CN113515835B - Stress-strain response calculation method of metal matrix composite under spectrum load - Google Patents

Stress-strain response calculation method of metal matrix composite under spectrum load Download PDF

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CN113515835B
CN113515835B CN202110284300.7A CN202110284300A CN113515835B CN 113515835 B CN113515835 B CN 113515835B CN 202110284300 A CN202110284300 A CN 202110284300A CN 113515835 B CN113515835 B CN 113515835B
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strain
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CN113515835A (en
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孙志刚
姚旭博
牛序铭
肖瀚民
刘炎
宋迎东
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Nanjing University of Aeronautics and Astronautics
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    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F30/00—Computer-aided design [CAD]
    • G06F30/20—Design optimisation, verification or simulation
    • C—CHEMISTRY; METALLURGY
    • C22—METALLURGY; FERROUS OR NON-FERROUS ALLOYS; TREATMENT OF ALLOYS OR NON-FERROUS METALS
    • C22F—CHANGING THE PHYSICAL STRUCTURE OF NON-FERROUS METALS AND NON-FERROUS ALLOYS
    • C22F1/00—Changing the physical structure of non-ferrous metals or alloys by heat treatment or by hot or cold working
    • C22F1/16—Changing the physical structure of non-ferrous metals or alloys by heat treatment or by hot or cold working of other metals or alloys based thereon
    • C22F1/18—High-melting or refractory metals or alloys based thereon
    • C22F1/183—High-melting or refractory metals or alloys based thereon of titanium or alloys based thereon
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    • G16C—COMPUTATIONAL CHEMISTRY; CHEMOINFORMATICS; COMPUTATIONAL MATERIALS SCIENCE
    • G16C60/00—Computational materials science, i.e. ICT specially adapted for investigating the physical or chemical properties of materials or phenomena associated with their design, synthesis, processing, characterisation or utilisation
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Abstract

The invention discloses a stress-strain response calculation method of a metal matrix composite under spectrum load, which is used for solving the stress born by a composite matrix and fibers at a crack plane of a single cell model based on the single cell model with the crack length of the matrix; dividing a matrix and fibers of the single cell model with the cracks into n units; establishing a friction sliding model containing the crack length of the matrix; determining the yield condition of the SiC/Ti composite material matrix, judging whether the matrix enters plastic yield, and solving the stress born by the composite material matrix and the fiber at the crack plane after the single cell model with the crack enters the plastic yield; when solving the yield of the non-boundary unit of the matrix, the interfacial shear stress and the stress strain of the fiber and the matrix; performing coordination judgment on the strain of the matrix and the fiber; solving the average stress strain of the composite material characteristic unit, the matrix and the fiber; and calculating the stress-strain response of the metal matrix composite material under the spectrum load based on the friction slip model containing the matrix crack length and the matrix plastic yield in the steps.

Description

Stress-strain response calculation method of metal matrix composite under spectrum load
Technical Field
The invention belongs to the technical field of materials, and particularly relates to a stress-strain response calculation method of a metal matrix composite under a spectrum load.
Background
The continuous silicon carbide fiber reinforced titanium-based composite material (Titanium Matrix Composites, hereinafter referred to as SiC/Ti) has the characteristics of high specific strength, high specific stiffness and the like, and is mainly applied to aeroengine rotor components. When the SiC/Ti structure is subjected to centrifugal stress, the SiC/Ti composite material is usually subjected to a pulling load, and in an actual service environment, the pulling load is irregular and complex, so that the fatigue performance of the SiC/Ti composite material under the spectral load is necessary to be studied. By researching the interfacial micromechanics of the metal matrix composite and analyzing the influencing factors, the deformation and fracture mechanism of the metal matrix composite can be known more deeply, and the method has important significance for researching the fatigue performance of the metal matrix composite. In order to more reliably apply the metal matrix composite material to a practical structure, it is necessary to study its stress-strain behavior. And the metal matrix composite material is used as a composite material, complex microscopic damage failure can occur under the action of spectrum load, and the stress-strain behavior of the material shows obvious nonlinearity. Therefore, the stress-strain behavior calculation method under the spectral load loading of the metal matrix composite material is necessary to develop and research, and a good foundation is laid for researching the fatigue life of the composite material in the service environment.
In the prior art, a method for calculating stress-strain behavior of a unidirectional ceramic matrix composite material under any loading and unloading is proposed by a patent CN 111400922A, namely a method for calculating stress-strain behavior of the unidirectional ceramic matrix composite material under any loading and unloading by using a friction sliding model and based on the model. The document Hysteresis loop model of unidirectional carbon fiber-reinforced ceramic matrix composites under an arbitrary cyclic load proposes a unidirectional fiber reinforced ceramic matrix composite micromechanics model based on friction theory, which realizes the simulation of the complex load history stress-strain response of the unidirectional fiber reinforced ceramic matrix composite, however, the cracking of the SiC/Ti composite metal matrix is a slow-expansion process and is not a penetration crack of the brittle matrix of the ceramic matrix composite, moreover, the metal matrix is an elastoplastic material, and plastic yield can occur in the loading process. At present, a series of SiC/Ti composite material micromechanics models are developed at home and abroad, but most of the models adopt a shear hysteresis theory to solve stress-strain response of the composite material, and the traditional shear hysteresis model only can analyze the stress-strain response under simple load and cannot analyze the stress-strain response under any loading and unloading.
In view of the foregoing, it is desirable to establish a method for efficiently calculating the stress-strain response of metal matrix composites under spectral loading.
