A numerical calculation method of cohesive element based on integral point stress-displacement average

Abstract

This invention relates to the field of numerical simulation technology in fracture mechanics, specifically disclosing a numerical calculation method for cohesive elements based on stress-displacement averaging at integration points. The method includes: establishing a unit-thickness cohesive element model incorporating cohesive behavior, and setting at least two integration points within it; calculating the corresponding mechanical response at each integration point; performing equivalent processing on the tangential relative displacement to obtain the equivalent tangential displacement; calculating the equivalent cohesive response and damage variables based on the equivalent tangential displacement and the cohesive constitutive relation; performing equivalent processing on the tangential stress based on the equivalent cohesive response to obtain the equivalent tangential stress; and calculating the nodal force matrix to obtain the mechanical response of the structure during crack propagation. This invention reduces the influence of response differences between different integration points on the element stiffness matrix and nodal force vectors, improves the stability of the overall solution process, and enhances the smoothness and reliability of the results during crack propagation.

Description

Technical Field

[0001] This invention relates to the field of numerical simulation technology of fracture mechanics, and in particular to a numerical calculation method for cohesive elements based on stress-displacement averaging at integral points. Background Technology

[0002] In the field of fracture analysis of engineering structures, the finite element method based on cohesion models is widely used to simulate the crack initiation and propagation process of materials. By introducing cohesion relationships at element interfaces or within elements, the mechanical response characteristics during crack propagation can be effectively described.

[0003] In existing technologies, cohesive elements are typically based on numerical integration methods, calculating stress and relative displacement at each integration point, and constructing the element stiffness matrix and nodal force vectors based on the local response at each integration point. This type of method has been widely used in engineering analysis and is adopted by various commercial finite element software.

[0004] However, in practical applications, it has been found that the point-by-point calculation method based on integration points has certain limitations when crack propagation or when the element enters the damage evolution stage. Due to the significant differences in response at different integration points, the internal mechanical state of the element may exhibit inconsistencies, leading to local fluctuations in the calculation results. Simultaneously, the overall element response is highly dependent on individual integration points, easily causing instability in numerical calculations and exhibiting high sensitivity to mesh generation. Furthermore, relying on the local damage state of integration points to characterize the overall damage behavior of the element under partial failure or non-uniform damage conditions can affect the accuracy of damage region identification and calculation during crack propagation.

[0005] Therefore, it is necessary to propose a new numerical calculation method for cohesive elements to improve the above-mentioned problems and enhance the stability and reliability of crack propagation analysis. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the prior art and provide a numerical calculation method based on the equivalent treatment of stress and displacement at integral points, which is applicable to fracture elements containing cohesion in crack propagation analysis.

[0007] To achieve the above objectives, this invention provides a numerical calculation method for cohesive elements based on stress-displacement averaging at integration points, comprising the following steps: S1. Element Construction: Construct a cohesive element model with a unit thickness that includes cohesive behavior. Set at least two integration points in the cohesive element model. The integration points characterize the local mechanical response inside the cohesive element model. The initial state of the element is either the elastic stage or the damage evolution stage. S2. Obtaining physical quantities at integration points: Under external loading, calculate the corresponding mechanical response quantities at each integration point. The mechanical response quantities include stress and the corresponding relative displacement, which are denoted as the local displacement response quantities at each integration point. S3. Equivalent Tangential Displacement Construction: Define the cohesive constitutive relation at the integration point, and based on the local displacement response at each integration point, perform equivalent processing on the tangential relative displacement to obtain the equivalent tangential displacement at the element level. S4. Calculate the equivalent cohesive response: Based on the equivalent tangential displacement and combined with the cohesive constitutive relation, calculate the equivalent cohesive response and damage variables; S5. Equivalent tangential stress construction: Based on the equivalent cohesive response, the tangential stress is equivalently processed to obtain the equivalent tangential stress at the unit level. S6. Calculate the nodal forces of the unit: Based on the equivalent cohesive response, combined with the finite element discretization method, calculate the nodal force matrix to obtain the mechanical response of the structure during crack propagation.

