CN109033742B - It is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue - Google Patents
It is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue Download PDFInfo
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Abstract
The present invention is a kind of for simulating the shear-deformable hyperelastic model of soft tissue, belongs to biological tissue's modeling field.Biologic soft tissue has the characteristics such as compressibility, anisotropy, super-elasticity, however existing model shows mechanical characteristic identical with isotropic material when simulation biologic soft tissue is shear-deformable.It in order to solve this problem, include I the invention proposes one kind5, I7The strain energy equation of strain invariant.In order to intuitively find out when shear-deformable, biologic soft tissue shows anisotropic, therefore linearizes to stress-strain relation, and the physical background of linearisation is: under conditions of small deformation, biologic soft tissue stress-strain relation is linear.Be calculated by derivation: when shear-deformable, model proposed by the present invention shows anisotropic characteristic aligned.The present invention proposes that hyperelastic model has truly simulated Constitutive Relation of Soft Tissue, has profound influence to the modeling and simulation of biologic soft tissue.
Description
Technical field:
The present invention is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue, belongs to biological tissue
Modeling field.
Background technique:
Biologic soft tissue be compressibility, anisotropy, nonlinearity material, the modeling and simulation of biologic soft tissue
It is extremely complex process, but is widely used, such as: organizational project, bioengineering, rehabilitation medical, aids drug transport, it is raw
The hot spot of the mechanical characteristic of object tissue being modeled to for research.Up to now, imitated biological tissue's mechanical characteristic is most thought
It is hyperelastic model, and hyperelastic model has been developed to diversified forms, such as: polynomial form, exponential form, to number form
Formula, power function form, mixed form.However existing compressible anisotropy hyperelastic model is shown when simulating shear-deformable
Characteristic identical with every material in the same direction.In order to solve this problem with the biomethanics of more accurate simulation soft tissue
Characteristic includes I the invention proposes one kind5, I7The strain energy equation of strain invariant.When simulation soft tissue is shear-deformable, this
The stress-strain relation that the strain energy equation that invention proposes is shown is more close to objective physical phenomenon.The present invention proposes super-elasticity
Model has truly simulated Constitutive Relation of Soft Tissue, has profound influence to the modeling and simulation of biologic soft tissue.
Summary of the invention:
It is an object of the invention in order to overcome the shortcomings of existing compressible anisotropy hyperelastic model, it may be assumed that is simulating
When shear-deformable, anisotropic material shows characteristic identical with every material in the same direction, and the one kind provided includes I5, I7
The hyperelastic model of strain invariant can truly simulate Constitutive Relation of Soft Tissue.Since biologic soft tissue is in small change
Under shape, ess-strain is linear relationship, in order to intuitively illustrate that strain energy equation proposed by the present invention can be such that soft tissue model is cutting
Anisotropy is shown under shear deformation, the present invention is calculated under linear deformation by the linearisation to strain stress relationship, soft
Tissue model is different in the modulus of shearing of three axial directions, that is, anisotropy is embodied when shear-deformable.By pushing away
Lead calculating, it was demonstrated that the present invention shear-deformable shows anisotropy simulating.
Technical solution:
It is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue, which is characterized in that described one
Kind is described for simulating orthotropy fibre reinforced materials for simulating the shear-deformable hyperelastic model of soft tissue
Fibre reinforced materials includes two cluster fibers and matrix, and in cartesian coordinate system, two cluster fibers are distributed in x/y plane, and about
Y-axis is symmetrical, it is described it is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue the following steps are included:
Step 1: proposing strain energy equation；
Step 2: calculating cauchy stress；
Step 3: the linearisation of stress-strain relation；
Step 4: calculating modulus of shearing；
The concrete form of strain energy equation in the step 1 are as follows:
Wherein κ is bulk modulus, c1, c2, c3It is material parameter, k is dimensionless group, and F is deformation gradient, Zuo Kexi lattice
Woods strain tensor C=FTF, right Cauchy's Green strain tensor B=FFT, Ii(i=1,4,5,6,7) is strain invariant, I1=tr
(C), J is volume ratio,I4=MCM, I5=MC2M, I6=M ' CM ', I7=M ' C2M ', wherein
M, M ' are that the optimal direction of two cluster fibers can obtain M=(cos θ sin θ 0), M '=(- cos θ if M and the angle of x-axis are θ
sinθ 0)；
The calculation formula of cauchy stress in the step 2 are as follows:
Formula (1), which is brought into formula (2), can obtain the concrete form of cauchy stress are as follows:
Wherein
The step 3 is to carry out linearization process to formula (2), and in order to meet under small deformation, ess-strain is in line
Sexual intercourse, displacement gradient H=F-I, I are unit tensor, infinitesimal strain tensorIf ∈=| | ∈ | |, suddenly
Slightly higher order term is truncated ∈ single order, can obtain B=C=I+ ∈, I1=3+2 ∈t, I3=1+2 ∈t, I4=1+2 ∈M, I5=1+4
∈M, I6=1+2 ∈M′, I7=1+4 ∈M′, wherein ∈t=tr (∈), ∈M=M ∈ M, ∈M′=M ' ∈ M '；Then formula (3)
The concrete form of the partial derivative of middle W are as follows:
Formula (4) is updated in formula (3), the expression formula of cauchy stress are as follows:
Wherein
The step 4, which calculates modulus of shearing, to be derived from formula (5):
It is obtained from formula (7) and formula (8), works as m4≠ 0,When, μxz≠μyz, described one kind is for simulating soft group
Knit shear-deformable hyperelastic model, which is characterized in that contain I in strain energy equation5, I7Strain invariant, and m4≠ 0, by
InThen the material of the modeling can it is shear-deformable be performance anisotropy, meet it is orthogonal respectively to
The objective requirement of unlike material.
