JP5006724B2 - Finite element analysis method for anisotropic members - Google Patents

Description

  The present invention relates to a finite element method analysis method for anisotropic members.

  Conventionally, a finite element method (FEM) has been used to analyze deformation and stress of members made of various materials such as rubber (see, for example, Patent Document 1 and Patent Document 2). In the analysis, physical properties such as the shape of the member and the elastic coefficient of the material of the member are given, and numerical calculation is performed.

  By the way, in a member mainly composed of a material such as rubber, one containing a reinforcing fiber inside is often used. In the rubber member containing this fiber, when the orientation direction of the fiber is not uniform in all directions, the anisotropy in which the elastic modulus becomes a different value depending on the direction is shown. In order to show non-linearity whose values are greatly different between the direction of compression and the direction of extension, an analysis considering this anisotropy (and non-linearity) is required.

  In Patent Document 1, the analysis object is a composite in which a plurality of cords are arranged in the same direction on the same surface (in the form of a film) inside the rubber. In addition to dividing the portion into solid elements, the cord is modeled as a membrane element having a different Young's modulus in the longitudinal direction of the cord and the direction perpendicular thereto, and the Young's modulus for compression and the tensile A method for creating a finite element model of a composite that gives Young's modulus is disclosed.

  Further, in Patent Document 2, in consideration of the orientation direction of the short fibers contained in the rib rubber layer of the V-ribbed belt, the physical property values (longitudinal elastic modulus, transverse elastic modulus, etc.) are different from those in the orientation direction and other directions. A belt approach angle calculation method is disclosed.

  The one disclosed in Patent Document 1 constitutes a calculation model in which a complex is divided into a plurality of element types and is close to actuality. Therefore, although the analysis accuracy is improved, the scale of the calculation model increases and the calculation cost increases. In addition, the finite element model (that is, the calculation model) becomes complicated, which increases the cost of model creation.

  Further, the one disclosed in Patent Document 2 does not increase the scale of the calculation model unlike the one disclosed in Patent Document 1 described above, but is defined as a linear material. In particular, the analysis value is not actual because it cannot reflect the property that the elastic modulus differs greatly between the stretch direction and the compression direction in the orientation direction of the fiber contained in the rubber. It was very different from the value.

  Therefore, the strain energy function is used as a finite element method analysis method for anisotropic members that can consider the anisotropy and nonlinearity of members without dividing the analysis target into multiple element types and constructing a calculation model. An analysis method using is used.

  Various formulas have been proposed as the strain energy function, and examples of the following (formula 1) and (formula 2) are given here.

In the above (formula 1) and (formula 2), I ₁ , I ₂ , and I ₄ are invariants of strain, and are represented by the following (formula 3) to (formula 5).

From the above (Formula 3) and (Formula 4), it can be seen that I ₁ and I ₂ represent strain states of the isotropic substrate material rubber. In (Formula 5), I ₄ is the orientation vector of the fiber. It can be understood that the strain state in the fiber orientation direction is represented.

  In the analysis method using the strain energy function, the stress is expressed as a derivative of the strain of the strain energy function, and the physical properties of the material having anisotropy and nonlinearity such as an elastomer such as rubber can be reflected. It can be done.

However, in the analysis method using the strain energy function, although the anisotropy and nonlinearity of the member can be reflected to some extent, C _{1 to} C ₄ as shown in the above (Expression 1) and (Expression 2). Is fixed as a constant, and cannot be reflected when the physical properties such as the elastic modulus differ greatly between the stretch side and the compression side near the zero point of strain as in the case of actual rubber materials. The analysis accuracy was limited.
JP 2003-94916 A JP 2006-292020 A

  The present invention has been made in view of the above points, and the object of the present invention is to use a strain energy function to increase the anisotropy and non-linearity of a member, in particular, a large physical property value between the expansion side and the compression side. It is an object of the present invention to provide a finite element method analysis method for anisotropic members that can improve the analysis accuracy by considering the difference.

In order to solve the above problems, the invention according to claim 1 is directed to an analysis model creation step for creating an analysis model, an analysis execution step for executing a finite element method analysis on the analysis model, and an analysis result of the analysis model is output. and an output step of by computer execution, an finite element analysis method of the anisotropic member for reflecting the anisotropy by strain energy function, wherein the analysis execution step, in increments loop to vary the load conditions A stiffness matrix creating step for creating a stiffness matrix, and a stress calculating step for calculating a stress from the created stiffness matrix and a load condition. In the stiffness matrix creating step, the strain extension side and the compression side are used as strain energy functions. Two strain energy functions with different coefficients , It is characterized in applying a corresponding strain energy function of the two strain energy function depending on whether extension side or the contraction side.

