CN114781182B - Interaction integration method for solving thermal fracture problem of piezoelectric piezomagnetic composite material - Google Patents
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Abstract
The invention discloses an interaction integration method for solving the thermal fracture problem of a piezoelectric piezomagnetic composite material, which considers the influence of thermal load on the constitutive equation of the piezoelectric piezomagnetic material and the change of the interaction integration form, and obtains a new form of the interaction integration method of the piezoelectric piezomagnetic material containing a complex material interface under the thermal load through strict theoretical derivation, thereby providing a method for solving the strength factor of the piezoelectric piezomagnetic composite material under the thermal load. The interaction integration method is suitable for piezoelectric piezomagnetic materials with complex interfaces, and strict theoretical derivation proves that the material interfaces do not influence the value of interaction integration, so that the application range of the traditional interaction integration method is expanded to a great extent. Through the setting of the composite material attribute, the calculation of the thermal fracture problem of the piezoelectric piezomagnetic composite materials with different arrangement modes can be realized.
Description
Technical Field
The invention belongs to the technical field of fracture mechanics, relates to an interaction integration method, and particularly relates to an interaction integration method for solving a stress intensity factor, an electric displacement intensity factor and a magnetic induction intensity factor of a piezoelectric piezomagnetic composite material crack tip under the action of thermal load.
Background
Piezoelectric piezomagnetic materials have been widely used in the field of intelligent structures, such as sensors, brakes, aerospace, and the like, due to their excellent magnetoelectric coupling effect. The piezoelectric piezomagnetic material is a composite material containing both piezoelectric material phase and piezomagnetic material phase, and has a new material property, namely electromagnetic property, besides the piezoelectric and piezomagnetic properties of the component phases. However, due to the limitation of the manufacturing process, a large number of defects exist in the material, and in addition, the intrinsic brittleness of the piezoelectric phase and the piezomagnetic phase is very easy to break and damage in the service process. In addition, severe temperature loading further exacerbates fracture failure of piezomagnetic materials. Therefore, for the reliability requirement of the piezoelectric piezomagnetic material, the fracture mechanism of the piezoelectric piezomagnetic composite material containing a complex material interface under the thermal load needs to be deeply researched.
In the piezoelectric piezomagnetic material line elastic fracture mechanics, the intensity factor is an important parameter for evaluating the fracture behavior of the material, and comprises a stress intensity factor, an electric displacement intensity factor and a magnetic induction intensity factor. The current main methods for solving the stress intensity factor comprise a displacement method, a stress method, J integral and interaction integral. The displacement method and the stress method are high in experience, and accuracy of a calculation result is difficult to evaluate. Although J integration can be very effective in calculating the stress intensity factor, it is not easy to separate the stress intensity factors of type I and type II using J integration for the mixed crack problem. The interaction integration method well solves the problem and is a known method for solving the stress intensity factor with higher accuracy.
However, the current fracture mechanics has no suitable method for obtaining the strength factor of the crack tip of the piezoelectric piezomagnetic material containing the complex material interface under the thermal load. Therefore, in order to accurately evaluate the fracture behavior of the piezoelectric piezomagnetic composite material under the thermal load, it is very important to establish an interaction integration method of the piezoelectric piezomagnetic material containing the complex interface under the thermal load to solve the intensity factor.
Disclosure of Invention
Aiming at solving the defects existing in the research in the background technology, the invention provides an interaction integration method for solving the thermal fracture problem of the piezoelectric-magnetic composite material under the condition of heat load. The invention considers the influence of the thermal load on the constitutive equation of the piezoelectric piezomagnetic material and the change of the interaction integral form, and obtains a new form of the interaction integral method of the piezoelectric piezomagnetic material containing a complex material interface under the thermal load through strict theoretical derivation, thereby providing a method capable of solving the strength factor of the piezoelectric piezomagnetic composite material under the thermal load.
