CN113077854B - Construction and solving method of elasto-viscous plastic material fractal constitutive model - Google Patents

Construction and solving method of elasto-viscous plastic material fractal constitutive model Download PDF

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CN113077854B
CN113077854B CN202110628530.0A CN202110628530A CN113077854B CN 113077854 B CN113077854 B CN 113077854B CN 202110628530 A CN202110628530 A CN 202110628530A CN 113077854 B CN113077854 B CN 113077854B
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李德建
饶远昊
刘校麟
丁涛
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Abstract

The invention discloses a construction and solving method of an elasto-viscoplastic material fractal constitutive model, which comprises the steps of forming an elementary module by combining and connecting basic mechanical element modules, iterating the basic mechanical element modules by using the elementary module on the basis of the elementary module to obtain the fractal element constitutive model, wherein in the iteration process, the parameters of each basic mechanical element module in a newly added part of the first iteration are preset multiples of the parameters of each basic mechanical element module in the elementary module; and starting from the second iteration, the parameters of each basic mechanical element module in the next iteration newly-added part are preset multiples of the parameters of each basic mechanical element module in the previous iteration newly-added part. The model constructed by the invention not only can well describe the elastoviscoplasticity mechanical properties of the material, but also can describe the very complicated actual physical structure of the material in a microscopic view; the complexity of the fractal constitutive model can be expressed, the number of parameters can not be increased, and the solution of the constructed fractal constitutive model can be well realized.

Description

Construction and solving method of elasto-viscous plastic material fractal constitutive model
Technical Field
The invention relates to the technical field of material mechanics, in particular to a method for constructing and solving an elasto-viscous plastic material fractal constitutive model.
Background
Materials of nature have complex physicomechanical properties, including various forms of elasticity, viscosity, plasticity, and combinations thereof. The component model simply and intuitively describes the elasticity, viscosity and plasticity of the material by using a plurality of basic mechanical components, and is an important means for researching the mechanical properties of the material.
The mathematical substance of the element model is a differential equation, the order of the element model is related to the complexity of the model, the order of the differential equation is increased by one order every time a basic mechanical element is added in the model, and when the model is more complex, the equation is difficult to solve, and even an analytic solution cannot be obtained.
At present, common element models such as Maxwell and Burgers have fewer elements, although the models are easy to solve, the expressive force of the models is insufficient, the elastoviscoplasticity mechanical properties of the materials cannot be well described, and the addition of elements can cause that a differential equation cannot be solved. On the other hand, the actual physical structure of the material in a microscopic view is very complex, and the element model simplifies the physical structure of the material into a series-parallel combination of several basic mechanical elements, which is not in accordance with the actual structure.
Disclosure of Invention
In order to solve the problems in the prior art, the invention provides the following technical scheme.
The invention provides a method for constructing an elasto-viscous plastic material fractal constitutive model on one hand, which comprises the following steps:
respectively constructing basic mechanical element modules for describing elasticity, viscosity and plasticity of the material;
combining and connecting the basic mechanical element modules to obtain primitive modules;
on the basis of the primitive module, the primitive module is utilized to iterate the basic mechanical element module to obtain a fractal element constitutive model; the parameters of each basic mechanical element module in the newly added part of the first iteration are preset multiples of the parameters of each basic mechanical element module in the primitive module; and starting from the second iteration, the parameters of each basic mechanical element module in the next iteration newly-added part are preset multiples of the parameters of each basic mechanical element module in the previous iteration newly-added part.
Preferably, the elastoviscous plastic material fractal constitutive model is constructed on the basis of a Matlab/Simulink platform.
Preferably, the combining and connecting the basic mechanical element modules to obtain a primitive module comprises:
reserving an input port and an output port of the basic mechanical element module, and packaging the basic mechanical element module;
and combining and connecting the packaged basic mechanical element modules according to the combined connection rule of the basic mechanical elements to obtain the primitive modules.
Preferably, the rules of the combined connection of the basic mechanical elements include:
the basic mechanical element describing the viscosity of the material is connected in parallel with the basic mechanical element describing the plasticity of the material and then connected in series with the basic mechanical element describing the elasticity of the material.
