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CN110705076A - Method for solving fracture problem of functional gradient piezoelectric material with arbitrary attributes - Google Patents

Method for solving fracture problem of functional gradient piezoelectric material with arbitrary attributes Download PDF

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CN110705076A
CN110705076A CN201910909283.4A CN201910909283A CN110705076A CN 110705076 A CN110705076 A CN 110705076A CN 201910909283 A CN201910909283 A CN 201910909283A CN 110705076 A CN110705076 A CN 110705076A
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piezoelectric material
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孔园洁
王志海
果立成
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Harbin University of Science and Technology
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Abstract

A method for solving the fracture problem of a functional gradient piezoelectric material with any attribute belongs to the technical field of fracture mechanics. The method solves the problems that the obtained crack parameters are inaccurate and the application of the existing solving method is limited because the arbitrary attribute of the functional gradient piezoelectric material is not considered in the solving of the fracture problem of the existing functional gradient piezoelectric material. The functional gradient piezoelectric material is divided into a plurality of layers, and the material property of each sub-layer is assumed to be distributed according to a special function form, which is equivalent to a curve for approximating the real material property of the functional gradient piezoelectric material by using a special piecewise function curve. The method not only can reflect the influence of the inhomogeneous parameters of the material properties on the stress field at the tip of the crack, but also can fully consider the influence of the distribution form of the material properties on the fracture behavior of the functional gradient piezoelectric material, thereby solving the fracture problem of the functional gradient piezoelectric material with any properties. The method can be applied to solving the fracture problem of the functional gradient piezoelectric material.

Description

一种求解具有任意属性的功能梯度压电材料断裂问题的方法A method for solving the fracture problem of functionally graded piezoelectric materials with arbitrary properties

技术领域technical field

本发明属于断裂力学技术领域，具体涉及一种求解具有任意属性的功能梯度压电材料断裂问题的方法。The invention belongs to the technical field of fracture mechanics, and in particular relates to a method for solving the fracture problem of functionally gradient piezoelectric materials with arbitrary properties.

背景技术Background technique

目前为止，压电材料在新兴技术中发挥着至关重要的作用，特别是在航空航天、电子和生物等领域。由于压电材料具有降低应力集中、提高断裂韧性的潜力，为满足压电材料在寿命和可靠性方面的要求，将功能梯度材料推广到压电材料中，这种具有连续变化特性的材料被称为功能梯度压电材料。由于功能梯度压电材料的制造、加工及成型过程，导致其产生各种裂纹和缺陷。近年来，很多学者研究分析了一系列的功能梯度压电材料断裂问题。通常，为分析解决功能梯度压电材料断裂问题，假定材料属性为特殊函数，例如指数函数和线性函数。但由于功能梯度压电材料具有任意属性，采用特殊函数求解功能梯度压电材料断裂问题的缺点是：功能梯度压电材料属性的分布形式限制这类方法的应用。因此，目前需要一种方法求解具有任意属性的功能梯度压电材料的断裂问题。Piezoelectric materials have played a crucial role in emerging technologies so far, especially in the fields of aerospace, electronics, and biology. Because piezoelectric materials have the potential to reduce stress concentration and improve fracture toughness, in order to meet the requirements of piezoelectric materials in terms of life and reliability, functionally graded materials are extended to piezoelectric materials. This material with continuously changing properties is called It is a functionally graded piezoelectric material. Due to the manufacturing, processing and molding process of functionally graded piezoelectric materials, various cracks and defects occur. In recent years, many scholars have studied and analyzed a series of fracture problems of functionally graded piezoelectric materials. Typically, to analytically solve the fracture problem of functionally graded piezoelectric materials, the material properties are assumed to be special functions, such as exponential and linear functions. However, due to the arbitrary properties of functionally gradient piezoelectric materials, the disadvantage of using special functions to solve the fracture problem of functionally gradient piezoelectric materials is that the distribution of the properties of functionally gradient piezoelectric materials limits the application of such methods. Therefore, there is currently a need for a method to solve the fracture problem of functionally graded piezoelectric materials with arbitrary properties.

