A crack dynamic propagation model considering both inertial force and phase-field inertia

Abstract

The application discloses a crack dynamic expansion model considering inertial force and phase field inertia simultaneously, and introduces the contribution of inertial force and phase field inertia of a dynamic system on the basis of a variation framework of a classical phase field fracture theory, and constructs a dynamic damage fracture phase field model suitable for quasi-brittle materials such as rocks. In the model, the control equation is coupled with a momentum conservation equation considering an inertial term and a phase field evolution equation describing damage evolution, and a high-efficiency and robust numerical implementation scheme is developed. The application provides a new and powerful numerical analysis tool for understanding the dynamic damage and fracture mechanism of rock mass under deep high ground stress and strong engineering disturbance environment, and thus provides a more reliable theoretical basis for the prediction, early warning and prevention and control design of dynamic disasters of deep underground engineering.

Description

Technical Field

[0001] This invention belongs to the field of crack dynamic propagation technology, specifically relating to a crack dynamic propagation model that simultaneously considers inertial force and phase field inertia. Background Technology

[0002] Rock mass failure in engineering practice often exhibits strong dynamic properties. Whether it's stress wave propagation from blasting and excavation, rockbursts / rockbursts, or cyclical effects like seismic loads and mechanical vibrations, all place the rock mass in a dynamic environment with high strain rates or complex loading paths. Under these conditions, the mechanical response of rock materials displays significant inertial effects, rate dependence, and energy propagation characteristics. The classic quasi-static phase-field fracture model neglects the inertial terms in the governing equations, thus fundamentally failing to capture key dynamic behaviors such as the interaction between stress waves and cracks, the dynamic propagation velocity limit of cracks, dynamic bifurcation phenomena, and the competition mechanism between kinetic energy and fracture energy. This results in fundamental limitations in simulating rock fragmentation, multi-crack nucleation, and high-speed propagation caused by transient loads such as impacts and explosions. Currently, the forefront of rock dynamics research is gradually moving from traditional stress wave propagation laws and macroscopic dynamic constitutive relations to quantitative descriptions of microscopic damage accumulation, dynamic crack initiation, and propagation mechanisms. However, existing theories still struggle to couple inertial effects with complex fracture processes, necessitating the construction of a more comprehensive dynamic fracture theory framework. Summary of the Invention

[0003] To address the aforementioned problems, embodiments of the present invention propose a crack dynamic propagation model that simultaneously considers inertial force and phase field inertia.

[0004] The crack dynamic propagation model of the present invention, which simultaneously considers inertial forces and phase field inertia, is generally expressed in the form of the total energy functional of the crack dynamic propagation model as follows:

[0005]

[0006] In the formula, The general form of the system's total energy functional. For phase field inertial properties, For displacement field inertial properties, For elastic properties, For fracture energy, The work done by external forces on the system. For displacement, For phase field variables, The first time derivative of the displacement. The first time derivative of the phase field variable;

[0007] The total energy functional considering phase field inertia is specifically expressed as follows:

[0008]

[0009] In the formula, L represents the specific form of the total energy functional considering the phase field inertia. For continuous materials, the overall solution domain is... For phase field variables, For displacement, The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain are determined based on the wave velocity and the critical energy release rate. Let be the square of the first-order time derivative of the phase field variable. The derivative of the position function, For the density of the material, The first time derivative of the displacement is the square of the displacement. It is a degenerate function. For tensile stress, It is compressive stress. For the total linear strain, Here are the phase field regularization parameters. The critical energy release rate. For the spatial derivatives of the phase field variables, For physical strength, To act on the boundary The traction force on the top, The boundary of the traction force. This indicates the location of the crack.

[0010] The strong formal governing equations of the crack dynamic propagation model are as follows:

[0011]

[0012] In the formula, For continuous materials, the overall solution domain is... The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain are determined based on the wave velocity and the critical energy release rate. For displacement, For phase field variables, For the second-order time derivative of the phase field variable, and These are the weighting functions for displacement and phase field variables, respectively. For the density of the material, The second time derivative of the displacement. For tensile stress, It is compressive stress. For the total linear strain, and Let represent the weight functions for the displacement gradient and the phase field variable gradient, respectively. For physical strength, For the degenerate function with respect to the phase field variables The first derivative, The critical energy release rate. Here are the phase field regularization parameters. It is a degenerate function. For spatial differentiation operations, To act on the boundary The traction force on the top, The boundary of the traction force. This indicates the location of the crack.

