CN113936752A - Bionic staggered structure containing viscoelastic matrix and dynamics modeling method thereof - Google Patents

Bionic staggered structure containing viscoelastic matrix and dynamics modeling method thereof Download PDF

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CN113936752A
CN113936752A CN202111069204.7A CN202111069204A CN113936752A CN 113936752 A CN113936752 A CN 113936752A CN 202111069204 A CN202111069204 A CN 202111069204A CN 113936752 A CN113936752 A CN 113936752A
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聂云清
罗青
季浩然
刘畅
段富强
张涵
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National University of Defense Technology
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Abstract

The application provides a bionic staggered structure containing a viscoelastic matrix and a dynamics modeling method thereof, wherein the bionic staggered structure containing the viscoelastic matrix comprises a plurality of hard reinforcements arranged in parallel and the viscoelastic matrix; the bionic staggered structure has periodicity and repeatability, can be obtained by a unit cell structure in a mirror image and array mode, the unit cell structure is a minimum analysis unit of an integral structure, and comprises reinforcement bodies positioned at the top and the bottom and a base body positioned in the middle, and the influence of the base body between the length directions of the reinforcement bodies is ignored. Compared with the prior art, the invention establishes the bionic staggered structure kinetic theory model containing the viscoelastic matrix, obtains the theoretical solution of kinetic analysis through theoretical analysis, and avoids the defects of complex modeling and long analysis time when the traditional finite element method is used.

Description

Bionic staggered structure containing viscoelastic matrix and dynamics modeling method thereof
Technical Field
The invention belongs to the technical field of bionic composite material structure mechanics, and particularly relates to a bionic staggered structure containing a viscoelastic matrix and a dynamics modeling method thereof.
Background
After millions of years of evolution, many organisms have developed biological compounds with higher strength and better toughness, such as bones, teeth, deer horns and various shells, which are called load-bearing biomaterials, are composed of minerals and biological polymers, and have special mechanical properties, so that the biological compounds can support the weight of animals, resist external force and impact and protect internal soft organs.
The bionic staggered structure designed based on the structure has the characteristics of light weight, high rigidity and high toughness, and has wide application prospect. The current research mainly focuses on the static analysis of the bionic staggered structure, and the rigidity and strength characteristics under the static load are researched based on the theory and the finite element method; the mechanical characteristics of the bionic staggered structure under the dynamic load are mainly analyzed by a finite element method. However, the finite element method has the disadvantages of complex modeling, time-consuming calculation and the like, is not suitable for the initial design of the bionic staggered structure, and a new theoretical method for dynamic modeling of the bionic staggered structure needs to be provided urgently to analyze the mechanical characteristics of the bionic staggered structure under dynamic load.
Disclosure of Invention
The invention aims to provide a bionic staggered structure containing a viscoelastic matrix and a dynamics modeling method thereof, and the method solves the dynamics analysis problem of the bionic staggered structure under the action of a dynamic load.
The invention provides a bionic staggered structure containing a viscoelastic matrix, which has the following specific technical scheme:
a bionic staggered structure containing a viscoelastic matrix comprises a plurality of hard reinforcements arranged in parallel and the viscoelastic matrix; the bionic staggered structure has periodicity and repeatability, can be obtained by a single cell structure in a mirror image and array mode, the single cell structure is a minimum analysis unit of an integral structure, and comprises reinforcement bodies positioned at the top and the bottom and a base body positioned in the middle, and the influence of the base body between the length directions of the reinforcement bodies is ignored;
the hard reinforcement is made of linear elastic material, the constitutive relation of the hard reinforcement conforms to Hooke's law,
Figure BDA0003259487920000011
in the formula sigmaiFor normal stress, E is the modulus of elasticity, uiIs displacement;
the viscoelastic matrix is a viscoelastic material, the constitutive relation of the viscoelastic matrix conforms to a Kelvin-Voigt constitutive model, and the shearing force tau can be expressed as
Figure BDA0003259487920000021
Wherein G is the shear modulus of the matrix, gamma is the shear strain, eta is the viscosity coefficient,
Figure BDA0003259487920000022
is the matrix shear strain rate.
