Near-field dynamics simulation method for low-speed impact fracture of ultra-high performance concrete

Abstract

The invention discloses a near-field dynamics simulation method for low-speed impact fracture of ultra-high performance concrete, which comprises the following steps: establishing a completely discrete model of a UHPC component, and realizing coupling of a concrete gel matrix and steel fibers through an Embedded method; a transition unit is introduced, a classic finite element unit and a PeriFEM local unit are connected through keys, force on the keys is modified, and coupling of the two methods is achieved; a continuous local unit is introduced, a motion equation of the local unit is deduced by adopting an explicit central difference algorithm, and an explicit algorithm based on the continuous local unit is constructed and used for solving the motion equation and calculating the damage condition of the UHPC matrix. According to the simulation method provided by the invention, the fracture process of the UHPC under low-speed impact can be effectively simulated, and a new solution of near-field dynamics and finite element coupling is provided.

Description

Technical Field

[0001] The present invention belongs to the field of numerical simulation of concrete materials, and particularly relates to a peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete. Background Art

[0002] In recent years, with the continuous advancement of China's urbanization process, the construction of major infrastructure such as roads, bridges, and marine engineering has developed rapidly, and these large-scale projects are closely related to concrete materials. The modern engineering construction also puts forward higher requirements for the performance of concrete. Ultra High Performance Concrete (UHPC), as a new type of building material, has broad application prospects in the engineering field due to its excellent properties such as high compressive strength and low permeability. However, the application progress of UHPC in actual engineering is relatively slow. One of the main reasons is that there is a lack of systematic theoretical support and standardized guidance for key performance indicators such as the ultimate strength and residual strength after cracking of UHPC. Therefore, in-depth study of the damage and failure mechanism of UHPC materials is of great significance for engineering applications and designs.

[0003] Currently, the modeling of steel fiber ultra-high performance concrete is generally based on the classical finite element method. The classical finite element method performs well in dealing with conventional mechanical problems of linear and homogeneous materials, but when dealing with fracture problems, it is necessary to introduce failure criteria for crack initiation and development, which increases the complexity and uncertainty of calculations. In contrast, peridynamics (PD), as a meshless method, processes the non-local interactions between material points through an integral form of interaction model, and can naturally handle discontinuous problems, especially showing significant advantages in the simulation of crack initiation, propagation, and multi-crack interaction. However, the computational cost of PD is relatively high, especially when dealing with large-scale engineering problems, and the computational efficiency becomes its bottleneck. Therefore, how to combine the advantages of peridynamics and the finite element method, and organically combine the non-local interaction ability of PD with the computational efficiency of FEM, is one of the important directions for the study of damage and failure processes and fracture mechanisms. Summary of the Invention

[0004] The present invention provides a peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete. Based on the PeriFEM (peridynamics-based finite element method), this method uses continuous local elements to establish an explicit algorithm of PeriFEM based on continuous finite elements. On this basis, the PeriFEM method is coupled with the classical finite element method to further improve the computational efficiency without affecting the computational accuracy.

[0005] The technical solution adopted by the present invention to solve its technical problems is as follows:

[0006] A peridynamic simulation method for the low-velocity impact fracture of ultra-high performance concrete, specifically including the following steps:

[0007] S1: Establish a fully discrete model of the ultra-high performance concrete component;

[0008] S2: Couple the PeriFEM region and the classical finite element region according to the region division;

[0009] S3: Define the material properties, add a virtual boundary layer, and apply boundary conditions and external force loads according to the working conditions;

[0010] S4: Establish an explicit algorithm of PeriFEM based on continuous local elements, and give the motion equation of local elements based on the explicit central difference algorithm;

[0011] S5: Solve the motion equation of UHPC material points and calculate UHPC damage according to the explicit algorithm of PeriFEM based on continuous local elements proposed in step S4.

