JP3809374B2 - Stress-strain relationship simulation method and method for obtaining yield point in unloading process - Google Patents

Description

[0001]
BACKGROUND OF THE INVENTION
The present invention relates to a method for simulating a stress-strain relationship of an elastoplastic material and a method for obtaining a yield point in an unloading process.
[0002]
[Prior art]
When the material is deformed to the plastic region by applying an external force by pressing in the mold, and then the product is taken out from the mold, the amount of deformation of the material is somewhat restored at the time of unloading, where the external force is removed after the plastic working. A so-called springback phenomenon occurs. Since the amount of springback that occurs during this unloading process affects the outer dimensions of the product, it is necessary to predict the amount of springback in advance and incorporate it into the mold shape and processing conditions.
[0003]
The state of the springback phenomenon is shown using FIG. 6 in which a stress-strain curve of a general elastic-plastic material is shown with strain on the horizontal axis and stress on the vertical axis. When a tensile external force is applied to the material, plastic deformation occurs at the yield point A through the elastic deformation region. Since this yield point A is the yield point when plastic deformation of the material is started for the first time, it is particularly named the initial yield point, and the stress at that time is the initial yield stress Y.₀That is to say. The external force is continuously applied beyond the initial yield point A, the material is plastically deformed, and the processing is stopped at a point B that reaches a predetermined strain corresponding to a desired shape, where the external force is unloaded and removed. When the external force is removed from the plastically deformed material, some strain returns to the state of the balance point D where the residual stress in the material balances, and thus the springback phenomenon occurs.
[0004]
The springback amount is given by the difference between the strain amount at the unloading point B and the strain amount at the balance point D. After the unloading point B, the material first returns in the opposite direction according to the elastic properties. In the model called the isotropic hardening model, the point C where the stress is zero is considered to be an elastic region from the unloading point B and the target point E. Therefore, from the strain d1 shown on the elastic characteristic curve corresponding to the balance point D, The amount of springback will be predicted.
[0005]
However, practically most materials yield at point F under stress less than point E and deviate from elastic properties. This yield is distinguished from the initial yield point and is referred to as the yield point F in the unloading process. The stress-strain curve of the material after the yield point F in the unloading process has a larger slope than the stress-strain curve between the initial yield point A and the unloading point B during the first tensile plastic deformation. Thus, after unloading, the phenomenon that the yield point decreases and the slope of the stress-strain curve becomes larger is called the Bauschinger effect. Even if the Bauschinger effect is taken into account, depending on the evaluation, the amount of strain at the balance point D differs depending on d2 or d3 shown in the figure, so that a difference in prediction of the springback amount occurs.
[0006]
Therefore, in order to accurately predict the springback amount, it is necessary to accurately simulate the stress-strain relationship that can express the Bauschinger effect. Japanese Patent Laid-Open No. 2000-275154 approximates the stress-strain relationship of an elastoplastic material with a composite hardening model that combines an isotropic hardening model and a kinematic hardening model. And a method of simulating a stress-strain relationship that defines the amount of isotropic hardening and back stress as variables and expresses the Bauschinger effect of the material. Here, the back stress includes a linear kinematic hardening component and a non-linear kinematic hardening component, and is defined by a back stress function having a coefficient function with the equivalent plastic strain of the material as a variable.
[0007]
That is, a secondary yield function is used as the yield function (f) expressed by the equation (1).
[Expression 4]
f = (1/2) * [(S_x−α ′_x)²+ (S_y−α ′_y)²+ (S_z−α ′_z)²+2 (S_xy−α ′_xy)²+2 (S_yz−α ′_yz)²+2 (S_zx−α ′_zx)²]-(1/3) * (Y₀+ R)²              (1)
here,
S_i(I = x, y, z) is a normal stress σ acting on a plane perpendicular to the coordinate system corresponding to the index i._iDeviation stress,
S_ij(I, j = x, y, z) is the shear stress σ acting on the plane perpendicular to the coordinate system corresponding to the index i in the direction of the coordinate axis corresponding to the index j._ijDeviation stress,
α´_i(I = x, y, z) is a deviation component related to the back stress acting on a plane perpendicular to the coordinate system corresponding to the index i,
α´_ij(I, j = x, y, z) is a deviation component related to the back stress acting in the direction of the coordinate axis corresponding to the index j on a plane perpendicular to the coordinate system corresponding to the index i,
Y₀Is the initial yield stress, which is the yield stress when plastic deformation of the material is first started,
R is the amount of isotropic curing.