A bionic staggered structure dynamic modeling method containing a viscoelastic matrix comprises a plurality of hard reinforcements arranged in parallel and the viscoelastic matrix; the bionic staggered structure has periodicity and repeatability, can be obtained by a single cell structure in a mirror image and array mode, the single cell structure is a minimum analysis unit of an integral structure, and comprises reinforcement bodies positioned at the top and the bottom and a base body positioned in the middle, and the influence of the base body between the length directions of the reinforcement bodies is ignored;
the reinforcement is made of a linear elastic material, the constitutive relation of the reinforcement conforms to Hooke's law,
Figure BDA0003259487920000023
in the formula sigmaiFor enhancing the normal stress of the body, E is the modulus of elasticity, uiIn order to be able to displace,
the matrix is a viscoelastic material, the constitutive relation of the matrix conforms to a Kelvin-Voigt constitutive model, and the matrix shear force tau can be expressed as
Figure BDA0003259487920000024
Wherein G is the shear modulus of the matrix, gamma is the shear strain of the matrix, eta is the viscosity coefficient of the matrix,
Figure BDA0003259487920000025
is the matrix shear strain rate;
the dynamics modeling method comprises the following steps:
s1, constructing a kinetic equation set of the unit cell structure by utilizing a kinetic equilibrium relation,
Figure BDA0003259487920000026
Figure BDA0003259487920000027
in the formula u1And u2The displacement of the reinforcement, respectively top and bottom, is related to the time t and the abscissa x, i.e. can be expressed as u1(x, t) and u2(x, t), G, η, and h are the shear modulus, viscosity coefficient, and thickness of the matrix, E, b is the elastic modulus and thickness of the reinforcement, respectively,cin order to enhance the wave velocity of the body,xis the abscissa and t is time;
the boundary conditions and initial conditions of the equations are as follows,
Figure BDA0003259487920000028
Figure BDA0003259487920000031
where σ (t) is the impact load on the unit cell, in relation to time t, L is the unit cell length, u1(x,0)，u2(x,0) are the displacement functions of the reinforcement at the top and bottom at time 0, respectively, x being the abscissa;
s2, defining an intermediate variable related to time t and abscissa x, and defining an elastic wave function v (x, t) ═ u1(x,t)+u2(x, t) and a dissipation function w (x, t) ═ u1(x,t)-u2(x, t) equations (1) and (2) in the system of kinetic equations described in S1 can be added and subtracted, respectively, and decoupled into two independent partial differential equations,
Figure BDA0003259487920000032
Figure BDA0003259487920000033
in the formula
Figure BDA0003259487920000034
And
Figure BDA0003259487920000035
for the coefficients related to the material and the geometrical parameters,
Figure BDA0003259487920000036
the boundary condition and the initial condition are respectively,
Figure BDA0003259487920000037
Figure BDA0003259487920000038
wherein w (x,0) and v (x,0) are the elastic wave function and the dissipative wave function at the time of 0 respectively, and x is the abscissa;
s3, solving the decoupled partial differential equation (5) to obtain the following solution,
Figure BDA0003259487920000039
in the formula Tw0(T) and Twn(t) intermediate variables related to the geometrical parameters, the material parameters, the impact load and the time t, ∑ is the sum sign,
Figure BDA0003259487920000041
Figure BDA0003259487920000042
w0and wnIs a coefficient related to a material parameter,
Figure BDA0003259487920000043
awnare coefficients that are related to the geometric parameters,
Figure BDA0003259487920000044
psi is an integral variable;
s4, solving the partial differential equation (6) after decoupling to obtain the following solution,
Figure BDA0003259487920000045
in the formula Tvn(t) intermediate variables related to the geometrical parameters, the material parameters, the impact load and the time t, ∑ is the sum sign,
Figure BDA0003259487920000046
s5, using the analytic solutions of v (x, t) and w (x, t) described in S3 and S4, the displacement expression of the kinetic equation set described in S1 can be obtained as follows,
Figure BDA0003259487920000047
s6, obtaining the strain epsilon of the top reinforcement body in the bionic staggered structure through the displacement expression of the kinetic equation system in S51Stress σ1Strain epsilon of bottom reinforcement2Stress σ2The matrix shear strain gamma and the shear force tau are respectively,
Figure BDA0003259487920000048
Figure BDA0003259487920000049
Figure BDA00032594879200000410
Figure BDA00032594879200000411
Figure BDA00032594879200000412
Figure BDA00032594879200000413
further, the step S1 specifically includes the following sub-steps S11 to S12:
s11, based on Newton' S second law, it can be derived that the dynamic equilibrium equation of the top and bottom hard reinforcements is as follows,
Figure BDA0003259487920000051
Figure BDA0003259487920000052
in the formula sigma1Positive stress of the tip reinforcement, σ2The normal stress of the bottom reinforcement, tau is the shearing force and rho is the reinforcement density;
s12, reinforcing the body to be in normal stress
Figure BDA0003259487920000053
Shear stress of substrate
Figure BDA0003259487920000054
Shear strain of the substrate
Figure BDA0003259487920000055
Substituting the formula into the dynamic equilibrium equation of S11 to obtain
Figure BDA0003259487920000056
Figure BDA0003259487920000057
Further, the solving of the partial differential equation (5) in the step S3 specifically includes the following sub-steps S31 to S34:
s31, making the dissipation wave function w equal to w1+w2Wherein w is1And w2Is an intermediate variable related to time t and abscissa x, and w2For the smoothing function, only the boundary conditions are satisfied, take
Figure BDA0003259487920000058
In the formula, L is the unit cell length, and is substituted into the equation (1), and the known term at the right end is subjected to Fourier decomposition to obtain the product
Figure BDA0003259487920000059
Wherein L is the unit cell length, pw(t)、pw0(t) and pwn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t, pw(t) is derived by shifting the term of equation (1) by
Figure BDA00032594879200000510
pw0(t) and pwn(t) is pw(t) coefficients in Fourier series decomposition of
Figure BDA00032594879200000511
Figure BDA00032594879200000512
S32, solving equation (17) by using a separation variational method, and assuming that
Figure BDA0003259487920000061
Substituting into partial differential equation (17), simplifying into two ordinary differential equations with equal terms at two ends,
Figure BDA0003259487920000062
Figure BDA0003259487920000063
in the formula Tw0(T) and Twn(t) is a function of the geometric parameter, the material parameter, the impact load and the time t for solving for w1Intermediate variable of (1), with w1In a relationship of
Figure BDA0003259487920000064
pw0(t) and pwn(t) is the same as the geometric parameter, materialMaterial parameters, impact load and time t related intermediate variables,
Figure BDA0003259487920000065
Figure BDA0003259487920000066
decomposing the initial condition (7) of the equation by using Fourier series,
Figure BDA0003259487920000067
Figure BDA0003259487920000068
in the formula qw0And q iswnIs a coefficient related to geometric parameters, material parameters and initial moment impact load, and is obtained by performing Fourier decomposition on the formula (21) including
Figure BDA0003259487920000069
The initial conditions under which ordinary differential equations (18) and (19) can be obtained are as follows,
Figure BDA00032594879200000610
Figure BDA00032594879200000611
s33, transforming the equations (18) and (19) and solving the transformed equations by using Laplace method,
the equation (18) is first transformed into the form
Figure BDA00032594879200000612
In the formula yw(t) and gw(t) is an intermediate variable related to time t,
Figure BDA0003259487920000071
Figure BDA0003259487920000072
ewnand awnFor the coefficients related to the material and the geometrical parameters,
Figure BDA0003259487920000073
Figure BDA0003259487920000074
the formula (24) is subjected to a Laplace forward transform to obtain
Figure BDA0003259487920000075
Where s is a complex variable in the Laplace transform and Yw(s) is yw(t) a laplace transform of the image,
Figure BDA0003259487920000076
Gw(s) is gw(t) a laplace transform of the image,
Figure BDA0003259487920000077
the compound can be obtained by the formula,
Figure BDA0003259487920000078
inverse Laplace transform on the above equation yields the solution of ordinary differential equation (18) as follows
Figure BDA0003259487920000079
In the formula wnAnd awnTo be made of materials and tablesThe coefficients with which the parameters are related,
Figure BDA00032594879200000710
Figure BDA00032594879200000711
psi is an integral variable.