The purpose of the invention is realized by the following technical scheme:
an interaction integration method for solving a piezoelectric piezomagnetic composite material thermal fracture problem comprises the following steps:
the method comprises the following steps: in consideration of the influence of the heat load, obtaining a constitutive equation, a kinematic equation and a balance equation of the piezoelectric piezomagnetic material related to the heat load; establishing a J integral form of the piezoelectric piezomagnetic material under the thermal load, and substituting the uniform material crack tip field serving as an auxiliary field into the J integral form;
step two: the method comprises the steps of obtaining a line integral form of interaction integration by extracting an interaction part of a real field and an auxiliary field, converting the line integral into area integration, dividing the interaction integral form into a uniform term and a non-uniform term, and deducing the non-uniform term by substituting a balance equation, definitions of a strain, an electric field and a magnetic field and a definition of the auxiliary field to obtain the mutual integral form of the piezoelectric piezomagnetic material under the heat load;
step three: the influence of the introduction of an integration area into a material interface on interaction integration needs to be researched, so that the integration area is divided into two different material parts, a line integration form along the material interface is given, the characteristic of good bonding of the material interface is set, and a curve coordinate system is given based on the material interface;
step four: introducing the characteristics on the material interface into an interface integral term, deducing line integral along the material interface by using a chain rule, giving a line integral form along the material interface under the thermal load, and obtaining the influence on an interaction integral form;
step five: and solving the corresponding stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor by taking different values of the auxiliary intensity factor according to the relationship between the interaction integral and the intensity factor in the piezoelectric piezomagnetic material.
Compared with the prior art, the invention has the following advantages:
1. the invention provides an interaction integration method for solving the intensity factor of the crack tip of the piezoelectric piezomagnetic composite material under the thermal load by introducing the influence of the thermal load and the material interface, enlarges the application range of the interaction integration method, and establishes the solution method for the intensity factor of the piezoelectric piezomagnetic composite material containing the complex interface under the thermal load.
2. The interaction integration method is suitable for piezoelectric piezomagnetic materials with complex interfaces, and strict theoretical derivation proves that the material interfaces do not influence the value of interaction integration, so that the application range of the traditional interaction integration method is expanded to a great extent. Through setting the properties of the piezoelectric magnetic composite material, the calculation of the thermal fracture problem of the piezoelectric magnetic composite material with different arrangement modes can be realized.
3. The method has good applicability and stability, can be combined with the existing calculation methods such as finite element, finite element expansion and the like, realizes the solution of the strength factors of different types of piezoelectric piezomagnetic composite materials, and is developed into a commercial program to flexibly adapt to the change of the required problems.
Drawings
FIG. 1 is a block diagram of a process for calculating an intensity factor based on an integral of an interaction involving a complex interface piezoelectric piezomagnetic composite under thermal load;
FIG. 2 is a graph of the integrated area of a magnetoelectroelastic material having an interface;
FIG. 3 is a diagram of a laminated piezoelectric magnetic material sheet containing an oblique crack;
FIG. 4 is a schematic diagram of finite element mesh size and different integration regions;
FIG. 5 is a graph showing the calculation results of the stress intensity factor, the electric displacement intensity factor and the magnetic induction intensity factor for different integration regions
FIG. 6 is a graph showing the variation of stress intensity factor, electric displacement intensity factor and magnetic induction with piezomagnetic phase volume fraction.
Detailed Description
The technical solutions of the present invention are further described below with reference to the drawings, but the present invention is not limited thereto, and any modifications or equivalent substitutions may be made to the technical solutions of the present invention without departing from the spirit and scope of the technical solutions of the present invention.