[0008]
Also, the deviation component of back stress (α ′_i, Α ′_ij) Is a back stress function that can be expressed by Equation (2). Here, the equivalent plastic strain (ε^p _eq) Is used.
[Equation 5]
[α ′] = [α′1] + [α′2]
[dα′1] = C [2/3 (a * [dε^p])-Dε^p _eq* [Α'1]]
[dα′2] = 2/3 (H * [dε^p])
C = C ′ / 2 * (ε^p _eq+ Ε₀)^1/2        (2)
here,
[α′1] is a tensor representing a first back stress deviation component which is a component depending on the nonlinearity of kinematic hardening in the back stress,
[α′2] is a tensor representing a second back stress deviation component which is a component depending on the linearity of kinematic hardening in the back stress,
[dε^p] Is the tensor representing the plastic strain increment,
dε^p _eqIs the equivalent plastic strain increment,
ε₀Is equivalent plastic strain ε^p _eqA variable for preventing C from becoming infinite when
C is a coefficient representing the convergence speed of nonlinear kinematic hardening,
C ′ is a constant,
(2/3) a is the convergence value of nonlinear kinematic hardening,
H is a coefficient representing the magnitude of linear kinematic hardening.
[0009]
Further, the coefficient H representing the magnitude of the linear kinematic hardening is equivalent plastic strain (ε^p _eq) To obtain a function represented by equation (3).
[Formula 6]
ε^p _eq<When ε1 H = H1
ε1 <ε^p _eq<Ε2 H = p * ln (ε^p _eq) + Q
ε2 <ε^p _eqWhen H = H2 (3)
[0010]
Furthermore, the amount of isotropic cure (R) was a function of equation (4).
[Expression 7]
dR = b * (dε^p _eq(4)
[0011]
As described above, in the method of simulating the stress-strain relationship disclosed in Japanese Patent Application Laid-Open No. 2000-275154, the coefficient included in the yield function as a plastic constitutive equation that gives the stress-strain relationship is expressed as the equivalent plastic strain (ε^p _eq) Is a coefficient function. Then, experimental values of the stress-strain relationship are acquired as a plurality of discrete values, a coefficient function is identified based on the acquired experimental values, and the stress-strain relationship is determined using a plastic constitutive equation determined by the identification. Simulate.
[0012]
[Problems to be solved by the invention]
By using the above-mentioned conventional technology, a stress-strain relationship experiment is performed on a test piece using a standard test piece that is substantially the same as the material used for processing, and a coefficient function is identified from a plurality of discrete experimental values. Thus, a plastic constitutive equation for the test piece can be obtained. Then, using the obtained plastic constitutive equation and the finite element method, it is possible to simulate each stage of processing a complex three-dimensional shape of an actual workpiece material and predict the amount of springback.
[0013]
In using the above prior art, it is necessary to obtain the yield point in the unloading process. Equivalent plastic strain after the yield point in this unloading process (ε^p _eqThis is because each coefficient function is defined as a function of). A typical example of the method for determining the yield point of a material is shown in FIGS. In the first method, as shown in FIG. 7, when remarkable primary yield X and secondary yield Y are observed in the stress-strain curve, these are used as yield points. However, the yield in the unloading process cannot be used because such remarkable primary yield X and secondary yield Y are not observed. In the second method, as shown in FIG. 8, the applied stress point Z at which the plastic strain remaining after unloading becomes 0.2% is taken as the yield point for the material in which no remarkable primary yield X and secondary yield Y are observed. It is a method to do. However, in the unloading process where the Bauschinger effect appears, the value of 0.2% plastic strain is after entering the plastic region, and when simulating this point as the yield point, the springback amount is accurately predicted. It cannot be done and is inappropriate.
[0014]
For these conventional methods, the present inventors have devised a third method shown in FIG. That is, the third method is a method in which the yield point is a point where the slope of the characteristic curve changes in the stress-strain curve. According to this method, the disadvantages of the first and second methods can be solved. That is, as shown in the figure, the stress-strain curve in the unloading process has a linear characteristic at the initial stage from the unloading point. Therefore, the point where the slope of the characteristic curve deviates from this straight line is the yield in the unloading process. Used as point X0.