[0008] Preferably, the parameters of the cohesive element model include initial stiffness, interface strength parameters, and damage evolution parameters; The initial stiffness includes normal stiffness and tangential stiffness; the interface strength parameters include normal peak stress and tangential peak stress; the damage evolution parameters include normal ultimate separation displacement and tangential ultimate separation displacement.

[0009] Preferably, in step S3, the equivalent tangential displacement is the arithmetic mean of the relative tangential displacements at each integration point.

[0010] Preferably, in step S4, the damage variable is determined by the initial damage criterion and the failure criterion under mixed loading.

[0011] Preferably, in step S4, the calculation of the equivalent cohesive response and damage variables specifically involves: Elastic phase: When the separation displacement has not reached the damage initiation displacement, the cohesive force and the separation displacement satisfy a linear relationship. The separation displacement includes the normal displacement and the tangential displacement. The expression for the cohesive force at the integration point is: ; ; in, and These represent the cohesive forces in the normal and tangential directions, respectively. and These represent the normal and tangential displacements of the node, respectively. Damage evolution stages: When the equivalent separation displacement under hybrid loading reaches the damage initiation displacement, the interface enters the damage evolution stage, and the damage variables... It evolves gradually with the increase of separation displacement, and the expression is: ; in, ; For equivalent separation displacement, ; For the equivalent initial damage displacement, , The modal mixing coefficient is... This represents the normal displacement at the onset of cohesive damage. This is the equivalent failure displacement. This is the normal displacement when the cohesive force completely fails. During the damage evolution stage, the cohesive force expression at the integration point is: ; .

[0012] Preferably, in step S5, the equivalent tangential stress is obtained by averaging the tangential stress corresponding to each integration point.

[0013] Preferably, the equivalent processing method includes a simple average form and a weighted average form, wherein the weights in the weighted average are set according to the node position, damage variables, or geometric features.

[0014] Preferably, the method is applicable to situations where the cohesive unit is in a state of partial failure or non-uniform damage under mixed-mode loading conditions.

[0015] The present invention employs the above-mentioned numerical calculation method for cohesive elements based on stress-displacement averaging at integral points, and its beneficial effects are as follows: (1) Improve numerical stability: The present invention adopts an equivalent processing method to reduce the influence of response differences between different integration points on the element stiffness matrix and nodal force vector, thereby effectively reducing the possibility of numerical oscillation during the calculation process and improving the stability of the overall solution process; (2) Improved response smoothness and reliability: This invention improves the abrupt change in damage distribution, making the mechanical state inside the unit more continuous, effectively avoiding the discontinuity problem caused by local damage, thereby improving the smoothness and reliability of the results during crack propagation. (3) Reduced grid sensitivity: Compared with traditional methods that rely on the local response of a single integration point, this invention reduces the dependence of the calculation results on the distribution of integration points and grid division, so that analysis results with good consistency can still be obtained under different grid density conditions; (4) Enhanced applicability under mixed-mode loading conditions: Under mixed-mode loading conditions, the response differences at each integration point are more obvious. This invention reduces the impact of local response inconsistency through equivalent processing, making the calculation results more stable and applicable to complex crack propagation analysis problems. Attached Figure Description

[0016] Figure 1 This is a flowchart of a numerical calculation method for cohesive elements based on the average stress-displacement at integral points according to an embodiment of the present invention; wherein, (a) is a schematic diagram of the initial structure and nodal displacements of the cohesive element, (b) is a schematic diagram of the deformation of the cohesive element under loading and the relative displacement components of each node, (c) is a schematic diagram of the equivalent tangential displacement based on the integral point displacement, and (d) is a schematic diagram of the equivalent tangential stress based on the integral point stress, and 1, 2, 3, and 4 are nodes; Detailed Implementation

[0017] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to specific embodiments and the accompanying drawings. It should be understood that these descriptions are merely exemplary and not intended to limit the scope of the invention. Furthermore, descriptions of well-known structures and techniques are omitted in the following description to avoid unnecessarily obscuring the concept of the invention.