The invention provides an interaction integration method for solving a piezoelectric piezomagnetic composite material thermal fracture problem, which comprises the following steps of:
the method comprises the following steps: in consideration of the influence of the heat load, obtaining a constitutive equation, a kinematic equation and a balance equation of the piezoelectric piezomagnetic material related to the heat load; and establishing a J integral form of the piezoelectric magnetic material under the thermal load, and substituting the uniform material crack tip field as an auxiliary field into the J integral form. The method comprises the following specific steps:
under the conditions of no physical force, no free charge, no magnetic flux and no heat source, the balance equation of the piezoelectric piezomagnetic material is as follows:
σ ij,j ＝0,D i,i ＝0,B i,i ＝q i,i ＝0；
the kinematic equation is:
for a linear piezomagnetic material subjected to a thermal load, the total strain ε kl Total electric field E l And the total magnetic field H l Two parts are divided, one caused by the magnetoelectric elastic load (marked with the superscript "m") and one caused by the thermal load (marked with the superscript "th"):
the constitutive equation of a piezomagnetic material can be expressed as:
the constitutive equation of a piezomagnetic material can also be expressed as:
wherein u is i 、σ ij And ε ij Elastic displacement, stress and total strain; phi, D i And E i Electric potential, electrokinetic displacement and electric field, respectively;
B i and H i Respectively magnetic potential, magnetic induction intensity and magnetic field; material parameter C ijkl 、e lij 、h lij 、κ ij 、μ il And gamma ij Respectively, the elastic rigidity, the piezoelectric coefficient, the piezomagnetic coefficient, the dielectric constant, the electromagnetic coefficient and the magnetic conductivity; s ijkl 、η lij 、g lij 、β ij 、α il And λ ij Respectively, elastic flexibility coefficient, piezoelectric strain coefficient, piezomagnetic strain coefficient, dielectric isolation rate, magnetoelectric coefficient and magnetic resistance rate; f. of kl 、υ i And ω l Respectively is a thermal expansion coefficient, a thermal electric field constant and a thermal magnetic field constant; the heat flux defines q by the change in temperature i ＝-k i ΔT，k i And Δ T is the heat transfer coefficient and the absolute temperature difference between the current temperature and the stress-free initial temperature. Variables labeled with the symbol indices i, j, k, and l (i, j, k, l =1,2, 3) are components of a vector or tensor.
Assisted displacement in a polar coordinate system of the crack tip
Auxiliary potential phi aux And an auxiliary magnetic potential->
Is defined as: />
Auxiliary stress
Auxiliary potential shift->
And auxiliary magnetic induction->
Is defined as follows:
auxiliary strain
Auxiliary electric field->
And an auxiliary magnetic field->
The following equation is used to obtain:
wherein,
and &>
Respectively as type I, type II and type III auxiliary stress intensity factors, auxiliary electric displacement intensity factors and auxiliary magnetic induction intensity factors;
v N (theta) and w N (θ) is an angular function of the uniform material crack tip field, and the subscript N = { II, I, III, IV, V } corresponds to different cracking modes.
For a two-dimensional non-uniform crack-containing voltage magnet, the J-integral is defined as follows:
wherein F is the density of electromagnetic enthalpy, and the expression is
δ ij Is a kronecker symbol; n is j Is a contour line gamma 0 The unit outer normal vector of (1); sigma ij 、D j And B j Stress, electric displacement and magnetic induction intensity respectively; u. of i Phi and/or>
Respectively displacement, potential and magnetic potential.
According to the condition that the crack surface is set to have no traction force and the electromagnetism is not conducted, the J integral can be written as:
wherein m is j Is a contour line
Q is an arbitrary smooth weight function with a value of Γ 0 1 is taken up at gamma 1 The upper value is 0.
Superimposing the auxiliary field on the actual field results in a new state, and then the J integral corresponding to the superimposed state is represented as:
wherein, J act+aux Is a J integral form of superposition of the real field and the auxiliary field.
Step two: the method comprises the steps of obtaining a line integral form of interaction integration by extracting interaction parts of a real field and an auxiliary field, converting the line integral into area integration, dividing the interaction integral form into a uniform term and a non-uniform term, and deducing the non-uniform term by utilizing a balance equation, definitions of a strain, an electric field and a magnetic field and a definition of the auxiliary field to obtain the mutual integral form of the piezoelectric piezomagnetic material under the heat load. The method comprises the following specific steps:
the J integration corresponding to the superimposed state can be divided into three parts, namely:
J act+aux ＝J act +J aux +I；
wherein, J act Is a form of J integration where only real fields exist, J aux Is a J integral form in which only the auxiliary field is present, and the terms in which both the real field and the auxiliary field are present are interaction terms, i.e. interaction integrals:
to avoid a potential source of inaccuracy in the numerical calculations, the line integrals are converted to area integrals. By the law of divergence, the form of area integral can be obtained:
I＝I h +I nonh ；
wherein, I h Is defined as a homogeneous term, I nonh Defined as a non-uniform term.
