CN106777595B

CN106777595B - A kind of method of definite ceramic matric composite nonlinear vibration response

Info

Publication number: CN106777595B
Application number: CN201611092274.3A
Authority: CN
Inventors: 宋迎东; 高希光; 陈晶
Original assignee: Nanjing University of Aeronautics and Astronautics
Current assignee: Nanjing University of Aeronautics and Astronautics
Priority date: 2016-11-29
Filing date: 2016-11-29
Publication date: 2018-05-22
Anticipated expiration: 2036-11-29
Also published as: CN106777595A

Abstract

The present invention provides a kind of method of definite ceramic matric composite nonlinear vibration response, the described method includes：Test obtains the stress-strain diagram of ceramic matric composite CMCs in advance；The first formula, the second formula and the 3rd formula is utilized respectively to be fitted stress-strain diagram, and according to the magnitude relationship between the location of upper stress-strain state and parameters, the stress-strain diagram being fitted in subring；Each stress-strain diagram that above-mentioned fitting is obtained is applied in the finite element model of CMCs, to determine the vibratory response of CMCs.The method of definite ceramic matric composite nonlinear vibration response provided by the invention, stress-strain diagram can be fitted under arbitrary plus unloading, and finite element model is combined, it can realize the calculating process of CMCs nonlinear vibration responses, improve the efficiency of calculating.

Description

Method for determining nonlinear vibration response of ceramic matrix composite

Technical Field

The invention belongs to the technical field of composite material mechanical analysis, and particularly relates to a method for determining nonlinear vibration response of a ceramic matrix composite material.

Background

The Ceramic Matrix Composite (CMCs) has excellent high-temperature mechanical properties and is a key material for designing and manufacturing high-temperature structures. However, the stiffness of the CMCs changes with the change of the external load level, and a significant hysteresis loop exists under the action of cyclic load, so that the response of the CMCs under the vibration load is greatly different from that of a linear elastic material. The variable stiffness and the hysteresis characteristics can cause the amplitude-frequency characteristic curve of the CMCs to jump under a specific load, the jump of the amplitude-frequency characteristic curve is very harmful to the structure, and if the phenomenon cannot be accurately predicted and calculated, the structure can suddenly resonate under a vibration load to cause structural damage.

The existing research mainly focuses on damping test and calculation of CMCs and elastic vibration calculation of beams and plates of CMCs, but lacks on research on nonlinear vibration of the CMCs and mainly lacks on a constitutive model capable of describing the CMCs under any loading and unloading condition. Some scholars establish a CMCs constitutive model by using a mesomechanics method and a finite element method, and are not suitable for nonlinear vibration calculation of the CMCs structure due to overlarge calculated amount.

Disclosure of Invention

The invention aims to provide a method for determining the nonlinear vibration response of a ceramic matrix composite material.

To achieve the above objects, the present invention provides a method for determining the nonlinear vibrational response of a ceramic matrix composite material, the method comprising: testing in advance to obtain a stress-strain curve of the ceramic matrix composite CMCs; the stress-strain curve comprises a compression section, a tension section and at least one loading and unloading cycle; dividing the stress-strain curve into a linear section, a nonlinear section and a hysteresis loop; wherein the nonlinear section includes a transition region and a second linear region, the hysteresis loop includes a main loop and a sub loop of uniform shape, and a division between the linear section and the nonlinear sectionThe boundary point is (epsilon)_p,σ_p) The boundary point of the transition region and the second linear region is (epsilon)_s,σ_s) The minimum stress strain point of the hysteresis loop is (epsilon)_c,σ_c) The current stress-strain state of the CMCs is: (^tε,^tσ) last stress-strain state of (^t-Δtε,^t-Δtσ) of the stress-strain curve, the historical maximum strain in the stress-strain curve being ε_max(ii) a When epsilon_maxLess than epsilon_pTime or epsilon_maxGreater than epsilon_pAnd is^tEpsilon is less than epsilon_cFitting a stress-strain curve of the CMCs according to a first formula, wherein the first formula comprises^tEpsilon and a predetermined stress E₁₁(ii) a When epsilon_maxIs greater than or equal to epsilon_pAnd is^tEpsilon is greater than epsilon_maxFitting a stress-strain curve of the CMCs according to a second formula, wherein the second formula comprises the slope E of the second linear segment₁′₁、^tε、ε_s、σ_sAnd presetting curve parameters; when epsilon_maxIs greater than or equal to epsilon_pAnd epsilon_c≤^tε＜ε_maxFitting a stress-strain curve on the main ring according to a third formula; according to the position of the last stress-strain state and^tε and^t-Δtfitting a stress-strain curve on the sub-ring according to the magnitude relation of epsilon; and applying each stress-strain curve obtained by fitting to a finite element model of the CMCs to determine the vibration response of the CMCs.