发明内容SUMMARY OF THE INVENTION

本发明的目的是为解决现有功能梯度压电材料断裂问题的求解中未考虑功能梯度压电材料具有任意属性，导致获得的裂纹参数不准确，限制了现有功能梯度压电材料断裂问题的求解方法的应用的问题，而提出了一种求解具有任意属性的功能梯度压电材料断裂问题的方法。The purpose of the present invention is to solve the fracture problem of the existing functionally graded piezoelectric material without considering that the functionally graded piezoelectric material has arbitrary properties, resulting in inaccurate crack parameters obtained and limiting the fracture problem of the existing functionally graded piezoelectric material. A method for solving the fracture problem of functionally graded piezoelectric materials with arbitrary properties is proposed.

本发明为解决上述技术问题采取的技术方案是：，该方法包括以下步骤：The technical scheme adopted by the present invention for solving the above-mentioned technical problems is: the method comprises the following steps:

步骤一、建立功能梯度压电材料的本构方程和平衡方程；Step 1. Establish the constitutive equation and equilibrium equation of the functionally graded piezoelectric material;

步骤二、将功能梯度压电材料沿厚度方向均匀分为若干层，假设每层材料属性均随指数函数形式变化，且相邻层材料属性在界面处是连续的，由步骤一建立的本构方程和平衡方程，获得每层材料的本构方程和平衡方程；Step 2. Divide the functionally graded piezoelectric material into several layers uniformly along the thickness direction. It is assumed that the material properties of each layer change with the exponential function, and the material properties of adjacent layers are continuous at the interface. Equations and equilibrium equations to obtain the constitutive and equilibrium equations of each layer of material;

步骤三、在功能梯度压电材料的裂纹表面引入位错函数，并利用傅里叶变换和叠加原理，获得每层材料的本构方程和平衡方程的解析解；Step 3: Introduce a dislocation function on the crack surface of the functionally graded piezoelectric material, and use Fourier transform and superposition principle to obtain analytical solutions of the constitutive equation and equilibrium equation of each layer of material;

步骤四、根据步骤二的每层材料的本构方程和平衡方程以及步骤三的解析解，并结合每层材料的本构方程和平衡方程的边界条件以及相邻层材料的位移在界面处的连续条件，获得奇异积分方程组；Step 4: According to the constitutive equation and equilibrium equation of each layer of material in step 2 and the analytical solution of step 3, and combine the boundary conditions of the constitutive equation and equilibrium equation of each layer of material and the displacement of adjacent layer materials at the interface. Continuity condition, obtain singular integral equation system;

步骤五、求解步骤四获得的奇异积分方程组，获得表征裂纹位置和长度参数的应力强度因子，获得功能梯度压电材料的材料参数和几何参数与应力强度因子的关系；Step 5: Solve the singular integral equation system obtained in Step 4, obtain the stress intensity factor representing the crack position and length parameters, and obtain the relationship between the material parameters and geometric parameters of the functionally graded piezoelectric material and the stress intensity factor;

步骤六、根据待测功能梯度压电材料的材料参数和几何参数，获得待测功能梯度压电材料的应力强度因子，再根据应力强度因子获得待测功能梯度压电材料裂纹的位置和长度参数。Step 6: Obtain the stress intensity factor of the functionally graded piezoelectric material to be tested according to the material parameters and geometric parameters of the functionally graded piezoelectric material to be tested, and then obtain the position and length parameters of the crack of the functionally graded piezoelectric material to be tested according to the stress intensity factor .

本发明的有益效果是：本发明提出了一种求解具有任意属性的功能梯度压电材料断裂问题的方法，由于功能梯度压电材料属性随空间位置而变化，本发明将功能梯度压电材料分为若干层，假定每子层的材料属性按特殊的函数形式分布，相当于用特殊的分段函数曲线来逼近功能梯度压电材料的真实材料属性的曲线。本发明方法不仅可以反应材料属性的非均匀参数对裂纹尖端应力场的影响，还可以充分考虑材料属性的分布形式对功能梯度压电材料断裂行为的影响，进而能够求解具有任意属性的功能梯度压电材料的断裂问题。克服了现有功能梯度压电材料断裂问题的求解中未考虑功能梯度压电材料具有任意属性，导致现有功能梯度压电材料断裂问题的求解方法的应用受到限制的问题，本发明的求解方法更符合实际情况，应用范围广泛，求解结果具有可靠性。The beneficial effects of the present invention are as follows: the present invention proposes a method for solving the fracture problem of functionally graded piezoelectric materials with arbitrary properties. Since the properties of functionally graded piezoelectric materials vary with spatial positions, the present invention divides the functionally graded piezoelectric materials into For several layers, it is assumed that the material properties of each sublayer are distributed according to a special function form, which is equivalent to using a special piecewise function curve to approximate the curve of the real material properties of the functionally graded piezoelectric material. The method of the invention can not only reflect the influence of the non-uniform parameters of the material properties on the stress field at the crack tip, but also fully consider the influence of the distribution form of the material properties on the fracture behavior of the functionally graded piezoelectric material, so as to solve the functionally graded piezoelectric material with arbitrary properties. Fracture problems of electrical materials. Overcoming the problem that the functional gradient piezoelectric material has arbitrary properties not considered in the solution to the fracture problem of the existing functionally graded piezoelectric material, which leads to the limitation of the application of the existing method for solving the fracture problem of the functionally graded piezoelectric material, the solution method of the present invention It is more in line with the actual situation, has a wide range of applications, and the solution results are reliable.