[0015]
FIG. 10 shows the result of simulating the stress-strain relationship based on the plastic constitutive equation according to the prior art using the yield point in the unloading process obtained by the third method, and comparing it with the experimental value. Thus, by this method, a good approximation was obtained between the actual Bauschinger effect data 1 and the simulation result 3 in the region of large plastic strain in the unloading process.
[0016]
However, in the third method, when the point at which the slope of the stress-strain curve changes during the unloading process is used as the yield point in the unloading process, the obtained yield point varies. Was found out. In the stress-strain curve, in the region where the plastic strain increases from the yield point in the unloading process, there is a considerable difference between the actual Bausinger effect data 1 and the simulation result 3, and the springback amount is predicted. It became clear that there was a big madness.
[0017]
The present invention solves the problems of the prior art, obtains the yield point in the unloading process with less variation, and includes the experimental results including the region where the plastic strain increases from the yield point in the unloading process in the stress-strain curve. It is to provide a method for simulating the stress-strain relationship of an elastoplastic material and a method for obtaining a yield point in the unloading process, which can be matched with high accuracy.
[0018]
[Means for Solving the Problems]
The present invention is based on the fact that two new findings have been obtained by carefully analyzing the unloading process, in particular, the transition region from the elastic region to the plastic region, as a problem of the prior art. One of them is that in this transition region, the change in the tangential gradient of the stress-strain curve is characteristic, and from this, it has been found that the yield point in the unloading process can be obtained with little variation. Second, it is clear that the coefficient function of the prior art is not sufficient to express the actual Bauschinger effect when using the yield point in the unloading process with low variability and high accuracy obtained in this way. Thus, an alternative coefficient function has been found. First, these contents will be described.
[0019]
1. Yield point in unloading process
FIG. 1 is a diagram showing a stress σ-strain ε curve in the unloading process, in which a tangential gradient δσ / δε is calculated for each stress value, stress is plotted on the horizontal axis and tangential gradient δσ / δε is plotted on the vertical axis. is there. As is clear from this figure, the behavior of the tangential gradient δσ / δε can be divided into four regions. The first region (1) is a region in the stress state near the unloading point, and the tangential gradient δσ / δε has a high value, but changes linearly with a small change rate as the stress changes. In the second region (2), the value of the tangential gradient δσ / δε decreases nonlinearly as the difference in stress from the unloading point increases. In the third region (3), the tangential gradient δσ / δε decreases almost linearly with a large change rate with respect to the change in stress. In the fourth region (4), the tangential gradient δσ / δε decreases nonlinearly toward zero as the stress changes.
[0020]
In the conventional method in which the point at which the stress-strain curve gradient in the unloading process changes is the yield point in the unloading process, the region where the tangential gradient δσ / δε is constant is assumed. There is no significant area. Therefore, even if an attempt is made to obtain a point at which the slope of the stress-strain curve changes, the result varies.
[0021]
Therefore, if the relationship between the tangential gradient δσ / δε and the stress σ in FIG. 1 is carefully analyzed, the tangential gradient δσ / δε decreases as the stress changes, but the change is small at the initial stage and changes linearly. Then, the linear change becomes non-linear, and the tangential gradient δσ / δε rapidly decreases greatly. Therefore, a point where the relationship between the tangential gradient δσ / δε and the stress σ deviates from linearity can be considered as a point where the stress-strain relationship changes substantially, and this can be considered as a yield point in the unloading process. That is, the boundary point between the first region and the second region in FIG. 1 is determined as the yield point in the unloading process.
[0022]
If the stress σ-strain ε curve in the material unloading process is given, the point where the relationship between the tangential gradient δσ / δε and the stress σ deviates from linearity can be uniquely determined. Yield point can be obtained with little variation.
[0023]
2. About coefficient functions
FIG. 2 shows the yield point in the unloading process with small variation and high accuracy obtained in this way. The plastic constitutive equation is based on the plastic constitutive equation having the coefficient function identified by the experimental value. The result of simulating the relationship is shown. There is a difference between the actual Bauschinger effect data 5 and the simulation result 7. As described above, when the yield point in the unloading process is uniquely determined, the actual Bauschinger effect cannot be expressed sufficiently by using the conventional coefficient function.
[0024]
Therefore, it was found that the actual Bausinger effect can be expressed sufficiently by carefully analyzing the coefficient function and using the following coefficient function. That is, the amount of isotropic hardening (R) and the coefficient of the back stress function are expressed as equivalent plastic strain (ε^p _eq) And equivalent plastic strain at the start of unloading (ε^p _T) As a variable function.