Further, the first formula is:

^tσ＝E₁₁·^tε

wherein,^tσ denotes the stress in the current stress-strain state, E₁₁The pre-set stress is represented by the pre-set stress,^tepsilon represents the strain in the current stress-strain state.

Further, the second formula is:

wherein,^tσ denotes the stress in the current stress-strain state, c₁、c₂、c₃、c₄、c₅Representing said predetermined curve parameter, E₁′₁Represents the slope, ε, of the second linear segment_pRepresenting the strain at the point of demarcation between the linear and non-linear sections,^tε represents the strain in the current stress-strain regime, ε_sRepresenting the strain, σ, at the point of demarcation of the transition region and the second linear region_sRepresenting the stress at the point of demarcation of the transition region and the second linear region.

Further, the third formula is:

wherein p is_diAnd p_uiCurve parameters representing the unloading and loading paths, respectively, i 1,2, 4,^tsigma denotes the stress in the current stress-strain state,^tepsilon represents the strain in the current stress-strain state,^t-Δtepsilon represents the strain in the last stress-strain state.

Further, according to the position of the last stress strain state and^tε and^t-Δtthe fitting of the stress-strain curve on the subring specifically comprises the following steps:

when the last stress-strain state is in the first preset position and^tepsilon is less than^t-ΔtWhen epsilon, the stress-strain curve on the sub-ring is fitted according to the following formula:

^tσ＝q_d1+q_d2 ^tε+q_d3 ^tε²+q_d4 ^tε³

wherein,

^tsigma denotes the stress in the current stress-strain state,^tε represents the strain in the current stress-strain regime, p_diCurve parameters representing the unloading path, i 1,2, 4, epsilon_cRepresenting the strain, σ, corresponding to the minimum stress-strain point of the hysteresis loop_cRepresenting the stress, epsilon, corresponding to the minimum stress-strain point of said hysteresis loop_maxRepresenting the historical maximum strain, σ, in the stress-strain curve_maxRepresenting the historical maximum stress in the stress-strain curve,^t-Δtepsilon represents the strain in the last stress-strain state,^t-Δtσ represents the stress in the last stress-strain state;

when the last stress-strain state is in the second preset position and^tepsilon is greater than^t-ΔtWhen epsilon, the stress-strain curve on the sub-ring is fitted according to the following formula:

^tσ＝q_u1+q_u2 ^tε+q_u3 ^tε²+q_u4 ^tε³

wherein,

wherein p is_uiCurve parameters representing the loading path, i 1,2, 4.

Therefore, the method can fit the stress-strain curve under any loading and unloading, and can realize the calculation process of the CMCs nonlinear vibration response by combining the finite element model, thereby improving the calculation efficiency.

Drawings

FIG. 1(a) is a first schematic diagram of fitting a stress-strain curve on a sub-ring in an embodiment of the present invention;

FIG. 1(b) is a second schematic diagram of fitting a stress-strain curve on a sub-ring in an embodiment of the present invention;

FIG. 2 is a schematic diagram of a nonlinear vibrational response in an embodiment of the present invention.

Detailed Description

In order to make those skilled in the art better understand the technical solutions in the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application, and it is obvious that the described embodiments are only a part of the embodiments of the present application, and not all of the embodiments. All other embodiments obtained by a person of ordinary skill in the art without any inventive work based on the embodiments in the present application shall fall within the scope of protection of the present application.

The present embodiments provide a method for determining the nonlinear vibrational response of a ceramic matrix composite material, the method comprising the following steps.

S1: testing in advance to obtain a stress-strain curve of the ceramic matrix composite CMCs; the stress-strain curve comprises a compression section, a tension section and at least one loading and unloading cycle.

In this embodiment, a stress-strain curve of the CMCs can be obtained by testing using a universal tester.