Preferably, the assembling and connecting the packaged basic mechanical element modules to obtain a basic module comprises:
the method comprises the steps of connecting the load of an input module to a basic mechanical element module describing elasticity of the material and a basic mechanical element module describing plasticity of the material respectively, connecting the output of the basic mechanical element module describing plasticity of the material to a basic mechanical element module describing viscosity of the material, and adding the outputs of the basic mechanical element module describing elasticity of the material and the basic mechanical element module describing viscosity of the material.
Preferably, the basic mechanical elements describing elasticity, viscosity and plasticity of the material are a spring, a sticky kettle and a slip sheet respectively, and the corresponding modules are a spring module, a sticky kettle module and a slip sheet module respectively.
Preferably, the construction of the spring module comprises: selecting a digital module, an import module and an export module in a Library browser of a Matlab/Simulink platform, and describing the operation process of dividing an input stress signal by an elastic modulus.
Preferably, the construction of the clay pot module comprises: a Divide module, an Integrator module, an import module and an export module are selected from a Library browser of a Matlab/Simulink platform, and the operation process of dividing an input stress signal by a viscosity coefficient and then integrating the viscosity coefficient with time is described.
Preferably, the construction of the slider module comprises: selecting Switch, relative Operator, Constant, Add, subtrect, Product, import and output modules in the Library browser of the Matlab/Simulink platform, and describing an operation process of obtaining an output value by comparing an input stress signal with yield stress.
The invention further provides a solving method of the elasto-viscous plastic material fractal constitutive model, and the elasto-viscous plastic material fractal constitutive model is constructed according to the method.
The invention has the beneficial effects that: the invention provides a construction and solving method of an elasto-viscoplastic material fractal constitutive model, which comprises the steps of forming an elementary module by combining and connecting basic mechanical element modules, iterating the basic mechanical element modules by using the elementary module on the basis of the elementary module to obtain the fractal element constitutive model, wherein in the iteration process, the parameters of each basic mechanical element module in a newly added part of the first iteration are preset multiples of the parameters of each basic mechanical element module in the elementary module; and starting from the second iteration, the parameters of each basic mechanical element module in the next iteration newly-added part are preset multiples of the parameters of each basic mechanical element module in the previous iteration newly-added part. Therefore, the number of basic mechanical element modules is increased in an iterative mode, so that the elastoviscoplastic mechanical properties of the material and the very complex actual physical structure of the material in a microscopic view can be well described; in the iteration process, the complexity of the fractal constitutive model can be represented without increasing the number of parameters by setting the parameters of each basic mechanical element module in the next iteration newly-added part as the preset multiples of the parameters of each basic mechanical element module in the last iteration newly-added part, so that the constructed fractal constitutive model can be well solved.
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FIG. 1 is a schematic flow chart of a method for constructing an elasto-viscous plastic material fractal constitutive model according to the present invention;
FIG. 2 is a schematic view of a spring module according to an embodiment of the present invention;
FIG. 3 is a schematic view of a stick kettle module according to an embodiment of the present invention;
FIG. 4 is a schematic view of a slider module according to an embodiment of the present invention;
FIG. 5 is a schematic diagram of a module for encapsulating mechanical elements according to an embodiment of the invention;
FIG. 6 is a diagram illustrating a primitive model according to an embodiment of the present invention;
FIG. 7 is a diagram illustrating a primitive module according to an embodiment of the present invention;
fig. 8 is a schematic diagram of an n-order fractal element constitutive model construction process expressed by a primitive model iteration process according to an embodiment of the present invention;
fig. 9 is a parameter display diagram of basic mechanical element modules of each part in the fourth-order fractal element constitutive model according to the embodiment of the present invention.
Detailed Description
In order to better understand the technical solution, the technical solution will be described in detail with reference to the drawings and the specific embodiments.
The method provided by the invention can be implemented in the following terminal environment, and the terminal can comprise one or more of the following components: a processor, a memory, and a display screen. Wherein the memory has stored therein at least one instruction that is loaded and executed by the processor to implement the methods described in the embodiments described below.