The present invention has the advantages that for simulating Constitutive Relation of Soft Tissue
1) the invention proposes a kind of new strain energy equation, for simulating the anisotropy of biomaterial and non-linear answering
Stress-strain relationship.
2) strain energy equation proposed by the present invention can be such that biologic soft tissue shows in imitated biological tissue when shear-deformable
Modulus of shearing in anisotropy out, that is, three axis is different.
3) strain energy equation proposed by the present invention can simulate compressible material, i.e. biological tissue is in deformation process, volume
It is variable, that is, J ≠ 1.
Detailed description of the invention:
Fig. 1 is fibre reinforced materials structural schematic diagram.
Fig. 2 is fiber distribution schematic diagram.
Drawing reference numeral title: 1, fiber；2, matrix.
Specific embodiment:
The present invention is described in detail with reference to the accompanying drawing.
It is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue, which is characterized in that described one
Kind is described for simulating orthotropy fibre reinforced materials for simulating the shear-deformable hyperelastic model of soft tissue
Fibre reinforced materials includes two cluster fibers and matrix, as shown in Figure 1, it is flat that two cluster fibers are distributed in xy in cartesian coordinate system
Face, as shown in Fig. 2, two cluster fibers are symmetrical about y-axis, described is a kind of for simulating the shear-deformable hyperelastic model of soft tissue
Method for building up the following steps are included:
Step 1: proposing strain energy equation；
Step 2: calculating cauchy stress；
Step 3: the linearisation of stress-strain relation；
Step 4: calculating modulus of shearing；
The concrete form of strain energy equation in the step 1 are as follows:
Wherein κ is bulk modulus, c1, c2, c3It is material parameter, k is dimensionless group, and F is deformation gradient, Zuo Kexi lattice
Woods strain tensor C=FTF, right Cauchy's Green strain tensor B=FFT, Ii(i=1,4,5,6,7) is strain invariant, I1=tr
(C), J is volume ratio,I4=MCM, I5=MC2M, I6=M ' CM ', I7=M ' C2IM ', wherein
M, M ' are that the optimal direction of two cluster fibers can obtain M=(cos θ sin θ 0), M '=(- cos θ if M and the angle of x-axis are θ
sinθ 0)；
The calculation formula of cauchy stress in the step 2 are as follows:
Formula (1), which is brought into formula (2), can obtain the concrete form of cauchy stress are as follows:
Wherein
The step 3 is to carry out linearization process to formula (2), and in order to meet under small deformation, ess-strain is in line
Sexual intercourse, displacement gradient H=F-I, I are unit tensor, infinitesimal strain tensorIf ∈=| | ∈ | |, suddenly
Slightly higher order term is truncated ∈ single order, can obtain B=C=I+ ∈, I1=3+2 ∈t, I3=1+2 ∈t, I4=1+2 ∈M, I5=1+4
∈M, I6=1+2 ∈M′, I7=1+4 ∈M′, wherein ∈t=tr (∈), ∈M=M ∈ M, ∈M′=M ' ∈ M '；Then formula (3)
The concrete form of the partial derivative of middle W are as follows:
Formula (4) is updated in formula (3), the expression formula of cauchy stress are as follows:
Wherein
The step 4, which calculates modulus of shearing, to be derived from formula (5):
It is obtained from formula (7) and formula (8), works as m4≠ 0,When, μxz≠μyz, described one kind is for simulating soft group
Knit shear-deformable hyperelastic model, which is characterized in that contain I in strain energy equation5, I7Strain invariant, and m4≠ 0, by
InThen the material of the modeling can it is shear-deformable be performance anisotropy, meet it is orthogonal respectively to
The objective requirement of unlike material.
