CN-121859673-B - Dynamic fracture phase field calculation method for hybrid unit beam structure

CN121859673BCN 121859673 BCN121859673 BCN 121859673BCN-121859673-B

Abstract

The invention belongs to the technical field of computational fracture mechanics, and particularly relates to a method for calculating a dynamic fracture phase field of a hybrid unit beam structure. The method comprises the steps of determining material parameters required by calculation of a dynamic fracture phase field of a mixed unit beam structure, establishing a mathematical model of the dynamic fracture phase field of the mixed unit beam structure according to a Lagrange action amount principle, wherein the mathematical model comprises a displacement field control equation and a phase field evolution control equation, performing numerical dispersion on the mathematical model to establish a discrete model of the mixed unit beam structure, and performing numerical solution on the mathematical model after the numerical dispersion by combining a Newmark method with an interleaving algorithm until the displacement field and the phase field meet convergence conditions, so as to obtain an evolution result of the dynamic fracture phase field of the mixed unit beam structure. The invention solves the problem that the existing fracture phase field model can not accurately simulate the crack evolution of the beam structure in the thickness under the dynamic load, and simultaneously obviously reduces the calculation cost of simulating the dynamic fracture of the beam structure.

Inventors

CHENG FEI
YU XINHAO
XU PENGKAI
ZHANG YUMENG
YIN YUE
ZUO WENJIE

Assignees

吉林大学

Dates

Publication Date: 20260508
Application Date: 20260317

Claims (10)