附图说明Description of drawings

图1是本发明方法的流程图；Fig. 1 is the flow chart of the inventive method;

图2为具有任意属性功能梯度压电材料的几何结构图；Fig. 2 is the geometric structure diagram of functionally graded piezoelectric material with arbitrary properties;

y′代表局部直角坐标系的y′轴，θ代表x轴正向与x′轴正向的夹角；y' represents the y' axis of the local Cartesian coordinate system, and θ represents the angle between the positive x axis and the positive x' axis;

图3为具有任意属性功能梯度压电材料的分层结构图；Fig. 3 is a layered structure diagram of functionally graded piezoelectric material with arbitrary properties;

h2代表前2层材料的总厚度，hm-1代表前m-1层材料的总厚度，hm代表m层材料的总厚度，即hm＝h。h 2 represents the total thickness of the first two layers of materials, h m-1 represents the total thickness of the first m-1 layers of materials, and h m represents the total thickness of the m layers of materials, that is, h m =h.

具体实施方式Detailed ways

具体实施方式一：如图1所示，本实施方式所述的一种求解具有任意属性的功能梯度压电材料断裂问题的方法，该方法包括以下步骤：Embodiment 1: As shown in FIG. 1 , a method for solving the fracture problem of functionally graded piezoelectric materials with arbitrary properties described in this embodiment includes the following steps:

步骤一、建立功能梯度压电材料的本构方程和平衡方程；Step 1. Establish the constitutive equation and equilibrium equation of the functionally graded piezoelectric material;

步骤二、如图2所示，为求解具有任意属性的功能梯度压电材料断裂问题，针对功能梯度压电材料属性的任意性：Step 2, as shown in Figure 2, in order to solve the fracture problem of functionally gradient piezoelectric materials with arbitrary properties, for the arbitrary properties of functionally gradient piezoelectric materials:

将功能梯度压电材料沿厚度方向均匀分为若干层，假设每层材料属性均随指数函数形式变化，且相邻层材料属性在界面处是连续的，由步骤一建立的本构方程和平衡方程，获得每层材料的本构方程和平衡方程；The functionally graded piezoelectric material is evenly divided into several layers along the thickness direction. It is assumed that the material properties of each layer change in the form of an exponential function, and the material properties of adjacent layers are continuous at the interface. The constitutive equation and equilibrium established in step 1 Equations to obtain the constitutive equation and equilibrium equation of each layer of material;

步骤三、在功能梯度压电材料的裂纹表面引入位错函数，并利用傅里叶变换和叠加原理，获得每层材料的本构方程和平衡方程的解析解；Step 3: Introduce a dislocation function on the crack surface of the functionally graded piezoelectric material, and use Fourier transform and superposition principle to obtain analytical solutions of the constitutive equation and equilibrium equation of each layer of material;

步骤四、根据步骤二的每层材料的本构方程和平衡方程以及步骤三的解析解，并结合每层材料的本构方程和平衡方程的边界条件以及相邻层材料的位移在界面处的连续条件，获得奇异积分方程组；Step 4: According to the constitutive equation and equilibrium equation of each layer of material in step 2 and the analytical solution of step 3, and combine the boundary conditions of the constitutive equation and equilibrium equation of each layer of material and the displacement of adjacent layer materials at the interface. Continuity condition, obtain singular integral equation system;