A processor may include one or more processing cores. The processor connects various parts within the overall terminal using various interfaces and lines, performs various functions of the terminal and processes data by executing or executing instructions, programs, code sets, or instruction sets stored in the memory, and calling data stored in the memory.
The Memory may include a Random Access Memory (RAM) or a Read-Only Memory (ROM). The memory may be used to store instructions, programs, code sets, or instructions.
The display screen is used for displaying user interfaces of all the application programs.
In addition, those skilled in the art will appreciate that the above-described terminal configurations are not intended to be limiting, and that the terminal may include more or fewer components, or some components may be combined, or a different arrangement of components. For example, the terminal further includes a radio frequency circuit, an input unit, a sensor, an audio circuit, a power supply, and other components, which are not described herein again.
Example one
As shown in fig. 1, an embodiment of the present invention provides a method for constructing an elasto-viscous plastic material fractal constitutive model, including:
s101, respectively constructing basic mechanical element modules for describing elasticity, viscosity and plasticity of materials;
s102, combining and connecting the basic mechanical element modules to obtain primitive modules;
s103, on the basis of the primitive module, the primitive module is utilized to iterate the basic mechanical element module to obtain a fractal element constitutive model; the parameters of each basic mechanical element module in the newly added part of the first iteration are preset multiples of the parameters of each basic mechanical element module in the primitive module; and starting from the second iteration, the parameters of each basic mechanical element module in the next iteration newly-added part are preset multiples of the parameters of each basic mechanical element module in the previous iteration newly-added part.
Fractal is an organizational set used to describe a highly chaotic and complex, unordered arrangement. Fractal sets have some form of self-similarity, including approximate self-similarity or statistical self-similarity, which can also be used to describe complex physicomechanical properties of materials. The invention describes the complex physical and mechanical properties of the material, such as elasticity, viscosity, plasticity and the like, by constructing a fractal element constitutive model.
In a preferred embodiment of the invention, the elastoviscous plastic material fractal constitutive model is constructed based on a Matlab/Simulink platform.
The Matlab/Simulink platform is an integrated software package for dynamic system modeling, simulation and comprehensive analysis, has the functions of modularization, visualization and simulation modeling, can conveniently construct a constitutive model with a complex structure, and can obtain a simulation result without analyzing and solving the integral model.
Step S101 is executed to build a basic mechanical element module describing elasticity, viscosity and plasticity of the material. As an example, a spring module, a stick-can module, and a slide module are constructed to describe elasticity, viscosity, and plasticity of the material, respectively.
Wherein, according to the physical characteristics of the spring, the constitutive equation can be expressed as:
Figure 467587DEST_PATH_IMAGE001
Figure 587989DEST_PATH_IMAGE002
the stress is represented by the expression of,εwhich is indicative of the strain,Erepresenting the modulus of elasticity of the spring element.
The construction of the spring module comprises: selecting a digital module, an import module and an export module in a Library browser of a Matlab/Simulink platform, and describing the operation process of dividing an input stress signal by an elastic modulus. The specific construction process can be as follows: create In1 (input port 1) to represent stress, In2 (input port 2) to represent elastic modulus, divide input port 1 by input port 2 with Dvide module, resulting In strain output to Out1 (output port 1). The constructed spring module can be as shown in fig. 2.
According to the physical characteristics of the sticky kettle, the constitutive equation can be expressed as follows:
Figure 570989DEST_PATH_IMAGE003
Figure 435040DEST_PATH_IMAGE002
the stress is represented by the expression of,
Figure 240226DEST_PATH_IMAGE004
which is indicative of the rate of strain,ηindicating the viscosity coefficient of the viscous kettle element.
The construction of the sticky pot module comprises the following steps: a Divide module, an Integrator module, an import module and an export module are selected from a Library browser of a Matlab/Simulink platform, and the operation process of dividing an input stress signal by a viscosity coefficient and then integrating the viscosity coefficient with time is described. The specific construction process can be as follows: creating In1 (input port 1) to represent stress, In2 (input port 2) to represent viscosity coefficient, dividing the input port 1 by the input port 2 by a Dvide module, and integrating the time by an Integrator module to obtain a strain rate output to Out1 (output port 1). The constructed clay pot module can be as shown in fig. 3.
