Multi-scale method for delamination failure of impregnated REBCO superconducting coils

Abstract

The present invention relates to the technical field of impregnated REBCO superconducting coils, and provides a same-level multi-scale method for delamination damage of impregnated REBCO superconducting coils, comprising: 1) establishing a same-level multi-scale model; 1.1) impregnated REBCO superconducting coils are represented by macroscopic orthotropic homogenized materials, and a linear elastic solution is obtained; 1.2) defining an initial delamination criterion δ, and checking whether there is an initial delamination criterion δ≥1 on an integral point at each moment of the macroscopic linear elastic solution; 1.3) if there is an initial delamination criterion δ≥1 on an integral point at any moment, the grid of the macroscopic region is replaced with the grid of the mesoscopic region, and a cohesive interface is introduced in the mesoscopic region; 1.4) a coupling interface is introduced between the macroscopic region and the mesoscopic region to connect the macro-mesoscopic region, and a same-level multi-scale model is established; 2) solving the same-level multi-scale model, and outputting and storing the corresponding solution. The present invention can better perform local damage assessment.

Description

Technical Field

[0001] The invention relates to the technical field of impregnated REBCO superconducting coils, and in particular to a same-level multi-scale method for delamination destruction of impregnated REBCO superconducting coils. Background Art

[0002] REBCO coated conductors have become one of the most promising superconducting materials due to their superior current carrying performance and thermal stability. Currently, REBCO coated conductors have shown great application potential in the fields of high field science and high energy physics.

[0003] REBCO coated high-temperature superconducting tape is strain-sensitive, and its current-carrying performance is affected by stress / strain. Excessive stress / strain will cause a sharp and irreversible degradation of the superconducting critical current. Although the high-strength Hastelloy substrate enhances the tensile strength of REBCO coated high-temperature superconducting tape, allowing it to withstand a circumferential tensile stress of more than 700MPa under strong magnetic fields and high current conditions, its layered composite structure also limits the carrying capacity and mechanical properties of the tape in the thickness direction to a certain extent. A large number of experimental tests have shown the peeling strength of the second-generation high-temperature superconducting composite tape under transverse tensile load. The electromagnetic, thermal and mechanical properties of each composite layer are different. The thermal mismatch stress of the high-temperature superconducting tape during the cooling process and the electromagnetic force under high field will cause peeling damage, leading to the destruction and electromechanical failure of the superconducting tape.

[0004] Although the non-insulated technology has been widely used in the field of high-temperature superconducting magnets in recent years due to its quench self-protection characteristics, the epoxy impregnation process still plays an important role in superconducting magnets because it can ensure the structural stability of the coil winding when subjected to huge electromagnetic forces. However, due to the large thermal expansion coefficient of epoxy resin, the superconducting coil will be delaminated due to thermal mismatch stress during the process of cooling from room temperature to low-temperature operating environment. For example, some scholars have conducted a study on the critical current characteristics of REBCO double pancake coils in liquid nitrogen environment (77K) under epoxy impregnation wet winding and dry winding modes. The results show that the epoxy impregnated REBCO double pancake coils cause interface delamination failure of the REBCO coated superconductor itself due to thermal mismatch stress, resulting in a significant reduction in critical current (~8A), which is only 18% of the dry-wound coil. Some scholars have studied the resin strength, resin-copper interface strength and internal interface strength of the strip to determine the possible damage sites in the impregnated coil. The experiment shows that the strength of the resin itself and the resin-copper interface strength are generally higher than the internal interface strength of the strip, which further clarifies the strength competition relationship in the resin-strip-resin system. In order to prevent the delamination of REBCOCC tape in epoxy impregnated windings, people have developed polyimide coating manufacturing technology, because the plastic deformation of polyimide can absorb thermal shrinkage, and the debonding between polyimide and epoxy resin can replace delamination failure. In addition, a method of coating REBCO samples with a release agent to reduce delamination failure is proposed. This is a cost-effective and simple method suitable for epoxy resin impregnated REBCO windings. In addition, the outer diameter / inner diameter ratio of each self-coil is reduced by group impregnation, thereby reducing radial stress, which can also effectively reduce the risk of peeling failure of the coil. However, these methods have a negative impact on the structural stability of the winding and lead to a decrease in the overall structural stability of the coil. Summary of the invention

[0005] The present invention provides a same-level multi-scale method for delamination destruction of impregnated REBCO superconducting coils, which can overcome certain or some defects of the prior art.