步骤五、求解步骤四获得的奇异积分方程组，获得表征裂纹位置和长度参数的应力强度因子，获得功能梯度压电材料的材料参数和几何参数与应力强度因子的关系；Step 5: Solve the singular integral equation system obtained in Step 4, obtain the stress intensity factor representing the crack position and length parameters, and obtain the relationship between the material parameters and geometric parameters of the functionally graded piezoelectric material and the stress intensity factor;

步骤六、根据待测功能梯度压电材料的材料参数和几何参数，获得待测功能梯度压电材料的应力强度因子，再根据应力强度因子获得待测功能梯度压电材料裂纹的位置和长度参数。Step 6: Obtain the stress intensity factor of the functionally graded piezoelectric material to be tested according to the material parameters and geometric parameters of the functionally graded piezoelectric material to be tested, and then obtain the position and length parameters of the crack of the functionally graded piezoelectric material to be tested according to the stress intensity factor .

现有的解析求解功能梯度压电材料断裂问题的方法，将材料性质假定为特殊函数(线性、常数、指数函数)形式，这类方法忽略材料属性的分布形式，在求解具有任意属性功能梯度压电材料断裂问题时存在较大的困难，本实施方式提出了一种新的方法，有效克服了现有方法中的不足。The existing analytical methods for solving the fracture problem of functionally graded piezoelectric materials assume that the material properties are in the form of special functions (linear, constant, exponential functions). There is a great difficulty in the problem of electrical material fracture. This embodiment proposes a new method, which effectively overcomes the shortcomings of the existing methods.

具体实施方式二：本实施方式与具体实施方式一不同的是：所述步骤一的具体过程为：Embodiment 2: The difference between this embodiment and Embodiment 1 is that the specific process of step 1 is:

对于厚度为h的功能梯度压电材料，建立功能梯度压电材料的本构方程为：For a functionally graded piezoelectric material with a thickness of h, the constitutive equation for establishing the functionally graded piezoelectric material is:

Figure BDA0002214240280000031
Figure BDA0002214240280000031

式中：τxz代表功能梯度压电材料在垂直于x轴平面上的z轴方向剪应力，τyz代表功能梯度压电材料在垂直于y轴平面上的z轴方向剪应力，Dx代表功能梯度压电材料在x轴方向的电位移，Dy代表功能梯度压电材料在y轴方向的电位移；x轴、y轴和z轴分别为空间直角坐标系的三轴；where: τ xz represents the shear stress in the z-axis direction of the functionally graded piezoelectric material on the plane perpendicular to the x-axis, τ yz represents the shear stress in the z-axis direction of the functionally graded piezoelectric material on the plane perpendicular to the y-axis, and D x represents The electrical displacement of the functionally graded piezoelectric material in the x-axis direction, D y represents the electrical displacement of the functionally-graded piezoelectric material in the y-axis direction; the x-axis, the y-axis and the z-axis are the three axes of the spatial Cartesian coordinate system respectively;

w代表功能梯度压电材料的机械位移；φ代表电势；c44代表剪切模量；e15代表压电系数；ε11代表介电常数；w represents the mechanical displacement of the functionally graded piezoelectric material; φ represents the electric potential; c 44 represents the shear modulus; e 15 represents the piezoelectric coefficient; ε 11 represents the dielectric constant;

建立功能梯度压电材料的平衡方程为：The equilibrium equation for establishing functionally graded piezoelectric materials is:

Figure BDA0002214240280000041
Figure BDA0002214240280000041

功能梯度压电材料的本构方程和平衡方程的边界条件为：The boundary conditions of the constitutive equation and equilibrium equation of the functionally graded piezoelectric material are:

Figure BDA0002214240280000042
Figure BDA0002214240280000042

其中：τxz(0,y)代表功能梯度压电材料在x＝0平面处的剪应力，τxz(h,y)代表功能梯度压电材料在x＝h平面处的剪应力；Dx(0,y)＝0代表功能梯度压电材料在x＝0平面处的电位移；Dx(h,y)＝0代表功能梯度压电材料在x＝h平面处的电位移；where: τ xz (0, y) represents the shear stress of the functionally graded piezoelectric material at the x=0 plane, τ xz (h, y) represents the shear stress of the functionally graded piezoelectric material at the x=h plane; D x (0, y)=0 represents the electrical displacement of the functionally graded piezoelectric material at the x=0 plane; D x (h, y)=0 represents the electrical displacement of the functionally graded piezoelectric material at the x=h plane;

