A method for predicting and calculating elastic modulus of ultra-high performance concrete under multi-scale

Abstract

The application discloses a kind of elastic modulus prediction calculation methods under the multiscale of ultra-high performance concrete, the prediction calculation method includes (1) according to material composition and representative volume element (RVE) size, the structure type of determining solution object;(2) according to the structure type of solution object, RVE unit internal component is divided into matrix phase and inclusion phase;(3) according to the volume content of matrix phase, elastic parameter and the morphology of inclusion phase, elastic parameter, volume content, the local strain concentration tensor of all inclusion phase is calculated;(4) according to the local strain concentration tensor of all inclusion phase, the elastic parameter of RVE is calculated using mean field theory.Compared with conventional test methods such as empirical formula method, finite element calculation method, this method is not only more suitable for a variety of doped ultra-high performance concrete composite materials, but also can improve the calculation efficiency of concrete elastic parameters under different scales.

Description

TECHNICAL FIELD

[0001] The application relates to a method for predicting and calculating the elastic modulus of super high performance concrete in a multi-scale mode, and belongs to the field of building materials. BACKGROUND

[0002] When concrete materials are applied in engineering, the mechanical properties of the materials need to be fully understood to ensure the safety of structures, and the elastic properties have an important influence on the deformation of the structures. Since the elastic properties of concrete are the macroscopic performance of the performance contribution of the materials in different scales, the elastic properties are difficult to understand. In particular, the elastic properties of super high performance concrete formed by adding fibers are difficult to predict and calculate.

[0003] The prediction of the elastic modulus of traditional super high performance concrete is based on empirical formula, but this method has the following problems: on the one hand, the component size is required, and it is difficult to deeply understand the influence of the internal microstructure of concrete on the evolution law of the mechanical properties; on the other hand, a large amount of test cost is consumed, and the material design-elastic modulus quantitative relationship formed by relying on data accumulation lacks sufficient reliability for the prediction of the elastic properties of concrete. Therefore, it is urgent to develop a multi-scale elastic modulus prediction and calculation framework of super high performance concrete composite materials with high prediction accuracy, so as to provide technical support for the realization of the mechanical property design of concrete based on material proportioning. SUMMARY

[0004] The purpose of the application is to provide a method for predicting and calculating the elastic modulus of super high performance concrete in a multi-scale mode, which solves the problems of high test cost and low quantitative precision of the existing empirical formula.

[0005] Technical scheme: The method for predicting and calculating the elastic modulus of super high performance concrete in a multi-scale mode comprises the following steps:

[0006] (1) determining the structure type of the solving object according to the components of the super high performance concrete and the size interval of the representative volume (RVE) unit, wherein the structure type of the solving object is four scales: nanoscale calcium silicate hydrate (C-S-H) structure, microscale cement stone structure, mesoscale mortar structure and macroscale super high performance concrete structure;

[0007] (2) dividing the components in the RVE unit into matrix phase and inclusion phase according to the structure type of the solving object;

[0008] (3) calculating the local strain concentration tensor of all inclusion phases according to the volume content, elastic parameters of the matrix phase and the morphology, elastic parameters and volume content of the inclusion phase;

[0009] (4) calculating the elastic parameters of the RVE by using the average field theory according to the local strain concentration tensor of all inclusion phases.

[0010] wherein, in step (1), when the components of the ultra-high performance concrete are high density C-S-H and low density C-S-H, and the RVE cell size interval is 0.001-0.1 μm, the structure type of the solving object is nanoscale C-S-H structure.

[0011] wherein, in step (1), when the components of the ultra-high performance concrete are C-S-H, unhydrated particles, and fine sand, and the RVE cell size interval is 0.1 μm-0.1 mm, the structure type of the solving object is microscale cement stone structure.

[0012] wherein, in step (1), when the components of the ultra-high performance concrete are cement stone and fine aggregate, and the RVE cell size interval is 0.1 mm-10 mm, the structure type of the solving object is fine mortar structure.

[0013] wherein, in step (1), when the components of the ultra-high performance concrete are mortar and fiber, and the RVE cell size interval is >10 mm, the structure type of the solving object is macroscopic ultra-high performance concrete structure.

[0014] wherein, in step (2), when the solving object is nanoscale C-S-H structure, the matrix phase in the RVE cell is high density C-S-H, and the inclusion phase is low density C-S-H.