Disclosure of Invention
In order to solve the problems in the prior art, considering non-penetrating cracks of the SiC/Ti composite material and plastic yield of a matrix, the invention provides a stress-strain response calculation method of the metal matrix composite material under a spectrum load, establishes a unidirectional fiber reinforced metal matrix composite material micro-mechanics model, and realizes simulation of stress-strain response of the SiC/Ti composite material under the spectrum load.
In order to achieve the above purpose, the present invention adopts the following technical scheme:
a stress-strain response calculation method of a metal matrix composite under a spectrum load comprises the following steps:
step one: solving the stress born by the composite material matrix and the fiber at the crack plane of the unit cell model based on the unit cell model with the matrix crack length;
step two: dividing a matrix and fibers of the single cell model with the cracks into n units;
step three: establishing a friction sliding model containing the crack length of the matrix based on the first step and the second step;
step four: determining the yield condition of the SiC/Ti composite material matrix, judging whether the matrix enters plastic yield, and solving the stress born by the composite material matrix and the fiber at the crack plane after the single cell model with the crack enters the plastic yield;
step five: based on the third step, the fourth step is to solve the interfacial shear stress and the stress strain of the fiber and the matrix when the non-boundary unit of the matrix yields;
step six: performing coordination judgment on the strain of the matrix and the fiber;
step seven: solving the average stress strain of the composite material characteristic unit, the matrix and the fiber;
step eight: and calculating the stress-strain response of the metal matrix composite material under the spectrum load based on the friction slip model containing the matrix crack length and the matrix plastic yield in the steps.
Further, the specific steps of the first step are as follows:
solving the stress born by a matrix and fibers at the crack plane of the metal matrix composite unit cell model based on the metal matrix composite unit cell model with the crack length;
assuming that the stress borne by the matrix is linear with the cross-sectional area of the matrix, when a tensile load of the magnitude sigma is applied to both ends of the composite material, the stress sigma borne by the intact matrix is at the crack plane m1 In order to achieve this, the first and second,
wherein: e (E) m Modulus of elasticity, E, of the matrix c For the modulus of elasticity of the composite material, σ represents the applied stress, according to the mixing ratio E c ＝V f E f +V m E m ，E f Is the elastic modulus of the fiber, V f ,V m The volume fractions of the fiber and the matrix, V m ＝1-V f ；r 0 For average crack length r f Is the radius of the fiber, r m The radius of the matrix is Q, and the ratio of the intact matrix at the crack plane of the matrix to the whole matrix area is the radius of the matrix;
balanced by the axial stress at the crack plane,
σπr m 2 ＝σ f1 πr f 2 +σ m1 π(r m 2 -r 0 2 )
wherein: pi is the circumference ratio, sigma f1 Is the stress born by the fiber on the crack plane of the matrix,
the stress of the fiber at the crack plane is obtained as,
wherein: p is an intermediate quantity, which is intended to simplify the formula and has no substantial meaning;
further, the specific steps of the second step are as follows:
dividing a matrix and fibers in a single cell model with a matrix crack length into n units, numbering the n units into 1,2,3, …, i, … n, i representing an i-th unit, and connecting the units by springs; wherein the unit cell length L/2 is half the distance L between two adjacent cracks, each unit cell length
Compliance c between substrates f The method comprises the following steps:
compliance c between fibers m The method comprises the following steps:
further, the specific steps of the third step are as follows:
the sliding treatment of the fiber and the matrix in the loading process is divided into a quasi-static balance process, namely a balance state and an increment state; when an external stress delta sigma is applied on the composite material, if the matrix at the crack plane yields, the matrix and the fiber bear the stress delta sigma m1 、Δσ f1 The two kinds of the materials are respectively that,
Δσ m1 ＝σ m (max)-σ m
wherein: sigma (sigma) m (max) is the maximum axial stress that the matrix can bear, σ m Is the stress borne by the externally applied load matrix;
the fibres now assuming an increasing forceIncremental force exerted on the base body->
In equilibrium, the fibers and matrix of the ith cell satisfy the equilibrium equation of force:
wherein: i represents the unit number, j represents the load step number, F f (i, j) represents the axial tension to which the fiber of the ith unit is subjected, F m (i, j) represents the axial tension to which the matrix of the ith unit is subjected, f i,j Represents interfacial shear force of the ith unit, F f (i+1, j) is an axial tension force representing the axial tension force to which the fiber of the (i+1) th unit is subjected, F m (i+1, j) represents an axial tension to which the base of the (i+1) th cell is subjected;
the incremental force Δf is applied in an incremental state, the fibers and matrix also produce a corresponding force increment, where the fibers and matrix of the ith cell satisfy the equilibrium equation of force:
ΔF f (i,j)＝Δf i,j +ΔF f (i+1,j)
ΔF m (i,j)＝-Δf i,j +ΔF m (i+1,j)
