A fully Lagrangian explicit computational modeling method for internal forces at nodes of hyperelastic materials

Abstract

The present application relates to a fully Lagrangian explicit computational modeling method for the internal forces of hyperelastic material nodes. The method decomposes the energy density function in the second Piola-Kirchhoff stress of the hyperelastic material into a volume-related part and a deflection-related part according to the characteristics of the hyperelastic material, and constructs a second Piola-Kirchhoff stress expansion model of the hyperelastic material; re-expresses the right Cauchy-Green strain tensor in the second Piola-Kirchhoff stress expansion model using the calculation formula of the Jacobian matrix, and expands the re-expressed right Cauchy-Green strain tensor into a form composed of time-varying quantities and invariants, reconstructs the second Piola-Kirchhoff stress expansion model, and calculates the invariants therein in advance; and constructs a new fully Lagrangian explicit computational model for the internal forces of the hyperelastic material nodes based on the reconstructed second Piola-Kirchhoff stress expansion model and the deformation gradient, strain matrix, and initial unit volume of the hyperelastic material.

Description

Technical Field

[0001] The present application relates to the technical field of numerical simulation analysis, and in particular to a fully Lagrangian explicit computational modeling method for the internal forces of hyperelastic material nodes. Background Art

[0002] Hyperelastic materials are widely used in the automotive, electronics, and mechanical equipment industries. Their high-precision modeling capabilities are key to product innovation and reliability. Hyperelastic materials exhibit complex properties such as large deformation and recoverable hyperelasticity. Calculating the internal forces at their nodes allows for accurate prediction of their mechanical response, reducing trial-and-error costs and improving product design efficiency.

[0003] At present, most of the calculation models for the internal forces of hyperelastic materials at the nodes focus on improving the accuracy and convergence of the numerical algorithms, and there are very few reports on efficiency improvements. The reason is that the calculation models for the internal forces of hyperelastic materials at the nodes are usually based on the full Lagrangian calculation model. The format of the full Lagrangian calculation model is relatively fixed, and the calculation process is also basically fixed. In explicit analysis, all quantities related to the deformation gradient of the hyperelastic material usually need to be updated at each time step. The numerical implementation is relatively complex, which affects the analysis of the mechanical properties of the hyperelastic material and leads to low design efficiency of the final product. Summary of the Invention

[0004] Based on this, it is necessary to provide a fully Lagrangian explicit computational modeling method for the internal forces of hyperelastic material nodes, including:

[0005] S1: Based on the characteristics of a hyperelastic material, an energy density function in the second Piola-Kirchhoff stress of the hyperelastic material is decomposed into a volume-related part and a deflection-related part, and a second Piola-Kirchhoff stress expansion model of the hyperelastic material is constructed; the hyperelastic material is rubber;

[0006] S2: Re-express the right Cauchy-Green strain tensor in the second Piola-Kirchhoff stress expansion model using the calculation formula of the Jacobian matrix, and expand the re-expressed right Cauchy-Green strain tensor into a form composed of time-varying quantities and invariants, reconstruct the second Piola-Kirchhoff stress expansion model, and calculate the invariants in it in advance;

[0007] S3: Based on the reconstructed Second Piola-Kirchhoff stress expansion model and the deformation gradient, strain matrix and initial unit volume of the hyperelastic material, a new fully Lagrangian explicit calculation model for the internal forces of the hyperelastic material nodes is constructed.

[0008] Preferably, constructing a second Piola-Kirchhoff stress expansion model of a hyperelastic material includes:

[0009] According to the characteristics of hyperelastic materials, the energy density function of the second Piola-Kirchhoff stress of hyperelastic materials is decomposed into a volume-related part and a deflection-related part, and a preliminary expanded second Piola-Kirchhoff stress model is obtained.

[0010] For the second Piola-Kirchhoff stress model with step expansion, the equivalent right Cauchy-Green strain tensor in the deflection-related part is replaced by the algebraic expression of the second-order unit tensor and the trace of the second-order unit tensor to obtain the second Piola-Kirchhoff stress expansion model.

[0011] Preferably, the second Piola-Kirchhoff stress of the hyperelastic material is expressed as:

[0012] ;

[0013] in, Represents hyperelastic material the second Piola-Kirchhoff stress at moment 2; represents the energy density function; represents partial derivative; Indicates the The right Cauchy-Green strain tensor at time .