Similarly, the solution of ordinary differential equation (19) can be obtained by a similar method
Figure BDA00032594879200000712
In the formula w0In order to be the correlation coefficient,
Figure BDA00032594879200000713
psi is an integral variable.
S34, the solution to partial differential equation (5) can be obtained
Figure BDA00032594879200000714
Further, the step S4 specifically includes the following sub-steps S41 to S44:
s41, let the elastic wave function v be v1+v2，v1And v2Is an intermediate variable related to time t and abscissa x, and v2For the smoothing function, only the boundary conditions are satisfied, take
Figure BDA00032594879200000715
Substituting the known term into equation (2), and carrying out Fourier decomposition on the known term at the right end to obtain the final product
Figure BDA0003259487920000081
Wherein L is the unit cell length, pv0(t) and pvn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure BDA0003259487920000082
s42, solving equation (25) by using a separation variable method, and assuming that
Figure BDA0003259487920000083
The partial differential equation (25) is substituted, the corresponding terms at the two ends of the equation are equal, the partial differential equation is simplified into the following two ordinary differential equations,
Figure BDA0003259487920000084
Figure BDA0003259487920000085
in the formula Tv0(T) and Tvn(t) is a solution for v in relation to the geometrical parameters, the material parameters, the impact load and the time t1Intermediate variable of, and v1The relationship is as follows:
Figure BDA0003259487920000086
the initial condition (8) of the equation is decomposed by Fourier series,
Figure BDA0003259487920000087
Figure BDA0003259487920000088
in the formula qv0And q isvnIs a coefficient related to a geometrical parameter, a material parameter and an impact load at an initial moment, is obtained by performing Fourier decomposition on the formula (29),
Figure BDA0003259487920000089
the initial conditions under which ordinary differential equations (26) and (27) can be obtained are as follows,
Figure BDA00032594879200000810
Figure BDA00032594879200000811
s43, transforming the equations (26) and (27) and solving the transformed equations by using Laplace method,
the equation (26) is first transformed into the form
Figure BDA00032594879200000812
In the formula yv(t) and gv(t) is an intermediate variable related to time t, yv(t)＝Tvn(t)+c2avnσ(t)，
Figure BDA0003259487920000091
evnAnd avnFor the coefficients related to the material and the geometrical parameters,
Figure BDA0003259487920000092
the formula (32) is subjected to a Laplace forward transform to obtain
Figure BDA0003259487920000093
Where s is a complex variable in the Laplace transform and Yv(s) is yv(t) a laplace transform of the image,
Figure BDA0003259487920000094
Gv(s) is gv(t) a laplace transform of the image,
Figure BDA0003259487920000095
the compound can be obtained by the formula,
Figure BDA0003259487920000096
inverse laplace transform of the above equation yields a solution to ordinary differential equation (26) as follows,
Figure BDA0003259487920000097
in the formula avnAnd evnAre coefficients related to the material and geometric parameters,
Figure BDA0003259487920000098
psi is an integral variable.
Similarly, the solution of equation (27) can be solved in a similar manner,
Tv0(t)＝-c2av0σ(t)
in the formula av0Is a coefficient related to material and geometric parameters, av0＝L/2Ec2；
S44, the solution to partial differential equation (6) can be obtained
Figure BDA0003259487920000099
The invention has the beneficial technical effects that:
1. compared with the prior art, the invention establishes the bionic staggered structure kinetic theory model containing the viscoelastic matrix, obtains the theoretical solution of kinetic analysis through theoretical analysis, and avoids the defects of complex modeling and long analysis time when the traditional finite element method is used.
2. The top reinforcement strain epsilon of the bionic staggered structure containing the viscoelastic matrix is obtained by the dynamic modeling method1Stress σ1Strain epsilon of bottom reinforcement2Stress σ2Radical ofThe strain gamma and the shear force tau of the body shear can effectively analyze the strain and stress distribution states of the bionic staggered structure containing the viscoelastic matrix.
