particle orientation within a sphere of a given radius or from the slope of a radial distribution function.It was reported in the literature [5]that not only the mass of aerogel but also other properties,such as vibrational dynamics,scale according to aerogel's fractal dimension.In general,due to its inherent porosity,the aerogel morphology represents a fractal dimension of less than 3;and its fractal dimension decreases with decreasing cluster densities,as presented later in this chapter. In this study,a three-dimensional distinct element analysis (DEA) simulation was performed to determine the structure-property relationship of nanostructured x-aerogel material.The model attempted to incorporate microscale effects-such as particle bond stiffness,bond strength,particle frictional coefficient,initial cluster porosity (or density),and density of
Fig. 10.5. SEM images of crosslinked silica aerogels. The clusters of secondary nanoparticles (round particles) and the mesopores (dark spots) are clearly visible: (a) 30% strain – no appreciable change in mesoporous structure, (b) 45% strain – gradual decrease in the mesoporosity, few dark spots, and (c) 77% strain – appreciable loss of porosity, particles are squeezed closer to one another In this study, a three-dimensional distinct element analysis (DEA) simulation was performed to determine the structure–property relationship of nanostructured x-aerogel material. The model attempted to incorporate microscale effects – such as particle bond stiffness, bond strength, particle frictional coefficient, initial cluster porosity (or density), and density of Chapter 10: Modeling of Stiffness, Strength, and Structure The shape of clusters or cluster configuration, the existence of voids of all sizes, and the gradual loss of connectivity among mesoporous particles suggest that a fractal dimension can be attributed to the x-aerogel structures as a useful descriptive parameter [5]. The fractal dimension of a mesoporous structure can be determined from its particle orientation within a sphere of a given radius or from the slope of a radial distribution function. It was reported in the literature [5] that not only the mass of aerogel but also other properties, such as vibrational dynamics, scale according to aerogel’s fractal dimension. In general, due to its inherent porosity, the aerogel morphology represents a fractal dimension of less than 3; and its fractal dimension decreases with decreasing cluster densities, as presented later in this chapter. 467
468 S.Roy and A.Hossain secondary silica particles -into a macroscale structure-property relation- ship for the prediction of Young's modulus and strength.In addition, numerical analyses were carried out to determine the fractal dimension of the aerogel structure while varying its initial cluster porosity or density. Modeling methodology will provide insights for both stiffening and strengthening mechanisms and how these mechanisms can be optimized with minimum weight penalty.Therefore,it is envisioned that numerical modeling will greatly reduce the number of"trial-and-error"experiments necessary to further enhance the properties of this novel material. 10.3 Particle Mechanics for Numerical Modeling of Aerogels Various researchers have developed different cluster aggregation algo- rithms for simulating structural characterizations of mesoporous materials. Diffusion-limited cluster aggregation(DLCA)and reaction-limited cluster aggregation(RLCA)algorithms are some examples [7].A DLCA tech- nique was first developed by a research group at Harvard University to interpret scattering experiments and subsequently used for understanding different phenomenon related to porous media,such as gelation,fractal studies,and scattering spectroscopy.The DLCA algorithm proceeds with random filling of a three-dimensional cubic volume with nonintersecting spheres.The diameters of these spheres are chosen from a Gaussian dis- tribution function.These spheres are then set to diffuse inside the cubic boundary.When in motion,a particle is tested against overlapping with neighboring particles.If an overlap is detected,then that particle is merged to neighboring particles to form a new cluster.The diffusive motion is completed once all the particles have merged to form a single cluster.The process of aggregation is depicted schematically in Fig.10.6.A cluster of three particles is moved and tested for an overlap with neighboring particle as shown in Fig.10.6a.Once an overlap is detected,this cluster is aligned to the neighboring particle as shown in Fig.10.6b.Then,this cluster and particle are merged to form another cluster of four particles as shown in Fig.10.6c.This process continues until all particles have merged to form a final single cluster.After the network connectivity has been determined among the particles using DLCA,this information is transferred to the DEA software.Finally,the DEA can be used to determine the structure- property relationship through numerical simulation