[0025]
In the prior art, the amount of isotropic hardening (R) is equivalent to the equivalent plastic strain (ε^p _eq) Is a non-linear function represented by Expression (5).
[Equation 8]
R = K * (ε^p _eq)ⁿ                (5)
Here, K and n are constants determined by the material.
[0026]
Also, the coefficient (H) representing the magnitude of linear kinematic hardening is equivalent to the equivalent plastic strain (ε^p _eq) Depending on the size of the three cases, the equivalent plastic strain (ε^p _eq) In the first case where the constant is small, the equivalent plastic strain (ε^p _eq) Function. For example, Expression (6) can be used as the function.
[Equation 9]
H = p * (ε^p _eq) + Q (6)
[0027]
Further, when the coefficient C representing the convergence speed of the nonlinear kinematic hardening is expressed by the equation (7), ε₀Is the constant plastic strain at the time of unloading (ε^p _T) Function.
[Expression 10]
C = C ′ / 2 * (ε^p _eq+ Ε₀)^1/2      (7)
[0028]
FIG. 3 shows the amount of isotropic hardening (R), the coefficient representing the magnitude of linear kinematic hardening (H), and ε in the equation representing the coefficient C representing the speed of convergence of nonlinear kinematic hardening.₀The results of simulating the stress-strain relationship by applying the above-described function are shown below. The actual Bauschinger effect data 9 and the simulation result 11 are in good agreement. In this way, the actual Bauschinger effect can be sufficiently expressed by using an appropriate coefficient function.
[0029]
  3. Problem solving means
In order to achieve the object of the present invention, the stress-strain relationship simulation method according to the present invention approximates the stress-strain relationship of an elastoplastic material with a composite curing model combining an isotropic curing model and a kinematic curing model. The yield function as a plastic constitutive equation that gives the stress-strain relationship is defined as a function with the amount of isotropic hardening and the back stress as variables, where the back stress is a component of linear kinematic hardening and a component of non-linear kinematic hardening. A stress-strain relationship that expresses the Bauschinger effect of an elastoplastic material by defining a back stress function having a coefficient function with the equivalent plastic strain of the material as a variable. The amount of hardening R and the coefficient of the back stress function are expressed as equivalent plastic strain (ε^p _eq) And equivalent plastic strain at the start of unloading (ε^p _T) As a variableTheThe isotropic cure amount R isε ^p _eq Nonlinear functionR = K * (ε^p _eq)ⁿ,However, n is a number other than 1 and is one of the coefficients of the back stress function.The coefficient H representing the magnitude of linear kinematic hardening is the equivalent plastic strain ε^p _eqFunctionsage,One of the coefficients of the back stress functionA coefficient C representing the convergence speed of nonlinear kinematic hardening is expressed as C = C ′ / 2 * (ε^p _eq+ Ε₀)^1/2Ε₀IsEquivalent plastic strain at the start of unloadingε^p _T Is a variableSekiNumber andAnd an experimental value acquisition step for acquiring stress-strain-related experimental values as a plurality of discrete values, and a coefficient function for identifying the coefficient function including K, n, and C ′ determined by the material based on the plurality of experimental values. An identification step, and using the identified coefficient function to simulate a stress-strain relationship.
[0030]
Further, in the stress-strain relationship simulating method according to the present invention, in the experimental value acquisition step, after applying tensile stress to generate plastic strain, in the unloading process of unloading the tensile stress, A yield point calculation step of calculating a tangential gradient δσ / δε in a strain curve with each stress value, and calculating a point where the relationship between the tangential gradient δσ / δε and the stress σ deviates from linearity as a yield point in the unloading process. Equivalent plastic strain ε from the strain after the calculated yield point^p _eqIt is characterized by calculating | requiring.
[0031]
In the stress-strain relationship simulation method according to the present invention, S_i(I = x, y, z) is a normal stress σ acting on a plane perpendicular to the coordinate system corresponding to the index i._iDeviation stress with respect to S_ij(I, j = x, y, z) is applied to the plane perpendicular to the coordinate system corresponding to the index i, and the shear stress σ acting in the direction of the coordinate axis corresponding to the index j_ijDeviation stress with respect to α ′_i(I = x, y, z) is a deviation component relating to a back stress acting on a plane perpendicular to the coordinate system corresponding to the index i, α ′_ij(I, j = x, y, z) is a deviation component related to the back stress acting in the direction of the coordinate axis corresponding to the index j on a plane perpendicular to the coordinate system corresponding to the index i, Y₀Is the initial yield stress, which is the yield stress when plastic deformation of the material is first started, the yield function (f) is
## EQU11 ##
f = (1/2) * [(S_x−α ′_x)²+ (S_y−α ′_y)²+ (S_z−α ′_z)²+2 (S_xy−α ′_xy)²+2 (S_yz−α ′_yz)²+2 (S_zx−α ′_zx)²]-(1/3) * (Y₀+ R)²
It is characterized by being.