wherein: ΔF (delta F) f (i，j)，ΔF m (i, j) represents the force increment, ΔF, respectively, experienced by the fiber and matrix of the ith cell f (i+1,j)，ΔF m (i+1, j) represents the force increment, Δf, experienced by the fiber and matrix of the (i+1) th cell, respectively i,j An interfacial shear force increment for the i-th cell, wherein: Δf i,j The subscript i of (a) denotes the unit number, Δf i,j The subscript j of (1) denotes the number of load steps;
according to the displacement superposition principle, the displacement increment of the unit n:
wherein:indicating the incremental force carried by the fiber,/->Represents the incremental force borne by the matrix, Δσ represents the incremental applied stress applied to the composite, k represents the kth element, Δf k,j Represents the interfacial shear force increment, deltau, of the kth unit m (i,j)，Δu f (i, j) represents the displacement increment of the matrix and the fiber of the ith unit, deltau, respectively f (i+1,j)，Δu m (i+1, j) represents the displacement increment of the fiber and the matrix of the (i+1) th unit, respectively;
since in the incremental state the displacement produced by the elementary fibres and the matrix, which are not slipping, is equal:
Δu f (i,j)＝Δu m (i, j), i.e { units without slipping }
Solving to obtain:
for the solved ith unit interface shear force increment delta f i,j Based on the maximum shear stress criterion, once the shear stress of the unit is larger than the shear strength of the interface, the interface unit is considered to be debonded and slipped, and the debonded fiber transmits the stress to the matrix through the interface friction force; by introducing interfacial maximum friction shear force f max ＝2πr f l e τ i,max Interfacial bond strength τ ult Wherein τ i,max For the interfacial maximum friction shear stress, it is assumed that interfacial maximum friction shear stress τ i,max ＝τ ult The following slip criteria are established:
if Δf is satisfied i,j +f i,j ＞f max Will generate forward slip to increase the interfacial shear force delta f of the unit i,j ＝f max -f i,j ；
If Δf is satisfied i,j +f i,j ＜-f max Will slip in opposite directions to increase the interfacial shear force by Δf i,j ＝-f max -f i,j ；
At this time, the equation of the ith cell is deleted, and since the cell has slipped, the equation of n-1 cells is reconstructed, and Δf is solved i,j Repeatedly judging whether slipping occurs until no new unit slips, and updating the interface shearing force f i,j ＝Δf i,j +f i,j (instead of the equality relationship, assignment) and counting the number of units that slip and the slip direction; strain epsilon of matrix and fiber units m (i,j)、ε f (i, j) is:
wherein: f (f) k,j Represents interfacial shear stress of the kth unit, f k,j The subscript k of (f) denotes the unit number, f k,j The subscript j of (1) denotes the number of load steps;
the stress assumed by the matrix unit is expressed as:
wherein: sigma (sigma) m (i, j) is the matrix stress, σ, of the ith cell m (n, j) is the matrix stress of the nth cell.
Further, the specific steps of the fourth step are as follows:
the main stress of any point of the matrix is as follows:
σ 3 ＝0
wherein: sigma (sigma) 1 、σ 2 Sum sigma 3 First, second and third principal stresses, sigma, respectively, at any point in the matrix m In order to bear the stress of the externally applied load matrix,for the axial thermal residual stress of the matrix +.>Radial thermal residual stress of the substrate is respectively:
wherein: alpha f ,α m ,α c Coefficients of thermal expansion, v, of the fiber, matrix and composite material, respectively f ,υ m For poisson ratio of the respective fiber and matrix, Δt is the temperature difference, γ is an intermediate quantity, which is aimed at simplifying the formula without practical significance;
according to Von Mises yield criterion:
(σ 1 -σ 2 ) 2 +(σ 2 -σ 3 ) 2 +(σ 3 -σ 1 ) 2 ＝2σ ms
wherein: sigma (sigma) ms Solving the maximum axial stress sigma which can be born by the matrix for the yield strength of the matrix m (max) is:
if the matrix is an ideal elastoplastic material, when sigma m1 ≥σ m (max) at which the matrix is plastically yielding, the stress experienced by it will not increase, the strain being that of the fibres, at which time the crack face of the matrix is subjected to stress sigma m1 ＝σ m (max) stress σ borne by the fibre in the plane of the crack in the matrix f1 The method comprises the following steps:
further, the specific steps of the fifth step are as follows:
if sigma m (i,j)≥σ m (max) the matrix unit yields, at which time the matrix undergoes plastic flow, releasing the shear stress transmitted by the fiber through the interface to the matrix, the interfacial shear force df that needs to be released i,j The method comprises the following steps:
after the interface shear stress is released, the stress sigma born by the matrix unit m (i,j)＝σ m (max), the strain of the matrix unit follows the strain of the fibre, i.e. ε m (i,j)＝ε f (i, j) and subsequently updating the interfacial shear force f of the cell i,j ＝f i,j -df i,j Since the stresses need to be balanced, this portion of the released interfacial shear force is taken up by the following interface unit, which has an interfacial shear force of f i+1,j ＝f i+1,j +df i,j If the interfacial shear stress exceeds the load limit which can be transmitted by the interface, transmitting the rest force to the subsequent unit until the interfacial shear stress does not exceed the limit load; and so on, until all matrix units have a stress less than or equal to sigma m (max) at which the strain of the fiber and matrix unit becomes:
wherein: epsilon p (i, j) is the plastic strain of the matrix;
the stress that the fibres and matrix units are subjected to is,
wherein sigma m (i,j)，σ f (i, j) stresses borne by the matrix and the fibers, respectively.