[0013] The discrete form of the strongly-form governing equations is:

[0014]

[0015] In the formula, and These represent the unknown displacement vector and phase field variable vector, respectively, assembled from the global node degrees of freedom. It is a displacement shape function. For phase field variable shape functions, The second-order time derivative of the unknown phase field variable vector assembled from the global node degrees of freedom. The second-order time derivative of the unknown displacement vector assembled from the global node degrees of freedom. The gradient of the displacement shape function, The gradient of the shape function of the phase field variable. This represents the elastic stiffness matrix affected by the degradation of phase field variables. For the transpose of the shape function of the phase field variable, This is the transpose of the displacement shape function. For the transpose of the gradient of the shape function of the phase field variable, This is the transpose of the gradient of the displacement shape function.

[0016] The beneficial effects of this invention are that, based on the variational framework of classical phase-field fracture theory, it systematically introduces the contributions of inertial forces of the dynamic system and phase-field inertia, constructing a dynamic damage fracture phase-field model applicable to quasi-brittle materials such as rocks. In this model, the governing equations are coupled with the momentum conservation equation considering the inertial term and the phase-field evolution equation describing damage evolution, and an efficient and robust numerical implementation scheme has been developed. This invention provides a new and powerful numerical analysis tool for understanding the dynamic damage and fracture mechanism of rock masses under deep high ground stress and strong engineering disturbance environments, thus providing a more reliable theoretical basis for the prediction, early warning, and prevention and control design of dynamic disasters in deep underground engineering. Attached Figure Description

[0017] Figure 1 The geometry and boundary conditions of the prefabricated notched rectangular plate of the present invention; Figure 2 The crack propagation characteristics of the prefabricated notched rectangular plate of the present invention under different boundary loads are described. Figure 3 This invention describes the variation law of the total kinetic energy of the prefabricated notched rectangular plate under different boundary loads. Detailed Implementation

[0018] Embodiments of the present invention are described in detail below, examples of which are illustrated in the accompanying drawings. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain the present invention, and should not be construed as limiting the present invention.

[0019] 1. Establishment of a dynamic phase-field model for crack propagation

[0020] Consider a fracture mechanics model defined within a global solution domain B of a continuous medium, containing a discrete crack Γ. The boundary of the global solution domain B is denoted as... Within this solution domain, the action of The surface traction force is t, and the solution domain is also affected by distributed body forces b. A phase field variable is introduced. ∈[0,1], serving as a damage field to describe the gradual degradation process of a material from intact to completely fractured. Wherein, =0 corresponds to the complete and undamaged state of the material. =1 indicates that the material has completely fractured, while values ​​in between represent fractures. The value characterizes the transition state of partial damage, thus achieving a regularized characterization of cracks within the continuous medium framework. For orthotropic materials with linear elastic constitutive relations, the total energy of the system can be expressed using a Lagrangian energy functional, which provides a fundamental variational basis for establishing the dynamic phase-field fracture model and deriving the governing equations. When describing the dynamic fracture of materials, in addition to considering the influence of the energy corresponding to the displacement field inertial force on the fracture process, this invention also considers the influence of the energy corresponding to the phase-field inertial force on the fracture process. Therefore, the general form of the total energy functional of the system is expressed as:

[0021] (1)

[0022] In the formula, The general form of the system's total energy functional. For phase field inertial properties, For displacement field inertial properties, For elastic properties, For fracture energy, The work done by external forces on the system. For displacement, For phase field variables, The first time derivative of the displacement. Let be the first time derivative of the phase field variable.

[0023] The dynamic phase-field fracture model developed after considering the phase-field inertia value can reasonably describe the dynamic fracture behavior of materials and accurately predict the crack propagation path and the failure characteristics of materials. The phase-field inertia value defined in this invention... The expression is:

[0024] (2)

[0025] In the formula, The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain can be determined based on the wave velocity and the critical energy release rate. Let be the square of the first-order time derivative of the phase field variable. The derivative of the position function, For continuous materials, the overall solution domain is... Let be the first time derivative of the phase field variable.

[0026] Displacement field inertial performance The expression is:

[0027] (3)

[0028] In the formula, For the density of the material, The first time derivative of the displacement is the square of the displacement. The derivative of the position function, For continuous materials, the overall solution domain is... It is the first time derivative of the displacement.