secondary silica particles – into a macroscale structure–property relationship for the prediction of Young’s modulus and strength. In addition, numerical analyses were carried out to determine the fractal dimension of the aerogel structure while varying its initial cluster porosity or density. Modeling methodology will provide insights for both stiffening and strengthening mechanisms and how these mechanisms can be optimized with minimum weight penalty. Therefore, it is envisioned that numerical modeling will greatly reduce the number of “trial-and-error” experiments necessary to further enhance the properties of this novel material. 10.3 Particle Mechanics for Numerical Modeling of Aerogels Various researchers have developed different cluster aggregation algorithms for simulating structural characterizations of mesoporous materials. Diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA) algorithms are some examples [7]. A DLCA technique was first developed by a research group at Harvard University to interpret scattering experiments and subsequently used for understanding different phenomenon related to porous media, such as gelation, fractal studies, and scattering spectroscopy. The DLCA algorithm proceeds with random filling of a three-dimensional cubic volume with nonintersecting spheres. The diameters of these spheres are chosen from a Gaussian distribution function. These spheres are then set to diffuse inside the cubic boundary. When in motion, a particle is tested against overlapping with neighboring particles. If an overlap is detected, then that particle is merged to neighboring particles to form a new cluster. The diffusive motion is completed once all the particles have merged to form a single cluster. The process of aggregation is depicted schematically in Fig. 10.6. A cluster of three particles is moved and tested for an overlap with neighboring particle as shown in Fig. 10.6a. Once an overlap is detected, this cluster is aligned to the neighboring particle as shown in Fig. 10.6b. Then, this cluster and particle are merged to form another cluster of four particles as shown in Fig. 10.6c. This process continues until all particles have merged to form a final single cluster. After the network connectivity has been determined among the particles using DLCA, this information is transferred to the DEA software. Finally, the DEA can be used to determine the structure– property relationship through numerical simulation. 468 S. Roy and A. Hossain
Chapter 10:Modeling of Stiffness,Strength,and Structure 469 (e) Fig.10.6.Aggregation process for particles in DLCA A particle flow code in three dimensions(PFC3)[3]simulates mecha- nical behavior of mesoporous structures when a bonded assembly of spherical particles is available from DLCA,as described earlier.PFC3D is classified as a discrete or distinct element analysis code as it allows finite displacements and rotations of discrete bodies and recognizes new contacts automatically.PFC models are categorized as direct damage-type numerical models in which deformation is not a function of prescribed relationships between stress and strain but of changing microstructure.The numerical model is composed of distinct particles that displace independently from one another and interact only at contacts.Newton's laws of motion provide the fundamental relations between particle motion and forces.The com- plex nature of mesoporous structures can be modeled by bonding particles together at their contact points and allowing the bond breakage for excessive loading to exceed the bond strength.The PFC3 conducts a particle flow model with the following assumptions: 1.The particles are treated as rigid bodies and are spherical in shape. The deformation of a packed-particle assembly results primarily from the sliding and rotation of rigid particles and not from the individual particle deformation. 2.Particles are in contact with each other.Contacts among the particles occur over a vanishingly small area,i.e.,at a point,where bonds can exist. 3.Behavior at contacts uses a soft-contact approach wherein the particles are allowed to overlap one another at contact points. 4.The magnitude of the overlap is related with contact force via the force-displacement law,and all overlaps are small enough compared to particle sizes. In addition to spherical particles (referred to as "balls"),the PFC3D includes"walls"to apply velocity boundary conditions for compaction and confinement of particle assemblies.The balls and walls interact with one another via forces that arise at contacts.The equations of motion are satisfied
Fig. 10.6. Aggregation process for particles in DLCA A particle flow code in three dimensions (PFC3D) [3] simulates mechanical behavior of mesoporous structures when a bonded assembly of spherical particles is available from DLCA, as described earlier. PFC3D is classified as a discrete or distinct element analysis code as it allows finite displacements and rotations of discrete bodies and recognizes new contacts automatically. PFC models are categorized as direct damage-type numerical models in which deformation is not a function of prescribed relationships between stress and strain but of changing microstructure. The numerical model is composed of distinct particles that displace independently from one another and interact only at contacts. Newton’s laws of motion provide the fundamental relations between particle motion and forces. The complex nature of mesoporous structures can be modeled by bonding particles together at their contact points and allowing the bond breakage for excessive loading to exceed the bond strength. The PFC3D conducts a particle flow model with the following assumptions: 1. The particles are treated as rigid bodies and are spherical in shape. The deformation of a packed-particle assembly results primarily from the sliding and rotation of rigid particles and not from the individual particle deformation. 2. Particles are in contact with each other. Contacts among the particles occur over a vanishingly small area, i.e., at a point, where bonds can exist. 3. Behavior at contacts uses a soft-contact approach wherein the particles are allowed to overlap one another at contact points. 4. The magnitude of the overlap is related with contact force via the force–displacement law, and all overlaps are small enough compared to particle sizes. In addition to spherical particles (referred to as “balls”), the PFC3D includes “walls” to apply velocity boundary conditions for compaction and confinement of particle assemblies. The balls and walls interact with one another via forces that arise at contacts. The equations of motion are satisfied Chapter 10: Modeling of Stiffness, Strength, and Structure 469