[0006] The same-level multi-scale method for delamination destruction of impregnated REBCO superconducting coils according to the present invention comprises the following steps:

[0007] 1) Establish a multi-scale model at the same level;

[0008] 1.1) The impregnated REBCO superconducting coil is represented by a macroscopic orthotropic homogenized material and a linear elastic solution is obtained;

[0009] 1.2) Define the initial delamination criterion δ, and check whether there is an initial delamination criterion δ≥1 at each integration point at each moment of the macroscopic linear elastic solution;

[0010] 1.3) If at any time there is an initial delamination criterion δ≥1 at the integration point, the grid of the macroscopic region at this time is replaced by the grid of the mesoscopic region, and a cohesive interface is introduced between the superconducting layer and the silver layer in the mesoscopic region;

[0011] 1.4) Introduce a coupling interface between the macroscopic region and the mesoscopic region to connect the macroscopic and mesoscopic regions and establish a multi-scale model at the same level;

[0012] 2) Solve the multi-scale model at the same level and output and store the corresponding solution.

[0013] Preferably, in step 1.1), the representative unit RVE method based on micromechanics is used to obtain the equivalent thermoelastic parameters of the homogenized material, specifically:

[0014] The six different unit strain tensors Applied to RVE:

[0015]

[0016] ε 0 represents unit strain;

[0017] Considering the periodic boundary conditions, the corresponding stress σ in the RVE is ij and strain ε kl Solve the following basic equations, including equilibrium equations, strain-displacement equations, and material constitutive relations:

[0018] σ ij +f i =0

[0019]

[0020] where f i represents the body force, u k,l K represents the displacement component derived from the k direction with respect to the l direction. ijkl Represents the stiffness matrix of each material;

[0021] Equivalent stiffness of homogeneous orthotropic materials Through the mean field theory, we can get:

[0022]

[0023] in, and denote the volume average stress and strain respectively:

[0024]

[0025] V represents the volume of RVE;

[0026] When a temperature change ΔT is applied to the RVE to allow it to expand freely, the corresponding equivalent average thermoelastic strain The temperature change satisfies the following relationship:

[0027]

[0028] Therefore, the equivalent stiffness and linear thermal expansion coefficient In cylindrical coordinate system, it is expressed as:

[0029]

[0030] Represents equivalent stiffness The components in the matrix, represents the equivalent shear modulus; Represents the components in the coefficient of linear thermal expansion matrix.

[0031] Preferably, in step 1.2), the secondary nominal stress failure criterion is used as the initial delamination criterion for predicting the occurrence of damage, as shown below:

[0032]

[0033] in and Represent the interfacial cohesive strength in the normal and tangential directions, σ n and σ s represent the stress in the normal and tangential directions of the interface, respectively. The symbol <> represents the Macaulay operator. <σ n >Indicates that normal compressive stress will not cause damage.

[0034] Preferably, in step 1.3), in the mesoscopic region, the multilayer structure inside the epoxy-impregnated REBCO coil is retained; and then the stratification behavior on the internal interface of the REBCO-coated conductor of the superconducting coil during operation is simulated based on the bilinear cohesive force CZM model;

[0035] For a given fracture mode, the bilinear CZM is defined by three independent parameters: the critical fracture energy release rate G ic , interfacial cohesive strength and initial penalty stiffness P; the initial and final separation displacements of the contact surface are expressed as and

[0036]

[0037] Where the subscript i = n, s represents the normal and tangent directions;

[0038] Spatial variables used to determine the load condition. Maximum displacement after calculation in the current step. Defined as the current maximum separation displacement of each point on the interface:

[0039]

[0040]

[0041] Among them, u n and u s are the normal and tangential displacements on the interface of the current step, and are the corresponding maximum separation displacements recorded in the normal and tangential directions at the previous time step, respectively. Only when the current displacement is greater than the previous displacement, Will be updated;

[0042] The interface damage process is described by the following constitutive equation:

[0043] σ i =P,u i ,i=n,s

[0044]

[0045] Where P' is the penalty stiffness, D i is the damage evolution function defined by the bilinear traction-separation relation.