[0032]
In the stress-strain relationship simulation method according to the present invention, S_i(I = x, y, z) is a normal stress σ acting on a plane perpendicular to the coordinate system corresponding to the index i._iExtended deviation stress with respect to S_ij(I, j = x, y, z) is applied to the plane perpendicular to the coordinate system corresponding to the index i, and the shear stress σ acting in the direction of the coordinate axis corresponding to the index j_ijExtended deviation stress with respect to α ′_i(I = x, y, z) is a deviation component relating to a back stress acting on a plane perpendicular to the coordinate system corresponding to the index i, α ′_ij(I, j = x, y, z) is a deviation component related to the back stress acting in the direction of the coordinate axis corresponding to the index j on a plane perpendicular to the coordinate system corresponding to the index i, Y₀Is the initial yield stress that is the yield stress when plastic deformation of the material is first started, the yield function (f) is
[Expression 12]
f = (1/4) * [(S_x−α ′_x)^Four+ (S_y−α ′_y)^Four+ (S_z−α ′_z)^Four+2 (S_xy−α ′_xy)^Four+2 (S_yz−α ′_yz)^Four+2 (S_zx−α ′_zx)^Four]-(1/9) * (Y₀+ R)^Four
It is characterized by being.
[0033]
In the stress-strain relationship simulating method according to the present invention, [α′1] is a tensor representing a first back stress deviation component that is a component depending on the nonlinearity of kinematic hardening in the back stress, [α ′ 2] is a tensor representing a second back stress deviation component which is a component depending on the linearity of kinematic hardening in the back stress, [dε^p], A tensor representing the plastic strain increment, dε^p _eqIs an equivalent plastic strain increment, C is a coefficient representing the convergence speed of nonlinear kinematic hardening, C ′ is a constant, (2/3) a is a convergence value of non-linear kinematic hardening, ε₀Equivalent plastic strain ε^p _eqRepresents a deviation component (α ′) of the back stress when a variable for preventing C from being infinite when H is 0, and H is a coefficient representing the magnitude of linear kinematic hardening. The tensor [α´] is
[Formula 13]
[α ′] = [α′1] + [α′2]
[dα′1] = C [2/3 (a * [dε^p])-Dε^p _eq* [Α'1]]
[dα′2] = 2/3 (H * [dε^p])
C = C ′ / 2 * (ε^p _eq+ Ε₀)^1/2
It is characterized by being.
[0034]
In addition, the method for obtaining the yield point in the unloading process according to the present invention is that the yield point in the unloading process of unloading the tensile stress after applying a tensile stress in the stress-strain relationship of the material to cause plastic strain. A tangential gradient calculating step for calculating a tangential gradient δσ / δε in each stress-strain curve with respect to each stress value, and a point where the relationship between the tangential gradient δσ / δε and the stress σ deviates from linearity. A yield point calculating step of calculating as a yield point in the unloading process.
[0035]
The stress-strain relationship simulation method according to the present invention includes an experimental value acquisition step for acquiring stress-strain relationship experimental values as a plurality of discrete values, and the coefficient function including K, n, and C ′ determined by the material. A coefficient function identifying step for identifying based on the plurality of experimental values, and using the identified coefficient function, a stress-strain relationship is simulated. Then, the amount of isotropic hardening and the coefficient of the back stress function are expressed as equivalent plastic strain (ε^p _eq) And equivalent plastic strain at the start of unloading (ε^p _T) As a variable function, and the isotropic hardening amount R is R = K * (ε^p _eq)ⁿAnd the coefficient H representing the magnitude of linear kinematic hardening is equivalent plastic strain (ε^p _eq) And a coefficient C representing the convergence speed of nonlinear kinematic hardening, C = C ′ / 2 * (ε^p _eq+ Ε₀)^1/2Ε₀Is ε^p _TIs a function of As a result, the stress-strain curve can be matched with the experimental result with high accuracy including the region where the plastic strain increases from the yield point in the unloading process.