3. The model established by the existing commercial analysis software is a finite element model, and an approximate solution is obtained, while the model established by the dynamic modeling method is an analytic model, and an accurate expression can be obtained by adopting the analytic model, so the dynamic modeling method has higher calculation precision.
Drawings
FIG. 1 is a schematic diagram of a bionic staggered structure;
FIG. 2 is a schematic diagram of a biomimetic cross-linked structure;
FIG. 3 is a schematic view of the dynamic load applied to the bionic staggered structure;
FIG. 4 shows the tip end hard reinforcement at t0The moment positive stress distribution;
FIG. 5 shows the hard reinforcement at the bottom end at t0The moment positive stress distribution;
FIG. 6 shows a soft matrix at t0And (5) distributing the shear stress at any moment.
Detailed Description
In order to make the objects, technical solutions and advantages of the present application more apparent, the present application is further described with reference to fig. 1-6.
As can be seen from the accompanying drawings 1-2 in the specification, the invention provides a bionic staggered structure containing a viscoelastic matrix, and the specific technical scheme is as follows:
a bionic staggered structure containing a viscoelastic matrix comprises a plurality of hard reinforcements arranged in parallel and the viscoelastic matrix; the bionic staggered structure has periodicity and repeatability, can be obtained by a single cell structure in a mirror image and array mode, the single cell structure is a minimum analysis unit of an integral structure, and comprises reinforcement bodies positioned at the top and the bottom and a base body positioned in the middle, and the influence of the base body between the length directions of the reinforcement bodies is ignored;
the hard reinforcement is made of linear elastic material, the constitutive relation of the hard reinforcement conforms to Hooke's law,
Figure BDA0003259487920000101
in the formula sigmaiFor normal stress, E is the modulus of elasticity, uiIs displacement;
the viscoelastic matrix is a viscoelastic material, the constitutive relation of the viscoelastic matrix conforms to a Kelvin-Voigt constitutive model, and the shearing force tau can be expressed as
Figure BDA0003259487920000102
Wherein G is the shear modulus of the matrix, gamma is the shear strain, eta is the viscosity coefficient,
Figure BDA0003259487920000103
is the matrix shear strain rate.
The invention also provides a bionic staggered structure dynamics modeling method containing the viscoelastic matrix, which comprises the following steps and technical principles:
the schematic diagram of the mode staggered structure containing the viscoelastic matrix is shown in fig. 1, and the mode staggered structure containing the viscoelastic matrix consists of hard reinforcements and soft matrixes which are staggered, wherein the hard reinforcements are regularly staggered and embedded in the soft matrixes. Because the structure has periodicity, the structure can be regarded as being formed by symmetrical and arrayed single cell structures, a schematic diagram of the single cell structure is shown in figure 2, and the single cell structure is composed of three parts because a soft matrix mainly bears shearing force and neglects the stretching effect of the matrix in a discontinuous band between hard reinforcements: a top hard reinforcement, a bottom hard body and a middle soft substrate. The mechanical properties of the overall structure can be investigated by kinetic analysis of the single-cell structure.
S1, constructing a kinetic equation set of the unit cell structure by utilizing a kinetic equilibrium relation,
Figure BDA0003259487920000111
Figure BDA0003259487920000112
in the formula u1And u2Displacement of the reinforcement at the top and bottom, respectively, in relation to time t and x, abscissa, G, η, and h are shear modulus, viscosity coefficient, and thickness of the matrix, E, b is elastic modulus and thickness of the reinforcement, c is wave velocity of the reinforcement, x is the abscissa, and t is time;
the boundary conditions and initial conditions of the equations are as follows,
Figure BDA0003259487920000113
Figure BDA0003259487920000114
wherein sigma (t) is the impact load on the unit cell, and is related to time t, and L is the unit cell length;
s2, defining an intermediate variable related to the time t and the abscissa x, the elastic wave function v ═ u1+u2And the dissipation wave function w ═ u1-u2Equations (1) and (2) in the kinetic equation system of S1 can be added and subtracted, respectively, and are decoupled into the following two independent partial differential equations,
Figure BDA0003259487920000115
Figure BDA0003259487920000116
in the formula
Figure BDA0003259487920000121
And
Figure BDA0003259487920000122
is related to material and geometryThe coefficient of the number-dependent,
Figure BDA0003259487920000123
the boundary condition and the initial condition are respectively,
Figure BDA0003259487920000124
Figure BDA0003259487920000125
s3, solving the decoupled partial differential equation (5) to obtain the following solution,
Figure BDA0003259487920000126
in the formula Tw0(T) and Twn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure BDA0003259487920000127
Figure BDA0003259487920000128
w0and wnIs a coefficient related to a material parameter,
Figure BDA0003259487920000129
awnare coefficients that are related to the geometric parameters,
Figure BDA00032594879200001210
tau is an integral time variable;
s4, solving the partial differential equation (6) after decoupling to obtain the following solution,
Figure BDA00032594879200001211
in the formula Tvn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure BDA00032594879200001212
s5, using the analytic solutions of v (x, t) and w (x, t) described in S3 and S4, the displacement expression of the kinetic equation set described in S1 can be obtained as follows,
Figure BDA0003259487920000131
s6, obtaining the strain epsilon of the top reinforcement body in the bionic staggered structure through the displacement expression of the kinetic equation system in S51Stress σ1Strain epsilon of bottom reinforcement2Stress σ2Shear strain gamma and shear force of matrixτRespectively, are as follows,
Figure BDA0003259487920000132
Figure BDA0003259487920000133
Figure BDA0003259487920000134
Figure BDA0003259487920000135
Figure BDA0003259487920000136
Figure BDA0003259487920000137
further, the step S1 specifically includes the following sub-steps S11 to S12:
s11, based on Newton' S second law, it can be derived that the dynamic equilibrium equation of the top and bottom hard reinforcements is as follows,
Figure BDA0003259487920000138
Figure BDA0003259487920000139
in the formula sigma1Positive stress of the tip reinforcement, σ2The normal stress of the bottom reinforcement, tau is the shearing force and rho is the reinforcement density;
s12, reinforcing the body to be in normal stress
Figure BDA00032594879200001310
Shear stress of substrate
Figure BDA00032594879200001311
Shear strain of the substrate
Figure BDA00032594879200001312
Substituting the formula into the dynamic equilibrium equation of S11 to obtain
Figure BDA00032594879200001313
Figure BDA00032594879200001314
Further, the solving of the partial differential equation (5) in the step S3 specifically includes the following sub-steps S31 to S34:
s31, making the dissipation wave function w equal to w1+w2Wherein w is1And w2Is related to time t and horizontalCoordinate x is related to an intermediate variable, and w2For smoothing functions, only boundary conditions are satisfied, preferably
Figure BDA0003259487920000141
In the formula, L is the unit cell length, and is substituted into the equation (1), and the known term at the right end is subjected to Fourier decomposition to obtain the product
Figure BDA0003259487920000142
Wherein L is the unit cell length, pw(t)、pw0(t) and pwn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t, pw(t) is derived by shifting the term of equation (1) by
Figure BDA0003259487920000143
pw0(t) and pwn(t) is pw(t) coefficients in Fourier series decomposition of
Figure BDA0003259487920000144
Figure BDA0003259487920000145
S32, solving equation (17) by using a separation variational method, and assuming that
Figure BDA0003259487920000146
Substituting into partial differential equation (17), simplifying into two ordinary differential equations with equal terms at two ends,
Figure BDA0003259487920000147
Figure BDA0003259487920000148
in the formula Tw0(T) and Twn(t) isRelating to geometrical parameters, material parameters, impact load and time t for solving for w1Intermediate variable of (1), with w1In a relationship of
Figure BDA0003259487920000149
pw0(t) and pwn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure BDA00032594879200001410
Figure BDA00032594879200001411
decomposing the initial condition (7) of the equation by using Fourier series,
Figure BDA00032594879200001412
Figure BDA0003259487920000151
in the formula qw0And q iswnCoefficients relating to geometrical parameters, material parameters, impact load at the initial moment,
Figure BDA0003259487920000152
Figure BDA0003259487920000153
the initial conditions under which ordinary differential equations (18) and (19) can be obtained are as follows,
Figure BDA0003259487920000154
Figure BDA0003259487920000155
s33, transforming the equations (18) and (19) and solving the transformed equations by using Laplace method,
the equation (18) is first transformed into the form
Figure BDA0003259487920000156
In the formula yw(t) and gw(t) is an intermediate variable related to time t,
Figure BDA0003259487920000157
Figure BDA0003259487920000158
ewnand awnFor the coefficients related to the material and the geometrical parameters,
Figure BDA0003259487920000159
Figure BDA00032594879200001510
the formula (24) is subjected to a Laplace forward transform to obtain
Figure BDA00032594879200001511
Where s is a complex variable in the Laplace transform, Gw(s) is gw(t) a laplace transform of the image,
Figure BDA00032594879200001512
the compound can be obtained by the formula,
Figure BDA00032594879200001513
inverse Laplace transform on the above equation yields the solution of ordinary differential equation (18) as follows
Figure BDA00032594879200001514
In the formula wnAnd awnFor the coefficients related to the material and the geometrical parameters,
Figure BDA00032594879200001515
Figure BDA0003259487920000161
similarly, the solution of ordinary differential equation (19) can be obtained by a similar method
Figure BDA0003259487920000162
In the formula w0In order to be the correlation coefficient,
Figure BDA0003259487920000163
τ is a time integral variable.
S34, the solution to partial differential equation (5) can be obtained
Figure BDA0003259487920000164
Further, the step S4 specifically includes the following sub-steps S41 to S44:
s41, let the elastic wave function v be v1+v2，v1And v2Is an intermediate variable related to time t and abscissa x, and v2For smoothing functions, only boundary conditions are satisfied, preferably
Figure BDA0003259487920000165
Substituting the known term into equation (2), and carrying out Fourier decomposition on the known term at the right end to obtain the final product
Figure BDA0003259487920000166
Wherein L is the unit cell length, pv0(t) and pvn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure BDA0003259487920000167
s42, solving equation (25) by using a separation variable method, and assuming that
Figure BDA0003259487920000168
The partial differential equation (25) is substituted, the corresponding terms at the two ends of the equation are equal, the partial differential equation is simplified into the following two ordinary differential equations,
Figure BDA0003259487920000169
Figure BDA00032594879200001610
in the formula Tv0(T) and Tvn(t) is a solution for v in relation to the geometrical parameters, the material parameters, the impact load and the time t1Intermediate variable of, and v1The relationship is as follows:
Figure BDA00032594879200001611
the initial condition (8) of the equation is decomposed by Fourier series,
Figure BDA0003259487920000171
Figure BDA0003259487920000172
in the formula qv0And q isvnCoefficients relating to geometrical parameters, material parameters, impact load at the initial moment,
Figure BDA0003259487920000173
Figure BDA0003259487920000174
the initial conditions under which ordinary differential equations (26) and (27) can be obtained are as follows,
Figure BDA0003259487920000175
Figure BDA0003259487920000176
s43, transforming the equations (26) and (27) and solving the transformed equations by using Laplace method,
the equation (26) is first transformed into the form
Figure BDA0003259487920000177
In the formula yv(t) and gv(t) is an intermediate variable related to time t, yv(t)＝Tvn(t)+c2avnσ(t)，
Figure BDA0003259487920000178
evnAnd avnFor the coefficients related to the material and the geometrical parameters,
Figure BDA0003259487920000179
the formula (32) is subjected to a Laplace forward transform to obtain
Figure BDA00032594879200001710
Where s is the complex variable in the laplace transform.