470 S.Roy and A.Hossain for each ball;however,they are not satisfied for each wall,i.e.,forces acting on a wall do not influence its motion.Instead,its motion is specified by the user and remains constant regardless of the contact forces acting upon it. The calculation cycle in PFC3D is a time-stepping algorithm that requires repeated applications of the laws of motion to each particle,a force-displacement law to each contact,and an updating of wall positions. Contact among particles forms and breaks automatically during the course of a simulation.The calculation cycle is shown in Fig.10.7.At the start of each time step,a set of contacts is updated from known particle and wall positions.The force-displacement law is then applied to each contact to update contact forces.Next,the law of motion is applied to each particle to update its velocity and position.The constitutive behavior used in PFC3D is mainly represented by contact models which,in essence,describe physical behavior at each contact by stiffness,slips,and bonding models. Update particle and wall positions and set of contacts Force-Displacement law Law of motion (Applied to each contact) (Applied to each particle) *Relative motion *Resultant force and moment *Constitutive law Contact forces Fig.10.7.Calculation cycle used in PFC3D The contact stiffness relates contact forces and relative displacements in normal and shear directions.The normal stiffness is a secant stiffness since it relates total normal force to total normal displacement.The shear stiffness represents a tangent stiffness as it relates the shear force and displacement in incremental form.The normal and shear stiffness are expressed in(10.1)and (10.2),respectively P=kovo? (10.1) △P=kAy (10.2) Here,P,k,and v indicate force,stiffness,and particle velocity, respectively.Subscripts n and s represent normal and shear components, respectively
for each ball; however, they are not satisfied for each wall, i.e., forces acting on a wall do not influence its motion. Instead, its motion is specified by the user and remains constant regardless of the contact forces acting upon it. The calculation cycle in PFC3D is a time-stepping algorithm that requires repeated applications of the laws of motion to each particle, a force–displacement law to each contact, and an updating of wall positions. Contact among particles forms and breaks automatically during the course of a simulation. The calculation cycle is shown in Fig. 10.7. At the start of each time step, a set of contacts is updated from known particle and wall positions. The force–displacement law is then applied to each contact to update contact forces. Next, the law of motion is applied to each particle to update its velocity and position. The constitutive behavior used in PFC3D is mainly represented by contact models which, in essence, describe physical behavior at each contact by stiffness, slips, and bonding models. Fig. 10.7. Calculation cycle used in PFC3D [3] The contact stiffness relates contact forces and relative displacements in normal and shear directions. The normal stiffness is a secant stiffness since it relates total normal force to total normal displacement. The shear stiffness represents a tangent stiffness as it relates the shear force and displacement in incremental form. The normal and shear stiffness are expressed in (10.1) and (10.2), respectively n nn P = k v , (10.1) s ss P kv = . (10.2) Here, P, k, and v indicate force, stiffness, and particle velocity, respectively. Subscripts n and s represent normal and shear components, respectively. S. Roy and A. Hossain ∆ ∆ 470
Chapter 10:Modeling of Stiffness,Strength,and Structure 471 If the contact normal stiffness is altered during the course of simulation, there will be an immediate effect upon the entire assembly.Whereas,if the shear stiffness is altered,it will only affect the new increment of shear force.Two stiffness models,linear and Hertz-Mindlin,are available in PFC3D for representing linear and nonlinear relations between force and displacement,respectively.The slip model in PFC3D allows two entities in contact to slide relative to one another.A separation occurs if they are not bonded and a tensile force develops between them.The slip condition exists when the shear component of force reaches its maximum limit PFC3D allows particles to be bonded together at contacts and supports two types of bonding models:a contact-bond model and a parallel-bond model.Both bonds can be envisioned as a kind of glue joining two particles. The contact-bond glue is of a vanishingly small radius that acts only at the contact point,while the parallel-bond glue is of a finite radius that acts over a circular cross-section lying between the particles.The contact bond can only transmit a force,while the parallel bond can transmit both a force and a moment.Both types of bonds may be active at the same time; however,the presence of a contact bond inactivates the slip model.The bonding logic is illustrated in Fig.10.8. Contact Bond Deformation is assumed to occur at Contact bond exists over vanishingly small area contact point only. of contact point.It does not resist moment and Linear contact law: breaks if the nommal and shear force exceeds the bond strength. Pn=kv,and AP,=kAv, Parallel Bond P=Contact force,k=Contact stiffness and v=Displacement Subscripts"n”and“s"represent normal and shear components,respectively. Hertz-Mindlin contact law: Nonlinear relationship between force and displacement Parallel bond exists over larger small area of Slip condition: contact point.It does resist both force and moment and breaks if the normal and shear P.>uP.where represents friction coefficient stress exceeds the bond strength Fig.10.8.Contact and parallel bonding logics used in PFC3 [3]