[0015] wherein, in step (2), when the solving object is microscale cement stone structure, the matrix phase in the RVE cell is C-S-H, and the inclusion phase is unhydrated particles, fine sand, and pores.

[0016] wherein, in step (2), when the solving object is fine mortar structure, the matrix phase in the RVE cell is cement stone, and the inclusion phase is fine aggregate, aggregate-cement stone ITZ transition zone, and air pores.

[0017] wherein, in step (2), when the solving object is macroscopic ultra-high performance concrete structure, the matrix phase in the RVE cell is mortar, and the inclusion phase is fiber.

[0018] wherein, in step (3), the elastic parameters of the matrix phase and the inclusion phase both include elastic modulus and Poisson's ratio; the volume content of the matrix phase and the inclusion phase are both the ratio of the volume of each phase to the total phase volume under the RVE cell at the structure.

[0019] wherein, in step (3), the morphology of the inclusion phase is obtained by approximating the three-dimensional shape thereof, and the influence of the inclusion morphology on the component is embodied by the Eshelby tensor in the local strain concentration tensor T r .

[0020] Wherein, low density C-S-H, pore, unhydrated particles, fine aggregate, fine sand and the like are considered in the form of circular inclusions to their influence on the representative volume element, and the fiber is considered in the form of long ellipsoidal inclusions according to the aspect ratio to its influence on the representative volume element.

[0021] Wherein, in step (3), the morphology of the inclusion phase is approximated by processing its three-dimensional shape, and the influence of the inclusion morphology on the component is embodied by the Eshelby tensor in the local strain concentration tensor T r :

[0022] When the inclusion phase is a spherical inclusion, the local strain concentration tensor T r is as formula (1):

[0023]

[0024] In the formula, S r is the Eshelby tensor, L r is the stiffness tensor of the inclusion phase, L0 is the stiffness tensor of the matrix phase, and I is the unit tensor matrix.

[0025] When the inclusion phase is an ellipsoidal inclusion, the local strain concentration tensor T r needs to be averaged to obtain, as formula (2):

[0026] T r = [(1-α)(T lo ) -1 +α(T up ) -1 ] -1 (2)

[0027] In the formula, is the lower limit of the inclusion local strain concentration tensor, is the upper limit of the inclusion local strain concentration tensor, and α=x r (1+c r ) / 2 is a smoothing coefficient, wherein x r is the volume content of the inclusion phase, S r is the ellipsoidal Eshelby tensor of the inclusion phase, L r is the stiffness tensor of the inclusion phase, L0 is the stiffness tensor of the matrix phase, and I is the unit tensor matrix.

[0028] Wherein, in step (4), the inclusion phase local strain concentration tensor is brought into the average field, and the elastic parameters of the RVE are calculated by formula (3):

[0029]

[0030] In the formula, c ris the volume content of each phase (matrix phase or each inclusion phase) in the RVE unit; L r is the stiffness tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, T r is the local strain concentration tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, c n is the volume content of each phase (matrix phase or each inclusion phase) in the RVE unit, T n is the local strain concentration tensor of each phase (matrix phase or each inclusion phase) in the RVE unit. Wherein, when the local strain concentration tensor of the matrix phase is T0, then T0=I.

[0031] Beneficial effects: Compared with the prior art, the present application has the following remarkable advantages:

[0032] The present application can predict the elastic modulus of concrete structure under different scales based on the raw material ratio, fully considering the influence of material ratio and mechanical properties on the component. Compared with the conventional test methods such as empirical formula method and finite element calculation method, the present application is more suitable for various doped ultra-high performance concrete composite materials, and can improve the calculation efficiency of concrete elastic parameters under different scales, and realize the accurate prediction of multi-scale elastic properties of ultra-high performance concrete. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 is a schematic diagram of the composition of ultra-high performance concrete under different scales. DETAILED DESCRIPTION

[0034] The technical solutions of the present application will be further described below in combination with the drawings.