  In such an analysis method, an analysis method capable of reflecting the anisotropy and nonlinearity of a member using a strain energy function so that it is not necessary to construct a calculation model by dividing the analysis target into a plurality of element types. Even when the physical property values are greatly different between the expansion side and the compression side, the analysis can be performed with high accuracy.

Further, the invention according to claim 2 is the invention according to claim 1, wherein the strain energy function is expressed by a primary main invariant I ₁ and a secondary main invariant I ₂ of the right Cauchy green deformation tensor C, and a right Cauchy green deformation. and the product Ca ₀ the orientation vector a ₀ anisotropy tensor C and the anisotropic member and the product I ₄ of the orientation vector a _0, and the polynomials whose variable, and the order of I ₄ 3 or higher order It is characterized by this.

  With such a configuration, the accuracy can be further improved.

  In the present invention, the anisotropy and non-linearity of the anisotropic member are shown, and the analysis accuracy is improved even in the case where the physical property value is greatly different between the extension direction and the compression direction.

  The best mode for carrying out the present invention will be described below with reference to the accompanying drawings.

  In this embodiment, the anisotropic member to be analyzed is assumed to be a rubber containing reinforcing fibers inside, and exhibits anisotropy having different physical property values (elastic modulus) depending on the fiber orientation direction. Non-linearity is exhibited, and the physical property values are greatly different between the compression direction and the extension direction in the fiber orientation direction. The material to be analyzed is not particularly limited as long as it is a member exhibiting anisotropy and nonlinearity.

  The finite element method analysis method includes an analysis model creation step of creating an analysis model by dividing an anisotropic member to be analyzed into a plurality of elements, an analysis execution step of executing a finite element method analysis on the analysis model, The main body is composed of an output step for outputting the analysis result of the analysis model. In the present invention, as described above, an anisotropic member such as rubber containing fibers is used as an analysis target, and thus a strain energy function is used so that anisotropy and nonlinearity can be reflected.

  The analysis execution step includes a stiffness matrix creation step for creating a stiffness matrix in an increment loop for changing the load condition, and a stress calculation step for calculating stress from the created stiffness matrix and the load condition. In this embodiment, the output step is executed in the increment loop of the analysis execution step, and an analysis result is output for each increment.

  As the strain energy function, various formulas including the above (Formula 1) and (Formula 2) have been proposed. The strain energy function W used in the present embodiment is shown in (Formula 6) and (Formula 7).

In the formula, I ₁ , I ₂ , and I ₄ are invariants of strain, and are represented by the above (formula 3) to (formula 5). Note that the higher the order of (I ₄ −1) in the equation, the more the analysis accuracy is improved. However, since the amount of calculation increases, in the present embodiment, the maximum is the third order.

  Here, (Equation 6) shows a strain energy function in the case of expansion, and (Equation 7) shows a strain energy function in the case of compression.

  The flow of analysis will be described based on the flowchart shown in FIG.

First, in the analysis model creation step, the analysis target is divided into a plurality of elements to create an analysis model (element division data such as dimensions and shapes, constraint conditions, etc.), and the energy function and physical property values ( C ₁ , C ₂ , C _43P , C _42P , C _43M , C _42M ) and load conditions and other data are created and preprocessed such as reading into the storage unit (S1).

  Next, an analysis execution step is executed, and an increment loop process for gradually increasing the load condition is started (S2). In the nonlinear analysis as in the present invention, since force and displacement are not in a proportional relationship, an accurate solution cannot be obtained even if analysis is performed with a predetermined load applied at a time. Therefore, the analysis is performed by dividing the predetermined load finely and gradually increasing the load. A repeating unit that divides the load in this way and gradually increases the load is called increment. In (S2), a predetermined load is applied in the first increment, and in the second and subsequent increments, a load obtained by adding the divided load to the load in the previous increment is applied as the load.

  In each increment loop, a load vector is created according to the load and the analysis model (S3).

  Next, rigidity matrix creation steps (S4) (S5) are executed. In the stiffness matrix creation step, the entire stiffness matrix is assembled (S5) by repeating the element stiffness matrix creation step (S4) for creating the stiffness matrix of each element shown in FIG. In each element stiffness matrix creation step (S4), a step (S41) of creating a B matrix representing the displacement-strain relationship from the analysis model data, and a stress-strain relationship from the analysis model, energy function, and physical property data. (S42), and a step (S43) of creating an element stiffness matrix representing the displacement-stress relationship from the B matrix and the D matrix. If there is an element for which no element stiffness matrix has been created, the next element stiffness matrix is created again in the element stiffness matrix creating step (S4), and this is repeated to create an element stiffness matrix for all elements. The step of creating the element stiffness matrix may be performed in parallel instead of the iterative loop.