The compound can be obtained by the formula,
Figure BDA00032594879200001711
inverse laplace transform of the above equation yields a solution to ordinary differential equation (26) as follows,
Figure BDA00032594879200001712
in the formula avnAnd evnAre coefficients related to the material and geometric parameters,
Figure BDA00032594879200001713
τ is a time integral variable.
Similarly, the solution of equation (27) can be solved in a similar manner,
Tv0(t)＝-c2av0σ(t)
in the formula av0Is a coefficient related to material and geometric parameters, av0＝L/2Ec2；
S44, the solution to partial differential equation (6) can be obtained
Figure BDA0003259487920000181
The following describes a specific implementation process of the bionic staggered structure dynamics modeling method containing the viscoelastic matrix by specific examples, and the calculation results of the dynamics modeling method are shown by the attached figures 3-6.
(1) The material parameters chosen in the examples are as follows: the elastic modulus E of the hard reinforcement is 105GPa, and the density rho is 2000kg/m3(ii) a The shear modulus G of the soft viscoelastic matrix is 1.4GPa and the viscosity coefficient η is 5Pa · s.
(2) The geometric parameters chosen in the examples are as follows: thickness b of hard reinforcing body is 5 x 10-4m, thickness of matrix h 1 × 10-4m, unit cell length L5 × 10-3m。
(3) The structure is subjected to dynamic load of a triangular impact load type shown in figure 3, and the load amplitude sigmamax30MPa, load duration t0＝1×10-6s。
FIGS. 4-5 show the data at t0At this time, the normal stress distribution inside the top and bottom hard-reinforcing members, as can be seen from fig. 4 to 5, the normal stress inside the top and bottom hard-reinforcing members gradually decreases from the load application end to the free end during the application of the external load.
FIG. 6 shows the data at t0As is clear from FIG. 6, the shear stress distribution in the soft matrix at this time point is such that the shear stress of the soft matrix gradually decreases from the ends of the hard reinforcing members toward the center.
The above description is only a preferred embodiment of the present application and is not intended to limit the present application, and it will be apparent to those skilled in the art that various modifications and variations can be made in the embodiment of the present application. Any modification, equivalent replacement, improvement and the like made within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims (5)

1. A bionic staggered structure containing a viscoelastic matrix is characterized in that,
the bionic staggered structure containing the viscoelastic matrix comprises a plurality of hard reinforcements arranged in parallel and the viscoelastic matrix; the bionic staggered structure has periodicity and repeatability, can be obtained by a single cell structure in a mirror image and array mode, the single cell structure is a minimum analysis unit of an integral structure, and comprises reinforcement bodies positioned at the top and the bottom and a base body positioned in the middle, and the influence of the base body between the length directions of the reinforcement bodies is ignored;
the hard reinforcement is made of linear elastic material, the constitutive relation of the hard reinforcement conforms to Hooke's law,
Figure FDA0003259487910000011
in the formula sigmaiFor normal stress, E is the modulus of elasticity, uiIs displacement;
the viscoelastic matrix is a viscoelastic material, the constitutive relation of the viscoelastic matrix conforms to a Kelvin-Voigt constitutive model, and the shearing force tau can be expressed as
Figure FDA0003259487910000012
Wherein G is the shear modulus of the matrix, gamma is the shear strain, eta is the viscosity coefficient of the matrix,
Figure FDA0003259487910000013
is the matrix shear strain rate.
