JP2014010047A - Program for outputting stress/strain curve equation and its device and method for evaluating physical property of elastic material - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a model for evaluating the physical property of an elastic material by using both visco-elasticity and super-elasticity.SOLUTION: The correlation equation of a stress and a strain to be searched on the basis of a Maxwell model in which elastic elements and viscous elements are serially arranged, that is, the correlation equation of a stress and a strain including a strain energy density function introducing the high order item of the invariant of a strain as an item related to elasticity is used to evaluate the physical property of an elastic material.

Description

  The present invention is a program for outputting a stress-strain curve equation necessary for expressing a large deformation behavior of an elastic material having physical properties having both viscoelasticity and rubber elasticity (superelasticity) with good quantitativeness on a simulation. The present invention also relates to a device for evaluating the physical properties of the elastic material.

  Conventionally, there is viscoelasticity as an index for evaluating the physical properties of elastic materials, and a generalized Maxwell model in which elastic elements and viscous elements are arranged in series is used as a model for expressing the viscoelasticity. In the generalized Maxwell model, a correlation equation between stress and strain, that is, a stress-strain curve equation can be obtained.

  However, in recent years, with the improvement in quality and added value of elastic materials (for example, pressure-sensitive adhesives), it has become impossible to sufficiently evaluate the physical properties of elastic materials only with viscoelasticity. Specifically, if the elastic material is nonlinear and causes large deformation, it is necessary to add not only viscoelasticity but also superelasticity to the index in order to evaluate the physical properties.

  Therefore, the applicant of the present application developed a generalized Maxwell model that is a viscoelastic model, created a model (generalized Maxwell development model) that can evaluate both physical properties of viscoelasticity and superelasticity, and formulated it. Then, the tool which can utilize it was proposed (patent document 1).

Japanese Patent No. 4852626

  The applicant of the present application creates a new model that can evaluate the physical properties of both viscoelasticity and superelasticity by an approach different from that described in Patent Document 1, and formulates the model. Suggest tools that can be used.

The program according to the present invention as a tool is:
A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a program for outputting a correlation equation of a strain as a stress-strain curve equation,
Computer
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the identified stress-strain correlation equation as a stress-strain curve equation,
It is made to function as.

  According to this configuration, by inputting different parameters according to physical properties (including other parameters as necessary), a stress-strain curve formula corresponding to the physical properties of the elastic material is output. For example, it is possible to expect an application in which a desired stress-strain curve equation is pursued by appropriately changing and inputting parameters, and a material design is performed so as to match it.

In the program according to the present invention, as an example,
The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p _s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I _B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II _B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c ₁ , c ₂ , c ₃ constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment When τα (τα> 0) and a constant determined by experiment are expressed by βα∞ (βα∞> 0), they are defined by the following equations.

In the program according to the present invention, as an example including temperature dependence,
The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p _s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I _B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II _B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c ₁ , c ₂ , c ₃ constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment τα (τα> 0), the constant determined by experiment is βα∞ (βα∞> 0), the temperature of the elastic material is θ, the reference temperature (20 ° C.) is θ _S , the glass transition temperature, and the temperature inherent to the material the theta _g, when expressed a constant determined by experiment d _1, d _2, defined by the following formula .

The device according to the present invention as another tool is
A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a strain correlation equation as a stress-strain curve equation,
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the specified correlation equation of stress and strain as a stress-strain curve equation.

In addition, the physical property evaluation method of the elastic material as another tool is
A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And the physical property of the elastic material is evaluated using a correlation equation of strain and strain.

  As described above, according to the present invention, it is possible to provide a tool capable of creating a model that can evaluate both physical properties of viscoelasticity and superelasticity, formulating the model, and using the model. In addition, by using the tool, large deformation behavior of an elastic material exhibiting physical properties having both viscoelasticity and superelasticity can be expressed on a simulation with good quantitativeness.

The conceptual diagram of a generalized Maxwell model is shown. A conceptual diagram of the Operator Split method is shown. (A) An actual stress-strain curve diagram actually measured at three types of tension speeds, and a stress-strain curve diagram expressed by a calculation solution based on a model according to a comparative form for each tension speed, b) An actual stress-strain curve diagram actually measured at the same three types of tension speeds and a stress-strain curve diagram expressed by a calculation solution based on the model according to this embodiment for each tension speed are shown. (A) An actual stress-strain curve diagram actually measured at three types of temperatures, and a stress-strain curve diagram expressed by a calculation solution based on a model according to a comparative form for each temperature are shown, (b) An actual stress-strain curve diagram actually measured at the same three types of temperatures and a stress-strain curve diagram expressed by a calculation solution based on the model according to the present embodiment for each temperature are shown.

