In this work a three-level model of ceramic composites materials based on a reaction bonded silicon carbide is developed. Numerical solution is based on the method of multiscale homogenization along with the finite element method. As a result a series of local problems on the periodical cells of 3 structure levels are solved. The calculations of stress concentration tensors in matrixes and weighing materials are presented. New criteria for matrixes and weighing materials is used to calculate the strength properties in multiaxis stressed condition. This criteria includes essential diferences (more than an order of magnitude) of ceramic properties under straining and compression. The model which includes scale efect of strength of ceramic composite materials is proposal. The computational research of sequential micro-destruction processes of ceramic composite until complete destruction is done. The results show that changing of concentration of larger fractions is less significant then content of smaller fractions in the presence of polydisperse structure in ceramics.
Composite materials based on reaction bonded silicon carbide matrix (RBSiC) and SiC disperse filler are perspective materials for creation of shockproof protecting systems because of their high strength, stiffness, destruction energy and relatively low cost. However characteristics of this materials significantly depend on manufacturing technological processes and on receipt of composite components. In addition during the hardening details can give strong shrink, giving significant residual stresses. This residual stresses can give deformation and even breakdown in the final product. To select the optimal content of ceramic composite components of SiC system and to calculate strength properties of such materials it is demanded to develop special mathematic model, which can forecast strength properties of composite materials including variation of content, form and disperse filling. This model also should take into account locked-up stresses, appearing in ceramic composite during agglomeration of particles.
Existing analytical and numerical models of composite materials, armed with particles, allow to forecast elastic properties with certain precision, however numerical calculation of strength properties is essentially more complex problem, because it is necessary to build the appropriate model of microcrack emission in heterogeneous structure. Attempts to build such models using simple concentration of finite-element mesh were not successful, because of dramatic increase of non-physical singularity effect of calculation. Widely known commercial software not always allow to get adequate results of microdestruction modeling of composite.
Nowadays the great attention is paid to the development of numerical
finite element methods of microstress modeling in composites [
This work continues the development cycle [
A composite based on reaction bonded silicon carbide consists of a filler and
a silicon carbide matrix. The filler is powder of silicon carbide of the different
fractions. The silicon carbide matrix is synthesized by chemical reaction of
liquid silicon, carbon and solid carbon, which is produced during the pyrolysis of
phenol-formaldehyde resin [
We consider a model of reaction bonded silicon carbide composite material
which has three structure levels [
All structural levels may be considered as independent according to the method
of multiscale homogenization [
Consider the solution of local problems for the periodicity cell of the second
level having the volume . It includes the matrix 2 and fine-disperse filler.
We believe that the PC2 has three-axial symmetry, therefore instead of a full
volume of PC2 we can consider its 1/8th part of volume ˜ . This volume ˜
consist of the components: − 1 pieces of fine-disperse particulate of filler of
the volume ˜ , = 1... − 1, and binding matrix 2 ( = ). For calculating
microstresses in PC2 by homogenization method [
System (1) is supplemented by the special boundary conditions at the surfaces ′ = { = 0.5} of the 1/8th part of PC ′ : ( ) = 1/2¯ , ( ) = 0, ( ) = 0, ̸= ̸= ̸= , ′ : ( ) = (1/4)¯ , ( ) = 0, ( ) = 0, , = {, }, (2) ′ : ( ) = 0, ( ) = 0, ( ) = 0, ̸= ̸= ̸= , where ¯ are the components of the averaged deformation tensor for PC, ( ) ≡ ( ) are the vectors of forces.
The boundary conditions at the symmetry planes = { = 0} are similar to relations (2), where we assume ¯ = 0. 4
Using the numerical solution of problems (1), (2) we find the fields of displacements ( ) and stresses ¯ ( )( ) in the PC2 at given values of average deformations ¯ . (1) These fields are used to find the average values of stress: ¯ =< >=
¯ ( ), ¯ ( ) = ⟨︀ ( )︀⟩ =
( )( ) . ︁∑ , 3 ︁∫ ˜ ¯ = ¯ ( ) ¯ , where Then the components of the tensor of effective elasticity moduli of the composite are calculated by the formulas (3) (4) , = (5) (6) (7) ¯
. effective Poisson constants
= − ¯ crostresses ( ) ( ) = ∑︀ , 3 tensor of elastic compliances ¯
, that is inverse to ¯ constants of the composite, such as effective Young moduli where there is no summation over and . After that we calculate the effective , and technical elastic , and effective shear moduli The components of the tensor of stress concentrations = 1/ ¯ ( ) connect mi ( )( ) in the matrix and the filler (the fine disperse particles SiC) with average stresses ¯ in the PC2 by the formulas ( ) ( ) = ( )
( )¯ , ∈ ˜ , = 1....