Using equilibrium equations, heterogeneous term integrals I nonh Can be simplified as follows:
using definition of strain, electric field and magnetic field
It is possible to obtain:
according to the definition of the auxiliary field, there are:
wherein,
and &>
Is the material parameter at the crack tip. Corresponding to (I) nonh Can be expressed as:
to further discuss the contribution of thermal load, we use an independent variable I thermal To represent the above formula I nonh The second integral term of (1). This integral I takes into account the change in temperature Δ T thermal Expressed as:
step three: the influence of the introduction of the integration region into the material interface on the interaction integration needs to be studied, so that the integration region is divided into two different material portions, a line integration form along the material interface is given, the property of good bonding of the material interface is set, and a curvilinear coordinate system is given based on the material interface. The method comprises the following specific steps:
if a material interface exists inside the integration region, the integration region needs to be divided into regions not including the material interface. As shown in FIG. 2, the integration region includes an arbitrary curve material interface Γ interface . The integration region A is bounded by an interface F interface Divided into closed circuits Γ B1 And Γ B2 Two enclosed sub-areas A 1 And A 2 The closed loop is respectively
And &>
The corresponding interaction integral is then of the form:
integral along material interface I interface The expression is as follows:
wherein for two areas A 1 And A 2 Corresponding different materials 1 and 2, corresponding different material areas are indicated by superscripts (1) and (2).
It is noted that the auxiliary displacement in the whole area
Auxiliary stress->
Auxiliary potential phi aux Auxiliary potential shift->
Auxiliary magnetic potential->
And auxiliary magnetic induction->
Is defined by the analytical solution of cracks in the homogeneous piezomagnetic solid. Thus, these auxiliary variables and their derivatives are continuous across the interface, with the following relationships:
as shown in FIG. 2, a curvilinear coordinate system ([ xi ] s) is established based on the material interface 1 ,ξ 2 ) The relationship between the coordinates of the curve and the point P in a material is:
wherein, Q (x) 10 ,x 20 ) Is a thin film transistor located at a material interface interface One point above and closest to point P.
The material interface is in an equilibrium state, and the gamma ray is arranged at the material interface interface The mechanical traction, electrical displacement and magnetic induction on both sides are equal, so:
because the material interface is set to be bonded completely, and relative slip does not occur between the interfaces, the displacement, the electric potential and the magnetic potential on the two sides of the interface are opposite to the curved coordinate xi 2 The derivatives of the directions are equal, as are:
step four: and introducing the characteristics on the material interface into an interface integral term, deducing the line integral along the material interface by using a chain rule, giving a line integral form along the material interface under the thermal load, and obtaining the influence on an interaction integral form. The method comprises the following specific steps:
depending on the continuity condition at the interface, the interface fraction can be expressed as:
according to stress-strain relationship and auxiliary stress
The first term of interface integration can be expressed as:
using chain rules and expressions
It is possible to obtain:
analogously, applying the electric displacement-potential relationship, the magnetic induction intensity-magnetic potential relationship, the chain-derivative rule and
it is possible to obtain:
substituting the expression obtained above into a line integral expression can obtain the interface area as:
the finally obtained integral expression of the interaction of the piezoelectric magnetic material containing the interface under the thermal load is as follows:
it can be seen that the new interaction integration method proposed by the present invention is effective for both non-uniform under thermal load and magnetoelectric elastic materials with complex interfaces, and does not need to consider the continuity of material properties at each interface.
Step five: and solving the corresponding stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor by taking different values of the auxiliary intensity factor according to the relationship between the interaction integral and the intensity factor in the piezoelectric piezomagnetic material. The method comprises the following specific steps:
the energy release rate at the crack tip is equal to the J integral, which is related to the intensity factor as follows:
here, K = [ K = II K I K III K IV K V ] T Y is a (5 × 5) Irwin matrix, which is a vector of five intensity factors.