  When the element stiffness matrix is created for all the elements, the entire stiffness matrix is created by adding all the element stiffness matrices in the overall stiffness matrix creating step (S5), and the load condition comprising the load vector in the overall stiffness matrix. (S6), and the stress is calculated (S7). Then, the convergence determination of the stress result is performed (S8), and if the stress result does not satisfy the predetermined convergence condition, the process returns to the step (S3) of creating the load vector and the stress is calculated again. If the result of the stress satisfies a predetermined convergence condition, an analysis result of the stress, strain, displacement, etc. is output in the output step (S9).

  Then, it is determined whether or not there is a next increment (S10). If there is, the process returns to the beginning of the increment loop (S2), and the analysis is performed by setting the new load condition described above. If not, the analysis is terminated (S11).

In the present invention, in the stiffness matrix creation step, a stiffness matrix is created in a state where the strain energy function in the case of extension and the strain energy function in the case of compression can be selected, and a load condition is given to the entire stiffness matrix (S6). When the solution is obtained by the above method, it is determined whether expansion or compression is performed for each calculation, and the corresponding one of the expansion side and the compression side is applied. More specifically, in the stiffness matrix creation step, at a position called an integration point that exists in each element, a strain energy function in the case of expansion of (Equation 6) and a strain energy function in the case of compression of (Equation 7). When a stiffness matrix is created and solved, the strain energy function is applied at each integration point by determining whether it is stretched or compressed at each integration point.
<Example>
A uniaxial extension test and a uniaxial compression test are performed on the fiber orientation direction and the orthogonal direction of the anisotropic member, and the fiber orientation direction and the orthogonal direction of the anisotropic member are the same as the conventional method as described above. Finite element analysis was performed with the method of the present invention. In this test and analysis, it is evaluated whether or not it is possible to analyze a sudden change in the stress-strain relationship in the low deformation region, that is, on the extension side and the compression side across the zero point of strain.

  As the anisotropic member, a material obtained by adding 20 phr of nylon short fibers, 30 phr of carbon, a vulcanizing agent, a vulcanization accelerator and the like to chloroprene rubber as a base material was used.

  The uniaxial extension test uses a 50 kN autograph as a measuring instrument, and as a test piece, two strips of JIS No. 1 in which the longitudinal direction of the anisotropic member is a fiber orientation direction and a direction perpendicular to the fiber orientation direction, respectively. Prepare a test piece (width 5 mm, length 100 mm, thickness 2.0 mm), and hold both ends of the test piece 11 with the chuck 2 as shown in FIG. A test of only one way of initial pulling was performed at a test speed of 50 ± 5 mm / min.

  In the uniaxial compression test, a 50 kN autograph was used as a measuring instrument, and the test piece was a two cylindrical test with a diameter of 29.0 ± 0.5 mm and a height of 12.5 ± 0.5 mm as shown in FIG. The pieces 12a and 12b are prepared, and the test piece 12a is such that the axial direction of the cylinder becomes the fiber orientation direction, and the test piece 12b has the axial direction of the cylinder perpendicular to the fiber orientation direction (the fiber orientation direction is the circumferential direction). Then, a test speed of 10 mm / min was conducted in accordance with JIS K6254 (stress / strain test method for low deformation of vulcanized rubber).

  FIG. 5 shows the results of the uniaxial extension test and the uniaxial compression test.

  Next, analysis by the finite element method will be described.

  In this analysis, the stress when forced displacement was applied to the analytical model was obtained by calculation. The analysis model was a cubic solid element with eight vertices as nodes. As shown in FIG. 6, the cube has a side length of 1 mm, an apex as an origin, X axis, Y axis, and Z axis are set along each side from the origin, and (1, 0, 0) 6 is forcibly displaced in the X-axis direction, and the other three nodes (1, 1, 0), (1, 0, 1) and (1, 1, 1) belonging to the plane A in FIG. Calculation was performed by setting the tying to connect the displacement in the direction. As a result, the displacement and reaction force of the node to which the forced displacement is applied coincide with the engineering strain and the nominal stress, respectively. Further, the direction in which the forced displacement is applied is the X-axis direction (that is, the fiber orientation direction) as shown in FIG. 6A, and the Y-axis direction (the direction orthogonal to the fiber orientation direction) as shown in FIG. 6B. The two types of analysis were analyzed.