Claims (3)
1. a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue, which is characterized in that described one kind
For simulating the shear-deformable hyperelastic model of soft tissue, for simulating orthotropy fibre reinforced materials, the fibre
Tieing up reinforcing material includes two cluster fibers and matrix, and in cartesian coordinate system, two cluster fibers are distributed in x/y plane, and about y
Axial symmetry, it is described it is a kind of for simulating the method for building up of the shear-deformable hyperelastic model of soft tissue the following steps are included:
Step 1: proposing strain energy equation；
Step 2: calculating cauchy stress；
Step 3: the linearisation of stress-strain relation；
Step 4: calculating modulus of shearing；
The concrete form of strain energy equation in the step 1 are as follows:
Wherein κ is bulk modulus, c1, c2, c3It is material parameter, k is dimensionless group, and F is deformation gradient, and right Cauchy Green answers
Become tensor C=FTF, Zuo Kexi Green strain tensor B=FFT, Ii(i=1,3,4,5,6,7) is strain invariant, I1=tr
(C), J is volume ratio, andI4=MCM, I5=MC2M, I6=M ' CM ', I7=M ' C2M ',
Middle M, M ' they are that the optimal direction of two cluster fibers can obtain M=(cos θ, sin θ, 0) if M and the angle of x-axis are θ, M '=(- cos θ,
Sin θ, 0)；
The calculation formula of cauchy stress in the step 2 are as follows:
Formula (1), which is brought into formula (2), can obtain the concrete form of cauchy stress are as follows:
Wherein
The step 3 is to carry out linearization process to formula (2), and in order to meet under small deformation, ess-strain is linearly closed
System, also for intuitively finding out under conditions of shear-deformable, model shows anisotropy, and displacement gradient H=F-I, I are single
Position tensor, infinitesimal strain tensorIf ∈=| | ∈ | |, ignore higher order term, ∈ single order is truncated, B can be obtained
=C=I+ ∈, I1=3+2 ∈t, I3=1+2 ∈t, I4=1+2 ∈M, I5=1+4 ∈M, I6=1+2 ∈M′, I7=1+4 ∈M′, wherein
∈t=tr (∈), ∈M=M ∈ M, ∈M′=M ' ∈ M '；Then in formula (3) partial derivative of W concrete form are as follows:
Formula (4) is updated in formula (3), the expression formula of cauchy stress are as follows:
Wherein
The step 4, which calculates modulus of shearing, to be derived from formula (5):
It is obtained from formula (7) and formula (8), works as m4≠ 0,When, μi′k′≠μj′k′, contain I in strain energy equation5, I7Strain
Invariant, and m4≠ 0, due toThen the material of the modeling is shown respectively when shear-deformable
Anisotropy meets the objective requirement of orthotropic material.
2. according to claim 1 a kind of for simulating the shear-deformable hyperelastic model of soft tissue, it is characterised in that: use
In the anisotropy and nonlinear stress strain relation of simulation biomaterial, in order to intuitively find out when shear-deformable, biology
Soft tissue shows anisotropic, linearizes to stress-strain relation, and the physical background of linearisation is: in the item of small deformation
Under part, biologic soft tissue stress-strain relation is linear.
3. according to claim 1 a kind of for simulating the shear-deformable hyperelastic model of soft tissue, it is characterised in that: institute
That states is a kind of for simulating the shear-deformable hyperelastic model of soft tissue suitable for simulation compressible material, i.e., in deformation process
In, the volume of material is variable, that is, J ≠ 1.
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| CN110991103B (en) * | 2019-11-21 | 2024-04-02 | 哈尔滨理工大学 | Method for establishing super-elastic model containing interaction of fibers and matrix |
| CN111210880B (en) * | 2019-12-30 | 2022-05-03 | 浙江大学 | Soft matter growth self-assembly method based on nonlinear force electric coupling growth model |
| CN113281172B (en) * | 2021-06-28 | 2021-12-17 | 哈尔滨理工大学 | Method for establishing anisotropic mechanical property model of tracheal tissue |
| CN113705039B (en) * | 2021-08-02 | 2023-11-03 | 南京信息工程大学 | High-fidelity lung deformation model integrated with biological characteristics and modeling method thereof |
| CN114056451B (en) * | 2021-12-16 | 2024-01-19 | 吉林大学 | Bionic foot pad with multidirectional braking stability and dynamic anti-fatigue characteristics |
| CN117497069B (en) * | 2023-10-23 | 2024-05-24 | 华中科技大学 | Construction method and device of super-elastic constitutive model of high polymer material |
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