[0018] Example 1: As Figure 1 As shown, the present invention provides a numerical calculation method for cohesive elements based on stress-displacement averaging at integration points, comprising the following steps: 1. For example Figure 1 As shown in (a), a cohesive element of unit thickness is constructed, consisting of four nodes 1, 2, 3, and 4. Nodes 1 and 3 form one node pair, and nodes 2 and 4 form another node pair, which are used to describe the cohesive behavior within the element.

[0019] 2. For example Figure 1 As shown in (b), under external loading, each node experiences relative displacement. , , , These are the tangential relative displacement components of each node; , , , For each node, the normal relative displacement components are: This represents the tangential relative displacement along the cohesive interface. This represents the relative displacement along the normal direction.

[0020] 3. For example Figure 1As shown in (c). In this embodiment, the tangential displacement of the node pairs is equivalently processed to characterize the overall element response, specifically as follows: For node pair 1-2, the corresponding equivalent tangential displacement Through the and The average is calculated as follows: ; For node pair 3-4, the corresponding equivalent tangential displacement Through the and The average is calculated as follows: ; Based on the equivalent tangential displacement and the original normal displacement, the cohesive constitutive relation is substituted to obtain the corresponding cohesive response and damage.

[0021] 4. For example Figure 1 As shown in (d), the shear stress at integration points 1 and 2 is averaged in the same way as the tangential displacement, and the nodal force is obtained by integrating the nodal stress.

[0022] Example 2: 1. In this embodiment, a unit thickness and a length with cohesive force of [missing information] are first established. The cohesive model with a diameter of 1 mm includes four nodes: 1, 2, 3, and 4. Nodes 1 and 3 form one pair, and nodes 2 and 4 form another pair. The initial state is that the displacement of each node is 0.

[0023] 2. Apply displacement load: Assuming nodes 3 and 4 are fixed, the tangential and normal displacements of node 1 are as follows: =0.0005mm =0.0002mm; the tangential and normal displacements of node 2 are respectively: =0.05mm =0.08mm; Based on the above displacement, a cohesive model with large gradient strain is constructed.

[0024] 3. Averaged tangential displacement and shear stress at the integration point: First, define the cohesive constitutive relation at the integration point. Taking the bilinear cohesive law as an example, describe the mechanical response of the interface under the separation of normal and tangential forces.

[0025] The parameters of the cohesive model include initial stiffness, interface strength parameters, and damage evolution parameters. The initial stiffness includes normal stiffness (…). ) and tangential stiffness ( Interface strength parameters include normal peak stress ( ); ) and tangential peak stress ( Damage evolution parameters include normal limit separation displacement (); ) and tangential limit separation displacement ( ).

[0026] During the elastic phase, when the separation displacement has not reached the damage initiation displacement, the cohesive force and the separation displacement satisfy a linear relationship, and the expression for the cohesive force at the integration point is: ; ; in, and These represent the cohesive forces in the normal and tangential directions of the node, respectively. and These represent the normal and tangential displacements of the node, respectively. When the equivalent separation displacement under hybrid loading reaches the damage initiation displacement, the interface enters the damage evolution stage, and the damage variables... It evolves gradually with the increase of separation displacement, and the expression is: ; in, ; For equivalent separation displacement, ; The equivalent initial damage displacement depends on the loading method and the initial damage criterion. , The modal mixing coefficient is... This represents the normal displacement at the onset of cohesive damage. The equivalent failure displacement is related to the loading method and failure criterion. This is the normal displacement when the cohesive force completely fails. During the damage evolution stage, the cohesive force expression at the integration point is: ; .