[0046] Preferably, in step 1.4), the displacement of the macro region is assumed to be The displacement of the mesoscopic region is Define the interface reaction force F related to the two displacements on the coupling interface i for:

[0047]

[0048] Among them C ii is the diagonal component of the elastic constant matrix C on the coupling interface, which is C rr and C zz Assume C ii is a maximum value to satisfy the continuity between the macroscopic region and the mesoscopic region. The elastic constant matrix on the coupling interface is:

[0049]

[0050] Preferably, in step 2), the electromagnetic force f=B×J is introduced, where B is the magnetic field intensity and J is the current density; and the multi-scale mechanical response of the overall structure is obtained by solving the equilibrium equation, strain-displacement equation and material constitutive relationship in cylindrical coordinates.

[0051] The present invention simplifies the composite material structure of the non-hazardous area into a single homogenized material; refines the modeling of the hazardous area and introduces a cohesive force model between interfaces, thereby obtaining results consistent with the fully refined model while greatly reducing the number of grids; the present invention is more suitable for local damage assessment of large-scale superconducting magnets. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 It is a flow chart of a same-level multi-scale method for delamination destruction of an immersed REBCO superconducting coil in an embodiment. DETAILED DESCRIPTION

[0053] In order to further understand the content of the present invention, the present invention is described in detail in conjunction with the accompanying drawings and embodiments. It should be understood that the embodiments are only for explaining the present invention and are not intended to limit it.

[0054] Example

[0055] like Figure 1 As shown, this embodiment provides a same-level multi-scale method for delamination damage of impregnated REBCO superconducting coils, which includes the following steps:

[0056] 1) Establish a multi-scale model at the same level;

[0057] 1.1) The impregnated REBCO superconducting coil is represented by a macroscopic orthotropic homogenized material and a linear elastic solution is obtained;

[0058] 1.2) Define the initial delamination criterion δ, and check whether there is an initial delamination criterion δ≥1 at each integration point at each moment of the macroscopic linear elastic solution;

[0059] 1.3) If at any time there is an initial delamination criterion δ≥1 at the integration point, the grid of the macroscopic region at this time is replaced by the grid of the mesoscopic region, and a cohesive interface is introduced between the superconducting layer and the silver layer in the mesoscopic region;

[0060] 1.4) Introduce a coupling interface between the macroscopic region and the mesoscopic region to connect the macroscopic and mesoscopic regions and establish a multi-scale model at the same level;

[0061] 2) Solve the multi-scale model at the same level and output and store the corresponding solution.

[0062] In step 1.1), in the macroscopic region, the internal multiphase structure needs to be equivalent to a single orthotropic homogenized material. Therefore, the representative element (RVE) method based on micromechanics is used to obtain the equivalent thermoelastic parameters of the homogenized material. The RVE includes epoxy resin, insulating layer, copper stabilization layer, silver protective layer, Hastelloy base layer and superconducting layer, specifically:

[0063] Six different unit strains Applied to RVE:

[0064]

[0065] ε 0 represents unit strain;

[0066] Considering the periodic boundary conditions, the corresponding stress σ in the RVE is ij and strain ε ij Solve the following basic equations, including equilibrium equations, strain-displacement equations, and material constitutive relations:

[0067] σ ij +f i =0

[0068]

[0069] where f i represents the body force, u k,l K represents the displacement component derived from the k direction with respect to the l direction. ijkl Represents the stiffness matrix of each material;

[0070] Equivalent stiffness of homogeneous orthotropic materials Through the mean field theory, we can get:

[0071]

[0072] in, and denote the volume average stress and strain respectively:

[0073]

[0074] V represents the volume of RVE;

[0075] When a temperature change ΔT is applied to the RVE to allow it to expand freely, the corresponding equivalent average thermoelastic strain The temperature change satisfies the following relationship:

[0076]

[0077] Therefore, the equivalent stiffness and linear thermal expansion coefficient In cylindrical coordinate system, it is expressed as:

[0078]

[0079] Represents equivalent stiffness The components in the matrix, represents the equivalent shear modulus; Represents the components in the coefficient of linear thermal expansion matrix.