Further, the specific steps of the step six are as follows:
when the interfacial shear force of the unit is |f i,j |<f max When the fiber and the matrix unit do not slip, the strains of the fiber and the matrix are equal, but when the fiber is unloaded to a certain time due to the unrecoverable strain generated by the plastic deformation of the matrix, the matrix is only pressed due to the plastic deformation, and the strain is inconsistent, at the moment, in order to realize the strain coordination, the fiber needs to be partially stressed through an interfaceThe force is released to the matrix, the fiber/matrix slides reversely, the released interfacial shear force is,
the interfacial shear force f of the cell is then updated i,j ＝f i,j -df i,j However, the interfacial shear stress cannot exceed the load limit that the interface can transmit, if so, the interfacial stress is equal to the maximum load that the interface can carry, and so on, until the fiber and the matrix are in strain coordination or the interface is completely slipped; the fiber and matrix unit strains and stresses are then updated:
further, the specific steps of the step seven are as follows:
solving the average stress strain of the composite material characteristic unit, the matrix and the fiber:
average strain epsilon of fiber axis f,j The method comprises the following steps:
mean stress sigma in the fiber axis direction f,j The method comprises the following steps:
σ f,j ＝E f ε f,j +E f (α c -α f )ΔT
average strain epsilon in axial direction of substrate m,j The method comprises the following steps:
matrix axial average stress sigma m,j The method comprises the following steps:
the axial strain of the composite material is equal to the axial strain epsilon of the fiber c,j The method comprises the following steps:
wherein: epsilon f (k,j),ε m (k, j) fiber and matrix strain for the kth cell, respectively, k representing cell number, j representing load step number, m representing total cell number, σ m (k, j) is the matrix stress of the kth cell.
Further, the specific steps of the step eight are as follows:
and calculating the stress-strain response of the metal matrix composite material under the spectral load based on the crack length of the belt matrix and the friction slip model of the plastic yield of the matrix.
Compared with the prior art, the invention has the following beneficial effects:
(1) The invention provides a method capable of effectively calculating stress-strain response of the SiC/Ti composite material under specific damage under the spectrum load, and can provide a theoretical basis for the research of the performance of the subsequent metal matrix composite material under the spectrum load.
(2) The invention is mainly based on a friction sliding model, considers non-penetrating cracks of a metal matrix composite material and plastic yield of a matrix, and provides a unidirectional fiber reinforced metal matrix composite material micro-mechanical model, which overcomes the defect that a traditional shear hysteresis model cannot simulate load spectrum loading and also overcomes the defect that the shear hysteresis model is used for the metal matrix composite material and does not consider plastic deformation of the matrix.
(3) The invention provides a theoretical basis for researching the fatigue life of the metal matrix composite in actual working conditions.
Drawings
FIG. 1 is a flow chart of a method of computing stress strain response micromechanics for a composite material under a spectral load;
FIG. 2 is a metal matrix composite feature;
FIG. 3 is a load spectrum;
FIG. 4 is a graph of the loading stress strain of 1500MPa spectrum for the maximum stress load at a given damage.
Detailed Description
The invention will be further illustrated with reference to examples.
The parameters used in this example are shown in the following table.
TABLE 3.1 basic Material parameters of SiC/Ti composite materials
A stress-strain response calculation method of a metal matrix composite under a spectrum load comprises the following steps:
step one: solving the stress born by the composite material matrix and the fiber at the crack plane of the unit cell model based on the unit cell model with the matrix crack length;
specifically, the specific steps of the first step are as follows:
solving the stress born by a matrix and fibers at the crack plane of the metal matrix composite unit cell model based on the metal matrix composite unit cell model with the crack length;
the invention takes the characteristic volume element containing single fiber and surrounding matrix in unidirectional silicon carbide fiber reinforced titanium-based composite material as a research object, introduces the crack length of the matrix, as shown in figure 2 (a), wherein r f Is the radius of the fiber, r m Radius of matrix, r 0 For average crack length, L represents the average crack spacing of the substrate, L d Represents the debonding length of the fiber/matrix interface;
assuming that the stress borne by the matrix is linear with the cross-sectional area of the matrix, when a tensile load of the magnitude sigma is applied to both ends of the composite material, the stress sigma borne by the intact matrix is at the crack plane m1 In order to achieve this, the first and second,
wherein: e (E) m Modulus of elasticity, E, of the matrix c For the modulus of elasticity of the composite material, σ represents the applied stress, according to the mixing ratio E c ＝V f E f +V m E m ，E f Is the elastic modulus of the fiber, V f ,V m The volume fractions of the fiber and the matrix, V m ＝1-V f ；r 0 For average crack length r f Is the radius of the fiber, r m The radius of the matrix is Q, and the ratio of the intact matrix at the crack plane of the matrix to the whole matrix area is the radius of the matrix;
balanced by the axial stress at the crack plane,