Considering the equilibrium state resulting from the superposition of the real field and the auxiliary field, the J integral is:
wherein, I is integrated as: i = K T YK aux 。
In the two-dimensional case, K III If K is taken out of K =0 aux ＝[1,0,0,0,0] T 、K aux ＝[0,1,0,0,0] T And K aux ＝[0,0,0,1,0] T And K aux ＝[0,0,0,0,1] T Then, it can be simplified to:
I (II) ＝K II Y 11 +K I Y 12 +K IV Y 14 +K V Y 15 ；
I (I) ＝K II Y 12 +K I Y 22 +K IV Y 24 +K V Y 25 ；
I (IV) ＝K II Y 41 +K I Y 42 +K IV Y 44 +K V Y 45 ；
I (V) ＝K II Y 51 +K I Y 52 +K IV Y 54 +K V Y 55 。
and solving four equations simultaneously, namely solving the stress intensity factor, the electric displacement intensity factor and the magnetic induction intensity factor respectively.
To illustrate the applicability of the above-described aspects of the present invention, the following description is further presented in conjunction with an example of the thermal loading of a magnetoelectronics elastic material with a complex interface.
Calculation example: research on thermal fracture problem of multi-interface-containing magnetoelectric elastic composite material plate
As shown in fig. 3, a square piezoelectric piezomagnetic composite plate has an oblique crack in the middle, the width of the plate is W =2, the length of the crack is 2a, and the oblique angle is θ. Left temperature of the panel is set to T 1 =0 ℃ and T on the right 2 = 20 ℃, initial temperature of the entire panel is set to T 0 =0 ℃. X at upper and lower ends of setting plate 2 Displacement in direction, limiting x by the midpoint between the upper and lower ends of the plate 1 The displacements in the directions are all 0. The model parameters used were: a (-0.304, -0.268), B (0.304, 0.268), v f ＝v f (V PM ) =2l/W =0.2 to 0.8. The influence of the boundary conditions of the crack surfaces of thermal insulation and thermal conduction in the piezoelectric magnetic material on the crack strength factor is researched. The type I and type II stress intensity factors, the electric displacement intensity factor and the magnetic induction intensity factor respectively pass through
And &>
Performing dimensionless, dimensionless parameters
And &>
To describe the relative proportions of the two materials in the piezomagnetic and piezoelectric phases occupied in the piezomagnetic material, the volume fraction is defined as:
v f ＝V PM /(V PE +V PM )＝V PM /V MEE ；
wherein v is f Volume fraction, V, occupied by magnetic phases in place of gauge pressure PM Represents the volume occupied by the piezomagnetic phase; v PE Representing the volume occupied by the piezoelectric phase; v MEE Representing the volume occupied by the entire piezoelectric magnetic phase.
As shown in fig. 4 for v f ＝v f (V PM ) The effectiveness of the interaction integration method provided by the invention on piezoelectric piezomagnetic materials with complex interfaces is tested in the case of = 50%. Respectively selecting 12 different integration regions R I /h e Zone R as an integration zone of =1 to 500 I /h e = 1-200 does not include a material interface and R I /h e =250 to 500 contains a material interface. The relative deviation of the calculation result of the intensity factor obtained from the calculation result of fig. 5 is within 0.8%, which shows that the interaction integration method provided by the invention is applicable to piezomagnetic materials containing complex material interfaces, the material interfaces do not influence the calculation accuracy, and the application range of the interaction integration method is greatly expanded.
Stress intensity factor, as shown in FIGS. 6 (a) - (d)
And &>
The intensity factor of the electric displacement of the crack tip in the piezomagnetic phase is greater as the volume fraction of the piezomagnetic phase increases>
First increasing and then decreasing with increasing volume fraction of the piezomagnetic phase, at volume fraction v f Peak is reached when = 0.5. Crack tip in piezomagnetic phase->
Close to 0, indicating a volume fraction pair +>
Has little effect. Magnetic induction factor of crack tip in piezomagnetic phase with increasing volume fraction
The change is significant, and the magnetic induction factor of the crack tip in the piezoelectric phase->
The variation is not significant.
The above calculation examples verify the correctness and applicability of the above scheme of the present invention.