According to the physical characteristics of the sliding sheet element, the constitutive equation is expressed as:
Figure 531530DEST_PATH_IMAGE005
Figure 532984DEST_PATH_IMAGE002
the stress is represented by the expression of,εwhich is indicative of the strain,
Figure 200726DEST_PATH_IMAGE006
representing the yield stress of the slider element.
The construction of the sliding piece module comprises the following steps: selecting Switch, relative Operator, Constant, Add, subtrect, Product, import and output modules in the Library browser of the Matlab/Simulink platform, and describing an operation process of comparing an input stress signal with yield stress to obtain a strain output value. The specific construction process can be as follows: creation of In1 (input port 1) represents stress and In2 (input port 2) represents yield stress. The input port 1 and the input port 2 are simultaneously connected to two Relational Operator (r.o.) modules for comparison: the upper left corner of the upper Switch module is connected with a subtrect module, the middle of the upper Switch module is connected with an R.O. (relational operation) module, and the lower left corner of the upper Switch module is connected with a Constant module, which means that when the value of an input port 1 is greater than that of an input port 2, the output value is the value of the input port 1 minus the value of the input port 2, otherwise, the output value is 0; the upper left corner of the Switch module below is connected with an Add module, the middle of the Switch module below is connected with an r.o. (relational operation) module, and the lower left corner is connected with a Constant module, which means that when the value of the input port 1 is smaller than the value of the negative input port 2, the output value is the value of the output port 1 minus the value of the input port 2, otherwise, the output value is 0. The output values of the two logical operations are added by an Add module to obtain the stress of the output to Out1 (output port 1). The constructed slider module may be as shown in FIG. 4.
And executing step S102, and combining and connecting the basic mechanical element modules to obtain a primitive module. The method specifically comprises the following steps:
and S1021, reserving an input port and an output port of the basic mechanical element module, and packaging the basic mechanical element module.
As an embodiment, for example, the spring module, the kettle sticking module and the slide module are packaged after an input port and an output port are reserved. The specific packaging process may be:
the block diagram of the spring module, the sticky kettle module or the sliding piece module shown in fig. 2, 3 and 4 is selected, and the Creat Subsystem is selected by clicking the right mouse button, so that the module of the packaging mechanical element (shown in fig. 5) can be obtained. In fig. 5, "input" in the packaged mechanical element module corresponds to an input port of the basic mechanical element module, "output" corresponds to an output port of the basic mechanical element module, and "parameter" corresponds to E, η, σS。
A load module (selected by classification from Source in the built-in library browser) is connected to the input, a Constant module is connected to the parameter, and a simulation result module (selected by classification from Sink in the built-in library browser) is connected to the output, so that the simulation result is displayed.
And S1022, combining and connecting the packaged basic mechanical element modules according to the combination and connection rule of the basic mechanical elements to obtain the primitive modules.
Wherein the combination connection rule of the basic mechanical elements comprises:
the basic mechanical element describing the viscosity of the material is connected in parallel with the basic mechanical element describing the plasticity of the material and then connected in series with the basic mechanical element describing the elasticity of the material.
The step of connecting the packaged basic mechanical element modules in a combined manner to obtain a primitive module comprises:
the method comprises the steps of connecting the load of an input module to a basic mechanical element module describing elasticity of the material and a basic mechanical element module describing plasticity of the material respectively, connecting the output of the basic mechanical element module describing plasticity of the material to a basic mechanical element module describing viscosity of the material, and adding the outputs of the basic mechanical element module describing elasticity of the material and the basic mechanical element module describing viscosity of the material.
As an embodiment, for example, the basic mechanical elements describing elasticity, viscosity and plasticity of the material are a spring, a sticky pot and a slide sheet respectively, and according to the combination connection rule of the basic mechanical elements, the sticky pot element and the slide sheet element can be connected in parallel and then connected in series with the spring element to obtain a cell model, which can be shown in fig. 6. Wherein, the serial connection means that two elements are connected end to end, which means that the stress of the two elements is the same, the strain is different and the sum of the two is the total strain; parallel connection means that two elements are connected end to end, and means that the strains of the two elements are the same, the stresses are different, and the sum of the two stresses is the total stress.