[0014] Preferably, the preliminary expanded second Piola-Kirchhoff stress model is expressed as:

[0015] ;

[0016] ;

[0017] ;

[0018] ;

[0019] in, Represents hyperelastic material the initial expansion of the second Piola-Kirchhoff stress model at the moment; represents the skew correlation part; Indicates the The equivalent right Cauchy-Green strain tensor at time ; Indicates volume-related parts; Determinant representing the deformation gradient of a hyperelastic material; represents the determinant; Indicates the The right Cauchy-Green strain tensor at time t; represents partial derivative; Represents hyperelastic material deformation gradient at a given moment; represents the skew mapping operator; express and Perform a double dot product operation.

[0020] Preferably, the second Piola-Kirchhoff stress expansion model is expressed as:

[0021] ;

[0022] in, Represents hyperelastic material The second Piola-Kirchhoff stress expansion model at moment ; Determinant representing the deformation gradient of a hyperelastic material; represents the energy density function; represents the trace of the second-order unit tensor; represents the second-order unit tensor; represents partial derivative; Indicates the The right Cauchy-Green strain tensor at time .

[0023] Preferably, the expression of the right Cauchy-Green strain tensor is:

[0024] ;

[0025] in, Indicates the The right Cauchy-Green strain tensor at time t; Represents hyperelastic material deformation gradient at a given moment; represents transpose;

[0026] The right Cauchy-Green strain tensor is re-expressed using the calculation formula of the Jacobian matrix. The re-expressed right Cauchy-Green strain tensor is recorded as:

[0027] ;

[0028] in, Indicates the The right Cauchy-Green strain tensor reformulated at the moment; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; Indicates transpose.

[0029] Preferably, expanding the reformulated right Cauchy-Green strain tensor into a form composed of time-varying quantities and invariants comprises:

[0030] The time-varying Jacobian matrix of the reformulated right Cauchy-Green strain tensor is rewritten as follows:

[0031] ;

[0032] in, Indicates the Jacobian matrix at time instant; represents transpose; ~ Represents matrices respectively The elements in

[0033] The values ​​of the off-diagonal elements symmetric along the diagonal in the rewritten time-varying Jacobian matrix are equal;

[0034] The reformulated right Cauchy-Green strain tensor is rewritten as a combination of six time-varying quantities and invariants, and the calculation formula is:

[0035] ;

[0036] ; ;

[0037] ; ;

[0038] ; ;

[0039] in, represents the rewritten right Cauchy–Green strain tensor; represents the initial Jacobian matrix; 、 、 、 、 、 They represent the first invariant, the second invariant, the third invariant, the fourth invariant, the fifth invariant, and the sixth invariant respectively.

[0040] Preferably, constructing a new fully Lagrangian explicit calculation model for the internal forces of hyperelastic material nodes includes:

[0041] The reconstructed second Piola-Kirchhoff stress expansion model is multiplied by the deformation gradient, strain matrix and initial unit volume of the hyperelastic material to obtain the preliminary nodal internal force calculation model of the hyperelastic material.

[0042] According to the expansion of deformation gradient, strain matrix and initial unit volume and the reconstructed second Piola-Kirchhoff stress expansion model, the preliminary nodal internal force calculation model of hyperelastic material is adjusted, and a new fully Lagrangian explicit calculation model of nodal internal force of hyperelastic material is constructed.

[0043] Preferably, the preliminary node internal force calculation model of the hyperelastic material is expressed as:

[0044] ;

[0045] ;

[0046] ;

[0047] in, Represents hyperelastic material The internal force of the node at the moment; Represents hyperelastic material deformation gradient at a given moment; Represents hyperelastic material The second Piola-Kirchhoff stress expansion model at moment ; represents the strain matrix; represents the initial unit volume of the hyperelastic material; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; represents transpose; represents the energy density function; represents partial derivative; Represents the matrix of partial derivatives of the shape function with respect to the natural coordinates in the unit natural coordinate system.

[0048] Preferably, the new fully Lagrangian explicit calculation model for the internal forces at the nodes of hyperelastic materials is expressed as:

[0049] ;

[0050] ;

[0051] in, Represents hyperelastic material The internal force of the node at the moment; Determinant representing the deformation gradient of a hyperelastic material; represents the initial unit volume of the hyperelastic material; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; represents transpose; represents the energy density function; represents partial derivative; represents the trace of the second-order unit tensor; represents the second-order unit tensor; Represents the partial derivative matrix of the shape function with respect to the natural coordinates in the unit natural coordinate system; Represents a quantity that does not change with time.