  Hereinafter, an embodiment of the present invention will be described. First, the flow until the correlation formula between stress and strain is derived will be described.

The stress (Cauchy stress) is represented by the following formula (1).

The first term on the right side of the equation (1) is a volume change term of the Cauchy stress and is represented by the following equation (2).
Here, p _s represents the pressure of the solid, and bold I represents the unit tensor on the second floor.

Further, the second term on the right side of the equation (1) is an equivalent product change term of the Cauchy stress, and the generalized Maxwell model shown in FIG. 1 is expanded to a finite deformation theory, and the elastic part is represented as a superelastic body. Based on this viscoelastic model, it is formulated and expressed by the following equation (3).

The first term on the right side of the equation (3) is the equilibrium stress of the elastic unit (see FIG. 1), and is represented by the following equation (4).
Where J is the volume change rate, Ψ _iso ∞ is the strain energy density function, I _B with the upper line is the first reduced invariant (first invariant of strain) of the left Cauchy-Green deformation tensor, and II _B with the upper line is left The second reduction invariant of the Cauchy-Green deformation tensor (second invariant of strain), B with an overline represents the reduction invariant of the left Cauchy-Green deformation tensor, and bold I represents the second-order unit tensor.

As the strain energy density function Ψ _iso ∞, the present inventor wrote in “Yoshihiro Yamashita, Tokio Kawabata: Approximate Formula of Strain Energy Density Function of Reinforced Rubber, Journal of Japan Rubber Association, 65 (9), 517-528, 1992.” The disclosed strain energy density function (the following equation (5) ′) was studied.
Here, the overlined I _B is the first reduced invariant of the left Cauchy-Green deformation tensor (first invariant of strain), and the overlined II _B is the second reduced invariant of the left Cauchy-Green deformation tensor (first strain invariant). Bivariate), c ₁ , c ₂ , and c ₃ are constants determined by experiment, and p is a constant determined by experiment (p> 0).

However, in this strain energy density function, as shown in FIG. 3A and FIG. 4A described later, the correlation with the actually measured value is not necessarily good. This is because the stress-strain curve of a superelastic material (for example, an adhesive) is more nonlinear than the stress-strain curve of carbon black reinforced rubber, which is the subject of the non-patent literature.
This non-linearity is caused by an equal product change term that depends on the past motion history. Therefore, the stress-strain curve of the superelastic material can be described with higher accuracy by adding the second invariant high-order term of the left Cauchy-Green deformation tensor that represents the change in equal volume and setting an independent coefficient. Can do. Therefore, the following equation (5) was devised as the strain energy density function Ψ _iso ∞.
Here, the overlined I _B is the first reduced invariant of the left Cauchy-Green deformation tensor (first invariant of strain), and the overlined II _B is the second reduced invariant of the left Cauchy-Green deformation tensor (first strain invariant). Bivariate), c ₁ , c ₂ , c ₃ are constants determined by experiment, and p, q are constants determined by experiment (p> 0, q> 0). As can be seen from comparison with equation (5) ′, the fourth term on the right side of equation (5) is the higher-order term of the second invariant of the added strain.

The second term on the right side of the equation (3) is the non-equilibrium stress of the α-th viscoelastic unit (see FIG. 1) and is represented by the following equation (6).
Here, Δt is a time increment, a (θ) is a time-temperature conversion factor (transfer factor), τα is a relaxation time determined by experiment (τα> 0), βα∞ is a constant determined by experiment (βα∞> 0) ).

In general, superelastic materials (such as adhesives) cause very large deformations of more than a few thousand percent at nominal strain. The stress-strain curve at that time rises into an S-shape due to superelasticity, and shows speed dependence due to viscosity. These viscous-superelastic behaviors are temperature dependent. The time-temperature conversion rule is generally established in viscoelastic materials, and it is known that time and temperature can be handled equivalently. That is, the temperature dependence can be expressed by converting the temperature change of the viscoelastic material into a time change. In this embodiment, the WLF (Williams-Landel-Ferry) formula (the following formula (7)) which is a typical time-temperature conversion rule is used.
Here, θ is the temperature of the elastic material, θ _S is the reference temperature (20 ° C.), θ _g is the glass transition temperature, and is a temperature specific to the material, and d ₁ and d ₂ are constants determined by experiments.