S2: dividing the stress-strain curve into a linear section, a nonlinear section and a hysteresis loop; wherein the nonlinear section comprises a transition region and a second linear region, the hysteresis loop comprises a main loop and a sub loop which are consistent in shape, and the dividing point between the linear section and the nonlinear section is (epsilon)_p,σ_p) The boundary point of the transition region and the second linear region is (epsilon)_s,σ_s) The minimum stress strain point of the hysteresis loop is (epsilon)_c,σ_c) The current stress-strain state of the CMCs is: (^tε,^tσ) last stress-strain state of (^t-Δtε,^t-Δtσ) of the stress-strain curve, the historical maximum strain in the stress-strain curve being ε_max。

In this embodiment,. epsilon._p＝2.0e-4，ε_c＝-1.044e-3，ε_s＝2.22e-3，σ_p＝1.94e7，σ_c＝-1.01e8，σ_s＝1.20e8。

S3: when epsilon_maxLess than epsilon_pTime or epsilon_maxGreater than epsilon_pAnd is^tEpsilon is less than epsilon_cFitting a stress-strain curve of the CMCs according to a first formula, wherein the first formula comprises^tEpsilon and a predetermined stress E₁₁。

In this embodiment, the first formula is:

^tσ＝E₁₁·^tε

S4: when epsilon_maxIs greater than or equal to epsilon_pAnd is^tEpsilon is greater than epsilon_maxFitting a stress-strain curve of the CMCs according to a second formula, wherein the second formula comprises the slope E of the second linear segment₁′₁、^tε、ε_s、σ_sAnd presetting curve parameters.

In this embodiment, the second formula is:

wherein,^tσ denotes the stress in the current stress-strain state, c₁、c₂、c₃、c₄、c₅Representing the preset curve parameter, E'₁₁Represents the slope, ε, of the second linear segment_pRepresenting the strain at the point of demarcation between the linear and non-linear sections,^tε represents the strain in the current stress-strain regime, ε_sRepresenting the strain, σ, at the point of demarcation of the transition region and the second linear region_sRepresenting the stress at the point of demarcation of the transition region and the second linear region.

S5: when epsilon_maxIs greater than or equal to epsilon_pAnd epsilon_c≤^tε＜ε_maxFitting a stress-strain curve on the main ring according to a third formula; according to the position of the last stress-strain state and^tε and^t-Δtand e, fitting a stress-strain curve on the sub-ring according to the magnitude relation of the epsilon.

In this embodiment, when ε_maxIs greater than or equal to epsilon_pAnd epsilon_c≤^tε＜ε_maxWhen the CMCs are in the stress-strain state at present, the CMCs fall on a hysteresis loop, and the starting point of a main loop is (epsilon)_max,σ_max) And ends in (epsilon)_c,σ_c). Referring to FIGS. 1(a) and 1(b), point A indicates (. epsilon.)_max,σ_max) The point C represents (. epsilon.)_c,σ_c). The main loop is composed of an unload path AC and a load path CA, so that the third formula can be:

wherein p is_diAnd p_uiCurve parameters representing the unloading and loading paths, respectively, i 1,2, 4,^tsigma denotes the stress in the current stress-strain state,^tepsilon represents the strain in the current stress-strain state,^t _- ^Δtepsilon represents the strain in the last stress-strain state.

The curve parameters of the unloading and loading paths may be obtained by fitting an experimental curve. In this embodiment, the parameters p and ε may be established using polynomials_maxThe relationship (c) is as follows:

wherein, b_dijAnd b_uijJ is 0,1,2.. is a curve parameter, which can be obtained by fitting an experimental curve.

In the present embodiment, if the last stress-strain state was at the first preset position point B, and^tepsilon is less than^t-ΔtEpsilon, then the point P of the current stress-strain state falls on the discharge path BC of the sub-ring, so that the expression for path BC can be obtained by transforming path AC. Specifically, the stress-strain curve on the sub-ring can be fitted according to the following formula:

^tσ＝q_d1+q_d2 ^tε+q_d3 ^tε²+q_d4 ^tε³

wherein,

^tsigma denotes the stress in the current stress-strain state,^tε represents the strain in the current stress-strain regime, p_diCurve parameter, i 1,2, representing the unloading path.,4，ε_cRepresenting the strain, σ, corresponding to the minimum stress-strain point of the hysteresis loop_cRepresenting the stress, epsilon, corresponding to the minimum stress-strain point of said hysteresis loop_maxRepresenting the historical maximum strain, σ, in the stress-strain curve_maxRepresenting the historical maximum stress in the stress-strain curve,^t-Δtepsilon represents the strain in the last stress-strain state,^t-Δtσ represents the stress in the last stress-strain state.