假定c44、e15、ε11的表达式为：Assume that the expressions of c 44 , e 15 , ε 11 are:

[c44,e15,ε11]＝[c440,e150,ε110]f(x) (4)[c 44 ,e 15 ,ε 11 ]=[c 440 ,e 150 ,ε 110 ]f(x) (4)

式中：c440、e150和ε110均为常数，f(x)为功能梯度压电材料属性函数。In the formula: c 440 , e 150 and ε 110 are all constants, and f(x) is the property function of functionally graded piezoelectric materials.

具体实施方式三：本实施方式与具体实施方式二不同的是：所述步骤二的具体过程为：Embodiment 3: The difference between this embodiment and Embodiment 2 is that the specific process of the second step is:

针对功能梯度压电材料属性的任意性，为求解材料断裂问题，本实施方式采用分段指数模型。In view of the arbitrary properties of functionally graded piezoelectric materials, in order to solve the material fracture problem, this embodiment adopts a piecewise exponential model.

将功能梯度压电材料沿厚度方向均匀分为m层，如图3所示，则第n层材料的厚度为hn–hn-1，其中：n＝1,2,…,m，hn代表第1层材料至第n层材料的厚度和，hn-1代表第1层材料至第n-1层材料的厚度和；The functionally graded piezoelectric material is evenly divided into m layers along the thickness direction, as shown in Figure 3, the thickness of the nth layer material is h n –h n-1 , where: n=1,2,...,m,h n represents the thickness sum of the 1st layer material to the nth layer material, h n-1 represents the thickness sum of the 1st layer material to the n-1th layer material;

假定每层材料属性均随指数函数形式变化：The material properties of each layer are assumed to vary exponentially in the form of:

Figure BDA0002214240280000043
Figure BDA0002214240280000043

cn44代表第n层材料的剪切模量；en15代表第n层材料的压电系数；εn11代表第n层材料的介电常数；

Figure BDA0002214240280000044
代表第n层材料的属性形式；c n44 represents the shear modulus of the nth layer material; e n15 represents the piezoelectric coefficient of the nth layer material; ε n11 represents the dielectric constant of the nth layer material;
Figure BDA0002214240280000044
represents the property form of the nth layer material;

相邻层材料属性在界面处连续的连续条件为：The continuous condition for the material properties of adjacent layers to be continuous at the interface is:

Figure BDA0002214240280000045
Figure BDA0002214240280000045

其中：c(n+1)44代表第n+1层材料的剪切模量；e(n+1)15代表第n+1层材料的压电系数；ε(n+1)11代表第n+1层材料的介电常数；

Figure BDA0002214240280000046
代表第n+1层材料的属性形式；f(hn)代表第n层材料的属性函数；Among them: c (n+1)44 represents the shear modulus of the material of the n+1th layer; e (n+1)15 represents the piezoelectric coefficient of the material of the n+1th layer; ε (n+1)11 represents the first The dielectric constant of the n+1 layer material;
Figure BDA0002214240280000046
Represents the property form of the n+1th layer material; f(h n ) represents the property function of the nth layer material;

由步骤一建立的本构方程和平衡方程，获得第n层材料的本构方程为：From the constitutive equation and equilibrium equation established in step 1, the constitutive equation obtained for the nth layer material is:

Figure BDA0002214240280000051
Figure BDA0002214240280000051

其中：τnxz代表第n层材料在垂直于x轴平面上的z轴方向剪应力，τnyz代表第n层材料在垂直于y轴平面上的z轴方向剪应力，Dnx代表第n层材料在x轴方向的电位移，Dny代表第n层材料在y轴方向的电位移；wn代表第n层材料的机械位移；φn代表第n层材料的电势；where: τ nxz represents the shear stress in the z-axis direction of the n-th layer material on the plane perpendicular to the x-axis, τ nyz represents the z-axis shear stress of the n-th layer material on the plane perpendicular to the y-axis, and D nx represents the nth layer. The electrical displacement of the material in the x-axis direction, D ny represents the electrical displacement of the n-th layer material in the y-axis direction; wn represents the mechanical displacement of the n-th layer material; φ n represents the potential of the n-th layer material;

第n层材料的平衡方程为：The equilibrium equation for the nth layer material is:

Figure BDA0002214240280000052
Figure BDA0002214240280000052

同理，获得每层材料的本构方程和平衡方程。In the same way, the constitutive and equilibrium equations of each layer of material are obtained.