Claims (4)
1. An interaction integration method for solving a piezoelectric piezomagnetic composite thermal fracture problem is characterized by comprising the following steps of:
the method comprises the following steps: in consideration of the influence of the heat load, obtaining a constitutive equation, a kinematic equation and a balance equation of the piezoelectric piezomagnetic material related to the heat load; establishing a J integral form of the piezoelectric piezomagnetic material under the thermal load, and substituting the uniform material crack tip field as an auxiliary field into the J integral form;
under the conditions of no physical force, no free charge, no magnetic flux and no heat source, the balance equation of the piezoelectric piezomagnetic material is as follows:
σ ij,j ＝0,D i,i ＝0,B i,i ＝q i,i ＝0；
the kinematic equation is:
for a linear piezomagnetic material subjected to a thermal load, the total strain ε kl Total electric field E l And the total magnetic field H l Divided into two parts, one part caused by the magnetoelectric elastic load, denoted by the superscript "m", and one part caused by the thermal load, denoted by the superscript "th", there are:
the constitutive equation of a piezomagnetic material can be expressed as:
can also be expressed as:
wherein u is i 、σ ij And ε ij Elastic displacement, stress and total strain; phi, D i And E i Electric potential, electrokinetic displacement and electric field, respectively;
B i and H i Respectively magnetic potential, magnetic induction intensity and magnetic field; material parameter C ijkl 、e lij 、h lij 、κ ij 、μ il And gamma ij Respectively, the elastic rigidity, the piezoelectric coefficient, the piezomagnetic coefficient, the dielectric constant, the electromagnetic coefficient and the magnetic conductivity; s ijkl 、η lij 、g lij 、β ij 、α il And λ ij Respectively, elastic flexibility coefficient, piezoelectric strain coefficient, piezomagnetic strain coefficient, dielectric isolation rate, magnetoelectric coefficient and magnetic resistance rate; f. of kl 、υ i And ω l Respectively is a thermal expansion coefficient, a thermal electric field constant and a thermal magnetic field constant; the heat flux defines q by the change in temperature i ＝-k i ΔT，k i And Δ T is the heat transfer coefficient and the absolute temperature difference between the current temperature and the stress-free initial temperature; the variables labeled with the symbol indices i, j, k, and l are components of a vector or tensor, i, j, k, l =1,2,3;
the method comprises the following specific steps of establishing a J integral form of the piezoelectric piezomagnetic material under the thermal load, and substituting the crack tip field of the uniform material as an auxiliary field into the J integral form:
for a two-dimensional non-uniform crack-containing voltage magnet, the J-integral is defined as follows:
wherein F is the electromagnetic enthalpy density; delta ij Is a kronecker symbol; n is j Is a contour line gamma 0 The unit outer normal vector of (1); sigma ij 、D j And B j Stress, electric displacement and magnetic induction intensity respectively; u. u i Phi and
respectively displacement, potential and magnetic potential;
depending on the condition that the crack plane is set to non-tractive, electromagnetically non-conductive, the J integral can be written as:
wherein m is j Is a contour line
Q is an arbitrary smooth weight function with a value of Γ 0 Taking 1 above gamma 1 The upper value is 0;
superimposing the auxiliary field on the actual field results in a new state, and then the J integral corresponding to the superimposed state is expressed as:
wherein, J act+aux Is a J integral form of superposition of the real field and the auxiliary field,
is an auxiliary displacement, phi aux Is the auxiliary potential(s) of the electric field,
is the auxiliary magnetic potential of the magnetic field,
it is the auxiliary stress that is applied,
is the auxiliary electric potential shift and the auxiliary electric potential shift,
is used for assisting the magnetic induction intensity,
it is the auxiliary strain that is applied,
is an auxiliary electric field, and the electric field,
is an auxiliary magnetic field;
step two: the method comprises the steps of obtaining a line integral form of interaction integration by extracting an interaction part of a real field and an auxiliary field, converting the line integral into area integration, dividing the interaction integral form into a uniform term and a non-uniform term, and deducing the non-uniform term by substituting a balance equation, definitions of a strain, an electric field and a magnetic field and a definition of the auxiliary field to obtain the mutual integral form of the piezoelectric piezomagnetic material under the heat load;
step three: dividing the integration area into two different material parts, giving a line integration form along a material interface, setting the characteristic of good bonding of the material interface, and giving a curve coordinate system based on the material interface;
step four: introducing the characteristics on the material interface into an interface integral term, deducing line integral along the material interface by using a chain rule, giving a line integral form along the material interface under the thermal load, and obtaining the influence on an interaction integral form;
step five: and solving the corresponding stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor by taking different values of the auxiliary intensity factor according to the relationship between the interaction integral and the intensity factor in the piezoelectric piezomagnetic material.