[0035] Example 1

[0036] The sample components of the ultra-high performance concrete material include C-S-H, fine sand and unhydrated particles, the RVE size interval is 0.1 μm-0.1 mm, and thus the solving object is the microcement stone structure;

[0037] The phase separation results of the internal components of the microcement stone RVE unit are that the matrix phase is C-S-H, the inclusion phase is unhydrated particles of inclusion phase 1, pores of inclusion phase 2 and fine sand of inclusion phase 3, the influence of inclusion phases 1-3 on the representative volume (RVE) unit is considered in the form of circular inclusions, which is shown as:

[0038]

[0039] In the formula, S i is the circular Eshelby tensor of the i-th phase, L i is the stiffness tensor of the i-th phase, i is the inclusion phase 1-3, L0 is the stiffness tensor of the matrix phase, and I is the unit tensor matrix.

[0040] The elastic parameters and volume content parameters of each phase are taken from the literature 1 “Multi-level homogenization for the prediction of the mechanical properties of ultra-high-performance concrete. Construction and Building Materials, 2019, 229: 116797” as shown in Table 1.

[0041] Table 1 Elastic parameters and volume content of each phase

[0042]

[0043]

[0044] The local strain concentration tensor T i Substituting into the mean field theory, the stiffness matrix of the cement stone RVE is obtained as:

[0045]

[0046] where c i (i = 0, 1, 2, 3) is the volume content of each phase (matrix phase or each inclusion phase) in the RVE unit; L i (i = 0, 1, 2, 3) is the stiffness tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, T i (i = 0, 1, 2, 3) is the local strain concentration tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, c n (n = 0, 1, 2, 3) is the volume content of each phase (matrix phase or each inclusion phase) in the RVE unit, T n (n = 0, 1, 2, 3) is the local strain concentration tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, and when the local strain concentration tensor of the matrix phase is T0, T0 = I.

[0047] Through calculation, the elastic modulus of the ultra-high performance concrete at the microstructure scale of the cement stone is 51.96 GPa, with an error of 0.173% compared with the predicted value of 52.05 GPa in literature 1.

[0048] Example 2

[0049] The material sample components of the ultra-high performance concrete include mortar and fibers, and the RVE size interval is > 10 mm, so the solving object is determined to be the macroscopic ultra-high performance concrete structure.

[0050] The phase separation results of the internal components of the macroscopic ultra-high performance concrete RVE unit are: the matrix phase is mortar, the inclusion phase is fiber, and the aspect ratio is 3000; the influence of the ellipsoidal inclusion on the representative volume element is considered, at this time, the local strain concentration tensor needs to be averaged, which is represented by:

[0051] T1=[(1-α)(T lo ) -1 +α(T up ) -1 ] -1

[0052] In the formula, is the lower limit of the inclusion local strain concentration tensor, is the upper limit of the inclusion local strain concentration tensor, alpha=c1(1+c1) / 2 is the smoothing coefficient, c1 is the volume content of the inclusion phase 1, S1 is the ellipsoidal Eshelby tensor of the inclusion phase 1, L1 is the stiffness tensor of the inclusion phase 1, L0 is the stiffness tensor of the matrix phase, and I is the unit tensor matrix.

[0053] The elastic parameters and volume content parameters of each phase are taken from the literature 1 "Multi-level homogenization for the prediction of the mechanical properties of ultra-high-performance concrete. Construction and Building Materials, 2019, 229: 116797", as shown in Table 2,

[0054] Table 2 Elastic parameters and volume content of each phase

[0055]

[0056]

[0057] The local strain concentration tensor T i of the inclusion phase is substituted into the average field theory, and the stiffness matrix of the ultra-high performance concrete RVE is obtained as follows:

[0058]

[0059] Where, c i (i=0,1) is the volume content of each phase (matrix phase or each inclusion phase) in the RVE unit; L i (i=0,1) is the stiffness tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, and T i(i = 0, 1) is the local strain concentration tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, c n (n = 0, 1) is the volume content of each phase (matrix phase or each inclusion phase) in the RVE unit, T n (n = 0, 1) is the local strain concentration tensor of each phase (matrix phase or each inclusion phase) in the RVE unit, the local strain concentration tensor of the matrix phase is T0 and T0 = I.

[0060] It is calculated that the elastic modulus of the ultra-high performance concrete is 58.13 GPa, and the error with the predicted value 58.28 GPa in document 1 is 0.257%.

[0061] Through the calculation of the elastic modulus of concrete at different scales in two embodiments, and compared with the test results, it can be seen that the application has good consistency compared with the test results.