[0029] To reasonably describe the tensile-compressive asymmetric fracture behavior of materials and avoid non-physical compressive damage evolution, a strain energy decomposition method is employed. The phase field variables are considered... The evolution of the strain should be primarily driven by elastic tensile strain, while the contribution of compressive strain components to damage should be suppressed or completely eliminated. Therefore, elastic strain energy... The expression is:

[0030] (4)

[0031] In the formula, For continuous materials, the overall solution domain is... The derivative of the position function, It is a degenerate function. For tensile stress, It is compressive stress. The total linear strain.

[0032] , , and The expression is:

[0033] (5)

[0034] In the formula, This is a numerical stability factor used to ensure the stability of numerical calculations; in this application, it is taken as... , and Let Lamé constant be . For the total linear strain, For tensile strain, The spatial derivative of the displacement. It is the transpose of the spatial derivative of the displacement. Represents trace operation. These are phase field variables.

[0035] Tensile strain The expression is:

[0036] (6)

[0037] In the formula, Indicates accumulation. and These represent the principal values ​​of the elastic strain tensor and their corresponding characteristic directions, respectively. This is the cross product operation. This is for absolute value operations.

[0038] In the phase-field fracture theory framework based on the variational principle, the original discrete crack is transformed by the phase-field variable. The dispersion characterization, through phase field regularization, transforms the sharp crack geometry into a smooth damage transition zone, thus avoiding the numerical difficulties of directly tracing discontinuous interfaces. Based on phase field variables... fracture energy of materials The expression is:

[0039] (7)

[0040] In the formula, Here are the phase field regularization parameters. The critical energy release rate. For the spatial derivatives of the phase field variables, For phase field variables, For continuous materials, the overall solution domain is... It is the differential of the position function.

[0041] Work done by external forces on the system The expression is:

[0042] (8)

[0043] In the formula, For physical strength, To act on the boundary The traction force on the top, The boundary of the traction force. The location of the crack. For continuous materials, the overall solution domain is... The derivative of the position function, For displacement.

[0044] Based on the above theoretical framework, the overall mechanical behavior of this fracture system can be completely described by its total Lagrange energy functional. This functional is the sum of all energy contributions of the system, representing the total potential energy corresponding to the system state under given loads and constraints. Its specific form, considering phase field inertia, is expressed as follows:

[0045]

[0046] In the formula, L represents the specific form of the total energy functional considering the phase field inertia. For phase field variables, For displacement, The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain can be determined based on the wave velocity and the critical energy release rate. Let be the square of the first-order time derivative of the phase field variable. The derivative of the position function, For continuous materials, the overall solution domain is... For the density of the material, The first time derivative of the displacement is the square of the displacement. It is a degenerate function. For tensile stress, It is compressive stress. For the total linear strain, Here are the phase field regularization parameters. The critical energy release rate. For the spatial derivatives of the phase field variables, For physical strength, To act on the boundary The traction force on the top, The boundary of the traction force. This indicates the location of the crack.

[0047] Within the constructed total Lagrangian energy functional framework, the displacement u and the phase field variables... These constitute two independent but coupled field variables describing the system state. Here, u characterizes the macroscopic deformation motion of the material, while... This characterizes the microstructural evolution of internal damage in the material. According to the variational principle of energy, the true state of the system at any given time should cause the functional to take a stationary value. Therefore, by considering the displacement u and phase field variables in the functional respectively... By performing independent variational operations and setting them to zero, the coupled control equations governing the system's mechanical response and damage evolution can be derived. The derived control equations are as follows:

[0048] (10)

[0049] In the formula, For the second-order time derivative of the phase field variable, The second time derivative of the displacement. For the degenerate function with respect to the phase field variables The first derivative, The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain can be determined based on the wave velocity and the critical energy release rate. For the density of the material, Here are the phase field regularization parameters. The critical energy release rate. For tensile stress, It is compressive stress. For the total linear strain, It is a degenerate function. For physical strength, For spatial differentiation operations, For displacement, These are phase field variables.