具体实施方式四：本实施方式与具体实施方式三不同的是：所述步骤三中，位错函数的表达式为：Embodiment 4: The difference between this embodiment and Embodiment 3 is that: in the third step, the expression of the dislocation function is:

Figure BDA0002214240280000053
Figure BDA0002214240280000053

其中：a和b代表裂纹的两尖端，以a和b所在直线为x′轴，建立局部直角坐标系，且x′轴的正方向与向量

Figure BDA0002214240280000054
的方向相同；g1(x′)代表关于机械位移的错位函数，g2(x′)代表关于电势的错位函数，+0代表从正向趋近于0，-0代表从负向趋近于0。Where: a and b represent the two tips of the crack, take the line where a and b are as the x' axis, and establish a local rectangular coordinate system, and the positive direction of the x' axis is the same as the vector
Figure BDA0002214240280000054
in the same direction; g 1 (x′) represents the dislocation function with respect to mechanical displacement, g 2 (x′) represents the dislocation function with respect to electric potential, +0 represents approaching 0 from positive, -0 represents approaching from negative at 0.

局部直角坐标系的坐标原点与空间直角坐标系的坐标原点为同一点，该点位于第1层材料的下表面处。The coordinate origin of the local rectangular coordinate system and the coordinate origin of the spatial rectangular coordinate system are the same point, and the point is located at the lower surface of the first layer material.

具体实施方式五：本实施方式与具体实施方式四不同的是：所述步骤四中，每层材料的本构方程和平衡方程的边界条件为：Embodiment 5: The difference between this embodiment and Embodiment 4 is: in the step 4, the boundary conditions of the constitutive equation and equilibrium equation of each layer of material are:

Figure BDA0002214240280000061
Figure BDA0002214240280000061

其中：τ1xz(0,y)代表第1层材料在x＝0平面处的剪应力，τmxz(h,y)代表第m层材料在x＝h平面处的剪应力；D1x(0,y)代表第1层材料在x＝0平面处的电位移，Dmx(h,y)代表第m层材料在x＝h平面处的电位移；Where: τ 1xz (0, y) represents the shear stress of the first layer of material at the x=0 plane, τ mxz (h, y) represents the shear stress of the mth layer of material at the x=h plane; D 1x (0 , y) represents the electrical displacement of the first layer material at the x=0 plane, D mx (h, y) represents the electrical displacement of the mth layer material at the x=h plane;

在相邻层材料的界面处，位移的连续条件为：At the interface of adjacent layer materials, the continuum condition for displacement is:

Figure BDA0002214240280000062
Figure BDA0002214240280000062

其中：τnxz(hn,y)代表第n层材料在x＝hn平面处的剪应力，τ(n+1)xz(hn,y)代表第n+1层材料在x＝hn平面处的剪应力，Dnx(hn,y)代表第n层材料在x＝hn平面处的电位移，D(n+1)x(hn,y)代表第n+1层材料在x＝hn平面处的电位移，wn(hn,y)代表第n层材料在x＝hn平面处的机械位移，wn+1(hn,y)代表第n+1层材料在x＝hn平面处的机械位移，φn(hn,y)代表第n层材料在x＝hn平面处的电势，φn+1(hn,y)代表第n+1层材料在x＝hn平面处的电势。Where: τ nxz (h n , y) represents the shear stress of the nth layer of material at x=h n plane, τ (n+1)xz (h n , y) represents the n+1th layer of material at x=h Shear stress at the n plane, D nx (h n ,y) represents the electrical displacement of the nth layer material at the x=h n plane, D (n+1)x (h n ,y) represents the n+1th layer The electrical displacement of the material at the x=h n plane, w n (h n , y) represents the mechanical displacement of the nth layer material at the x=h n plane, w n+1 (h n , y) represents the n+th The mechanical displacement of a layer of material at the x=h n plane, φ n (h n , y) represents the potential of the nth layer of material at the x=h n plane, φ n+1 (h n , y) represents the nth The potential of +1 layer material at the x=h n plane.