σπr m 2 ＝σ f1 πr f 2 +σ m1 π(r m 2 -r 0 2 )
wherein: pi is the circumference ratio, sigma f1 Is the stress born by the fiber on the crack plane of the matrix,
the stress of the fiber at the crack plane is obtained as,
wherein: p is an intermediate quantity, which is intended to simplify the formula and has no substantial meaning;
step two: dividing a matrix and fibers of the single cell model with the cracks into n units;
specifically, the specific steps of the second step are as follows:
as shown in FIG. 2 (b), the matrix and the fibers in the single-cell model with the matrix crack length are divided into n units, the n units are numbered sequentially as 1,2,3, …, i, … n, i represents the i-th unit, and the springs are usedTo connect the units; wherein the unit cell length L/2 is half the distance L between two adjacent cracks, each unit cell length
Compliance c between substrates f The method comprises the following steps:
compliance c between fibers m The method comprises the following steps:
step three: establishing a friction sliding model containing the crack length of the matrix based on the first step and the second step;
specifically, the specific steps of the third step are as follows:
the sliding treatment of the fiber and the matrix in the loading process is divided into a quasi-static balance process, namely a balance state and an increment state, as shown in fig. 2 (d); when an external stress delta sigma is applied on the composite material, if the matrix at the crack plane yields, the matrix and the fiber bear the stress delta sigma m1 、Δσ f1 The two kinds of the materials are respectively that,
Δσ m1 ＝σ m (max)-σ m
wherein: sigma (sigma) m (max) is the maximum axial stress that the matrix can bear, σ m Is the stress borne by the externally applied load matrix;
the fibres now assuming an increasing forceIncremental force exerted on the base body->
In equilibrium, the fibers and matrix of the ith cell satisfy the equilibrium equation of force:
wherein: i represents the unit number, j represents the load step number, F f (i, j) represents the axial tension to which the fiber of the ith unit is subjected, F m (i, j) represents the axial tension to which the matrix of the ith unit is subjected, f i,j Represents interfacial shear force of the ith unit, F f (i+1, j) is an axial tension force representing the axial tension force to which the fiber of the (i+1) th unit is subjected, F m (i+1, j) represents an axial tension to which the base of the (i+1) th cell is subjected;
the incremental force Δf is applied in an incremental state, the fibers and matrix also produce a corresponding force increment, where the fibers and matrix of the ith cell satisfy the equilibrium equation of force:
ΔF f (i,j)＝Δf i,j +ΔF f (i+1,j)
ΔF m (i,j)＝-Δf i,j +ΔF m (i+1,j)
wherein: ΔF (delta F) f (i，j)，ΔF m (i, j) represents the force increment, ΔF, respectively, experienced by the fiber and matrix of the ith cell f (i+1,j)，ΔF m (i+1, j) represents the force increment, Δf, experienced by the fiber and matrix of the (i+1) th cell, respectively i,j An interfacial shear force increment for the i-th cell, wherein: Δf i,j The subscript i of (a) denotes the unit number, Δf i,j The subscript j of (1) denotes the number of load steps;
according to the displacement superposition principle, the displacement increment of the unit n:
/>
wherein:indicating the incremental force carried by the fiber,/->Represents the incremental force borne by the matrix, Δσ represents the incremental applied stress applied to the composite, k represents the kth element, Δf k,j Represents the interfacial shear force increment, deltau, of the kth unit m (i，j)，Δu f (i, j) represents the displacement increment of the matrix and the fiber of the ith unit, deltau, respectively f (i+1,j)，Δu m (i+1, j) represents the displacement increment of the fiber and the matrix of the (i+1) th unit, respectively;
since in the incremental state the displacement produced by the elementary fibres and the matrix, which are not slipping, is equal:
Δu f (i,j)＝Δu m (i, j), i.e { units without slipping }
Solving to obtain:
for the solved ith unit interface shear force increment delta f i,j Based on the maximum shear stress criterion, once the shear stress of the unit is larger than the shear strength of the interface, the interface unit is considered to be debonded and slipped, and the debonded fiber transmits the stress to the matrix through the interface friction force; by introducing interfacial maximum friction shear force f max ＝2πr f l e τ i,max Interfacial bond strength τ ult Wherein τ i,max For the interfacial maximum friction shear stress, it is assumed that interfacial maximum friction shear stress τ i,max ＝τ ult The following slip criteria are established:
if Δf is satisfied i,j +f i,j ＞f max Will slip forward to make the unitInterfacial shear force increment Δf i,j ＝f max -f i,j ；
If Δf is satisfied i,j +f i,j ＜-f max Will slip in opposite directions to increase the interfacial shear force by Δf i,j ＝-f max -f i,j ；
At this time, the equation of the ith cell is deleted, and since the cell has slipped, the equation of n-1 cells is reconstructed, and Δf is solved i,j Repeatedly judging whether slipping occurs until no new unit slips, and updating the interface shearing force f i,j ＝Δf i,j +f i,j (instead of the equality relationship, assignment) and counting the number of units that slip and the slip direction; strain epsilon of matrix and fiber units m (i,j)、ε f (i, j) is:
wherein: f (f) k,j Represents interfacial shear stress of the kth unit, f k,j The subscript k of (f) denotes the unit number, f k,j The subscript j of (1) denotes the number of load steps;
the stress assumed by the matrix unit is expressed as:
wherein: sigma (sigma) m (i, j) is the matrix stress, σ, of the ith cell m (n, j) is the matrix stress of the nth cell.