According to the series-parallel connection rule of the basic mechanical elements in the elementary model, the spring has the same stress as the parallel connected kettle-sticking module and slip sheet, namely the total stress, and the sum of the strains is the total strain, and the packaged spring module, kettle-sticking module and slip sheet module are combined and connected to obtain the elementary module, as shown in fig. 7. The method is implemented as follows: the load of the input module is respectively connected to the spring module and the sliding sheet module, and the stress of the spring module and the stress of the sliding sheet module are the same. In FIG. 7, the stress output by the slider module is σ - σS(equivalent to the self-stress of the slider as σS) And inputting the stress into a kettle sticking module to meet the condition that the sum of the stress of the kettle sticking module and the kettle sticking module is the total stress. Finally, the strain output by the spring and the strain output by the sticking kettle are added to obtain the total strain.
And step S103 is executed, the basic mechanical element module is iterated by utilizing the primitive module on the basis of the primitive module, and a fractal element constitutive model is obtained.
In the embodiment of the invention, the execution process of the step is explained by using the iterative process of the primitive model: combining and connecting basic mechanical elements to form an elementary model, and then replacing each basic mechanical element in the initial elementary model by using a new elementary model on the basis of the initial elementary model to obtain a first-order fractal element constitutive model; then, on the basis of the first-order fractal element constitutive model, replacing each basic mechanical element in the first-order fractal element constitutive model by using a new primitive model to obtain a second-order fractal element constitutive model; and analogizing in turn to obtain an n-order fractal element constitutive model as shown in fig. 8. In the Matlab/Simulink platform, a primitive module is used for replacing a primitive model.
Therefore, by adopting the method provided by the invention, the number of basic mechanical element modules is increased in an iterative mode, so that the elastoviscoplasticity mechanical properties of the material and the very complex actual physical structure on the mesoscopic aspect of the material can be well described, and the complexity of the fractal constitutive model can be well expressed.
When the n-order model and the n-1-order model can obtain the same simulation result under the same condition, n can be considered as infinite. In practical application, the method provided by the invention is found to meet the condition when n = 3. Therefore, in the embodiment of the invention, the model construction, solving and simulation processes are simple and easy to implement.
In the iteration process, the parameters of each basic mechanical element module in the newly added part of the first iteration can be set as preset multiples of the parameters of each basic mechanical element module in the primitive module; and starting from the second iteration, the parameters of each basic mechanical element module in the next iteration newly-added part are preset multiples of the parameters of each basic mechanical element module in the previous iteration newly-added part. The parameters of the basic mechanical element module of each part in the four-order fractal element constitutive model can be shown in fig. 9. In fig. 9, the corresponding primitive modules are represented in a primitive model. Before iteration begins, parameters of each basic mechanical element in the elementary model are E, eta and sigmaSThe parameters of each basic mechanical element in the newly added part after the first iteration are aE, b eta and c sigmaSThe preset multiples a, b and c are respectively the preset multiples of the parameters of each basic mechanical element in the elementary model; the parameter of each basic mechanical element in the newly added part after the second iteration is a2E、b2η、c2σSThe preset multiples a, b and c are respectively the parameters of each basic mechanical element in the newly added part after the first iteration; after the third iterationThe parameter of each basic mechanical element in the newly added part is a3E、b3η、c3σSThe preset multiples a, b and c of the parameters of each basic mechanical element in the newly added part after the second iteration are respectively; the parameter of each basic mechanical element in the newly added part after the fourth iteration is a4E、b4η、c4σSAnd preset multiples a, b and c of the parameters of each basic mechanical element in the newly added part after the third iteration are respectively.
Therefore, by adopting the method provided by the invention, in the iteration process, the parameters of the fractal element constitutive model obtained by setting the parameters of each basic mechanical element module in the next iteration newly-added part as the preset multiples of the parameters of each basic mechanical element module in the last iteration newly-added part are not increased compared with the primitive modules, so that the solution of the constructed fractal constitutive model can be easily realized.