[0036]
In the stress-strain relationship simulation method according to the present invention, the tangential gradient δσ / δε in the stress-strain curve is calculated for each stress value, and the relationship between the tangential gradient δσ / δε and the stress σ is linear. The point that falls off is calculated as the yield point in the unloading process. Therefore, the yield point in the unloading process can be obtained with less variation, and the stress-strain curve can be matched with the experimental results with high accuracy including the region where the plastic strain increases from the yield point in the unloading process.
[0037]
In the stress-strain relationship simulating method according to the present invention, the yield function as the plastic constitutive equation is a quadratic function of the deviation component of the deviation stress and the back stress. The yield function as a plastic constitutive equation is a quartic function of the deviation component of the deviation stress and the back stress. As a result, when the function of the yield function is quadratic, the yield point in the unloading process is obtained with little variation in the case of the fourth order, and the plastic strain increases from the yield point in the unloading process in the stress-strain curve. It is possible to match the experimental results including the region with high accuracy.
[0038]
DETAILED DESCRIPTION OF THE INVENTION
4 and 5 are flowcharts according to the embodiment of the present invention. FIG. 4 shows a flowchart of a stress-strain relationship simulation method, and FIG. 5 shows a flowchart of a method for obtaining a yield point in the unloading process.
[0039]
S1 in FIG. 4 is an unloading process yield point calculating step, using a test piece made of substantially the same material as the workpiece to be simulated, first applying a tensile stress and unloading when a predetermined plastic strain is reached, This is a process of obtaining a stress-strain curve in the unloading process and obtaining a yield point in the unloading process. The detailed contents of this process will be described later with reference to FIG.
[0040]
S3 is an experimental value acquisition process other than the yield point, and using a test piece made of the same material as the unloading process yield point calculating process of S1, a plurality of stress-strain relationships other than the yield point of the unloading process are obtained through experiments. It is a process of acquiring as a discrete value. For example, the isotropic hardening amount R and the equivalent plastic strain (ε^p _eq), The coefficient H representing the magnitude of linear kinematic hardening and the equivalent plastic strain (ε^p _eq), The coefficient C representing the convergence speed of nonlinear kinematic hardening is expressed as C = C ′ / 2 * (ε^p _eq+ Ε₀)^1/2Ε₀And equivalent plastic strain at unloading ε^p _TA plurality of discrete values are acquired for the relationship.
[0041]
S5 is a coefficient function identification step, and a function of the coefficient function including K, n, C ′ determined by the material from a plurality of discrete values of the stress-strain relationship acquired in the experimental value acquisition step other than the yield point of S3. It is the process of identifying. For example, for the isotropic hardening amount R, a plurality of equivalent plastic strains (ε^p _eq), K and n are determined from experimental values, and R = K * (ε^p _eq)ⁿIs specifically identified. The coefficient H representing the magnitude of linear kinematic hardening is equivalent plastic strain (ε^p _eq) To identify the function itself and₀Is equivalent plastic strain ε at unloading^p _TThe function itself is identified from the experimental value for.
[0042]
S7 is a stress-strain relationship calculation step, which uses the coefficient function identified in S5 to determine a yield function, which is a plastic constitutive equation, and calculates a stress-plastic strain relationship. Since the calculation formula for the stress-strain relationship in the elastic region is easier than in the plastic region, the elastic region and the plastic region can be connected to calculate the stress-strain relationship for the entire elastic-plastic region.
[0043]
S9 is a spring back amount calculation step, and each step of processing a complex three-dimensional shape of the actual work material using the plastic constitutive equation obtained as a result of the stress-strain relationship calculation step of S7 and the finite element method. Is a step of calculating the amount of springback.
[0044]
FIG. 5 shows a detailed flowchart of the yield point calculation step of the unloading process in S1 of FIG. S21 is a stress-strain relationship acquisition step, using a test piece made of substantially the same material as the workpiece to be simulated, applying a tensile stress first, unloading when a predetermined plastic strain is reached, and stress in the unloading process. A step of obtaining a strain curve;
[0045]
S23 is a step of calculating a tangential gradient δσ / δε for each stress value from the stress σ-strain ε curve acquired in the stress-strain relationship acquisition step of S21 in the tangential gradient calculation step.