If the contact normal stiffness is altered during the course of simulation, there will be an immediate effect upon the entire assembly. Whereas, if the shear stiffness is altered, it will only affect the new increment of shear force. Two stiffness models, linear and Hertz–Mindlin, are available in PFC3D for representing linear and nonlinear relations between force and displacement, respectively. The slip model in PFC3D allows two entities in contact to slide relative to one another. A separation occurs if they are not bonded and a tensile force develops between them. The slip condition exists when the shear component of force reaches its maximum limit. PFC3D allows particles to be bonded together at contacts and supports two types of bonding models: a contact-bond model and a parallel-bond model. Both bonds can be envisioned as a kind of glue joining two particles. The contact-bond glue is of a vanishingly small radius that acts only at the contact point, while the parallel-bond glue is of a finite radius that acts over a circular cross-section lying between the particles. The contact bond can only transmit a force, while the parallel bond can transmit both a force and a moment. Both types of bonds may be active at the same time; however, the presence of a contact bond inactivates the slip model. The bonding logic is illustrated in Fig. 10.8. Fig. 10.8. Contact and parallel bonding logics used in PFC3D [3] Chapter 10: Modeling of Stiffness, Strength, and Structure 471
472 S.Roy and A.Hossain Properties related to particles and their corresponding contacts are required to perform a simulation in PFC3D.The response of a mesoporous material is mainly affected by particle size and packing arrangement. Therefore,the model parameters cannot be related directly to a set of relevant material properties.The relation between PFC model parameters and commonly measured material properties is only known a priori for certain simple packing arrangements.In case of arbitrary packing or particle assemblage,the relation is found by means of a calibration process where repeated simulations are required to mimic the true material responses. The user needs to specify parameters related with particle contact stiffness, particle friction coefficients,bond strengths,and others to simulate a cor- responding set of macroresponses,such as elastic constants and peak strength envelope,etc.To get a rough estimate of particle contact stiffness and bond strength,PFCprovides the following two equations E=- (10.3) 4R' 0= Sn 4R2 (10.4) where E is the Young's modulus of the particle assembly as obtained from laboratory tests,kn is the normal contact stiffness of the particles,R is the particle radius,o is the measured tensile strength of the particle assembly, and sn is the normal bond strength in particle contacts.The shear com- ponents of stiffness and bond strength ks and ss are taken to be some fraction or equal to their respective normal components.These two relations are derived for a cubic array of particles,which may not be a correct repre- sentation of the actual particle arrangement in laboratory samples.Neverthe- less,these equations provide useful information that could be used to obtain first estimates of the micromechanical parameters.A flowchart is provided in Fig.10.9 to help the reader better understand the simulation process. 10.4 Materials Characterization of X-Aerogel Through Compression Experiments Compression experiments were performed to obtain material responses of x-aerogel specimens,which were subsequently used to verify the numerical results obtained from PFC3D.During the experiments,aerogel samples were
Properties related to particles and their corresponding contacts are required to perform a simulation in PFC3D. The response of a mesoporous material is mainly affected by particle size and packing arrangement. Therefore, the model parameters cannot be related directly to a set of relevant material properties. The relation between PFC model parameters and commonly measured material properties is only known a priori for certain simple packing arrangements. In case of arbitrary packing or particle assemblage, the relation is found by means of a calibration process where repeated simulations are required to mimic the true material responses. The user needs to specify parameters related with particle contact stiffness, particle friction coefficients, bond strengths, and others to simulate a corresponding set of macroresponses, such as elastic constants and peak strength envelope, etc. To get a rough estimate of particle contact stiffness and bond strength, PFC3D provides the following two equations n , 4 k E R (10.3) = n t 2 , 4 s R σ = (10.4) where E is the Young’s modulus of the particle assembly as obtained from laboratory tests, kn is the normal contact stiffness of the particles, R is the particle radius, σt is the measured tensile strength of the particle assembly, and sn is the normal bond strength in particle contacts. The shear components of stiffness and bond strength ks and ss are taken to be some fraction or equal to their respective normal components. These two relations are derived for a cubic array of particles, which may not be a correct representation of the actual particle arrangement in laboratory samples. Nevertheless, these equations provide useful information that could be used to obtain first estimates of the micromechanical parameters. A flowchart is provided in Fig. 10.9 to help the reader better understand the simulation process. S. Roy and A. Hossain 10.4 Materials Characterization of X-Aerogel Through Compression Experiments Compression experiments were performed to obtain material responses of x-aerogel specimens, which were subsequently used to verify the numerical results obtained from PFC3D. During the experiments, aerogel samples were 472