[0052] Beneficial effects: According to the characteristics of hyperelastic materials, this method decomposes the energy density function in the second Piola-Kirchhoff stress of hyperelastic materials into a volume-related part and a deflection-related part, and constructs a second Piola-Kirchhoff stress expansion model of hyperelastic materials; the right Cauchy-Green strain tensor in the second Piola-Kirchhoff stress expansion model is re-expressed using the calculation formula of the Jacobian matrix, and the re-expressed right Cauchy-Green strain tensor is expanded into a form composed of time-varying quantities and invariants, reconstructing the second Piola-Kirchhoff stress expansion model, and calculating the invariants therein in advance; based on the reconstructed second Piola-Kirchhoff stress expansion model, ... right Cauchy-Green strain tensor is expanded into a form composed of time-varying quantities and invariants, reconstructing the second Piola-Kirchhoff stress expansion model, and calculating the invariants therein in advance; based on the reconstructed second Piola-Kirchhoff stress expansion model, the second Piola-Kirchhoff stress expansion model is re-expressed using the calculation formula of the Jacobian matrix, and the right Cauchy-Green strain tensor is expanded into a form composed of time-varying quantities and invariants, reconstructing the second Piola-Kirchhoff stress expansion model, and calculating the invariants therein in advance; based on the reconstructed second Piola-Kir The stress expansion model is combined with the deformation gradient, strain matrix and initial unit volume of the hyperelastic material to construct a new fully Lagrangian explicit calculation model for the internal force of the node of the hyperelastic material. This method is based on the original fully Lagrangian explicit calculation model and does not need to change the model framework of the explicit calculation. Only the parameter calculation model of the hyperelastic material is re-derived. During the explicit time step analysis process, the calculation model only needs to calculate the internal force based on the invariants calculated in advance at each time step. The node internal force obtained based on the calculation model is convenient for the subsequent analysis of the mechanical properties of the hyperelastic material, reduces the trial and error cost of the product, and improves the design efficiency of the product. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] In order to more clearly illustrate the embodiments of the present application or the technical solutions in the prior art, the following briefly introduces the drawings required for use in the embodiments or the description of the prior art. Obviously, the drawings described below are only some embodiments of the present application. For ordinary technicians in this field, other drawings can be obtained based on these drawings without any creative work.

[0054] Figure 1 This is a flow chart of the fully Lagrangian explicit calculation modeling method for the internal forces of hyperelastic material nodes in an embodiment of the present application. DETAILED DESCRIPTION

[0055] To make the above-mentioned objects, features, and advantages of the present application more clearly understood, the specific embodiments of the present application are described in detail below with reference to the accompanying drawings. The following description sets forth many specific details to facilitate a full understanding of the present application. However, the present application can be implemented in many other ways than those described herein, and those skilled in the art can make similar improvements without violating the scope of the present application. Therefore, the present application is not limited to the specific embodiments disclosed below.

[0056] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of the technical features being referred to. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of such features. Throughout the description of this application, "plurality" means at least two, for example, two, three, etc., unless otherwise specifically defined.

[0057] like Figure 1 As shown, this embodiment provides a fully Lagrangian explicit computational modeling method for the internal forces of hyperelastic material nodes, including:

[0058] S1: Based on the characteristics of the hyperelastic material, the energy density function in the second Piola-Kirchhoff stress of the hyperelastic material is decomposed into a volume-related part and a deflection-related part, and a second Piola-Kirchhoff stress expansion model of the hyperelastic material is constructed; in this embodiment, the hyperelastic material is rubber.

[0059] Specifically, the second Piola-Kirchhoff stress expansion model for hyperelastic materials includes:

[0060] According to the characteristics of hyperelastic materials, the energy density function of the second Piola-Kirchhoff stress of hyperelastic materials is decomposed into a volume-related part and a deflection-related part, and a preliminary expanded second Piola-Kirchhoff stress model is obtained.

[0061] For the second Piola-Kirchhoff stress model with step expansion, the equivalent right Cauchy-Green strain tensor in the deflection-related part is replaced by the algebraic expression of the second-order unit tensor and the trace of the second-order unit tensor to obtain the second Piola-Kirchhoff stress expansion model.

[0062] Furthermore, the second Piola-Kirchhoff stress of the hyperelastic material is expressed as:

[0063] ;

[0064] in, Represents hyperelastic material the second Piola-Kirchhoff stress at moment 2; represents the energy density function; represents partial derivative; Indicates the The right Cauchy-Green strain tensor at time .