[0012] Further, the specific steps of step S1 to establish a fully discrete model of the ultra-high performance concrete component are as follows:

[0013] S11: Determine the size of the UHPC specimen, uniformly discretize the specimen according to the discrete spacing, and store the coordinates of the UHPC matrix material points;

[0014] S12: Divide the model calculation region into two parts. The region where damage and fracture may occur is calculated by the peridynamic method, while the region where damage and fracture will not occur is calculated by the classical finite element method;

[0015] S13: Calculate the number of fibers to be placed, generate the midpoint and angle parameters of random fibers, and calculate the coordinates of the two endpoints of the fibers according to the fiber length;

[0016] S14: Use the Embedded method to couple the steel fibers on the ultra-high performance concrete gel matrix of steel fibers, so that the displacement of the matrix is exactly the same as the displacement of the steel fibers.

[0017] Further, step S2 is specifically as follows:

[0018] S21: When coupling the PeriFEM region and the classical finite element region, a series of transitional peridynamic elements are generated on the local elements of the PeriFEM method near the classical finite element region. One end of these transitional peridynamic elements is based on the local elements, and the other end is based on the classical finite element elements;

[0019] S22: Determine the particle distribution. The Gauss integration points on the finite element region are the particles in the peridynamic region.

[0020] S23: Establish the coupling method between the classical finite element cells and the local cells. The transitional peridynamic cells only act on the local cells with forces, while the residual for the classical finite element cells is zero. The displacements of the four particles of the transitional cells on the classical finite element cells are determined using the nodal displacements and shape functions of the classical finite element method.

[0021] Further, step S4 is specifically as follows:

[0022] S41: Construct peridynamic continuous local cells based on the classical finite element, with adjacent cells sharing nodes.

[0023] S42: Based on step 1, establish the peridynamic motion equations of the steel fiber ultra-high performance concrete material points using the explicit central difference algorithm. First, assume that at the nth time step, the displacement variable of the material point x is:

[0024]

[0025] where κ and ζ are positive real number and complex number respectively. The stability study requires that for any κ, |ζ| ≤ 1. If this condition is satisfied, then the wave will not grow unbounded over time, thus causing non-convergence.

[0026] Establish the motion equations using the explicit central difference algorithm:

[0027]

[0028] In the formula, ρ (k) represents the mass of the material point x (k) ; ξ (k)(j) represents the relative position of two material points; v c(j) and V (j) represent the volume correction coefficient and the volume of the material point respectively; in addition, in the formula and represent the volume strains of the material point x (k) and x (j) at the nth time step respectively;

[0029] Substitute the displacement variable, and replace the exponential term with sine and cosine, and after arrangement, we can get:

[0030]

[0031] where

[0032]

[0033] Where a, b, and d are peridynamic material parameters, which can be calibrated through simple loading conditions; t is time; and δ is the peridynamic range size.

[0034] Furthermore, step S5 is specifically as follows:

[0035] S51: First, solve the motion equation of the UHPC material points. Equation (3) is a quadratic equation of one variable. Solving it can obtain:

[0036]

[0037] Substitute into Equation (2) to solve the displacement variable at each time step. It should be noted that if the condition is to be satisfied for any κ, then:

[0038]

[0039] A safety factor less than 1 is introduced in Equation (6) The smaller this safety factor, the more accurate the data calculated by the explicit algorithm, and the more iteration steps required to complete the calculation. The safety factor is selected based on a comprehensive consideration of the accuracy requirements and calculation costs;

[0040] S52: During the time integration, calculate the elongation rate of the unbroken bonds within all peridynamic elements, compare it with the critical compressive elongation rate and the critical tensile elongation rate to determine whether these bonds are broken, and introduce a scalar function μ(t, ξ) to correct the force density vector to simulate the bond failure situation;

[0041]

[0042] According to the number of broken bonds, the effective damage at any damage point x can be defined as:

[0043]

[0044] In the formula, V is the volume of the material point, and H is the integration region;

[0045] S53: After the time integration is completed, generate the displacement contour map and the damage contour map, and post-process the results. The displacement contour map is generated based on the built-in function of ABAQUS, and the damage contour map is generated through the VUMAT subroutine of ABAQUS.

