Probabilistic models of construction materials Prof. Liudmyla Trykoz

Transcription

1 Probabilistic models of construction materials Prof. Liudmyla Trykoz Ukrainian State University of Railway Transport, Kharkiv, Ukraine The subject of our research is properties of building materials. The problem is a lack of reliable calculation models of designed predicted properties of materials. The considered models will ensure compliance with real material properties which will appear during the service life. A reason is an existing notions about materials as homogeneous whole substance which has identical properties through cross-section in all directions. Electronmicroscopic photos indicate a clearly marked discrete structure. Therefore the data of calculated models of unbroken substance differ from real materials. Clay Concrete

2 Steel Moreover, the existing testing methods of material properties don t allow to obtain sufficiently accurate results. It isn t always possibly to make the results extrapolation which were got on small samples to a big construction, for example soil massive, concrete structures. According to our notions the material structure determines its properties. The structure is a size, a shape, an interposition and a quantity ratio of structural elements that are containing inside. Each of structural levels is determined by the size of structural elements. а) b) B Щ ЩB B Щ ЩB B Щ S П S П c) d) S П П S B Щ

3 S П C Ц C Ц CЦ Ц C Ц C Ц П S ЦC The scheme of concrete structure at the macro- (a), meso- (b), micro- (c) and submicrolevel (d) B broken stone; S sand; C cement particles; G cement gel and crystals; H hollows with air and water As we can see the concrete is not homogeneous. The discrete model as an alternative to the whole substance model is on the initial stage of its development. We need to study a behavior of a material which consists of particles in three aggregation states. These are solid particles and hollows that are filled both with water and air. The main problem is a stochastic pattern of the structural elements distribution and hollows between them. The size of each particle is a random variable. Particle sizes vary within some limits from the minimum value d min to the maximum value d max. Most often the distribution value law is considered to be normal. For example, the distribution of broken stone particles size for a concrete is shown on the figure.

4 0,632 0,304 0,005 0,007 0,052 0, >30 Фракції, мм Fractions, mm Unimodal and bimodal distribution of sand particles according to their size is presented on figures. 0,432 0,260 0,038 0,121 0,092 0,028 0, >5 Фракції, мм The optic-microscopic photo of sand Unimodal and bimodal distribution of cement particles according to their size is given on figure.

5 0,143 0,062 0,103 0,093 0,097 0,081 0,11 4 0,074 0,088 0,052 0,043 0,017 0,019 0, The particle sizes affect hollowness. The bigger the particle the bigger the hollowness. Therefore, the distribution of particles size determines the distribution of hollow s size. Besides, an extent of hollows filling with water depends on actual humidity. The hollowness determines two principal properties of materials strength and density: the smaller hollowness the bigger strength and density. The bigger the

6 density the bigger the strength. The smaller the amount in hollows the bigger the strength and density. Stre ngt h, MP a Hollowness, % Strength dependence of hardened cement paste on its hollowness We need to build a mathematical model that allows to calculate the material strength based on distribution laws of the solid particle size and the extent of hollows filling with particles of smaller sizes, water and air. The second result of structure heterogeneity is the anisotropy of properties. The anisotropy leads to the various properties (for instance, strength, deformability, electrical resistance, thermal conductivity, permeability) in different directions. The heterogeneity can even appear on the stage of manufacture. For example, the concrete mixture is stratified during it compaction. Accordingly, the construction properties through cross-section become absolutely different. It changes the calculation construction scheme in which the cross-section characteristics are accepted to be equal in all the points. Accordingly strains and deformations of real systems do not correspond to the calculated ones which cause the decrease of durability.

7 Due to various particles sizes the final structure has a random pattern. While the construction is loaded the direct stress σ and tangential stress τ appear. Due to the action of a load the real material structure changes, the redistribution of the particles interposition occurs. The direct stresses compress the bulk, and the tangential stresses shear it. If we can imagine that the element consists of spherical particles connected in contact points then the direct stresses affect the particles compression and reinforce the ties between them. The tangential stresses lead to the relative shear of particles that is destruction. We know the equation which is called Coulomb s law. This equation connects the direct stress σ and the tangential stress τ: τ = σ tttttt φφ + c