2. A bionic staggered structure dynamic modeling method containing a viscoelastic matrix is characterized in that,
the bionic staggered structure containing the viscoelastic matrix comprises a plurality of hard reinforcements arranged in parallel and the viscoelastic matrix; the bionic staggered structure has periodicity and repeatability, can be obtained by a single cell structure in a mirror image and array mode, the single cell structure is a minimum analysis unit of an integral structure, and comprises reinforcement bodies positioned at the top and the bottom and a base body positioned in the middle, and the influence of the base body between the length directions of the reinforcement bodies is ignored;
the reinforcement is made of a linear elastic material, the constitutive relation of the reinforcement conforms to Hooke's law,
Figure FDA0003259487910000014
in the formula sigmaiFor enhancing the normal stress of the body, E is the modulus of elasticity, uiIn order to be able to displace,
the matrix is a viscoelastic material, the constitutive relation of the matrix conforms to a Kelvin-Voigt constitutive model, and the matrix shear force tau can be expressed as
Figure FDA0003259487910000015
Wherein G is the shear modulus of the matrix, gamma is the shear strain of the matrix, eta is the viscosity coefficient of the matrix,
Figure FDA0003259487910000016
is the matrix shear strain rate;
the dynamics modeling method comprises the following steps:
s1, constructing a kinetic equation set of the unit cell structure by utilizing a kinetic equilibrium relation,
Figure FDA0003259487910000017
Figure FDA0003259487910000021
in the formula u1And u2The displacement of the reinforcement, respectively top and bottom, is related to the time t and the abscissa x, i.e. can be expressed as u1(x, t) and u2(x, t), G, η, and h are the shear modulus, viscosity coefficient, and thickness of the matrix, E, b is the elastic modulus and thickness of the reinforcement, c is the wave velocity of the reinforcement, x is the abscissa, and t is the time, respectively;
the boundary conditions and initial conditions of the equations are as follows,
Figure FDA0003259487910000022
Figure FDA0003259487910000023
where σ (t) is the impact load on the unit cell, in relation to time t, L is the unit cell length, u1(x,0)，u2(x,0) are the displacement functions of the reinforcement at the top and bottom at time 0, respectively, x being the abscissa;
s2, defining the coordinate with time t and abscissax-related intermediate variable, elastic wave function v (x, t) ═ u1(x,t)+u2(x, t) and a dissipation function w (x, t) ═ u1(x,t)-u2(x, t) equations (1) and (2) in the system of kinetic equations described in S1 can be added and subtracted, respectively, and decoupled into two independent partial differential equations,
Figure FDA0003259487910000024
Figure FDA0003259487910000025
in the formula
Figure FDA0003259487910000026
And
Figure FDA0003259487910000027
for the coefficients related to the material and the geometrical parameters,
Figure FDA0003259487910000028
the boundary condition and the initial condition are respectively,
Figure FDA0003259487910000029
Figure FDA0003259487910000031
wherein w (x,0) and v (x,0) are the elastic wave function and the dissipative wave function at the time of 0 respectively, and x is the abscissa;
s3, solving the decoupled partial differential equation (5) to obtain the following solution,
Figure FDA0003259487910000032
in the formula Tw0(T) and Twn(t) intermediate variables related to the geometrical parameters, the material parameters, the impact load and the time t, ∑ is the sum sign,
Figure FDA0003259487910000033
Figure FDA0003259487910000034
w0and wnIs a coefficient related to a material parameter,
Figure FDA0003259487910000035
awnare coefficients that are related to the geometric parameters,
Figure FDA0003259487910000036
psi is an integral variable;
s4, solving the partial differential equation (6) after decoupling to obtain the following solution,
Figure FDA0003259487910000037
in the formula Tvn(t) intermediate variables related to the geometrical parameters, the material parameters, the impact load and the time t, ∑ is the sum sign,
Figure FDA0003259487910000038
s5, using the analytic solutions of v (x, t) and w (x, t) described in S3 and S4, the displacement expression of the kinetic equation set described in S1 can be obtained as follows,
Figure FDA0003259487910000039
s6, kinetic method of S5The displacement expression of the program group can obtain the strain epsilon of the top reinforcement body in the bionic staggered structure1Stress σ1Strain epsilon of bottom reinforcement2Stress σ2The matrix shear strain gamma and the shear force tau are respectively,
Figure FDA0003259487910000041
Figure FDA0003259487910000042
Figure FDA0003259487910000043
Figure FDA0003259487910000044
Figure FDA0003259487910000045
Figure FDA0003259487910000046
3. the method for modeling the dynamics of a biomimetic interlaced structure including a viscoelastic substrate according to claim 2, wherein the step S1 specifically includes the following substeps S11-S12:
s11, based on Newton' S second law, it can be derived that the dynamic equilibrium equation of the top and bottom hard reinforcements is as follows,
Figure FDA0003259487910000047
Figure FDA0003259487910000048
in the formula sigma1Positive stress of the tip reinforcement, σ2The normal stress of the bottom reinforcement, tau is the shearing force and rho is the reinforcement density;
s12, reinforcing the body to be in normal stress
Figure FDA0003259487910000049
Shear stress of substrate
Figure FDA00032594879100000410
Shear strain of the substrate
Figure FDA00032594879100000411
Substituting the formula into the dynamic equilibrium equation of S11 to obtain
Figure FDA00032594879100000412
Figure FDA00032594879100000413
4. The method according to claim 2, wherein the solution of the partial differential equation (5) in step S3 comprises the following substeps S31-S34:
s31, making the dissipation wave function w equal to w1+w2Wherein w is1And w2Is an intermediate variable related to time t and abscissa x, and w2For the smoothing function, only the boundary conditions are satisfied, take
Figure FDA0003259487910000051
In the formula, L is the unit cell length, and is substituted into the equation (1), and the known term at the right end is subjected to Fourier decomposition to obtain the product
Figure FDA0003259487910000052
Wherein L is the unit cell length, pw(x,t)、pw0(t) and pwn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t, pw(x, t) is derived by shifting the term of equation (1) by