Claims

1. A method for predicting and calculating elastic modulus of ultra-high performance concrete under multi-scale, characterized in that, The method comprises the following steps: (1) determining the structure type of the solving object according to the components of the ultra-high performance concrete and the size interval of the RVE unit, wherein the structure type of the solving object is four scales: nanoscale C-S-H structure, microscale cement stone structure, mesoscale mortar structure and macroscale ultra-high performance concrete structure; when the components of the ultra-high performance concrete are high-density C-S-H and low-density C-S-H and the size interval of the RVE unit is 0.001-0.1 μm, the structure type of the solving object is the nanoscale C-S-H structure; when the components of the ultra-high performance concrete are C-S-H, un-hydrated particles and fine sand and the size interval of the RVE unit is 0.1 μm-0.1 mm, the structure type of the solving object is the microscale cement stone structure; when the components of the ultra-high performance concrete are cement stone and fine aggregate and the size interval of the RVE unit is 0.1 mm-10 mm, the structure type of the solving object is the mesoscale mortar structure; and when the components of the ultra-high performance concrete are mortar and fiber and the size interval of the RVE unit is >10 mm, the structure type of the solving object is the macroscale ultra-high performance concrete structure; (2) dividing the components in the RVE unit into matrix phase and inclusion phase according to the structure type of the solving object; (3) According to the volume content, elastic parameters of the matrix phase, and the morphology, elastic parameters, volume content of the inclusion phase, the local strain concentration tensor of all inclusion phases is calculated; the morphology of the inclusion phase is obtained by approximating its three-dimensional shape, and the influence of the inclusion morphology on the component is embodied in the local strain concentration tensor T r in the Eshelby tensor: When the inclusion phase is a spherical inclusion, the local strain concentration tensor T r The formula is as formula (1): In the formula, S r is the Eshelby tensor, L r is the stiffness tensor of the inclusion phase, L0 is the stiffness tensor of the matrix phase, and I is a unit tensor matrix. When the inclusion phase is an ellipsoidal inclusion, the local strain concentration tensor T r The averaging is then performed to obtain, for example, equation (2): T r = [(1 - a)(T lo ) -1 + a(T up ) -1 ] -1 (2) wherein, is the lower bound of the local strain concentration tensor of the inclusion, is the upper bound of the local strain concentration tensor of the inclusion, α = c r (1 + c r ) / 2 is the smoothing coefficient, wherein, c r is the volume fraction of the inclusion phase, S r is the ellipsoidal Eshelby tensor of the inclusion phase, L r is the stiffness tensor of the inclusion phase, L0 is the stiffness tensor of the matrix phase, and I is the unit tensor matrix. (4) calculating the elastic parameters of the RVE by using the average field theory according to the local strain concentration tensors of all the inclusion phases.

2. The method of claim 1, wherein the method is characterized by, In step (2), when the solving object is the nanoscale C-S-H structure, the matrix phase in the RVE unit is high-density C-S-H and the inclusion phase is low-density C-S-H.

3. The method of claim 1, wherein the method is characterized by, In step (2), when the solving object is the microscale cement stone structure, the matrix phase in the RVE unit is C-S-H and the inclusion phase is un-hydrated particles, fine sand and pores.

4. The method of claim 1, wherein the method is characterized by, In step (2), when the solving object is the mesoscale mortar structure, the matrix phase in the RVE unit is cement stone and the inclusion phase is fine aggregate, aggregate-cement stone ITZ transition zone and air pores; and when the solving object is the macroscale ultra-high performance concrete structure, the matrix phase in the RVE unit is mortar and the inclusion phase is fiber.

5. The method of claim 1, wherein the method is characterized by: In step (3), the elastic parameters of the matrix phase and the inclusion phase both include elastic modulus and Poisson's ratio; and the volume content of the matrix phase and the inclusion phase is the ratio of the volume of each phase to the total phase volume of the RVE unit at the structure.

6. The method of claim 1, wherein, In step (4), the local strain concentration tensors of the inclusion phases are brought into the average field, and the elastic parameters of the RVE are calculated by formula (3): where c r is the volume fraction of each phase within the RVE element; L r is the stiffness tensor of each phase within the RVE element, T r is the local strain concentration tensor of each phase within the RVE element, c n is the volume fraction of each phase within the RVE element, T n is the local strain concentration tensor of each phase within the RVE element.

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