[0065] Furthermore, the preliminary expansion of the second Piola-Kirchhoff stress model is expressed as:

[0066] ;

[0067] ;

[0068] ;

[0069] ;

[0070] in, Represents hyperelastic material the initial expansion of the second Piola-Kirchhoff stress model at the moment; represents the skew correlation part; Indicates the The equivalent right Cauchy-Green strain tensor at time ; Indicates volume-related parts; Determinant representing the deformation gradient of a hyperelastic material; represents the determinant; Indicates the The right Cauchy-Green strain tensor at time t; represents partial derivative; Represents hyperelastic material deformation gradient at a given moment; represents the skew mapping operator; express and Perform a double dot product operation.

[0071] Furthermore, the second Piola-Kirchhoff stress expansion model is expressed as:

[0072] ;

[0073] in, Represents hyperelastic material The second Piola-Kirchhoff stress expansion model at moment ; Determinant representing the deformation gradient of a hyperelastic material; represents the energy density function; represents the trace of the second-order unit tensor; represents the second-order unit tensor; represents partial derivative; Indicates the The right Cauchy-Green strain tensor at time .

[0074] S2: Re-express the right Cauchy-Green strain tensor in the second Piola-Kirchhoff stress expansion model using the calculation formula of the Jacobian matrix, and expand the re-expressed right Cauchy-Green strain tensor into a form composed of time-varying quantities and invariants, reconstruct the second Piola-Kirchhoff stress expansion model, and calculate the invariants in it in advance.

[0075] Specifically, the expression of the right Cauchy-Green strain tensor is:

[0076] ;

[0077] in, Indicates the The right Cauchy-Green strain tensor at time t; Represents hyperelastic material deformation gradient at a given moment; represents transpose;

[0078] The right Cauchy-Green strain tensor is re-expressed using the calculation formula of the Jacobian matrix. The re-expressed right Cauchy-Green strain tensor is recorded as:

[0079] ;

[0080] in, Indicates the The right Cauchy-Green strain tensor reformulated at the moment; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; Indicates transpose.

[0081] Furthermore, the reformulated right Cauchy-Green strain tensor is expanded into a form consisting of time-varying quantities and invariants:

[0082] The time-varying Jacobian matrix of the reformulated right Cauchy-Green strain tensor is rewritten as follows:

[0083] ;

[0084] in, Indicates the Jacobian matrix at time instant; represents transpose; ~ Represents matrices respectively The elements in

[0085] The values ​​of the off-diagonal elements symmetric along the diagonal in the rewritten time-varying Jacobian matrix are equal;

[0086] The reformulated right Cauchy-Green strain tensor is rewritten as a combination of six time-varying quantities and invariants, and the calculation formula is:

[0087] ;

[0088] ; ;

[0089] ; ;

[0090] ; ;

[0091] in, represents the rewritten right Cauchy–Green strain tensor; represents the initial Jacobian matrix; 、 、 、 、 、 They represent the first invariant, the second invariant, the third invariant, the fourth invariant, the fifth invariant, and the sixth invariant respectively, and each invariant is calculated in advance.

[0092] S3: Based on the reconstructed Second Piola-Kirchhoff stress expansion model and the deformation gradient, strain matrix and initial unit volume of the hyperelastic material, a new fully Lagrangian explicit calculation model for the internal forces of the hyperelastic material nodes is constructed.

[0093] Specifically, the new fully Lagrangian explicit calculation model for the internal forces of hyperelastic material nodes includes:

[0094] The reconstructed second Piola-Kirchhoff stress expansion model is multiplied by the deformation gradient, strain matrix and initial unit volume of the hyperelastic material to obtain the preliminary nodal internal force calculation model of the hyperelastic material.

[0095] According to the expansion of deformation gradient, strain matrix and initial unit volume and the reconstructed second Piola-Kirchhoff stress expansion model, the preliminary nodal internal force calculation model of hyperelastic material is adjusted, and a new fully Lagrangian explicit calculation model of nodal internal force of hyperelastic material is constructed.

[0096] Furthermore, the preliminary calculation model of the node internal force of the hyperelastic material is expressed as:

[0097] ;

[0098] ;

[0099] ;

[0100] in, Represents hyperelastic material The internal force of the node at the moment; Represents hyperelastic material deformation gradient at a given moment; Represents hyperelastic material The second Piola-Kirchhoff stress expansion model at moment ; represents the strain matrix; represents the initial unit volume of the hyperelastic material; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; represents transpose; represents the energy density function; represents partial derivative; Represents the matrix of partial derivatives of the shape function with respect to the natural coordinates in the unit natural coordinate system.