[0080] In step 1.2), the secondary nominal stress failure criterion is used as the initial delamination criterion for predicting the occurrence of damage, as shown below:

[0081]

[0082] in and Represent the interfacial cohesive strength in the normal and tangential directions, σ n and σ s represent the stress in the normal and tangential directions of the interface, respectively. The symbol <> represents the Macaulay operator. <σ n >Indicates that normal compressive stress will not cause damage.

[0083] In step 1.3), in the mesoscopic region, the multilayer structure inside the epoxy-impregnated REBCO coil is preserved; then, based on the bilinear cohesive force CZM model, the delamination behavior on the internal interface of the REBCO-coated conductor of the superconducting coil during operation is simulated;

[0084] For a given fracture mode, the bilinear CZM is defined by three independent parameters: critical fracture energy release rate (fracture toughness) G ic , interfacial cohesive strength and initial penalty stiffness P; the initial and final separation displacements of the contact surface are expressed as and

[0085]

[0086] Where the subscript i = n, s represents the normal and tangent directions;

[0087] Spatial variables used to determine the load condition. Maximum displacement after calculation in the current step. Defined as the current maximum separation displacement of each point on the interface:

[0088]

[0089]

[0090] Among them, u n and u s are the normal and tangential displacements on the interface of the current step, and are the corresponding maximum separation displacements recorded in the normal and tangential directions at the previous time step, respectively. Only when the current displacement is greater than the previous displacement, Will be updated;

[0091] The interface damage process is described by the following constitutive equation:

[0092] σ i =P,u i ,i=n,s

[0093]

[0094] Where P' is the penalty stiffness, D i is the damage evolution function defined by the bilinear traction-separation relation.

[0095] In step 1.4), assume that the displacement of the macro region is The displacement of the mesoscopic region is Define the interface reaction force F related to the two displacements on the coupling interface i for:

[0096]

[0097] Among them C ii is the diagonal component of the elastic constant matrix C on the coupling interface, which is C rr and C zz Assume C ii is a maximum value to satisfy the continuity between the macroscopic region and the mesoscopic region. The elastic constant matrix on the coupling interface is:

[0098]

[0099] During the excitation process, there is no influence of the thermal expansion coefficient, and the stiffness of the coupling interface in the axial direction is restored.

[0100] In step 2), the electromagnetic force f=B×J is substituted, where B is the magnetic field intensity and J is the current density; by solving the equilibrium equation, strain-displacement equation and material constitutive relationship in cylindrical coordinates, the multi-scale mechanical response of the overall structure is obtained.

[0101] This embodiment simplifies the composite material structure of the non-hazardous area into a single homogenized material; refines the modeling of the hazardous area and introduces a cohesive force model between the interfaces, and can still obtain results consistent with the fully refined model while greatly reducing the number of grids; this embodiment is more suitable for local damage assessment of large-scale superconducting magnets.

[0102] The present invention and its embodiments are described schematically above, and the description is not restrictive. The drawings show only one embodiment of the present invention, and the actual structure is not limited thereto. Therefore, if a person skilled in the art is inspired by it and designs a structural method and an embodiment similar to the technical solution without creativity without departing from the purpose of the invention, they shall all fall within the protection scope of the present invention.