If the last stress-strain state was at the second predetermined location point D, and^tepsilon is greater than^t-Δtε, the point P of the current stress-strain state may fall on the load path DA. In this way, an expression of the path DA may be obtained by transforming the main ring loading path CA, and in particular, a stress-strain curve on the sub-ring may be fitted according to the following formula:

^tσ＝q_u1+q_u2 ^tε+q_u3 ^tε²+q_u4 ^tε³

wherein,

wherein p is_uiTo representThe curve parameters of the loading path, i, 1,2, 4.

S6: and applying each stress-strain curve obtained by fitting to a finite element model of the CMCs to determine the vibration response of the CMCs.

In this embodiment, a finite element model of the ceramic matrix composite may be established, each stress-strain curve obtained by the fitting may be applied to the finite element model of the CMCs, and a center difference method may be applied to obtain the nonlinear vibration response of the ceramic matrix composite, and the calculation result is shown in fig. 2. In fig. 2, the nonlinear vibration response can be represented by the magnitude of the displacement amount.

The foregoing description of various embodiments of the present application is provided for the purpose of illustration to those skilled in the art. It is not intended to be exhaustive or to limit the invention to a single disclosed embodiment. As described above, various alternatives and modifications of the present application will be apparent to those skilled in the art to which the above-described technology pertains. Thus, while some alternative embodiments have been discussed in detail, other embodiments will be apparent or relatively easy to derive by those of ordinary skill in the art. This application is intended to cover all alternatives, modifications, and variations of the invention that have been discussed herein, as well as other embodiments that fall within the spirit and scope of the above-described application.

The embodiments in the present specification are described in a progressive manner, and the same and similar parts among the embodiments can be referred to each other, and each embodiment focuses on the differences from the other embodiments.

Although the present application has been described in terms of embodiments, those of ordinary skill in the art will recognize that there are numerous variations and permutations of the present application without departing from the spirit of the application, and it is intended that the appended claims encompass such variations and permutations without departing from the spirit of the application.

Claims

1. A method for determining a non-linear vibrational response of a ceramic matrix composite, the method comprising:

testing in advance to obtain a stress-strain curve of the ceramic matrix composite CMCs; the stress-strain curve comprises a compression section, a tension section and at least one loading and unloading cycle;

dividing a stress-strain curve into a linear section, a nonlinear section and a hysteresis loop; wherein the nonlinear section comprises a transition region and a second linear region, the hysteresis loop comprises a main loop and a sub-loop with consistent shapes, and the dividing point between the linear section and the nonlinear section is(ε_p,σ_p)，ε_pRepresenting the strain at the point of demarcation between the linear and non-linear segments; the boundary point between the transition region and the second linear region is (epsilon)_s,σ_s)，ε_sRepresenting the strain, σ, at the boundary point of the transition region and the second linear region_sRepresenting the stress at the interface of the transition region and the second linear region; the minimum stress strain point of the hysteresis loop is (epsilon)_c,σ_c)，ε_cRepresenting the strain, σ, corresponding to the minimum stress-strain point of the hysteresis loop_cRepresenting the stress corresponding to the minimum stress strain point of the hysteresis loop; the current stress-strain state of the CMCs is (^tε,^tσ)，^tEpsilon represents the strain in the current stress-strain state,^tσ represents the stress in the current stress-strain state; the last state of stress-strain is^t-Δtε,^t-Δtσ)，^t-ΔtEpsilon represents the strain in the last stress-strain state,^t-Δtσ represents the stress in the last stress-strain state; historical maximum strain epsilon in stress-strain curve_max；

1) When epsilon_maxLess than epsilon_pTime or epsilon_maxGreater than epsilon_pAnd is and^tepsilon is less than epsilon_cAnd fitting the stress-strain curve of the CMCs according to a first formula:

the first formula is:^tσ＝E₁₁·^tε

wherein,^tσ denotes the stress in the current stress-strain state, E₁₁Which represents a pre-set stress that is,^tε represents the strain in the current stress-strain state;

2) when epsilon_maxIs greater than or equal to epsilon_pAnd is and^tepsilon is greater than epsilon_maxFitting a stress-strain curve of the CMCs according to a second formula;

the second formula is:

<mrow> <mmultiscripts> <mi>&sigma;</mi> <mi>t</mi> </mmultiscripts> <mo>=</mo> <mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <msub> <mi>c</mi> <mn>1</mn> </msub> <mo>+</mo> <msub> <mi>c</mi> <mn>2</mn> </msub> <mo>&CenterDot;</mo> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo>+</mo> <msub> <mi>c</mi> <mn>3</mn> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>c</mi> <mi>4</mi> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>3</mn> </msup> <mo>+</mo> <msub> <mi>c</mi> <mi>5</mi> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>4</mn> </msup> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&epsiv;</mi> <mi>p</mi> </msub> <mo><</mo> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo><</mo> <msub> <mi>&epsiv;</mi> <mi>s</mi> </msub> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>&sigma;</mi> <mi>s</mi> </msub> <mo>+</mo> <msubsup> <mi>E</mi> <mn>11</mn> <mo>&prime;</mo> </msubsup> <mo>&CenterDot;</mo> <mrow> <mo>(</mo> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>s</mi> </msub> <mo>)</mo> </mrow> </mrow> </mtd> <mtd> <mrow> <msub> <mi>&epsiv;</mi> <mi>s</mi> </msub> <mo><</mo> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> </mrow> </mtd> </mtr> </mtable> </mfenced> </mrow>

wherein,^tσ denotes the stress in the current stress-strain state, c₁、c₂、c₃、c₄、c₅Represents a preset curve parameter, E'₁₁Represents the slope, ε, of the second linear segment_pRepresenting the strain at the point of demarcation between the linear and non-linear segments,^tε represents the strain in the current stress-strain regime, ε_sRepresenting the strain, σ, at the boundary point of the transition region and the second linear region_sRepresenting the stress at the interface of the transition region and the second linear region;

3) when epsilon_maxIs greater than or equal to epsilon_pAnd is epsilon_c≤^tε＜ε_maxAccording to the third publicationFitting a stress-strain curve on the main ring;

the third formula is:

<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <mrow> <mmultiscripts> <mi>&sigma;</mi> <mi>t</mi> </mmultiscripts> <mo>=</mo> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>2</mn> </mrow> </msub> <mo>&CenterDot;</mo> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>3</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>4</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>3</mn> </msup> </mrow> </mtd> <mtd> <mrow> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo><</mo> <mmultiscripts> <mi>&epsiv;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mi>&Delta;</mi> <mi>t</mi> </mrow> </mmultiscripts> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mmultiscripts> <mi>&sigma;</mi> <mi>t</mi> </mmultiscripts> <mo>=</mo> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>2</mn> </mrow> </msub> <mo>&CenterDot;</mo> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>3</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>2</mn> </msup> <mo>+</mo> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> <mo>&CenterDot;</mo> <msup> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mn>3</mn> </msup> </mrow> </mtd> <mtd> <mrow> <mmultiscripts> <mi>&epsiv;</mi> <mi>t</mi> </mmultiscripts> <mo>&GreaterEqual;</mo> <mmultiscripts> <mi>&epsiv;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mi>&Delta;</mi> <mi>t</mi> </mrow> </mmultiscripts> </mrow> </mtd> </mtr> </mtable> </mfenced>

wherein p is_diAnd p_uiCurve parameters representing the unloading and loading paths, respectively, i 1,2, 4,^tsigma denotes the stress in the current stress-strain state,^tepsilon represents the strain in the current stress-strain state,^t-Δtε represents the strain in the last stress-strain state;

according to the position of the last stress-strain state and^tε and^t-▽tfitting a stress-strain curve on the sub-ring according to the magnitude relation of epsilon;

and applying each stress-strain curve obtained by fitting to a finite element model of the CMCs to determine the vibration response of the CMCs.