[0101] The preliminary calculation model of the node internal force of the hyperelastic material is expressed as follows:

[0102] ;

[0103] In order to express the direct Jacobian update calculation method more clearly, the time-invariant quantity in the above formula is replaced by the parameter Instead, this part of the time-invariant quantity can be directly calculated at the initial configuration, thereby obtaining a new fully Lagrangian explicit calculation model for the internal forces at the nodes of hyperelastic materials, which is expressed as:

[0104] ;

[0105] ;

[0106] in, Represents hyperelastic material The internal force of the node at the moment; Determinant representing the deformation gradient of a hyperelastic material; represents the initial unit volume of the hyperelastic material; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; represents transpose; represents the energy density function; represents partial derivative; represents the trace of the second-order unit tensor; represents the second-order unit tensor; Represents the partial derivative matrix of the shape function with respect to the natural coordinates in the unit natural coordinate system; Represents a quantity that does not change with time.

[0107] The fully Lagrangian explicit computational modeling method for the internal forces of hyperelastic material nodes provided in this embodiment has the following beneficial effects:

[0108] This method is based on the fully Lagrangian (TL) explicit calculation format of hyperelastic materials, and re-derives the calculation model under the constitutive calculation of hyperelastic materials. By directly calculating the Jacobian matrix at each time step and only updating the Jacobian matrix, other invariants can be pre-calculated in the initial configuration, without the need to update all quantities related to the deformation gradient as in the traditional fully Lagrangian format, thereby reducing the amount of calculation and further improving the computational efficiency of large deformation calculations of hyperelastic materials. The nodal internal forces obtained based on this calculation model facilitate the subsequent analysis of the mechanical properties of hyperelastic materials, reduce the trial and error cost of the product, and improve the design efficiency of the product.

[0109] The technical features of the above-mentioned embodiments can be combined arbitrarily. In order to make the description concise, not all possible combinations of the technical features in the above-mentioned embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0110] The above-described embodiments merely represent several implementation methods of the present application. While the descriptions are relatively specific and detailed, they should not be construed as limiting the scope of the patent application. It should be noted that a person of ordinary skill in the art may make various modifications and improvements without departing from the spirit of the present application, and these modifications and improvements fall within the scope of protection of the present application. Therefore, the scope of protection of the present patent application shall be determined by the appended claims.

Claims

1. A fully Lagrangian explicit computational modeling method for internal forces at nodes of hyperelastic materials, characterized by: include: S1: Based on the characteristics of a hyperelastic material, an energy density function in the second Piola-Kirchhoff stress of the hyperelastic material is decomposed into a volume-related part and a deflection-related part, and a second Piola-Kirchhoff stress expansion model of the hyperelastic material is constructed; the hyperelastic material is rubber; S2: Re-express the right Cauchy-Green strain tensor in the second Piola-Kirchhoff stress expansion model using the calculation formula of the Jacobian matrix, and expand the re-expressed right Cauchy-Green strain tensor into a form composed of time-varying quantities and invariants, reconstruct the second Piola-Kirchhoff stress expansion model, and calculate the invariants in it in advance; S3: Based on the reconstructed Second Piola-Kirchhoff stress expansion model and the deformation gradient, strain matrix and initial unit volume of the hyperelastic material, a new fully Lagrangian explicit calculation model for the internal forces of the hyperelastic material nodes is constructed.

2. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 1, characterized in that: The second Piola-Kirchhoff stress expansion model for constructing hyperelastic materials includes: According to the characteristics of hyperelastic materials, the energy density function of the second Piola-Kirchhoff stress of hyperelastic materials is decomposed into a volume-related part and a deflection-related part, and a preliminary expanded second Piola-Kirchhoff stress model is obtained. For the second Piola-Kirchhoff stress model with step expansion, the equivalent right Cauchy-Green strain tensor in the deflection-related part is replaced by the algebraic expression of the second-order unit tensor and the trace of the second-order unit tensor to obtain the second Piola-Kirchhoff stress expansion model.

3. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 2, characterized in that: The second Piola-Kirchhoff stress of a hyperelastic material is expressed as: ； in, Represents hyperelastic material the second Piola-Kirchhoff stress at moment ; represents the energy density function; represents partial derivative; Indicates the The right Cauchy-Green strain tensor at time .

4. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 3, characterized in that: The preliminary expansion of the second Piola-Kirchhoff stress model is expressed as: ； ； ； ； in, Represents hyperelastic material the initial expansion of the second Piola-Kirchhoff stress model at the moment; represents the skew correlation part; Indicates the The equivalent right Cauchy-Green strain tensor at time ; Indicates volume-related parts; Determinant representing the deformation gradient of a hyperelastic material; represents the determinant; Indicates the The right Cauchy-Green strain tensor at time t; represents partial derivative; Represents hyperelastic material deformation gradient at a given moment; represents the skew mapping operator; express and Perform a double dot product operation.

5. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 4, characterized in that: The second Piola-Kirchhoff stress expansion model is expressed as: ； in, Represents hyperelastic material The second Piola-Kirchhoff stress expansion model at moment ; Determinant representing the deformation gradient of a hyperelastic material; represents the energy density function; represents the trace of the second-order unit tensor; represents the second-order unit tensor; represents partial derivative; Indicates the The right Cauchy-Green strain tensor at time .

6. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 5, characterized in that: The expression of the right Cauchy-Green strain tensor is: ； in, Indicates the The right Cauchy-Green strain tensor at time t; Represents hyperelastic material deformation gradient at a given moment; represents transpose; The right Cauchy-Green strain tensor is re-expressed using the calculation formula of the Jacobian matrix. The re-expressed right Cauchy-Green strain tensor is recorded as: ； in, Indicates the The right Cauchy-Green strain tensor reformulated at the moment; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; Indicates transpose.

7. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 6, characterized in that: Expanding the reformulated right Cauchy-Green strain tensor into a combination of time-varying quantities and invariants consists of: The time-varying Jacobian matrix of the reformulated right Cauchy-Green strain tensor is rewritten as follows: ； in, Indicates the Jacobian matrix at time instant; represents transpose; ~ Represents matrices respectively The elements in The values ​​of the off-diagonal elements symmetric along the diagonal in the rewritten time-varying Jacobian matrix are equal; The reformulated right Cauchy-Green strain tensor is rewritten as a combination of six time-varying quantities and invariants, and the calculation formula is: ； ； ； ； ； ； ； in, represents the rewritten right Cauchy–Green strain tensor; represents the initial Jacobian matrix; 、 、 、 、 、 They represent the first invariant, the second invariant, the third invariant, the fourth invariant, the fifth invariant, and the sixth invariant respectively.

8. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 7, characterized in that: The new fully Lagrangian explicit computational model for the internal forces of hyperelastic material nodes includes: The reconstructed second Piola-Kirchhoff stress expansion model is multiplied by the deformation gradient, strain matrix and initial unit volume of the hyperelastic material to obtain the preliminary nodal internal force calculation model of the hyperelastic material. According to the expansion of deformation gradient, strain matrix and initial unit volume and the reconstructed second Piola-Kirchhoff stress expansion model, the preliminary nodal internal force calculation model of hyperelastic material is adjusted, and a new fully Lagrangian explicit calculation model of nodal internal force of hyperelastic material is constructed.

9. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 8, characterized in that: The preliminary calculation model of the node internal force of hyperelastic materials is expressed as: ； ； ； in, Represents hyperelastic material The internal force of the node at the moment; Represents hyperelastic material deformation gradient at a given moment; Represents hyperelastic material The second Piola-Kirchhoff stress expansion model at moment ; represents the strain matrix; represents the initial unit volume of the hyperelastic material; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; represents transpose; represents the energy density function; represents partial derivative; Represents the matrix of partial derivatives of the shape function with respect to the natural coordinates in the unit natural coordinate system.

10. The fully Lagrangian explicit computational modeling method for internal forces of hyperelastic material nodes according to claim 9, characterized in that: The new fully Lagrangian explicit calculation model for the internal forces at the nodes of hyperelastic materials is expressed as: ； ； in, Represents hyperelastic material The internal force of the node at the moment; Determinant representing the deformation gradient of a hyperelastic material; represents the initial unit volume of the hyperelastic material; Represents hyperelastic material Jacobian matrix at time instant; represents the initial Jacobian matrix of the hyperelastic material; represents transpose; represents the energy density function; represents partial derivative; represents the trace of the second-order unit tensor; represents the second-order unit tensor; Represents the partial derivative matrix of the shape function with respect to the natural coordinates in the unit natural coordinate system; Represents a quantity that does not change with time.

Images