The components
( ) in the matrix and the filler are calculated by the formulas ( ) ( ) = ( )( ) ¯ , ∈ ˜ , = 1.... 5
The strength criterion of ceramic materials should take into account the
significant differences in their properties in tension and compression. Therefore, we
introduce a failure criterion of isotropic matrix 2 and filler particles [
( )2 3 ( )2(1 + ( ) ( −( ))) , ︂( ( )2 3 ( )2 − 1
︂)
where ( ) = 12 (⃒⃒ ( )⃒⃒ − ( )), ( ) = 1(1) + 2(2) + 3(3) are the invariants of
−
the stress tensor in the matrix and fillers,
( ) are stress intensity [
( ) , , ( ) are the ultimate compression is introduced strength, ultimate tensile strength and ultimate shear strength. For ultimate strengths the following relationships should be taken into account: > 0, > 0. In (7) a continuous positive function of the 1st invariant ( −( )) > √ 3 , ( −( )) = ⎧ ⎨
− ⎩ ( ), 0, ( ) ,
( ) > 0, − ( ) < ( ) < 0, ( ) < − ( ).
The failure criterion , which is calculated by the formula (7), has the value 0 if the stress is absent in the composite. It is ranged within 0 < ( loaded condition if there is not damage. And it takes the values ( if the fracture initiation occurs at some point . If the failure criterion reaches the value = 1, then we obtain strength surface of a component ( ) ( ) ) ≤ 1 in the ( )) ≥ 1, ( )2 = 3 ( )2(1 + ( ) ( ( ))).
− where = * are coordinates of the point in the PC2, * is the time point at which the condition (11) is executed first, ¯ ( *) are limit stresses.
In the tensile area ( ) > 0 the strength surface is the von Mises ellipsoid ( )2 = 3 ( )2. In the compression area − ( ) < ( ) < 0 the tensile strength is increased. And in the ”supercompression” area ( ) < − surface again is the von Mises ellipsoid, but with the modified tensile strength: ( ) the strength ( )2 = ( )2 .
( ) If the condition (
( )) ≥ 1 is satisfied at the point or in a certain area PC2, there is no complete destruction. This is partial destruction of PC2, hereinafter called microdestruction. Introduce the dependence of the components of the elastic modulus of the failure criterion for accounting microdestruction of components in the model: ( , ) = (1 − ℎ ( ( ( ) ( ) − 1)) ( ) , ∈ ˜ , = 1..., (10) where point.
( ) are the components of the tensor of elasticity moduli of the composite components (they are constants). According to the formula (10) if microdestruction occurs at the point , elasticity modulus is equal to zero at this
To calculate the strength of the composite as a whole, we need to calculate the limit values of average stresses ¯ . At this stresses an initial microdestruction occurs at least in one of its components (fillers or matrix) in a point * ∈ ˜ at a time *, and then complete destruction occurs. For the calculation of limit values of stresses in experimental research usually implement a process of linear load, in which the average stresses are proportional the time: ¯ ( ) = ˜ , where ˜ are the components of the stress gradient tensor. Substituting (5) in the strength criterion of the matrix or fillers (7) we obtain the initial failure condition of composite m∈a ˜x { ( ( ) ( )¯ ( *))} = 1, (8) (9) (11)
After appearance of the initial failure elastic moduli are changed in the destroyed areas of the matrix and/or fillers in accordance with the model described above. With further increase in average stress values ¯ ( ) failure condition (11) is satisfied in a large number of points of PC2, that is, there is the process of propagation of microdestruction. Some area *( ) of the partial destruction of the composite is formed in the periodicity cell 2.
For modeling of effective elastic and strength properties of PC3 and PC1 is used a similar method. The stresses occur due to the thermal strain 0 = ( − *) of the ceramic composite during cooling after the laser sintering. 6
The local tasks (1), (2) are solved by a finite element method which is described in
[
Numerical simulation of microdestruction of ceramics for periodicity cell of the third level Consider the numerical simulation of microdestruction of ceramics for periodicity cell of the third level with the following properties of SiC matrix 3: elastic modulus = 320 GPa, Poisson’s ratio = 0.35, ultimate strength 0 = 0, 07 GPa; 0 = 4 GPa; 0 = 0, 06 GPa. We suppose that the pores have a spherical shape. Fig. 3 shows some of the results of microstresses calculations in the PC3. Fig. 3a) shows the distribution of component 1(11)1 of the stress concentration tensor in the PC3, where concentration of pore before the start failure is equal to 20%. Fig. 3b) shows the distribution of parameter of damage in the PC3 under tension in the direction of 1.
Fig. 4 shows the process of microdestruction in the PC3 (matrix with defect) under compression. The failure of the periodicity cell starts on the surface of the pore (defect) and at first is spread in a direction perpendicular to load direction, and then the failure zone is turned round and spread in the direction of load action to complete destruction of PC3.
a) b) c)
Numerical simulation of microdestruction of ceramics for periodicity cell of the second level Fig. 5 shows the results of numerical solution of the process of microdestruction in the periodicity cell of the second level under compression. These results are calculated taking into account initial technological stresses, which occurs from the application of laser sintering.
a) A mathematical model of microdestruction of reaction bonded silicon carbide has been developed. This model is based on the homogenization method and the finite element method for solution of local problems on periodicity cells. The new strength criterion of ceramic materials has been applied. The comparison with experimental data has shown that this criterion is applicable to solve the problem of microdestruction of the reaction-bonded silicon carbide. It is demonstrated that the developed model allows to simulate the processes of microdestruction of the ceramic composite and can be used as a tool for research and design of new materials with specified properties.