  By the way, the Lagrange type finite element method has been used for numerical analysis of elastic materials. However, the Lagrange finite element method is difficult to handle the large deformation behavior, exfoliation phenomenon and fracture phenomenon of superelastic materials (for example, adhesive). On the other hand, in the Euler type finite element method, since the material deforms beyond the mesh fixed in the space, arbitrary large deformation analysis is possible. In addition, since a new material interface can be easily generated, it is advantageous for fracture analysis. Therefore, in this embodiment, paying attention to the viscosity-superelasticity of the adhesive, its behavior was analyzed by the Euler type finite element method.

When a hyperelastic material is modeled as a continuum without considering the microscopic scale at the molecular level, the equation of motion of the continuum is expressed by the following formula (8) in the Euler description.
Here, ρ represents a mass density, v represents a velocity vector, σ represents a Cauchy stress tensor, and b represents an object force vector.

Next, the description method of the solid deformation in the Euler display will be described. In the case of Lagrange display, which is generally used in solid mechanics, various strain tensors are obtained from the displacement vector of a material point, and body deformation is described. On the other hand, in the case of Euler display, the solid point cannot be described from the displacement vector because the substance point is not tracked. Therefore, in order to describe the solid deformation from the velocity vector field, the definition formula of the left Cauchy-Green deformation tensor, that is, the following equation (9)
The following formula (10), which is obtained by differentiating the two sides of the substance with respect to material time, is introduced.
Here, L represents a velocity gradient tensor. That is, by using Equation (10), the left Cauchy-Green deformation tensor can be obtained from the velocity vector field.

Equation (8) (equation of motion) and equation (10) (time evolution equation of the left Cauchy-Green deformation tensor) are generalized as the following equation (11).
Here, φ and f represent arbitrary functions, and v with a hat represents a mesh speed. That is, the basic equation of Euler display is expanded to ALE (arbitrary Lagrangian-Eulerian) display. Since the Euler type solution of this method cannot give boundary conditions to the material boundary surface, the ALE type solution method is used for the problem that gives boundary conditions to the material boundary surface.

If the equation (11) is approximated to the forward difference in the time direction, the following equation (12) is obtained.

In this method, calculation is performed by the Operator Split method of Equation (12). The concept of the Operator Split method is shown in FIG. In the Operator Split method, the equation (12) is divided into two equations as in the following equations (13) and (14).
Equation (13) is a non-advection step that advances time and is discretized by an explicit finite element method. In addition, * mark with an upper right means the value after a non-advection step. Equation (14) is an advection step that stops time. The velocity advection is calculated by the central difference method, and the left Cauchy-Green deformation tensor is calculated by the MUSCL method, which is a second-order upwind method for numerical stabilization. The VOF (Volume of Fluid) method is adopted as the interface trapping method, and the advection equation of the VOF function is also calculated by the MUSCL method.

  Next, in order to verify the validity of the correlation equation of stress and strain obtained by the equations (1) to (7), a uniaxial tensile test simulation of an acrylic adhesive was performed under test conditions with different temperatures and tensile speeds. . FIG. 3B shows stress-strain curves obtained when the test specimen at a temperature of 20 ° C. is uniaxially stretched at speeds of 5 mm / min, 50 mm / min, and 500 mm / min, respectively. Moreover, FIG.4 (b) has each shown the stress-strain curve when the test body of temperature 0 degreeC, 20 degreeC, and 40 degreeC is extended uniaxially at 50 mm / min.

Using formula (5) ′ instead of formula (5), the stress at the time when the specimen at a temperature of 20 ° C. was uniaxially stretched at speeds of 5 mm / min, 50 mm / min, and 500 mm / min (FIG. 3A) The correlation between the strain curve and the measured value is
Using the formula (5) ′ in place of the formula (5), the stress when the test specimens at temperatures of 0 ° C., 20 ° C. and 40 ° C. were uniaxially stretched at 50 mm / min (FIG. 4 (a)) The correlation between the strain curve and the measured value is
In both cases, the correlation is not always good.

However, using the formulas (1) to (7), the stress-strain curve when the specimen at a temperature of 20 ° C. is uniaxially stretched at a speed of 5 mm / min, 50 mm / min, and 500 mm / min (FIG. 3B). And the correlation between the measured values and
Using the formulas (1) to (7), the stress-strain curve and actual measurement when the specimens at temperatures of 0 ° C., 20 ° C., and 40 ° C. were uniaxially stretched at 50 mm / min (FIG. 4A). The correlation with the value of
Both have good correlation. As a result, the speed dependency due to the viscosity and superelasticity is reproduced, and it can be seen that the increase in the stress value due to the decrease in the temperature of the specimen can be reproduced.