[0027] In this embodiment, the model parameters can be taken as follows: , , , , , The initial damage criterion is the maximum stress criterion, and the failure criterion is the exponential form of the mixed critical fracture energy. By setting the parameters described above, the cohesive model can fully reflect the influence of tangential displacement on the evolution of interface damage under mixed-mode loading conditions, thereby improving the accuracy and stability of crack propagation analysis.

[0028] It should be noted that the above parameters and displacement boundary conditions are only exemplary values. They can be adjusted according to the actual situation under different materials or loading conditions without affecting the implementation of the present invention.

[0029] After averaging mm, mm, mm, substituting the averaged nodal relative displacements into the defined cohesive relationship, the stresses at integration points 1 and 2 are obtained as follows: MPa MPa MPa MPa; The damage states at integration points 1 and 2 are as follows: , ; The tangential stress at the integration point is averaged to obtain: MPa MPa MPa MPa; The Von Mises stresses corresponding to integration points 1 and 2 are respectively: MPa MPa; 4. Finally, the nodal forces are obtained: N, N, N, N; , , , .

[0030] The Von Mises stresses at integration points 1 and 2 corresponding to the original unaveraged tangential displacement and tangential stress are as follows: MPa MPa; The damage values ​​at integration points 1 and 2 corresponding to the original unaveraged tangential displacement and tangential stress are as follows: , ; The nodal forces corresponding to the original unaveraged tangential displacement and tangential stress are as follows: N, N, N, N; , , , .

[0031] Comparing the results of the original unaveraged processing and the processing by the method of this invention, it can be seen that after tangential displacement and tangential stress averaging, the damage value at integration point 2 did not change significantly, from 0.9993 to 0.9992. However, the damage value at integration point 1 increased from 0 to 0.9792, significantly reducing the abrupt change in damage value. The Von Mises stress at the integration point changed from a nine-fold difference to almost uniformity, and the abrupt changes in the horizontal and normal support reactions of the nodes were further reduced. These indicators have a significant effect on improving numerical stability, response smoothness and reliability, and reducing mesh sensitivity. In addition, since only the relative displacement and stress in the tangential direction are averaged, the influence on the reverse support reaction of the element is small. In most cases, the tensile load plays a decisive role in the final failure. Therefore, the method described in this patent can minimize the influence on normal displacement and stress while maintaining the above advantages.

[0032] The equivalent processing method can be a simple average; in other embodiments, a weighted average can also be used, where the weights can be set according to the node position, damage variables, or geometric features.

[0033] Therefore, this invention adopts the above-mentioned numerical calculation method for cohesive elements based on stress-displacement averaging at integration points, which: 1) improves numerical stability: This invention uses an equivalent processing method to reduce the influence of response differences between different integration points on the element stiffness matrix and nodal force vectors, thereby effectively reducing the possibility of numerical oscillations during the calculation process and improving the overall stability of the solution process; 2) improves response smoothness and reliability: This invention improves the abrupt changes in damage distribution, making the mechanical state inside the element more continuous, effectively avoiding discontinuities caused by local damage, thereby improving the smoothness and reliability of the results during crack propagation; 3) reduces mesh sensitivity: Compared with traditional methods that rely on the local response of a single integration point, this invention reduces the dependence of the calculation results on the distribution of integration points and mesh generation, so that analysis results with good consistency can still be obtained under different mesh density conditions; 4) enhances applicability under mixed-mode loading conditions: Under mixed-mode loading conditions, the response differences of each integration point are more obvious. This invention reduces the influence of local response inconsistencies through equivalent processing, making the calculation results more stable and applicable to complex crack propagation analysis problems.

[0034] It is worth noting that all contents not described in detail in this invention are existing technologies and are well known to those skilled in the art.