  In the analysis, as a conventional method, the strain energy function W is expressed by the following (formula 8).

The above-described method is used in a fixed manner and the method of the present invention in which (Equation 6) and (Equation 7) described above are selectively applied.

The constants in (Equation 6) and (Equation 7) are C ₁ = 4.84, C ₂ = −2.70, C _43P = −7.42, C _42P = 10.44, C _43M = − in the present analysis. 0.53, C _42M = 1.08 (all units are MPa).

The constants in (Equation 8) were set to C ₁₁ = 2.83, C ₁₂ = −0.25, C ₄₃ = 5.73, and C ₄₂ = 2.77 (all units are MPa) in the present analysis.

  FIG. 7 shows an analysis result by the method of the present invention, and FIG. 8 shows an analysis result by the conventional method. 7 and 8 also show the test results of FIG.

First, the results of the conventional method of FIG. 8 will be examined. When the direction of forced displacement is a direction perpendicular to the fiber orientation direction (broken line in the figure), there is a fair match, but when the direction of forced displacement is the fiber orientation direction (solid line in the figure), the test There is a considerable difference from the results. Although this conventional method can reflect non-linearity to some extent, an attempt is made to reflect an abrupt change in the stress-strain relationship between the expansion side and compression side across the strain zero point in the analysis result ( The term of I ₄ -1) becomes higher order and the calculation becomes complicated and the accuracy is limited, which is not practical.

  Next, the results of the method of the present invention in FIG. Both the case where the direction of forced displacement is perpendicular to the fiber orientation direction (broken line in the figure) and the case where the direction of forced displacement is the fiber orientation direction (solid line in the figure) are in good agreement with the test results. Thus, it can be seen that abrupt changes in the stress-strain relationship between the stretching side and the compression side across the strain zero point are also reflected in the analysis results.

  From the above results, in the analysis method of the present invention, the anisotropy and nonlinearity of the analysis object, particularly, the abrupt change in the stress-strain relationship between the stretch side and the compression side across the strain zero point. In this case, even if the strain energy function is expressed by an equation whose order is not very high, rapid changes on the expansion side and compression side across the zero point are accurately detected. It can be expressed and the calculation of the analysis is facilitated.

It is a whole flowchart of a finite element method. It is a flowchart of an element rigidity matrix preparation step same as the above. It is explanatory drawing of a uniaxial elongation test. It is a perspective view of the test piece of a uniaxial compression test. It is a test result (strain-stress relationship figure) of a uniaxial extension test and a uniaxial compression test. An analysis model is shown, (a) is explanatory drawing when the direction which gives forced displacement is made into a fiber orientation direction, (b) is explanatory drawing when the direction which gives forced displacement is made into the orthogonal direction of fiber orientation. . It is an analysis result (strain-stress relationship diagram) by the analysis method of the present invention. It is an analysis result (strain-stress relationship diagram) by a conventional analysis method.

Explanation of symbols

11 Test piece of uniaxial extension test 12a Test piece of uniaxial compression test (fiber orientation direction is axial direction)
12b uniaxial compression test specimen (fiber orientation direction orthogonal to the axis)
2 Chuck

Claims (2)

An analysis model creation step for creating an analysis model, an analysis execution step for executing a finite element method analysis on the analysis model, and an output step for outputting the analysis result of the analysis model are executed by a computer , and a strain energy function is used. An anisotropic member finite element method analysis method for reflecting anisotropy, wherein the analysis execution step includes a stiffness matrix creation step of creating a stiffness matrix in an increment loop for changing a load condition, and the created A stress calculation step of calculating stress from the stiffness matrix and the load condition, and in the stiffness matrix creation step, the strain energy function includes two strain energy functions having different coefficients on the strain extension side and the compression side, Depending on whether it is the expansion side or the compression side, the two strain errors Finite element analysis method of the anisotropic member characterized by applying a corresponding strain energy function of Energy Function. The strain energy function is expressed as follows: the primary primary invariant I ₁ and the secondary primary invariant I ₂ of the right Cauchy green deformation tensor C, and the anisotropic orientation vector a ₀ of the right Cauchy green deformation tensor C and the anisotropic member. the product I ₄ and the product Ca ₀ the orientation vector a _0, and the polynomials whose variable, finite element method of the anisotropic member according to claim 1, wherein making the order of I ₄ 3 or higher order analysis method.

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