1. The method for calculating the dynamic fracture phase field of the mixed unit beam structure is characterized by comprising the following steps of: Step one, determining material parameters required by dynamic fracture phase field calculation of a mixed unit beam structure; Step two, establishing a mathematical model of the dynamic fracture phase field of the mixed unit beam structure according to the Lagrange action quantity principle, wherein the mathematical model comprises a displacement field control equation and a phase field evolution control equation; thirdly, performing numerical value dispersion on the mathematical model, and establishing a dispersion model of the mixed unit beam structure, wherein a displacement field is dispersed by using a beam unit grid, and a phase field is dispersed by using a quadrilateral unit grid; The discrete model is as follows: ; Wherein, the Is a quality matrix; is a displacement field stiffness matrix; Is acceleration; Is a displacement field; is an external force vector; Is a phase field stiffness matrix; Is a phase field; is a phase field driving force vector; Step four, solving a displacement field and a phase field in the discrete model by adopting a Newmark method and combining an interleaving algorithm until the displacement field and the phase field meet convergence conditions, and obtaining a dynamic fracture phase field evolution result of the mixed unit beam structure; The Newmark method is specifically expressed as that the beam structure is at the first position Displacement, velocity and acceleration of time steps, by The displacement, the speed and the acceleration of the time walking beam structure are obtained: ; Wherein, the 、 And Respectively the first Displacement, speed and acceleration of the individual time walking beam structure; 、 And Respectively the first Displacement, speed and acceleration of the individual time walking beam structure; Is a time step; is the time step; And Setting parameters for guaranteeing unconditional stability of numerical integration process in Newmark method , 。
2. The method for calculating dynamic fracture phase field of a hybrid cell beam structure according to claim 1, wherein in the first step, the material parameters required for calculating the dynamic fracture phase field of the hybrid cell beam structure are determined, specifically including Young's modulus Density of mass Critical energy release rate Length dimension parameter Width of beam Length of beam Thickness of sum beam 。
3. The method for calculating the dynamic fracture phase field of the hybrid cell beam structure according to claim 1, wherein in the second step, the lagrangian action amount is composed of kinetic energy, surface energy, strain energy and external force work, and specifically expressed as follows: ; Wherein, the As a Lagrange function of the amount of work; Is the kinetic energy of the system; Is the strain energy of the system; Is the surface energy of the system; Is external force work.
4. The method for calculating the dynamic fracture phase field of the hybrid cell beam structure according to claim 1, wherein in the second step, the mathematical model includes a displacement field control equation and a phase field evolution control equation, which are specifically expressed as follows: ; Wherein u is the displacement field; Is acceleration; Is mass density; is the axial stress of any point in the beam; axial strain at any point in the beam; And Representing the distributed load applied in the beam axial direction and the transverse direction, respectively; And The axial displacement and the transverse displacement of the neutral axis upper point of the beam are respectively; Is a phase field; is a history variable; Is the critical energy release rate; is a length dimension parameter; is a phase field gradient; is a variable division operator; Is virtual displacement; Is axially strained Any virtual strain of (2); And Representing imaginary displacement components of the neutral axis of the beam in the axial direction and the transverse direction respectively; is a virtual phase field; is a virtual phase field gradient; is a computational domain; is a volume integral infinitesimal; To calculate the domain volume fraction.
5. The method for calculating the dynamic fracture phase field of the mixed unit beam structure according to claim 1 is characterized in that in the third step, a mixed unit beam structure discrete model is built, beam unit discrete is adopted for a displacement field, quadrilateral unit discrete is adopted for a phase field, two layers of quadrilateral units are distributed in a projection area of each beam unit in the thickness direction, and information exchange between the displacement field and the phase field is achieved through building a geometric mapping relation between the beam units and the quadrilateral units.
6. The method for calculating the dynamic fracture phase field of the mixed unit beam structure according to claim 5 is characterized by comprising the steps of dividing a cross beam unit grid along the axial direction of the beam structure, wherein the node degrees of freedom of the beam units comprise axial displacement, transverse displacement and cross section corners, dispersing the displacement field, sequentially establishing an upper quadrilateral unit and a lower quadrilateral unit grid in the thickness direction of the cross section of each beam unit, wherein the upper quadrilateral unit and the lower quadrilateral unit are both two-dimensional plane quadrilateral units and have only the phase field degrees of freedom and dispersing the phase field, and establishing the coupling relation between the displacement field and the phase field through sharing the same Gaussian integration point by the upper quadrilateral unit, the lower quadrilateral unit and the corresponding beam units, so that the mixed unit beam structure discrete model is obtained.
7. The method for calculating the dynamic fracture phase field of the hybrid cell beam structure according to claim 6, wherein the geometric mapping relationship between the beam cells and the corresponding quadrilateral cells can be expressed as: with the ith beam unit From two adjacent beam joints And The beam unit corresponds to an upper quadrangular unit in the thickness direction And a lower quadrangular unit The corresponding quadrilateral unit node numbering rules are defined as follows: ; Wherein, the Numbering beam nodes; the four nodes of the upper quadrilateral unit corresponding to the beam unit are sequentially positioned at the left upper, left middle, right middle and right upper positions; The beam units are corresponding lower quadrilateral units, four nodes of the beam units are sequentially positioned at the left middle, the left lower, the right lower and the right middle, each beam unit is corresponding to the upper quadrilateral unit and the lower quadrilateral unit which comprise eight nodes, the bottom edge of the upper quadrilateral unit is coincident with the top edge of the lower quadrilateral unit, and the upper quadrilateral unit and the lower quadrilateral unit share a number on a neutral axis And Each beam unit actually corresponds to six quadrilateral unit nodes; set the ith beam node Is positioned on the neutral axis and has the coordinates of Then mapping into three quadrilateral element nodes in the beam thickness direction: ; Wherein, the Is the horizontal coordinate of the ith beam node; is the vertical coordinate of the ith beam node; is the top edge node of the upper quadrilateral unit; is the bottom edge node of the lower quadrilateral unit; is a common node at the neutral axis, which serves as both the bottom side node of the upper quadrilateral unit and the top side node of the lower quadrilateral unit; Is the thickness of the beam.
8. The method for calculating the dynamic fracture phase field of the hybrid cell beam structure according to claim 1, wherein in the fourth step, a nested loop structure is adopted by combining the Newmark method with an interleaving algorithm solving process, namely, the Newmark method is used as an outer loop, and interleaving iteration of a displacement field and a phase field is used as an inner loop; When the outer circulation of each time step starts, firstly calculating predicted values of displacement, speed and acceleration by utilizing a Newmark method, then entering the inner circulation, and executing the following steps: (1) Fixed phase field, solving displacement field control equation to obtain displacement field increment And updating the values of displacement, speed and acceleration; (2) Updating the history variable according to the updated displacement field; (3) Fixed displacement field, solving phase field evolution control equation to obtain phase field increment ; Cyclically executing (1) to (3) until the displacement field increment is performed at the time step Sum phase field delta The Euclidean norm of (2) meets the convergence condition, and then enters the outer loop of the next time step, wherein the inner loop is used for solving the increment of the displacement field Sum phase field delta The mathematical model of (a) is specifically expressed as: ; Wherein, the Is a stiffness matrix associated with the displacement field; Is a stiffness matrix related to the phase field; the residual vector after the displacement field control equation is discretized; a residual vector after the phase field evolution control equation is discretized; Is the displacement field increment; is a phase field increment.
9. The method for calculating the dynamic fracture phase field of the hybrid cell beam structure according to claim 8, wherein the history variable is the maximum tensile strain energy density experienced by the material during the loading history, namely: ; Wherein, the Is a position vector; max is the maximum operator; Is strain; Is tensile strain energy density; As a function of the time variable, Is a time range Any time in between.
10. The method for calculating dynamic fracture phase field of hybrid cell beam structure according to claim 1, wherein the convergence condition is displacement field increment in the current iteration step Sum phase field delta Is smaller than a predetermined tolerance The method comprises the following steps: ; ; Wherein, the Is the euclidean norm; an ith component that is the displacement field increment; An ith component that is a phase field increment; Is the total number of components.