[0050] Numerical implementation of the two-phase field equation

[0051] In phase-field fracture theory, the governing equations are typically highly nonlinear and coupled, making analytical solutions difficult to obtain. Therefore, numerical discretization and solution become crucial steps in realizing the predictive capabilities of this model. The core process begins by transforming the strongly-form partial differential equations into a weaker form suitable for numerical computation, followed by spatial and temporal discretization of the system. For the strongly-form governing equations of equation (10), multiplying by the displacement weight function... and phase field weight function By integrating, we can obtain:

[0052]

[0053] In the formula, and These are the weighting functions for displacement and phase field variables, respectively; and These represent the weighting functions for the displacement gradient and the phase field variable gradient, respectively.

[0054] In the numerical discretization process based on the variational principle, and Belongs to a suitable admissible function space: It must satisfy the homogeneous condition on the displacement boundary, and Typically, boundary values ​​do not need to be explicitly defined. In weak-form integrals, and These represent the possible virtual change directions of the system. To use the finite element method for numerical solution, the overall solution domain B needs to be discretized into a set of finite elements, and the field variables and their weighting functions need to be spatially discretized and interpolated. Let the node set of the finite element mesh be N, then the discretized forms of the displacement field, phase field, and their corresponding weighting functions are:

[0055] (12)

[0056] In the formula, and These represent the unknown displacement vector and the unknown phase field variable vector, respectively, assembled from the global node degrees of freedom. It is a displacement shape function. For phase field variable shape functions, The weight function for the unknown displacement vector assembled from the global node degrees of freedom. The weight function is the unknown phase field variable vector assembled from the global node degrees of freedom.

[0057] To achieve the transformation from continuous medium form to discrete algebraic form, it is necessary to establish the interpolation relationship between the field variables within the element and the nodal degrees of freedom using shape function matrices. The interpolation of displacement u is achieved using the matrix... Description, phase field variables The interpolation is performed by the matrix The specific expression of the displacement shape function is closely related to the element type, interpolation order, and isoparametric transformation used. and phase field variable shape function The matrix is:

[0058] , (13)

[0059] In the formula, n represents the number of nodes in each unit. Represents the shape function of each node in the system. It is a displacement shape function. For phase field variable shape functions.

[0060] Displacement shape function and phase field variable shape function The gradients are respectively:

[0061] , (14)

[0062] In the formula, The gradient of the displacement shape function, The gradient of the shape function of the phase field variable. The gradient of the shape function representing the x-coordinate of the number of nodes in the system. The gradient of the shape function representing the ordinate of the number of nodes in the system.

[0063] The precise definition and high-precision calculation of the gradient operator are the numerical foundation for ensuring the accuracy of the displacement solution, accurately capturing damage localization behavior, and ultimately obtaining reliable fracture path prediction. The discrete forms of the displacement field, phase field gradient, and their corresponding weighting functions are:

[0064] (15)

[0065] In the formula, The gradient of the displacement shape function, The gradient of the shape function of the phase field variable.

[0066] Substituting equations (12)–(15) into equation (11), we obtain the discrete form of the strong-form governing equations as follows:

[0067] (16)

[0068] In the formula, The second-order time derivative of the unknown phase field variable vector assembled from the global node degrees of freedom. The second-order time derivative of the unknown displacement vector assembled from the global node degrees of freedom. This represents the elastic stiffness matrix affected by the degradation of phase field variables. For the transpose of the shape function of the phase field variable, This is the transpose of the displacement shape function. For the transpose of the gradient of the shape function of the phase field variable, This is the transpose of the gradient of the displacement shape function.

[0069] According to equation (16), the dynamic propagation law of cracks that simultaneously consider inertial force and phase field inertia can be studied.

[0070] 3. Model Validation

[0071] The mechanism of dynamic tensile crack propagation in a precast notched rectangular plate is analyzed using the model of this invention. The model is as follows: Figure 1 As shown, the model size is 100×40mm. 2 It is subjected to a constant stress load σ = 1 MPa. The model parameters are: elastic modulus of 32 GPa, Poisson's ratio of 0.2, and mass density ρ = 2450 kg / m³. 3 Critical energy release rate G c=3J / m 2 The characteristic parameter ε = 5 × 10 - 4 m. The Rayleigh wave speed is v = 2125 m / s.

[0072] Figure 2 The crack propagation characteristics under different boundary loads reveal a strong dependence of crack propagation on load amplitude: below the critical stress, the crack remains completely stationary; near the critical stress, the material exhibits sensitivity to defect size; and far above the critical stress, the crack undergoes significant dynamic propagation. During dynamic propagation, stress wave effects, material inertia, and energy release rate jointly control crack evolution. Under a high load of 5.5 MPa, the crack accelerates rapidly after initiation, with its velocity proportional to the driving force but constrained by Rayleigh wave velocity. Due to the finite geometry of the plate, stress waves reflected from the boundary may repeatedly act on the crack tip, causing stress field oscillations at the crack tip, which in turn affects the stability of the propagation path and may even induce bifurcation.