[0046] Through the above technical solutions, compared with the prior art, the present invention has the following beneficial effects:

[0047] 1. In the present invention, the fibers are coupled to the steel fiber ultra-high performance concrete gel matrix by the Embedded method. This can make the displacement of the gel matrix exactly the same as that of the steel fibers. When the gel matrix generates strain, the steel fibers also generate strain accordingly. The stress generated by the strain of the steel fibers will be distributed on the surrounding gel matrix, enhancing the gel matrix. Based on this method, the interaction between the steel fibers and the concrete gel matrix can be well simulated;

[0048] 2. An explicit algorithm of PeriFEM based on continuous local elements is established by using the explicit central difference method. The local elements of this algorithm are continuous elements, which are simply and conveniently coupled with the classical finite element method;

[0049] 3. In the present invention, by introducing a transition element, the classical finite element and the PeriFEM local element are connected by a bond and the force on the bond is modified to achieve the coupling of the two methods. This coupling can effectively reduce the calculation cost without affecting the accuracy of the numerical simulation. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 It is a schematic diagram of the geometric model of the three-point bending drop hammer impact test for the embodiment of the present invention;

[0051] Figure 2 It is a schematic diagram of the coupling of the steel fibers and the gel matrix for the embodiment of the present invention;

[0052] Figure 3 It is a schematic diagram of the coupling of the PeriFEM method and the classical finite element method for the embodiment of the present invention;

[0053] Figure 4 It is a flowchart of the explicit algorithm of PeriFEM based on continuous local elements for the embodiment of the present invention;

[0054] Figure 5 It is the matrix damage nephogram at different time steps for the embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0055] Next, the present invention will be further described in combination with the drawings and specific embodiments. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments.

[0056] For a clearer understanding of the technical features, objectives, and effects of the present invention, the specific embodiments of the present invention will now be described in detail with reference to the drawings.

[0057] The present invention provides a method for simulating the low-velocity impact fracture of ultra-high performance concrete by peridynamics, referring to Figure 1Schematic diagram of the geometric model for the three-point bending drop hammer impact test. Taking the three-point bending drop hammer impact test of ultra-high performance concrete as an example, the model uses a rectangular cross-section beam with dimensions of 700 mm × 150 mm. The clear span of the beam is 600 mm, and the span-to-depth ratio is 4 / 1, meeting the requirements of the three-point bending specimen. A hammer head with a mass of 126.6 kg is arranged at the mid-span position. A prefabricated crack with a width of 4 mm and a depth of 45 mm is arranged at the mid-span.

[0058] First, establish a fully discrete model of the ultra-high performance concrete component. The fully discrete model includes a fiber model and a gel matrix. The establishment of the fully discrete model of the UHPC component is divided into the following steps:

[0059] 1. Determine the size of the UHPC specimen. The model uses a rectangular cross-section beam with dimensions of 700 mm × 150 mm. The clear span of the beam is 600 mm, and the span-to-depth ratio is 4 / 1. Discretize the specimen uniformly according to the discrete spacing and store the coordinates of the UHPC matrix material points.

[0060] 2. Refer to Figure 1 Divide the model calculation area. Divide the model calculation area into two parts. The 160-mm mid-span area where damage and fracture may occur is calculated using the peridynamic method, while the area where damage and fracture will not occur is calculated using the classical finite element method.

[0061] 3. Place fibers in the mid-span area. Refer to Figure 1 , in the three-point bending test, arranging fibers only at the mid-span can save a large amount of computational complexity while having no obvious impact on the calculation results. Therefore, random fibers are arranged in the 160-mm peridynamic area at the mid-span of the model, and the midpoint and angle parameters of the random fibers are generated, and the coordinates of the two endpoints of the fibers are calculated according to the fiber length.

[0062] 4. Use the Embedded method to couple the fibers to the steel fiber ultra-high performance concrete gel matrix. Figure 2 is the coupling schematic diagram of the steel fiber and the gel matrix. The Embedded method is a fully coupled method that can keep the displacement of the matrix completely consistent with the displacement of the steel fiber. When the gel matrix generates strain, the steel fiber also generates strain accordingly. The stress generated by the strain of the steel fiber will be distributed on the surrounding gel matrix, enhancing the gel matrix.

[0063] Couple the PeriFEM area and the classical finite element area according to the area division. Refer to Figure 3, a series of peridynamic cells are generated on the local cells of the PeriFEM method near the classical finite element region. One end of these peridynamic cells is based on the local cell, and the other end is based on the classical finite element cell. These cells are called transitional peridynamic cells. These transitional peridynamic cells only act on the local cells with forces, while the residual force on the classical finite element cells is zero, that is, there is no force acting. The displacements of the four mass points of these transitional peridynamic cells on the classical finite element cells are naturally determined only based on the nodal displacements and shape functions of the classical finite element method.