Description

Dynamic fracture phase field calculation method for hybrid unit beam structure Technical Field The invention belongs to the technical field of computational fracture mechanics, and particularly relates to a method for calculating a dynamic fracture phase field of a hybrid unit beam structure. Background The method can accurately and efficiently predict the damage mode of the beam structure under the action of dynamic load, and has important significance for safety design and reliability evaluation of large engineering structures. In the existing structure fracture prediction method, physical tests often face the limitations of complex devices, high cost and the like, so numerical simulation gradually becomes a main means for predicting the structure fracture behavior. Existing numerical methods for simulating fracture can be broadly divided into two categories, discrete and continuous. The discrete method is difficult to effectively describe complex behaviors such as branching and merging of cracks in an evolution process. As one of the continuous methods, a fracture phase field method, which is capable of naturally simulating initiation, propagation, branching, and merging of cracks by introducing a phase field to perform a diffused representation of the cracks, has been attracting attention in recent years. However, the existing dynamic fracture phase field method generally adopts a two-dimensional or three-dimensional continuum unit to discrete the structure, so that the number of the calculation degrees of freedom is large, and the calculation cost is high. In addition, for fracture analysis of the beam structure, the traditional phase field model based on the one-dimensional beam unit presumes that the phase field evolution in the thickness direction of the beam is consistent, and the presumption is difficult to accurately reflect the actual fracture behavior of the beam structure under the dynamic loading condition. In order to overcome the defects, the invention provides a method for calculating the dynamic fracture phase field of the mixed unit beam structure, which adopts a beam unit and a quadrilateral unit to disperse a displacement field of the beam structure and a phase field representing fracture respectively, so that the fracture evolution behavior of the beam structure in the thickness direction under the action of dynamic load can be accurately described, and the calculation cost is effectively reduced. Disclosure of Invention In order to overcome the defects in the prior art, the invention provides a method for calculating a dynamic fracture phase field of a hybrid unit beam structure, which comprises the following steps: Step one, determining material parameters required by dynamic fracture phase field calculation of a mixed unit beam structure; Step two, establishing a mathematical model of the dynamic fracture phase field of the mixed unit beam structure according to the Lagrange action quantity principle, wherein the mathematical model comprises a displacement field control equation and a phase field evolution control equation; thirdly, performing numerical value dispersion on the mathematical model, and establishing a dispersion model of the mixed unit beam structure, wherein a displacement field is dispersed by using a beam unit grid, and a phase field is dispersed by using a quadrilateral unit grid; The discrete model is as follows: ; Wherein, the Is a quality matrix; is a displacement field stiffness matrix; Is acceleration; Is a displacement field; is an external force vector; Is a phase field stiffness matrix; Is a phase field; is a phase field driving force vector; Step four, solving a displacement field and a phase field in the discrete model by adopting a Newmark method and combining an interleaving algorithm until the displacement field and the phase field meet convergence conditions, and obtaining a dynamic fracture phase field evolution result of the mixed unit beam structure; The Newmark method is specifically expressed as that the beam structure is at the first position Displacement, velocity and acceleration of time steps, byThe displacement, the speed and the acceleration of the time walking beam structure are obtained: ; Wherein, the 、AndRespectively the firstDisplacement, speed and acceleration of the individual time walking beam structure;、 And Respectively the firstDisplacement, speed and acceleration of the individual time walking beam structure; Is a time step; is the time step; And Setting parameters for guaranteeing unconditional stability of numerical integration process in Newmark method,。 The method has the beneficial effects that the method realizes the fracture evolution calculation of the beam structure under the action of dynamic load, solves the problem that the existing method is difficult to accurately describe the crack evolution of the beam structure in the thickness direction under the action of dynamic load, effectively reduces the calcul