Step four: determining the yield condition of the SiC/Ti composite material matrix, judging whether the matrix enters plastic yield, and solving the stress born by the composite material matrix and the fiber at the crack plane after the single cell model with the crack enters the plastic yield;
specifically, the specific steps of the fourth step are as follows:
the main stress of any point of the matrix is as follows:
σ 3 ＝0
wherein: sigma (sigma) 1 、σ 2 Sum sigma 3 First, second and third principal stresses, sigma, respectively, at any point in the matrix m In order to bear the stress of the externally applied load matrix,for the axial thermal residual stress of the matrix +.>Radial thermal residual stress of the substrate is respectively:
wherein: alpha f ,α m ,α c Coefficients of thermal expansion, v, of the fiber, matrix and composite material, respectively f ,υ m For poisson ratio of the respective fiber and matrix, Δt is the temperature difference, γ is an intermediate quantity, which is aimed at simplifying the formula without practical significance;
according to Von Mises yield criterion:
(σ 1 -σ 2 ) 2 +(σ 2 -σ 3 ) 2 +(σ 3 -σ 1 ) 2 ＝2σ ms
wherein: sigma (sigma) ms Solving the maximum axial stress sigma which can be born by the matrix for the yield strength of the matrix m (max) is:
/>
if the matrix is an ideal elastoplastic material, when sigma m1 ≥σ m (max) at which the matrix is plastically yielding, the stress experienced by it will not increase, the strain being that of the fibres, at which time the crack face of the matrix is subjected to stress sigma m1 ＝σ m (max) stress σ borne by the fibre in the plane of the crack in the matrix f1 The method comprises the following steps:
step five: based on the third step, the fourth step is to solve the interfacial shear stress and the stress strain of the fiber and the matrix when the non-boundary unit of the matrix yields;
specifically, the specific steps of the fifth step are as follows:
if sigma m (i,j)≥σ m (max) the matrix unit yields, at which time the matrix undergoes plastic flow, releasing the shear stress transmitted by the fiber through the interface to the matrix, the interfacial shear force df that needs to be released i,j The method comprises the following steps:
after the interface shear stress is released, the stress sigma born by the matrix unit m (i,j)＝σ m (max), the strain of the matrix unit follows the strain of the fibre, i.e. ε m (i,j)＝ε f (i, j) and subsequently updating the interfacial shear force f of the cell i,j ＝f i,j -df i,j Since the stresses need to be balanced, this portion of the released interfacial shear force is taken up by the following interface unit, which has an interfacial shear force of f i+1,j ＝f i+1,j +df i,j If the interfacial shear stress exceeds the load limit which can be transmitted by the interface, transmitting the rest force to the subsequent unit until the interfacial shear stress does not exceed the limit load; and so on, until all matrix units have a stress less than or equal to sigma m (max) at which the strain of the fiber and matrix unit becomes:
wherein: epsilon p (i, j) is the plastic strain of the matrix;
the stress that the fibres and matrix units are subjected to is,
wherein sigma m (i,j)，σ f (i, j) stresses borne by the matrix and the fibers, respectively.
Step six: performing coordination judgment on the strain of the matrix and the fiber;
specifically, the specific steps of the step six are as follows:
when the interfacial shear force of the unit is |f i,j |<f max When the fiber and matrix unit does not slip, the strain of the fiber and the matrix is equal, but in the unloading process, the unrecoverable strain of the matrix caused by plastic deformation can lead the fiber strain to be unloaded to a certain time, the matrix can only be pressed due to plastic deformation, and the strain is inconsistent, at the moment, in order to realize the strain consistency, the fiber must release a part of stress to the matrix through an interface, the fiber/matrix reversely slides, the released interfacial shear force is that,
the interfacial shear force f of the cell is then updated i,j ＝f i,j -df i,j However, the interfacial shear stress cannot exceed the load limit that the interface can transmit, if so, the interfacial stress is equal to the maximum load that the interface can carry, and so on, until the fiber and the matrix are in strain coordination or the interface is completely slipped; the fiber and matrix unit strains and stresses are then updated:
step seven: solving the average stress strain of the composite material characteristic unit, the matrix and the fiber;
specifically, the specific steps of the step seven are as follows:
solving the average stress strain of the composite material characteristic unit, the matrix and the fiber:
average strain epsilon of fiber axis f,j The method comprises the following steps:
mean stress sigma in the fiber axis direction f,j The method comprises the following steps:
σ f,j ＝E f ε f,j +E f (α c -α f )ΔT
average strain epsilon in axial direction of substrate m,j The method comprises the following steps:
matrix axial average stress sigma m,j The method comprises the following steps:
the axial strain of the composite material is equal to the axial strain epsilon of the fiber c,j The method comprises the following steps:
wherein: epsilon f (k,j),ε m (k, j) fiber and matrix strain for the kth cell, respectively, k representing cell number, j representing load step number, m representing total cell number, σ m (k, j) is the matrix stress of the kth cell.
Step eight: and calculating the stress-strain response of the metal matrix composite material under the spectrum load based on the friction slip model containing the matrix crack length and the matrix plastic yield in the steps.
The specific steps of the step eight are as follows:
specifically, the stress-strain response of the metal matrix composite under the spectrum load is calculated based on a friction slip model with the matrix crack length and the matrix plastic yield, and a flow chart of a micromechanics calculation method is shown in figure 1. The stress-strain response of a SiC/Ti composite material with specific damage under spectral load is simulated by the method. The profile of the applied load is shown in FIG. 3, and the stress-strain curve results are shown in FIG. 4.
The foregoing is only a preferred embodiment of the invention, it being noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the present invention, and such modifications and adaptations are intended to be comprehended within the scope of the invention.

Claims (9)

1. The stress-strain response calculation method of the metal matrix composite under the spectral load is characterized by comprising the following steps of:
step one: solving the stress born by the composite material matrix and the fiber at the crack plane of the unit cell model based on the unit cell model with the matrix crack length;
step two: dividing a matrix and fibers of the single cell model with the cracks into n units;
step three: establishing a friction sliding model containing the crack length of the matrix based on the first step and the second step;
step four: determining the yield condition of the SiC/Ti composite material matrix, judging whether the matrix enters plastic yield, and solving the stress born by the composite material matrix and the fiber at the crack plane after the single cell model with the crack enters the plastic yield;
step five: based on the third step, the fourth step is to solve the interfacial shear stress and the stress strain of the fiber and the matrix when the non-boundary unit of the matrix yields;
step six: performing coordination judgment on the strain of the matrix and the fiber;
step seven: solving the average stress strain of the composite material characteristic unit, the matrix and the fiber;
step eight: and calculating the stress-strain response of the metal matrix composite material under the spectrum load based on the friction slip model containing the matrix crack length and the matrix plastic yield in the steps.