[0064] Next, define the material properties, add a virtual boundary layer, and apply the initial boundary conditions. Apply displacement boundary conditions or other boundary conditions according to the working conditions. The mechanical parameters of the steel fiber and mortar matrix are shown in Table 1.

[0065] Table 1 Mechanical parameters of steel fiber and mortar matrix (a) Mechanical parameters of mortar matrix

[0066]

[0067] (b) Table of fiber mechanical parameters

[0068]

[0069] Next, establish an explicit algorithm for the PeriFEM method based on continuous local cells. Based on the explicit central difference algorithm, give the motion equation of the local cells. The specific steps are as follows:

[0070] 1. Construct peridynamic continuous local cells based on classical finite elements, and share nodes between adjacent cells.

[0071] 2. On the basis of step 1, establish the peridynamic motion equation of the material points of steel fiber ultra-high performance concrete. First, assume that at the nth time step, the displacement variable of the material point x is:

[0072]

[0073] where κ and ζ are positive real numbers and complex numbers respectively. The stability study requires that for any κ, |ζ| ≤ 1. If this condition is satisfied, then the wave will not grow unbounded with time, thus causing non-convergence.

[0074] Use the explicit central difference algorithm to establish the motion equation of the material points of steel fiber ultra-high performance concrete:

[0075]

[0076] In the formula, in the formula, ρ (k) represents the mass of the material point x (k) ; ξ (k)(j)represents the relative position of two material points; v c(j) and V (j) represent the volume correction coefficient and the volume of the material point respectively; in addition, in the formula and represent the volume strains of the material point x (k) and x (j) at the nth time step respectively;

[0077] Substitute the displacement variables, replace the exponential terms with sine and cosine, and after arrangement, we can get:

[0078]

[0079] where

[0080]

[0081] In the formula, a, b, and d are peridynamic material parameters, which can be calibrated through simple loading conditions; t is time; δ is the peridynamic range size.

[0082] Finally, solve the motion equation of the UHPC material point and calculate the UHPC damage. Figure 4 The flow chart is given, and the specific steps are as follows:

[0083] 1. First, solve the motion equation of the UHPC material point. Formula (3) is a quadratic equation of one variable. Solving it, we can get:

[0084]

[0085] Substitute it into formula (2) to solve the displacement variable at each time step. If it is required to satisfy the condition for any κ, then:

[0086]

[0087] A safety factor less than 1 is introduced in formula (6). The smaller this safety factor is, the more accurate the data calculated by the explicit algorithm is, and the more iteration steps are required to complete the calculation. The value of the safety factor is selected according to the comprehensive consideration of accuracy requirements and calculation costs;

[0088] 2. During the time integration, calculate the elongation rate of the unbroken bonds in all peridynamic units, compare it with the critical compressive elongation rate and the critical tensile elongation rate, and judge whether these bonds are broken. A scalar function μ(t, ξ) is introduced to correct the force density vector to simulate the failure of the bonds.

[0089]

[0090] According to the number of broken bonds, the effective damage of any damage point x can be defined as:

[0091]

[0092] where V is the volume of the material point and H is the integration region;

[0093] 3. After the time integration is completed, the displacement contour plot and the damage contour plot can be generated, and the results can be post-processed. Among them, the displacement contour plot is generated based on the built-in function of ABAQUS, and the damage contour plot is generated through the VUMAT subroutine of ABAQUS. In the PeriFEM method, the ratio of the number of broken bonds on this local element to the total number of bonds is defined as the damage value of this local element. Generally, it is considered that when the damage value > 0.5, a macroscopic crack is formed. Figure 5 The matrix damage contour plots of the specimen at 0.2 ms, 0.4 ms, 0.6 ms, 0.8 ms, and 1.0 ms are given. The fracture development of the gel matrix can clearly reflect the fracture behavior of the steel fiber ultra-high performance concrete after impact.

[0094] The above is only a preferred embodiment of the present invention, and does not limit the present invention in any other form. Any modification or equivalent change made according to the technical essence of the present invention still belongs to the scope protected by the present invention.