本发明的上述算例仅为详细地说明本发明的计算模型和计算流程，而并非是对本发明的实施方式的限定。对于所属领域的普通技术人员来说，在上述说明的基础上还可以做出其它不同形式的变化或变动，这里无法对所有的实施方式予以穷举，凡是属于本发明的技术方案所引伸出的显而易见的变化或变动仍处于本发明的保护范围之列。The above calculation examples of the present invention are only to illustrate the calculation model and calculation process of the present invention in detail, but are not intended to limit the embodiments of the present invention. For those of ordinary skill in the art, on the basis of the above description, other different forms of changes or changes can also be made, and it is impossible to list all the embodiments here. Obvious changes or modifications are still within the scope of the present invention.

Claims (5)

1. A method of solving a fracture problem for a functionally graded piezoelectric material having arbitrary properties, the method comprising the steps of:
step one, establishing a constitutive equation and a balance equation of the functionally graded piezoelectric material;
step two, uniformly dividing the functionally graded piezoelectric material into a plurality of layers along the thickness direction, and obtaining a constitutive equation and a balance equation of each layer of material according to the constitutive equation and the balance equation established in the step one on the assumption that the material attribute of each layer changes along with the exponential function form and the material attributes of adjacent layers are continuous at the interface;
introducing a dislocation function on the surface of the crack of the functional gradient piezoelectric material, and obtaining an analytic solution of a constitutive equation and an equilibrium equation of each layer of material by utilizing Fourier transform and superposition principles;
step four, obtaining a singular integral equation set according to the constitutive equation and the balance equation of each layer of material in the step two and the analytic solution in the step three and by combining the boundary conditions of the constitutive equation and the balance equation of each layer of material and the continuous conditions of the displacement of the materials of the adjacent layers at the interface;
solving the singular integral equation set obtained in the step four to obtain stress intensity factors representing crack positions and length parameters, and obtaining the relationship between material parameters and geometric parameters of the functional gradient piezoelectric material and the stress intensity factors;
and step six, obtaining a stress intensity factor of the functional gradient piezoelectric material to be detected according to the material parameters and the geometric parameters of the functional gradient piezoelectric material to be detected, and obtaining the position and the length parameters of the crack of the functional gradient piezoelectric material to be detected according to the stress intensity factor.
2. The method for solving the fracture problem of the functionally graded piezoelectric material with any property according to claim 1, wherein the specific process of the first step is as follows:
for the functionally graded piezoelectric material with the thickness of h, the constitutive equation for establishing the functionally graded piezoelectric material is as follows:
in the formula: tau isxzRepresents the shear stress of the functionally graded piezoelectric material in the direction of the z-axis in the plane vertical to the x-axisyzRepresenting the shear stress of the functionally graded piezoelectric material in the direction of the z-axis in a plane perpendicular to the y-axis, DxRepresenting the electrical displacement of the functionally graded piezoelectric material in the x-axis direction, DyRepresenting the electric displacement of the functional gradient piezoelectric material in the y-axis direction; the x axis, the y axis and the z axis are three axes of a space rectangular coordinate system respectively;
w represents the mechanical displacement of the functionally graded piezoelectric material; phi represents a potential; c. C44Representative of shear modulus；e15Represents the piezoelectric coefficient; epsilon11Represents a dielectric constant;
the equilibrium equation of the functionally graded piezoelectric material is established as follows:
Figure FDA0002214240270000021
the boundary conditions of the constitutive equation and the equilibrium equation of the functionally graded piezoelectric material are as follows:
Figure FDA0002214240270000022
wherein: tau isxz(0, y) represents the shear stress of the functionally graded piezoelectric material at the x-0 plane, τxz(h, y) represents the shear stress of the functionally graded piezoelectric material at the x ═ h plane; dx(0, y) ═ 0 represents the electrical displacement of the functionally graded piezoelectric material at the x ═ 0 plane; dx(h, y) ═ 0 represents the electrical displacement of the functionally graded piezoelectric material at the x ═ h plane;
suppose c44、e15、ε11The expression of (a) is:
[c44,e15,ε11]＝[c440,e150,ε110]f(x) (4)