2. The method of calculating stress-strain response of a metal matrix composite under a spectral load according to claim 1,
the specific steps of the first step are as follows:
solving the stress born by a matrix and fibers at the crack plane of the metal matrix composite unit cell model based on the metal matrix composite unit cell model with the crack length;
assuming that the stress borne by the matrix is linear with the cross-sectional area of the matrix, when a tensile load of the magnitude sigma is applied to both ends of the composite material, the stress sigma borne by the intact matrix is at the crack plane m1 In order to achieve this, the first and second,
wherein: e (E) m Modulus of elasticity, E, of the matrix c For the modulus of elasticity of the composite material, σ represents the applied stress, according to the mixing ratio E c ＝V f E f +V m E m ，E f Is the elastic modulus of the fiber, V f ,V m Bodies of fibres and matrix respectivelyIntegration number, V m ＝1-V f ；r 0 For average crack length r f Is the radius of the fiber, r m The radius of the matrix is Q, and the ratio of the intact matrix at the crack plane of the matrix to the whole matrix area is the radius of the matrix;
balanced by the axial stress at the crack plane,
σπr m 2 ＝σ f1 πr f 2 +σ m1 π(r m 2 -r 0 2 )
wherein: pi is the circumference ratio, sigma f1 Is the stress born by the fiber on the crack plane of the matrix,
the stress of the fiber at the crack plane is obtained as,
wherein: p is an intermediate quantity;
3. the method of calculating stress-strain response of a metal matrix composite under a spectral load according to claim 2,
the specific steps of the second step are as follows:
dividing a matrix and fibers in a single cell model with a matrix crack length into n units, numbering the n units into 1,2,3, …, i, … n, i representing an i-th unit, and connecting the units by springs; wherein the unit cell length L/2 is half the distance L between two adjacent cracks, each unit cell length
Compliance c between substrates f The method comprises the following steps:
compliance c between fibers m The method comprises the following steps:
4. the method for calculating stress-strain response of metal matrix composite according to claim 3, wherein the specific steps of the third step are as follows:
the sliding treatment of the fiber and the matrix in the loading process is divided into a quasi-static balance process, namely a balance state and an increment state; when an external stress delta sigma is applied on the composite material, if the matrix at the crack plane yields, the matrix and the fiber bear the stress delta sigma m1 、Δσ f1 The two kinds of the materials are respectively that,
Δσ m1 ＝σ m (max)-σ m
wherein: sigma (sigma) m (max) is the maximum axial stress that the matrix can bear, σ m Is the stress borne by the externally applied load matrix;
the fibres now assuming an increasing forceIncremental force exerted on the base body->
In equilibrium, the fibers and matrix of the ith cell satisfy the equilibrium equation of force:
wherein: i represents the unit number, j represents the load step number, F f (i, j) represents the axial tension to which the fiber of the ith unit is subjected, F m (i, j) represents the axial tension to which the matrix of the ith unit is subjected, f i，j Represents interfacial shear force of the ith unit, F f (i+1, j) is an axial tension force representing the axial tension force to which the fiber of the (i+1) th unit is subjected, F m (i+1, j) represents an axial tension to which the base of the (i+1) th cell is subjected;
the incremental force Δf is applied in an incremental state, the fibers and matrix also produce a corresponding force increment, where the fibers and matrix of the ith cell satisfy the equilibrium equation of force:
ΔF f (i,j)＝Δf i,j +ΔF f (i+1,j)
ΔF m (i,j)＝-Δf i,j +ΔF m (i+1,j)
wherein: ΔF (delta F) f (i,j)，ΔF m (i, j) represents the force increment, ΔF, respectively, experienced by the fiber and matrix of the ith cell f (i+1,j)，ΔF m (i+1, j) represents the force increment, Δf, experienced by the fiber and matrix of the (i+1) th cell, respectively i,j An interfacial shear force increment for the i-th cell, wherein: Δf i,j The subscript i of (a) denotes the unit number, Δf i,j The subscript j of (1) denotes the number of load steps;
according to the displacement superposition principle, the displacement increment of the unit n:
wherein the method comprises the steps of：Indicating the incremental force carried by the fiber,/->Represents the incremental force borne by the matrix, Δσ represents the incremental applied stress applied to the composite, k represents the kth element, Δf k,j Represents the interfacial shear force increment, deltau, of the kth unit m (i,j)，Δu f (i, j) represents the displacement increment of the matrix and the fiber of the ith unit, deltau, respectively f (i+1,j)，Δu m (i+1, j) represents the displacement increment of the fiber and the matrix of the (i+1) th unit, respectively;
since in the incremental state the displacement produced by the elementary fibres and the matrix, which are not slipping, is equal:
Δu f (i,j)＝Δu m (i, j), i.e { units without slipping }
Solving to obtain:
for the solved ith unit interface shear force increment delta f i,j Based on the maximum shear stress criterion, once the shear stress of the unit is larger than the shear strength of the interface, the interface unit is considered to be debonded and slipped, and the debonded fiber transmits the stress to the matrix through the interface friction force; by introducing interfacial maximum friction shear force f max ＝2πr f l e τ i,max Interfacial bond strength τ ult Wherein τ i,max For the interfacial maximum friction shear stress, it is assumed that interfacial maximum friction shear stress τ i,max ＝τ ult The following slip criteria are established:
if Δf is satisfied i,j +f i,j ＞f max Will generate forward slip to increase the interfacial shear force delta f of the unit i,j ＝f max -f i,j ；
If Δf is satisfied i,j +f i,j ＜-f max Will slip in opposite directions to increase the interfacial shear force by Δf i,j ＝-f max -f i,j ；
At this time, the equation of the ith cell is deleted, and since the cell has slipped, the equation of n-1 cells is reconstructed, and Δf is solved i,j Repeatedly judging whether slipping occurs until no new unit slips, and updating the interface shearing force f i,j ＝Δf i,j +f i,j Counting the number of units which slip and the slip direction; strain epsilon of matrix and fiber units m (i,j)、ε f (i, j) is:
wherein: f (f) k,j Represents interfacial shear stress of the kth unit, f k,j The subscript k of (f) denotes the unit number, f k,j The subscript j of (1) denotes the number of load steps;
the stress assumed by the matrix unit is expressed as:
wherein: sigma (sigma) m (i, j) is the matrix stress, σ, of the ith cell m (n, j) is the matrix stress of the nth cell.