Claims

1. A peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete, characterized in that: The specific steps include: S1: Establish a fully discrete model of UHPC components; S2: Couple the PeriFEM region and the classical finite element region according to the region division; S3: Define material properties, add virtual boundary layers, and apply boundary conditions and external loads according to the working conditions; S4: Establish the PeriFEM explicit algorithm based on continuous local elements, and give the motion equations of the local elements based on the explicit central difference algorithm; S5: Solve the motion equations of the UHPC material points and calculate the UHPC damage according to the PeriFEM explicit algorithm based on continuous local elements proposed in step S4.

2. The peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete according to claim 1, characterized in that: Step S1 is specifically as follows: S11: determine the size of the UHPC specimen, uniformly discretize the specimen according to the discrete spacing, and store the coordinates of the UHPC matrix material points; S12: Divide the model calculation area into two parts. The area where damage and fracture may occur is calculated using the peridynamic method, while the area where damage and fracture will not occur is calculated using the classical finite element method. S13: Calculate the number of fibers to be placed, generate the midpoint and angle parameters of the random fibers, and calculate the coordinates of the two end points of the fibers according to the fiber length; S14: The steel fiber is coupled to the steel fiber ultra-high performance concrete gel matrix using the embedded method, so that the displacement of the matrix is ​​completely consistent with the displacement of the steel fiber.

3. The peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete according to claim 1, characterized in that: Step S2 is specifically as follows: S21: When coupling the PeriFEM region and the classical finite element region, a series of transitional peridynamic elements are generated on the local elements of the PeriFEM method close to the classical finite element region. These transitional peridynamic elements are based on the local elements at one end and the classical finite element elements at the other end. S22: determining the mass point distribution, and determining the Gaussian integral points on the finite element region as the mass points in the peridynamic region; S23: The coupling mode between the classical finite element unit and the local unit is established. The transitional peridynamic unit has a strong effect only on the local unit, while the residual error for the classical finite element unit is zero. The displacements of the four particles of the transitional unit on the classical finite element unit are determined by the node displacement and shape function of the classical finite element method.

4. The peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete according to claim 1, characterized in that: Step S4 is specifically as follows: S41: Constructing peridynamic continuous local elements based on classical finite elements, with nodes shared between adjacent elements; S42: Based on step S41, the peridynamic motion equation of the material point of the steel fiber ultra-high performance concrete is established by using the explicit central difference algorithm. First, it is assumed that at the nth time step, the displacement variable of the material point x is: The peridynamic equations of motion are established using an explicit central difference algorithm; In the formula, ρ (k) Represents the material point x (k) The quality of (k)(j) Indicates the relative position of two material points; v c(j) and V (j) denote the volume correction coefficient and the material point volume respectively; κ and ζ are positive real numbers and complex numbers respectively; in addition, and Represent the material point x (k) and x (j) Volumetric strain at the nth time step; Substitute the displacement variables and replace the exponential terms with sine and cosine, and we can get: in Where a, b, d are peridynamic material parameters, which can be calibrated by simple load conditions; t is time; δ is the near field size.

5. The peridynamic simulation method for low-velocity impact fracture of ultra-high performance concrete according to claim 4, characterized in that: Step S5 is specifically as follows: S51: Solve the motion equation of the UHPC material point. Formula (3) is a quadratic equation system. Solving it, we can get: Substituting into formula (2) can solve the displacement variable at each time step. If the conditions are to be met for any κ, then: A safety factor less than 1 is introduced in formula (6) The smaller the safety factor, the more accurate the data calculated by the explicit algorithm, and the more iterations are required to complete the calculation; The value of is selected based on the comprehensive consideration of accuracy requirements and computational costs; S52: In the time integral, the elongation of the unbroken bonds in all peridynamic units is calculated and compared with the critical compressive elongation and critical tensile elongation to determine whether these bonds are broken. A scalar function μ( t, ξ) Correct the force density vector; According to the number of broken bonds, the effective damage at any damage point x is defined as: Where V is the volume of the material point and H is the integration area; S53: After the time integration is completed, a displacement cloud map and a damage cloud map are generated, and the results are post-processed.

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