2. The method for determining the nonlinear vibrational response of a ceramic matrix composite as in claim 1, wherein the position of the previous state of stress-strain is determined based on the position of the previous state of stress-strain and^tε and^t-Δtthe fitting of the stress-strain curve on the subring specifically comprises the following steps:

^tσ＝q_d1+q_d2 ^tε+q_d3 ^tε²+q_d4 ^tε³

wherein,

<mrow> <msub> <mi>q</mi> <mrow> <mi>d</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mi>&delta;</mi> </msub> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> </mrow> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>&epsiv;</mi> <mi>c</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>2</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>&epsiv;</mi> <mi>c</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&sigma;</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>q</mi> <mrow> <mi>d</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> </mrow> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>2</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>2</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>&epsiv;</mi> <mi>c</mi> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>4</mn> </mrow> </msub> </mrow>

<mrow> <msub> <mi>q</mi> <mrow> <mi>d</mi> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mfrac> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>2</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>3</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>4</mn> </mrow> </msub> </mrow>

<mrow> <msub> <mi>q</mi> <mrow> <mi>d</mi> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mfrac> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>d</mi> <mn>4</mn> </mrow> </msub> </mrow>

<mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mmultiscripts> <mi>&sigma;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mi>&Delta;</mi> <mi>t</mi> </mrow> </mmultiscripts> <mo>-</mo> <msub> <mi>&sigma;</mi> <mi>c</mi> </msub> </mrow> <mrow> <msub> <mi>&sigma;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mo>-</mo> <msub> <mi>&sigma;</mi> <mi>c</mi> </msub> </mrow> </mfrac> <mo>,</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mmultiscripts> <mi>&epsiv;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mi>&Delta;</mi> <mi>t</mi> </mrow> </mmultiscripts> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> </mrow> <mrow> <msub> <mi>&epsiv;</mi> <mi>max</mi> </msub> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> </mrow> </mfrac> </mrow>

^tsigma denotes the stress in the current stress-strain state,^tε represents the strain in the current stress-strain regime, p_diCurve parameters representing the unloading path, i 1,2, 4, epsilon_cRepresenting the strain, σ, corresponding to the minimum stress-strain point of the hysteresis loop_cRepresenting the stress, epsilon, corresponding to the minimum stress-strain point of the hysteresis loop_maxRepresenting the historical maximum strain, σ, in the stress-strain curve_maxRepresenting the historical maximum stress in the stress-strain curve,^t-Δtepsilon represents the strain in the last stress-strain state,^t-Δtσ represents the stress in the last stress-strain state;

^tσ＝q_u1+q_u2 ^tε+q_u3 ^tε²+q_u4 ^tε³

wherein,

<mrow> <msub> <mi>q</mi> <mrow> <mi>u</mi> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>1</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> </mrow> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>2</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>3</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mn>3</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> <mn>3</mn> </msup> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>&sigma;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <mo>)</mo> </mrow> </mrow>

<mrow> <msub> <mi>q</mi> <mrow> <mi>u</mi> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mfrac> <mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </mrow> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>2</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>2</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> <mn>2</mn> </msubsup> <msup> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> </mrow>

<mrow> <msub> <mi>q</mi> <mrow> <mi>u</mi> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mfrac> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>2</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>3</mn> </mrow> </msub> <mo>-</mo> <mfrac> <mrow> <mn>2</mn> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msub> <mi>&epsiv;</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> <mrow> <mo>(</mo> <mn>1</mn> <mo>-</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>)</mo> </mrow> </mrow> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> </mrow>

<mrow> <msub> <mi>q</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> <mo>=</mo> <mfrac> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <msubsup> <mi>s</mi> <mi>&epsiv;</mi> <mn>3</mn> </msubsup> </mfrac> <msub> <mi>p</mi> <mrow> <mi>u</mi> <mn>4</mn> </mrow> </msub> </mrow>

<mrow> <msub> <mi>s</mi> <mi>&sigma;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mmultiscripts> <mi>&sigma;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mi>&Delta;</mi> <mi>t</mi> </mrow> </mmultiscripts> <mo>-</mo> <msub> <mi>&sigma;</mi> <mi>max</mi> </msub> </mrow> <mrow> <msub> <mi>&sigma;</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>&sigma;</mi> <mi>max</mi> </msub> </mrow> </mfrac> <mo>,</mo> <msub> <mi>s</mi> <mi>&epsiv;</mi> </msub> <mo>=</mo> <mfrac> <mrow> <mmultiscripts> <mi>&epsiv;</mi> <mrow> <mi>t</mi> <mo>-</mo> <mi>&Delta;</mi> <mi>t</mi> </mrow> </mmultiscripts> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>max</mi> </msub> </mrow> <mrow> <msub> <mi>&epsiv;</mi> <mi>c</mi> </msub> <mo>-</mo> <msub> <mi>&epsiv;</mi> <mi>max</mi> </msub> </mrow> </mfrac> </mrow>

wherein p is_uiCurve parameters representing the loading path, i 1,2, 4.