Claims

1. A multi-scale method for delamination of impregnated REBCO superconducting coils, characterized by: The following steps are involved: 1) Establish a multi-scale model at the same level; 1.1) The impregnated REBCO superconducting coil is represented by a macroscopic orthotropic homogenized material and a linear elastic solution is obtained; 1.2) Define the initial delamination criterion δ, and check whether there is an initial delamination criterion δ≥1 at each integration point at each moment of the macroscopic linear elastic solution; In step 1.2), the secondary nominal stress failure criterion is used as the initial delamination criterion for predicting the occurrence of damage, as shown below: in and Represent the interfacial cohesive strength in the normal and tangential directions, σ n and σ s represent the stress in the normal and tangential directions of the interface, respectively. The symbol <> represents the Macaulay operator. <σ n >Shows that normal pressure stress does not induce damage; 1.3) If at any time there is an initial delamination criterion δ≥1 at the integration point, the grid of the macroscopic region at this time is replaced by the grid of the mesoscopic region, and a cohesive interface is introduced between the superconducting layer and the silver layer in the mesoscopic region; 1.4) Introduce a coupling interface between the macroscopic region and the mesoscopic region to connect the macroscopic and mesoscopic regions and establish a multi-scale model at the same level; 2) Solve the multi-scale model at the same level and output and store the corresponding solution.

2. The same-level multi-scale method for delamination failure of impregnated REBCO superconducting coils according to claim 1, characterized in that: In step 1.1), the representative unit RVE method based on micromechanics is used to obtain the equivalent thermoelastic parameters of the homogenized material, specifically: The six different unit strain tensors Applied to RVE: β＝1,2,...,6； ε 0 represents unit strain; Considering the periodic boundary conditions, the corresponding stress σ in the RVE ij and strain ε kl Solve the following basic equations, including equilibrium equations, strain-displacement equations, and material constitutive relations: s ij +f i =0 where f i represents the body force, u k,l K represents the displacement component derived from the k direction with respect to the l direction. ijkl Represents the stiffness matrix of each material; Equivalent stiffness of homogeneous orthotropic materials Through the mean field theory, we can get: in, and denote the volume average stress and strain respectively: V represents the volume of RVE; When a temperature change ΔT is applied to the RVE to allow it to expand freely, the corresponding equivalent average thermoelastic strain The temperature change satisfies the following relationship: Therefore, the equivalent stiffness and linear thermal expansion coefficient In cylindrical coordinate system, it is expressed as: Represents equivalent stiffness The components in the matrix, represents the equivalent shear modulus; Represents the components in the coefficient of linear thermal expansion matrix.

3. The same-level multi-scale method for delamination failure of impregnated REBCO superconducting coils according to claim 2, characterized in that: In step 1.3), in the mesoscopic region, the multilayer structure inside the epoxy-impregnated REBCO coil is preserved; then, based on the bilinear cohesive force CZM model, the delamination behavior on the internal interface of the REBCO-coated conductor of the superconducting coil during operation is simulated; For a given fracture mode, the bilinear CZM is defined by three independent parameters: the critical fracture energy release rate G ic , interfacial cohesive strength and initial penalty stiffness P; the initial and final separation displacements of the contact surface are expressed as and Where the subscript i = n, s represents the normal and tangent directions; Spatial variables used to determine the load condition. Maximum displacement after calculation in the current step. Defined as the current maximum separation displacement of each point on the interface: Among them, u n and u s are the normal and tangential displacements on the interface of the current step, and are the corresponding maximum separation displacements recorded in the normal and tangential directions at the previous time step, respectively. Only when the current displacement is greater than the previous displacement, Will be updated; The interface damage process is described by the following constitutive equation: σ i =P,u i ,i=n,s Where P' is the penalty stiffness, d i is the damage evolution function defined by the bilinear traction-separation relation.

4. The same-level multi-scale method for delamination destruction of impregnated REBCO superconducting coils according to claim 3, characterized in that: In step 1.4), assume that the displacement of the macro region is The displacement of the mesoscopic region is Define the interface reaction force F related to the two displacements on the coupling interface i for: Among them C ii is the diagonal component of the elastic constant matrix C on the coupling interface, which is C rr and C zz Assume C ii is a maximum value to satisfy the continuity between the macroscopic region and the mesoscopic region. The elastic constant matrix on the coupling interface is:

5. The same-level multi-scale method for delamination failure of impregnated REBCO superconducting coils according to claim 4, characterized in that: In step 2), the electromagnetic force f=B×J is substituted, where B is the magnetic field intensity and J is the current density; by solving the equilibrium equation, strain-displacement equation and material constitutive relationship in cylindrical coordinates, the multi-scale mechanical response of the overall structure is obtained.

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