Figure FDA0003259487910000053
pw0(t) and pwn(t) is pwThe coefficients in Fourier series decomposition of (x, t) are
Figure FDA0003259487910000054
Figure FDA0003259487910000055
S32, solving equation (17) by using a separation variational method, and assuming that
Figure FDA0003259487910000056
Substituting into partial differential equation (17), simplifying into two ordinary differential equations with equal terms at two ends,
Figure FDA0003259487910000057
in the formula Tw0(T) and Twn(t) is a function of the geometric parameter, the material parameter, the impact load and the time t for solving for w1Intermediate variable of (1), with w1In a relationship of
Figure FDA0003259487910000058
pw0(t) and pwn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure FDA0003259487910000061
Figure FDA0003259487910000062
decomposing the initial condition (7) of the equation by using Fourier series,
Figure FDA0003259487910000063
Figure FDA0003259487910000064
in the formula qw0And q iswnIs a coefficient related to geometric parameters, material parameters and initial moment impact load, and is obtained by performing Fourier decomposition on the formula (21) including
Figure FDA0003259487910000065
Figure FDA0003259487910000066
The initial conditions under which ordinary differential equations (18) and (19) can be obtained are as follows,
Figure FDA0003259487910000067
Figure FDA0003259487910000068
s33, transforming the equations (18) and (19) and solving the transformed equations by using Laplace method,
the equation (18) is first transformed into the form
Figure FDA0003259487910000069
In the formula yw(t) and gw(t) is an intermediate variable related to time t,
Figure FDA00032594879100000610
ewnand awnFor the coefficients related to the material and the geometrical parameters,
Figure FDA00032594879100000611
the formula (24) is subjected to a Laplace forward transform to obtain
Figure FDA00032594879100000612
Where s is a complex variable in the Laplace transform and Yw(s) is yw(t) a laplace transform of the image,
Figure FDA00032594879100000613
Gw(s) is gw(t) a laplace transform of the image,
Figure FDA0003259487910000071
the compound can be obtained by the formula,
Figure FDA0003259487910000072
inverse Laplace transform on the above equation yields the solution of ordinary differential equation (18) as follows
Figure FDA0003259487910000073
In the formula wnAnd awnFor the coefficients related to the material and the geometrical parameters,
Figure FDA0003259487910000074
Figure FDA0003259487910000075
psi is an integral variable.
Similarly, the solution of ordinary differential equation (19) can be obtained by a similar method
Figure FDA0003259487910000076
In the formula w0In order to be the correlation coefficient,
Figure FDA0003259487910000077
psi is an integral variable.
S34, the solution to partial differential equation (5) can be obtained
Figure FDA0003259487910000078
5. The method for modeling the dynamics of a biomimetic interlaced structure including a viscoelastic substrate according to claim 2, wherein the step S4 specifically includes the following substeps S41-S44:
s41, let the elastic wave function v be v1+v2，v1And v2Is an intermediate variable related to time t and abscissa x, and v2For the smoothing function, only the boundary conditions are satisfied, take
Figure FDA0003259487910000079
Substituting the known term into equation (2), and carrying out Fourier decomposition on the known term at the right end to obtain the final product
Figure FDA00032594879100000710
Wherein L is the unit cell length, pv(x,t)、pv0(t) and pvn(t) is an intermediate variable related to the geometrical parameters, the material parameters, the impact load and the time t,
Figure FDA00032594879100000711
Figure FDA00032594879100000712
s42, solving equation (25) by using a separation variable method, and assuming that
Figure FDA0003259487910000081
The partial differential equation (25) is substituted, the corresponding terms at the two ends of the equation are equal, the partial differential equation is simplified into the following two ordinary differential equations,
Figure FDA0003259487910000082
Figure FDA0003259487910000083
in the formula Tv0(T) and Tvn(t) is a solution for v in relation to the geometrical parameters, the material parameters, the impact load and the time t1Intermediate variable of, and v1The relationship is as follows:
Figure FDA0003259487910000084
the initial condition (8) of the equation is decomposed by Fourier series,
Figure FDA0003259487910000085
Figure FDA0003259487910000086
in the formula qv0And q isvnIs a coefficient related to a geometrical parameter, a material parameter and an impact load at an initial moment, is obtained by performing Fourier decomposition on the formula (29),
Figure FDA0003259487910000087
Figure FDA0003259487910000088
the initial conditions under which ordinary differential equations (26) and (27) can be obtained are as follows,
Figure FDA0003259487910000089
Figure FDA00032594879100000810
s43, transforming the equations (26) and (27) and solving the transformed equations by using Laplace method,
the equation (26) is first transformed into the form
Figure FDA00032594879100000811
In the formula yv(t) and gv(t) is an intermediate variable related to time t, yv(t)＝Tvn(t)+c2avnσ(t)，
Figure FDA00032594879100000812
evnAnd avnTo be related to material and geometric parametersThe coefficient of the correlation is such that,
Figure FDA00032594879100000813
the formula (32) is subjected to a Laplace forward transform to obtain
Figure FDA0003259487910000091
Where s is a complex variable in the Laplace transform and Yv(s) is yv(t) a laplace transform of the image,
Figure FDA0003259487910000092
Gv(s) is gv(t) a laplace transform of the image,
Figure FDA0003259487910000093
the compound can be obtained by the formula,
Figure FDA0003259487910000094
inverse laplace transform of the above equation yields a solution to ordinary differential equation (26) as follows,
Figure FDA0003259487910000095
in the formula avnAnd evnAre coefficients related to the material and geometric parameters,
Figure FDA0003259487910000096
psi is an integral variable.
Similarly, the solution of equation (27) can be solved in a similar manner,
Tv0(t)＝-c2av0σ(t)
in the formula av0Is made of a material andcoefficient related to geometric parameters, av0＝L/2Ec2；
S44, the solution to partial differential equation (6) can be obtained
Figure FDA0003259487910000097
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