  Thus, it has been confirmed that the model according to the present embodiment is a valid model as a model expressing the deformation of an adhesive (elastic material) exhibiting viscoelastic-superelastic properties. As described above, viscoelasticity is evaluated in the same manner as in the past, and superelasticity is evaluated by the method proposed here, so that an elastic material having physical properties having both viscoelasticity and superelasticity is large. Deformation behavior can be expressed with high quantitativeness on simulation.

  Here, the software according to the present embodiment is a correlation equation of stress and strain obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and a high-order term of an invariant of strain as a term related to elasticity. A program for outputting a stress-strain correlation equation including a strain energy density function introduced as a stress-strain curve equation, and a computer for inputting parameters in the stress-strain correlation equation; A specifying means for specifying a correlation equation of stress and strain according to the input parameters, and an output means for outputting the specified correlation equation of stress and strain as a stress-strain curve equation.

  The device according to the present embodiment is configured by a computer, and includes a CPU, a ROM, a working memory, a frame memory, a data input / output device, a hard disk, and a display, each connected to a bus. The ROM stores the program and various parameters, and the working memory is a memory necessary for the CPU to perform control, and includes a buffer, a register, and the like. The CPU performs various calculations and processes according to the computer program stored in the ROM. A data input / output device is the input means, and a control unit comprising a CPU, ROM and memory constitutes the specifying means. For the output means, if the display displays the stress-strain curve expressed by the stress-strain curve formula, the display constitutes the output means, and if the stress-strain curve formula is transmitted to the outside, the input is entered. If the output device constitutes the output means and has the stress-strain curve equation in the control unit (such as a case where an application using the same is installed in the control unit and the application is activated in the personal computer) ), The control unit itself constitutes the output means.

  Also, from the viewpoint of evaluating viscoelasticity together, it is preferable that the related one is programmed and held in the apparatus (installed in a computer).

  In addition, this invention is not limited to the said embodiment, A various change is possible in the range which does not deviate from the summary of this invention.

  For example, the stress-strain curve equation and the various correlation equations that serve as the premise thereof are not limited to those listed above, but are based on various correlation equations based on other appropriate approximation equations. It may be a derived stress-strain curve equation.

  The present invention can be used to design a high-quality or high-value-added elastic material exhibiting properties having both viscoelasticity and superelasticity.

Claims (5)

A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a program for outputting a correlation equation of a strain as a stress-strain curve equation,
Computer
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the identified stress-strain correlation equation as a stress-strain curve equation,
A program characterized by functioning as The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p _s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I _B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II _B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c ₁ , c ₂ , c ₃ constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment When τα (τα> 0), a constant determined by experiment is expressed as βα∞ (βα∞> 0),
The program according to claim 1, defined in The correlation between stress and strain is as follows: Cauchy stress is bold σ, solid pressure is p _s , second-order unit tensor is bold I, volume change rate is J, first Cauchy-Green deformation tensor first invariant ( The first invariant of strain) is overlined I _B , the second reduced invariant of the left Cauchy-Green deformation tensor (the second invariant of strain) is overlined II _B , and the reduced invariant of the left Cauchy-Green deformation tensor B with an overline, c ₁ , c ₂ , c ₃ constants determined by experiment, p, q (p> 0, q> 0) determined by experiment, Δt time increment, relaxation time determined by experiment τα (τα> 0), the constant determined by experiment is βα∞ (βα∞> 0), the temperature of the elastic material is θ, the reference temperature (20 ° C.) is θ _S , the glass transition temperature, and the temperature inherent to the material Is _represented by θ _g , and constants determined by experiments are represented by d ₁ and d ₂ ,
The program according to claim 1, defined in A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And a strain correlation equation as a stress-strain curve equation,
Input means for inputting parameters in the correlation equation of stress and strain;
A specifying means for specifying a correlation equation of the stress and strain according to the input parameters;
And an output means for outputting the specified correlation equation of stress and strain as a stress-strain curve equation.   A stress-strain correlation equation obtained based on the Maxwell model in which an elastic element and a viscous element are arranged in series, and includes a strain energy density function in which a high-order term of strain invariant is introduced as a term related to elasticity. And evaluating the physical properties of the elastic material using a correlation equation of the strain and the strain.