8 The plot is shown on figure. Pa Pa But actually this function is not linear. The 1 is the experimental curve; the 2 is its linear approximation. 1 the experimental curve; 2 the linear approximation Moreover, the values of shearing strength parameters are determined only in a lab. But the obtained results are depended very much on the testing methods. As we see in the table the three testing methods give the results which are ten times different for the same soil. Testing scheme Instantaneous shear without preliminary consolidation Fast shear without preliminary consolidation Parameter values φ, degree с, MPa 6 0,02 5 0,027 Slow shear with preliminary consolidation under cut loading 20 0,003

9 It should be mentioned these tests were made for dry consolidated soils. Practically we need to get the soil characteristics in their natural humid state. It is important for calculations of stability, for example foundation pit walls, embankment, and so on. In this case the tangential stresses τ are nonlinear function of the humidity W. We are very interested in an evaluation of stresses in each point of a bulk. According to Cauchy law the stresses in any points of the whole body are completely determined by nine components of stress tensors. Sometimes it is necessary to find the maximal direct stress and maximal tangential stress and the directions in which they act. To solve this problem it will be necessary to make stress tensor transformation that is known as Mohr s circles.

10 However, due to a number of assumptions the calculated values are very far from real ones. For prediction of real stress values it will be necessary to take into account the impact of hollows and humidity upon the final stress values. The deformation anisotropy means the various deformation values in three main axes. Linear deformations ε appear due to the direct stresses σ. Bulk deformations γ appear due to the tangential stresses τ. For simplification of calculations the connection between deformations and stresses is accepted as linear. We can find it from the following system of equations. εε xx = aa 11 σσ xx + aa 12 σσ yy + aa 13 σσ zz + aa 14 ττ yyyy + aa 15 ττ xxxx + aa 16 ττ xxxx εε yy = aa 21 σσ xx + aa 22 σσ yy + aa 23 σσ zz + aa 24 ττ yyyy + aa 25 ττ xxxx + aa 26 ττ xxxx εε zz = aa 31 σσ xx + aa 32 σσ yy + aa 33 σσ zz + aa 34 ττ yyyy + aa 35 ττ xxxx + aa 36 ττ xxxx γγ yyyy = aa 41 σσ xx + aa 42 σσ yy + aa 43 σσ zz + aa 44 ττ yyyy + aa 45 ττ xxxx + aa 46 ττ xxxx γγ xxxx = aa 51 σσ xx + aa 52 σσ yy + aa 53 σσ zz + aa 54 ττ yyyy + aa 55 ττ xxxx + aa 56 ττ xxxx γγ xxxx = aa 61 σσ xx + aa 62 σσ yy + aa 63 σσ zz + aa 64 ττ yyyy + aa 65 ττ xxxx + aa 66 ττ xxxx The coefficients a ij are obtained by the experimental way during the strength testing of materials. However under these conditions the material behaves as a brittle elastic body. Objectively elastic deformations don t develop when construction and structures work under the load in real life. In this case the plastic deformations appear which the connection with stresses is nonlinear. Deformations dependence on stresses As a result these methods does not exactly reflect the behaviour of the material under a load during maintenance time. The known solutions do not consider an influence of the outside conditions on the deformation value. For example, the increase of humidity leads to the deformation rise under the same load. Therefore, the

11 coefficients in these equations are not constants but they are the functions of time, temperature, humidity and other factors. The deformation value can depend on the load direction or the location specimen angle relating to the vertical load. The example of such an anisotropy is a piece of wood with various deformation properties along and across fibres. To design glued materials from wood waste it is necessary to predict in which direction the strength will be the biggest in order to place correctly a construction relating to action loads.

12 Compression strength of wood samples: a along the fibers; b across the fibers Such difference is observed for the swelling value along and across the fibers. Swelling, % Pine swelling: 1 along the fibers; 2, 3 across the fibers Humidity W, % So, there are some problems for the search of the common solutions. 1. The first problem is on the material design stage. We need to build a mathematical model which allows to determine the material strength as the function of the stochastic pattern of the structural elements distribution and hollows between them. 2. The second problem is on the construction design stage. It is necessary to develop a mathematical model which allows to determine the stresses and the deformations at any point of the heterogeneous material. It should take into account the stochastic distribution of the solid, liquid and gaseous phases in the cross-section. 3. The third problem is on the construction service life stage. We expect to have a mathematical model which allows to determine the stresses and the deformations at any point of the heterogeneous material. In this case the model should take into account that the temperature and humidity change affects the stresses and deformation values. It is important for the prediction of a construction durability.

13 In future we would like to get the same models for a description of other properties, for example, electrical resistance, thermal conductivity and so on.