in the formula: c. C440、e150And ε110Are constants, and f (x) is a function of the property of the functionally graded piezoelectric material.
3. The method for solving the fracture problem of the functionally graded piezoelectric material with any property according to claim 2, wherein the specific process of the second step is as follows:
the functionally gradient piezoelectric material is uniformly divided into m layers along the thickness direction, and the thickness of the nth layer of material is hn–hn-1Wherein: n is 1,2, …, m, hnRepresents the thickness sum of the 1 st layer material to the n th layer material, hn-1Represents the sum of the thicknesses of the 1 st layer material and the n-1 st layer material;
assuming that each layer material property varies with the form of an exponential function:
Figure FDA0002214240270000023
cn44represents the shear modulus of the nth layer material; e.g. of the typen15Represents the piezoelectric coefficient of the nth layer material; epsilonn11Represents the dielectric constant of the nth layer material;
Figure FDA0002214240270000024
representing the attribute form of the nth layer material;
the continuum condition for the material properties of adjacent layers to be continuous at the interface is:
Figure FDA0002214240270000025
wherein: c. C(n+1)44Represents the shear modulus of the (n + 1) th layer of material; e.g. of the type(n+1)15Represents the piezoelectric coefficient of the (n + 1) th layer of material; epsilon(n+1)11Represents the dielectric constant of the (n + 1) th layer of material;
Figure FDA0002214240270000026
representing the attribute form of the (n + 1) th layer of material; f (h)n) Representing a property function of the nth layer material;
and obtaining the constitutive equation of the nth layer material by the constitutive equation and the equilibrium equation established in the step one as follows:
Figure FDA0002214240270000031
wherein: tau isnxzRepresents the z-axis direction shear stress of the n layer material on the plane vertical to the x axisnyzRepresenting the z-axis shear stress of the n-th layer material in the plane perpendicular to the y-axis, DnxRepresenting the electric displacement of the n-th layer material in the x-axis direction, DnyRepresents the electric displacement of the nth layer material in the y-axis direction; w is anRepresents the mechanical displacement of the nth layer material; phi is anRepresents the potential of the nth layer material;
the equilibrium equation for the nth layer material is:
Figure FDA0002214240270000032
and obtaining the constitutive equation and the equilibrium equation of each layer of material in the same way.
4. The method for solving the fracture problem of the functionally graded piezoelectric material with arbitrary properties according to claim 3, wherein in the third step, the expression of the dislocation function is as follows:
wherein: a and b represent two tips of the crack, a straight line where the a and b are located is taken as an x 'axis, a local rectangular coordinate system is established, and the positive direction and the vector of the x' axisAre in the same direction; g1(x') represents a function of misalignment with respect to mechanical displacement, g2(x') represents a dislocation function with respect to potential, +0 represents approaching 0 from the positive direction, -0 represents approaching 0 from the negative direction.
5. The method for solving the fracture problem of the functionally graded piezoelectric material with arbitrary properties according to claim 4, wherein in the fourth step, the boundary conditions of the constitutive equation and the equilibrium equation of each layer of material are as follows:
Figure FDA0002214240270000035
wherein: tau is1xz(0, y) represents the shear stress of the layer 1 material at the x-0 plane, τmxz(h, y) represents the shear stress of the mth layer of material at the x ═ h plane; d1x(0, y) represents the electric displacement of the layer 1 material in the x-0 plane, Dmx(h, y) represents the m-th layer of material in x ═ xh electrical displacement at the plane;
at the interface of the materials of the adjacent layers, the continuous conditions of displacement are:
Figure FDA0002214240270000041
wherein: tau isnxz(hnY) represents the n-th layer material in the case of x ═ hnShear stress at the plane, τ(n+1)xz(hnY) represents the n +1 th layer of material when x ═ hnShear stress at the plane, Dnx(hnY) represents the n-th layer material in the case of x ═ hnElectric displacement at the plane, D(n+1)x(hnY) represents the n +1 th layer of material when x ═ hnElectric displacement at the plane, wn(hnY) represents the n-th layer material in the case of x ═ hnMechanical displacement at plane, wn+1(hnY) represents the n +1 th layer of material when x ═ hnMechanical displacement at the plane, phin(hnY) represents the n-th layer material in the case of x ═ hnPotential at the plane, phin+1(hnY) represents the n +1 th layer of material when x ═ hnThe potential at the plane.
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