2. The interaction integration method for solving the piezoelectric piezomagnetic composite thermal fracture problem according to claim 1, characterized in that the specific steps of the second step are as follows:
the J integration corresponding to the superimposed state can be divided into three parts, namely:
J act+aux ＝J act +J aux +I；
wherein, J act Is a form of J integration where only real fields exist, J aux Is a J integral form in which only the auxiliary field is present, and the terms in which both the real field and the auxiliary field are present are interaction terms, i.e. interaction integrals:
to avoid potential sources of inaccuracy in the numerical calculations, the line integrals are converted into area integrals, which, by the divergence theorem, can be obtained in the form of area integrals:
I＝I h +I nonh ；
wherein, I h Defined as a homogeneous term, I nonh Defined as a non-uniform term;
using equilibrium equations, heterogeneous term integrals I nonh Can be simplified as follows:
using definition of strain, electric field and magnetic field
It is possible to obtain:
according to the definition of the auxiliary field, there are:
wherein,
and
is the material parameter at the crack tip; corresponding to (I) nonh Can be expressed as:
3. the interaction integration method for solving the piezoelectric piezomagnetic composite thermal fracture problem according to claim 1, characterized in that the specific steps of the third step are as follows:
if a material interface is present in the integration region, the integration region needs to be divided into regions not including the material interface, and the integration region includes an arbitrary curve material interface Γ interface The integration region A is bounded by an interface F interface Divided into closed circuits Γ B1 And Γ B2 Two enclosed sub-areas A 1 And A 2 The closed loop is respectively
And
the corresponding interaction integral is of the form:
integration along material interfaceI interface The expression is as follows:
wherein for two areas A 1 And A 2 Corresponding dissimilar materials 1 and 2, with superscripts (1) and (2) denoting corresponding dissimilar material regions;
auxiliary displacement in the entire area
Auxiliary stress
Auxiliary potential phi aux Auxiliary electric potential shift
Auxiliary magnetic potential
And auxiliary magnetic induction
Defined by an analytical solution for cracks in a homogeneous piezomagnetic solid, these auxiliary variables and their derivatives are continuous at the interface, with the following relationships:
a curve coordinate system (xi) is established based on the material interface 1 ,ξ 2 ) The relationship between the coordinates of the curve and the point P in a material is:
wherein, Q (x) 10 ,x 20 ) Is a thin film transistor located at a material interface interface One point above and closest to point P;
the material interface is in an equilibrium state, and the gamma ray is arranged at the material interface interface The mechanical traction, electrical displacement and magnetic induction on both sides are equal, so:
because the material interface is set to be bonded completely, and relative slip does not occur between the interfaces, the displacement, the electric potential and the magnetic potential on the two sides of the interface are opposite to each other along the curve coordinate xi 2 The derivatives of the directions are equal, with:
4. the interaction integration method for solving the piezoelectric piezomagnetic composite thermal fracture problem according to claim 1, characterized in that the specific steps of the fourth step are as follows:
depending on the continuity condition at the interface, the interface fraction can be expressed as:
according to stress-strain relationship and auxiliary stress
The first term of interface integration can be expressed as:
using chain rules and expressions
It is possible to obtain:
using the electric displacement-potential relationship, magnetic induction intensity-magnetic potential relationship, chain-type derivation rule and
it is possible to obtain:
substituting the expression obtained above into a line integral expression can obtain the interface area as:
the finally obtained integral expression of the interaction of the piezoelectric magnetic material containing the interface under the thermal load is as follows:
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