Example two
The embodiment of the invention provides a solving method of an elasto-viscous plastic material fractal constitutive model, wherein the elasto-viscous plastic material fractal constitutive model is constructed according to the method in the embodiment I.
The method for constructing the elasto-viscous plastic material fractal constitutive model may specifically refer to the description in the first embodiment, and is not described herein again.
While preferred embodiments of the present invention have been described, additional variations and modifications in those embodiments may occur to those skilled in the art once they learn of the basic inventive concepts. Therefore, it is intended that the appended claims be interpreted as including preferred embodiments and all such alterations and modifications as fall within the scope of the invention. It will be apparent to those skilled in the art that various changes and modifications may be made in the present invention without departing from the spirit and scope of the invention. Thus, if such modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include such modifications and variations.

Claims (6)

1. A method for constructing an elasto-viscous plastic material fractal constitutive model is characterized by comprising the following steps:
respectively constructing basic mechanical element modules for describing elasticity, viscosity and plasticity of the material;
combining and connecting the basic mechanical element modules to obtain a primitive module, which specifically comprises: reserving an input port and an output port of the basic mechanical element module, and packaging the basic mechanical element module; and combining and connecting the packaged basic mechanical element modules according to a combined connection rule of the basic mechanical elements to obtain primitive modules, wherein the combined connection rule of the basic mechanical elements comprises the following steps: connecting a basic mechanical element for describing the viscosity of the material with a basic mechanical element for describing the plasticity of the material in parallel, and then connecting the basic mechanical elements with a basic mechanical element for describing the elasticity of the material in series; the step of connecting the packaged basic mechanical element modules in a combined manner to obtain a primitive module comprises: respectively connecting the load of an input module to a basic mechanical element module describing elasticity of the material and a basic mechanical element module describing plasticity of the material, connecting the output of the basic mechanical element module describing plasticity of the material to the basic mechanical element module describing viscosity of the material, and adding the outputs of the basic mechanical element module describing elasticity of the material and the basic mechanical element module describing viscosity of the material;
on the basis of the primitive module, the primitive module is utilized to iterate the basic mechanical element module to obtain a fractal element constitutive model; the parameters of each basic mechanical element module in the newly added part of the first iteration are preset multiples of the parameters of each basic mechanical element module in the primitive module; and starting from the second iteration, the parameters of each basic mechanical element module in the next iteration newly-added part are preset multiples of the parameters of each basic mechanical element module in the previous iteration newly-added part.
2. The method for constructing the elasto-viscoplastic material fractal constitutive model according to claim 1, wherein the elasto-viscoplastic material fractal constitutive model is constructed based on a Matlab/Simulink platform.
3. The method for constructing the fractal constitutive model of elasto-viscoplastic materials according to claim 1, wherein basic mechanical elements for describing elasticity, viscosity and plasticity of the materials are a spring, a viscous pot and a sliding sheet respectively, and corresponding modules are a spring module, a viscous pot module and a sliding sheet module respectively.
4. The method for constructing the elasto-viscous plastic material fractal constitutive model as claimed in claim 3, wherein the construction of the spring module comprises: selecting a digital module, an import module and an export module in a Library browser of a Matlab/Simulink platform, and describing the operation process of dividing an input stress signal by an elastic modulus.
5. The method for constructing the fractal constitutive model of elasto-viscous plastic material according to claim 3, wherein the construction of the clay pot module comprises: a Divide module, an Integrator module, an import module and an export module are selected from a Library browser of a Matlab/Simulink platform, and the operation process of dividing an input stress signal by a viscosity coefficient and then integrating the viscosity coefficient with time is described.
6. The method for constructing the elasto-viscous plastic material fractal constitutive model as claimed in claim 3, wherein the construction of the sliding sheet module comprises: selecting Switch, relative Operator, Constant, Add, subtrect, Product, import and output modules in the Library browser of the Matlab/Simulink platform, and describing an operation process of obtaining an output value by comparing an input stress signal with yield stress.
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