[0046]
S25 is a yield point calculation step, and the relationship between the tangent gradient δσ / δε and the stress σ obtained in the tangential gradient calculation step of S23 is a point where the change of the tangential gradient δσ / δε with respect to the change of stress deviates from the linearity. It is a process of calculating as a yield point in the unloading process. Specifically, the boundary point between the first region and the second region in FIG. 1 is calculated as the yield point in the unloading process.
[0047]
【The invention's effect】
The stress-strain relationship simulating method according to the present invention obtains the yield point in the unloading process with little variation, and in the stress-strain curve, including the region where the plastic strain increases from the yield point in the unloading process, Match with high accuracy. Moreover, the method for obtaining the yield point in the unloading process according to the present invention can obtain the yield point in the unloading process with little variation.
[Brief description of the drawings]
In the embodiment according to the present invention, the tangential gradient δσ / δε of the stress σ-strain ε curve in the unloading process is calculated with each stress value, the stress is plotted on the horizontal axis, and the tangential gradient δσ / FIG. 6 is a diagram showing δε.
FIG. 2 is a diagram showing a result of simulating a stress-strain relationship based on a conventional technique using a yield point in an unloading process according to an embodiment of the present invention.
FIG. 3 is a diagram showing a result of simulating a stress-strain relationship according to the embodiment of the present invention.
FIG. 4 is a flowchart of a stress-strain relationship simulation method according to an embodiment of the present invention.
FIG. 5 is a flowchart of a method for obtaining a yield point in an unloading process according to an embodiment of the present invention.
FIG. 6 is a diagram showing the state of the springback phenomenon, with strain on the horizontal axis and stress on the vertical axis.
FIG. 7 is a diagram showing a first method for determining a yield point of a material.
FIG. 8 is a diagram showing a second method for determining a yield point of a material.
FIG. 9 is a diagram showing a third method for determining the yield point of a material.
FIG. 10 shows the result of simulating the stress-strain relationship based on the plastic constitutive equation of the prior art, using the yield point in the unloading process obtained by the third method, and comparing the result with the experimental result. FIG.
[Explanation of symbols]
1, 5, 9 Actual bauschinger effect data, 3, 7, 11 Simulation results.

Claims (6)

Approximate the stress-strain relationship of an elastoplastic material with a composite hardening model that combines an isotropic hardening model and a kinematic hardening model,
The yield function as a plastic constitutive equation giving the stress-strain relationship is defined as a function with the amount of isotropic hardening and back stress as variables,
Where the back stress is
It is composed of a component of linear kinematic hardening and a component of non-linear kinematic hardening, and is defined by a back stress function having a coefficient function with the equivalent plastic strain of the material as a variable.
A method for simulating a stress-strain relationship expressing the Bauschinger effect of an elastoplastic material,
Said isotropic hardening amount R, the coefficient of the back stress function, and the equivalent plastic strain (ε ^p _eq) and unloading at the start of the equivalent plastic strain (epsilon ^{p _T)} the coefficient function whose variable,
The amount of isotropic hardening R is a nonlinear function of ε ^p _eq R = K * (ε ^p _eq ) ⁿ , where n is a number other than 1,
Factor H that represents the magnitude of the linear kinematic hardening, which is one of the coefficients of the back stress function, a function of the equivalent plastic strain epsilon ^p _eq,
A coefficient C representing the convergence speed of nonlinear kinematic hardening, which is one of the coefficients of the back stress function ,
C = C'/ 2 * (ε p eq + ε 0) ε 0 when expressed as ^1/2, the functions that the variable equivalent plastic strain epsilon ^p _T at the start of unloading,
An experimental value acquisition step of acquiring stress-strain-related experimental values as a plurality of discrete values;
A coefficient function identifying step for identifying the coefficient function including K, n, and C ′ determined by the material based on the plurality of experimental values;
Including
A stress-strain relationship simulation method, wherein the stress-strain relationship is simulated using the identified coefficient function. The stress-strain relationship simulation method according to claim 1,
In the experimental value acquisition step,