[0035] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit them. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the technical solutions of the present invention, and these modifications or equivalent substitutions cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A numerical calculation method for cohesive elements based on stress-displacement averaging at integral points, characterized in that, Includes the following steps: S1. Element Construction: Construct a cohesive element model with a unit thickness that includes cohesive behavior. Set at least two integration points in the cohesive element model. The integration points characterize the local mechanical response inside the cohesive element model. The initial state of the element is either the elastic stage or the damage evolution stage. S2. Obtaining physical quantities at integration points: Under external loading, calculate the corresponding mechanical response quantities at each integration point. The mechanical response quantities include stress and the corresponding relative displacement, which are denoted as the local displacement response quantities at each integration point. S3. Equivalent Tangential Displacement Construction: Define the cohesive constitutive relation at the integration point, and based on the local displacement response at each integration point, perform equivalent processing on the tangential relative displacement to obtain the equivalent tangential displacement at the element level. S4. Calculate the equivalent cohesive response: Based on the equivalent tangential displacement and combined with the cohesive constitutive relation, calculate the equivalent cohesive response and damage variables; S5. Equivalent tangential stress construction: Based on the equivalent cohesive response, the tangential stress is equivalently processed to obtain the equivalent tangential stress at the unit level. S6. Calculate the nodal forces of the unit: Based on the equivalent cohesive response, combined with the finite element discretization method, calculate the nodal force matrix to obtain the mechanical response of the structure during crack propagation.

2. The numerical calculation method for cohesive elements based on stress-displacement averaging at integral points according to claim 1, characterized in that, The parameters of the cohesive element model include initial stiffness, interface strength parameters, and damage evolution parameters. The initial stiffness includes normal stiffness and tangential stiffness; the interface strength parameters include normal peak stress and tangential peak stress; the damage evolution parameters include normal ultimate separation displacement and tangential ultimate separation displacement.

3. The numerical calculation method for cohesive elements based on stress-displacement averaging at integral points according to claim 1, characterized in that, In step S3, the equivalent tangential displacement is the arithmetic mean of the relative tangential displacements at each integration point.

4. The numerical calculation method for cohesive elements based on stress-displacement averaging at integration points according to claim 1, characterized in that, In step S4, the damage variable is determined by the initial damage criterion and failure criterion under mixed loading.

5. The numerical calculation method for cohesive elements based on stress-displacement averaging at integration points according to claim 4, characterized in that, In step S4, the calculation of the equivalent cohesive response and damage variables specifically involves: Elastic phase: When the separation displacement has not reached the damage initiation displacement, the cohesive force and the separation displacement satisfy a linear relationship. The separation displacement includes the normal displacement and the tangential displacement. The expression for the cohesive force at the integration point is: ； ； in, and These represent the cohesive forces in the normal and tangential directions, respectively. and These represent the normal and tangential displacements of the node, respectively. Damage evolution stages: When the equivalent separation displacement under hybrid loading reaches the damage initiation displacement, the interface enters the damage evolution stage, and the damage variables... It evolves gradually with the increase of separation displacement, and the expression is: ； in, ; For equivalent separation displacement, ; For the equivalent initial damage displacement, , The modal mixing coefficient is... This represents the normal displacement at the onset of cohesive damage. This is the equivalent failure displacement. This is the normal displacement when the cohesive force completely fails. During the damage evolution stage, the cohesive force expression at the integration point is: ； 。 6. The numerical calculation method for cohesive elements based on stress-displacement averaging at integration points according to claim 1, characterized in that, In step S5, the equivalent tangential stress is obtained by averaging the tangential stress corresponding to each integration point.

7. A numerical calculation method for cohesive elements based on stress-displacement averaging at integral points, as described in any one of claims 3 and 6, characterized in that... The equivalent processing methods include simple averaging and weighted averaging, where the weights in the weighted averaging are set according to node position, damage variables, or geometric features.

8. The numerical calculation method for cohesive elements based on stress-displacement averaging at integration points according to claim 1, characterized in that, The method is applicable to situations where the cohesive unit is in a state of partial failure or non-uniform damage under mixed-mode loading conditions.

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