5. The method for calculating stress-strain response of metal matrix composite material under spectral load according to claim 4, wherein the specific steps of the fourth step are:
the main stress of any point of the matrix is as follows:
σ 3 ＝0
wherein: sigma (sigma) 1 、σ 2 Sum sigma 3 First, second and third principal stresses, sigma, respectively, at any point in the matrix m In order to bear the stress of the externally applied load matrix,for the axial thermal residual stress of the matrix +.>Radial thermal residual stress of the substrate is respectively:
wherein: alpha f ,α m ,α c Coefficients of thermal expansion, v, of the fiber, matrix and composite material, respectively f ,υ m For poisson's ratio of the respective fiber and matrix, Δt is the temperature difference, γ is an intermediate quantity;
according to Von Mises yield criterion:
(σ 1 -σ 2 ) 2 +(σ 2 -σ 3 ) 2 +(σ 3 -σ 1 ) 2 ＝2σ ms
wherein: sigma (sigma) ms Solving the maximum axial stress sigma which can be born by the matrix for the yield strength of the matrix m (max) is:
if the matrix is an ideal elastoplastic material, when sigma m1 ≥σ m (max) at which the matrix is plastically yielding, the stress experienced by it will not increase, the strain being that of the fibres, at which time the crack face of the matrix is subjected to stress sigma m1 ＝σ m (max) stress σ borne by the fibre in the plane of the crack in the matrix f1 The method comprises the following steps:
6. the method of calculating stress-strain response of a metal matrix composite under a spectral load according to claim 5,
the specific steps of the fifth step are as follows:
if sigma m (i,j)≥σ m (max) the matrix unit yields, at which time the matrix undergoes plastic flow, releasing the shear stress transmitted by the fiber through the interface to the matrix, the interfacial shear force df that needs to be released i,j The method comprises the following steps:
after the interface shear stress is released, the stress sigma born by the matrix unit m (i,j)＝σ m (max), the strain of the matrix unit follows the strain of the fibre, i.e. ε m (i,j)＝ε f (i, j) and subsequently updating the interfacial shear force f of the cell i,j ＝f i,j -df i,j Since the stresses need to be balanced, this portion of the released interfacial shear force is taken up by the following interface unit, which has an interfacial shear force of f i+1,j ＝f i+1,j +df i,j If the interfacial shear stress exceeds the load limit which can be transmitted by the interface, transmitting the rest force to the subsequent unit until the interfacial shear stress does not exceed the limit load; and so on, until all matrix units have a stress less than or equal to sigma m (max) at which the strain of the fiber and matrix unit becomes:
wherein: epsilon p (i, j) is the plastic strain of the matrix;
the stress that the fibres and matrix units are subjected to is,
wherein sigma m (i,j)，σ f (i, j) stresses borne by the matrix and the fibers, respectively.
7. The method of calculating stress-strain response of a metal matrix composite under a spectral load according to claim 6,
the specific steps of the step six are as follows:
when the interfacial shear force of the unit is |f i,j |<f max When the fiber and matrix unit does not slip, the strain of the fiber and the matrix is equal, but in the unloading process, the unrecoverable strain of the matrix caused by plastic deformation can lead the fiber strain to be unloaded to a certain time, the matrix can only be pressed due to plastic deformation, and the strain is inconsistent, at the moment, in order to realize the strain consistency, the fiber must release a part of stress to the matrix through an interface, the fiber/matrix reversely slides, the released interfacial shear force is that,
the interfacial shear force f of the cell is then updated i,j ＝f i,j -df i,j However, the interfacial shear stress cannot exceed the load limit that the interface can transmit, if so, the interfacial stress is equal to the maximum load that the interface can carry, and so on, until the fiber and the matrix are in strain coordination or the interface is completely slipped; the fiber and matrix unit strains and stresses are then updated:
8. the method of calculating stress-strain response of a metal matrix composite under a spectral load according to claim 7,
the specific steps of the step seven are as follows:
solving the average stress strain of the composite material characteristic unit, the matrix and the fiber:
average strain epsilon of fiber axis f,j The method comprises the following steps:
mean stress sigma in the fiber axis direction f,j The method comprises the following steps:
σ f,j ＝E f ε f,j +E f (α c -α f )ΔT
average strain epsilon in axial direction of substrate m,j The method comprises the following steps:
matrix axial average stress sigma m,j The method comprises the following steps:
the axial strain of the composite material is equal to the axial strain epsilon of the fiber c,j The method comprises the following steps:
wherein: epsilon f (k,j),ε m (k, j) fiber and matrix strain for the kth cell, respectively, k representing cell number, j representing load step number, m representing total cell number, σ m (k, j) is the matrix stress of the kth cell.
9. The method of calculating stress-strain response of a metal matrix composite under a spectral load according to claim 8,
the specific steps of the step eight are as follows:
and calculating the stress-strain response of the metal matrix composite material under the spectral load based on the crack length of the belt matrix and the friction slip model of the plastic yield of the matrix.
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