In the unloading process in which tensile stress is applied and plastic strain is generated and then the tensile stress is unloaded, a tangential gradient δσ / δε in the stress-strain curve is calculated for each stress value, and the tangential gradient δσ / δε is calculated. A yield point calculating step of calculating a yield point in the unloading process at a point where the relationship with the stress σ deviates from linearity, and obtaining an equivalent plastic strain ε ^p _eq from the strain after the calculated yield point. To simulate stress-strain relationship. In the stress-strain relationship simulation method according to claim 1 or 2,
S _i (i = x, y, z) is a deviation stress with respect to a normal stress σ _i acting on a plane perpendicular to the coordinate system corresponding to the index i,
S _ij (i, j = x, y, z) is applied to the plane perpendicular to the coordinate system corresponding to the index i, the deviation stress relating to the shear stress σ _ij acting in the direction of the coordinate axis corresponding to the index j,
α ′ _i (i = x, y, z) is a deviation component related to the back stress acting on a plane perpendicular to the coordinate system corresponding to the index i,
α ′ _ij (i, j = x, y, z) is a deviation component related to the back stress acting in the direction of the coordinate axis corresponding to the index j on a plane perpendicular to the coordinate system corresponding to the index i,
Y ₀ is the initial yield stress, which is the yield stress when plastic deformation of the material is first started,
When
The yield function (f) is
(Equation 1)
f = (1/2) * [(S _x −α ′ _x ) ² + (S _y −α ′ _y ) ² + (S _z −α ′ _z ) ² +2 (S _xy −α ′ _xy ) ² +2 ( S _yz −α ′ _yz ) ² +2 (S _zx −α ′ _zx ) ² ] − (1/3) * (Y ₀ + R) ²
A stress-strain relationship simulation method characterized by: In the stress-strain relationship simulation method according to claim 1 or 2,
S _i (i = x, y, z) is an extended deviation stress with respect to a normal stress σ _i acting on a plane perpendicular to the coordinate system corresponding to the index i,
S _ij (i, j = x, y, z) is applied to the plane perpendicular to the coordinate system corresponding to the index i, the expanded deviation stress with respect to the shear stress σ _ij acting in the direction of the coordinate axis corresponding to the index j,
α ′ _i (i = x, y, z) is a deviation component related to the back stress acting on a plane perpendicular to the coordinate system corresponding to the index i,
α ′ _ij (i, j = x, y, z) is a deviation component related to the back stress acting in the direction of the coordinate axis corresponding to the index j on a plane perpendicular to the coordinate system corresponding to the index i,
Y ₀ is the initial yield stress, which is the yield stress when plastic deformation of the material is first started,
When
The yield function (f) is
(Equation 2)
f = (1/4) * [(S _x −α ′ _x ) ⁴ + (S _y −α ′ _y ) ⁴ + (S _z −α ′ _z ) ⁴ +2 (S _xy −α ′ _xy ) ⁴ +2 ( S _yz −α ′ _yz ) ⁴ +2 (S _zx −α ′ _zx ) ⁴ ] − (1/9) * (Y ₀ + R) ⁴
A stress-strain relationship simulation method characterized by: In the stress-strain relationship simulation method according to claim 3 or 4,
{α′1} is a tensor representing a first back stress deviation component that is a component of back stress that depends on kinematic nonlinearity;
{α′2} is a tensor representing a second back stress deviation component which is a component depending on linearity of kinematic hardening among back stresses,
{dε ^p } is a tensor representing the plastic strain increment,
dε ^p _eq is equivalent plastic strain increment,
C is a coefficient representing the convergence speed of nonlinear kinematic hardening,
C ′ is a constant,
(2/3) a is a convergence value of nonlinear kinematic hardening,
ε ₀ is a variable for preventing C from becoming infinite when the equivalent plastic strain ε ^p _eq is 0,
H is a coefficient representing the magnitude of linear kinematic hardening,
When
The tensor {α ′} representing the deviation component (α ′) of the back stress is:
(Equation 3)
{α ′} = {α′1} + {α′2}
{dα'1} = C [2/3 ( a * {dε p}) - dε p eq * {α'1}]
{dα′2} = 2/3 (H * {dε ^p })
C = C ′ / 2 * (ε ^p _eq + ε ₀ ) ^1/2
A stress-strain relationship simulation method characterized by: In the stress-strain relationship of the material, after applying tensile stress and generating plastic strain, it is a method for obtaining the yield point in the unloading process of unloading the tensile stress,
A tangential gradient calculating step of calculating a tangential gradient δσ / δε in the stress-strain curve with each stress value;
A yield point calculating step of calculating a point where the relationship between the tangential gradient δσ / δε and the stress σ deviates from linearity as a yield point in the unloading process;
A method for obtaining a yield point in an unloading process characterized by comprising:

Images