[0073] Figure 3 The total kinetic energy variation of the cracked plate under different boundary loads shows that under low stress (0.5 MPa and 1 MPa), the initiation energy is extremely low and increases slowly and linearly with time. This indicates that the crack hardly propagates, and the energy is mainly used for elastic deformation, with the kinetic energy at a low level, consistent with the description that the crack is completely stationary below the critical stress. Under medium stress (4.5 MPa), the crack propagates rapidly in the initial stage, and the kinetic energy increases rapidly; subsequently, due to energy release, stress wave reflection, or crack bifurcation, the kinetic energy fluctuates but remains at a relatively high level overall. Under high stress (5.5 MPa), the crack accelerates rapidly after initiation, and the kinetic energy increases sharply, consistent with the description that the crack undergoes significant dynamic propagation at stresses much higher than the critical stress, and the kinetic energy changes drastically during propagation, significantly affected by stress waves and inertia.

[0074] Although the above embodiments have been shown and described, it is understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Any changes, modifications, substitutions and variations made to the above embodiments by those skilled in the art are within the protection scope of the present invention.

Claims

1. A crack dynamic propagation model considering both inertial force and phase-field inertia, characterized in that, The general form of the total energy functional of the crack dynamic propagation model is as follows: ； In the formula, This is the general form of the system's total energy functional. For phase field inertial properties, For displacement field inertial properties, For elastic properties, For fracture energy, The work done by external forces on the system. For displacement, For phase field variables, The first time derivative of the displacement. The first time derivative of the phase field variable; The total energy functional considering phase field inertia is specifically expressed as follows: ； In the formula, L represents the specific expression of the total energy functional considering the phase field inertia. For continuous materials, the overall solution domain is... For phase field variables, For displacement, The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain are determined based on the wave velocity and the critical energy release rate. Let be the square of the first-order time derivative of the phase field variable. The derivative of the position function, For the density of the material, The first time derivative of the displacement is the square of the displacement. It is a degenerate function. For tensile stress, It is compressive stress. For the total linear strain, Here are the phase field regularization parameters. The critical energy release rate. For the spatial derivatives of the phase field variables, For physical strength, To act on the boundary The traction force on the top, The boundary of the traction force. This indicates the location of the crack.

2. The crack dynamic propagation model considering both inertial force and phase field inertia according to claim 1, characterized in that, The strong formal governing equations of the crack dynamic propagation model are as follows: ； In the formula, For continuous materials, the overall solution domain is... The parameters characterizing the magnitude of the phase field inertial effect throughout the entire material domain are determined based on the wave velocity and the critical energy release rate. For displacement, For phase field variables, For the second-order time derivative of the phase field variable, and These are the weighting functions for displacement and phase field variables, respectively. For the density of the material, The second time derivative of the displacement. For tensile stress, It is compressive stress. For the total linear strain, and Let represent the weight functions for the displacement gradient and the phase field variable gradient, respectively. For physical strength, For the degenerate function with respect to the phase field variables The first derivative, The critical energy release rate. Here are the phase field regularization parameters. It is a degenerate function. For spatial differentiation operations, To act on the boundary The traction force on the top, The boundary of the traction force. This indicates the location of the crack.

3. The crack dynamic propagation model considering both inertial force and phase field inertia according to claim 2, characterized in that, The discrete form of the strongly-form governing equations is: ； In the formula, and These represent the unknown displacement vector and phase field variable vector, respectively, assembled from the global node degrees of freedom. It is a displacement shape function. For phase field variable shape functions, The second-order time derivative of the unknown phase field variable vector assembled from the global node degrees of freedom. The second-order time derivative of the unknown displacement vector assembled from the global node degrees of freedom. The gradient of the displacement shape function, The gradient of the shape function of the phase field variable. This represents the elastic stiffness matrix affected by the degradation of phase field variables. For the transpose of the shape function of the phase field variable, This is the transpose of the displacement shape function. For the transpose of the gradient of the shape function of the phase field variable, This is the transpose of the gradient of the displacement shape function.

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