A method for evaluating nonlinear stress-strain relationship of ceramic matrix composites

Abstract

The application discloses a kind of ceramic matrix composite nonlinear stress-strain relationship evaluation methods, comprising the following steps: step 1. carry out the stress-strain relationship evaluation test of ceramic matrix composite;Step 2. according to test data, the parameters of anisotropic yield function and the evolution equation parameters of anisotropic back stress increment are determined;Step 3. by relevant plastic flow rule, inelastic strain is obtained;Step 4 and step 5 determine the elastic stress-strain relationship with damage;Step 6 selects the numerical simulation method based on incremental theory: complete implicit reflection algorithm is used as stress-strain relationship updating algorithm.The nonlinear stress-strain relationship evaluation method proposed in the application introduces back stress evolution to simulate the ratchet effect of ceramic matrix composite hysteresis loop under uniaxial cyclic loading, and the model is simple and easy to use, and the required test amount is small, which saves cost.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of fatigue strength, and particularly relates to a ceramic matrix composite nonlinear stress-strain relationship evaluation method. BACKGROUND

[0002] The ceramic matrix composite has excellent comprehensive performance at room temperature and high temperature, and has better high-temperature performance and strength than traditional alloys at high temperature, and has been widely applied in the fields of aerospace, military and the like. The ceramic matrix composite will produce various forms of internal micro-damage under external load, especially cyclic load, and the macro performance is the decline of mechanical properties and nonlinear characteristics. The previous isotropic linear elastic stress-strain relationship evaluation method is no longer applicable, so it is necessary to establish a new type of nonlinear stress-strain relationship evaluation method which can reflect the anisotropic mechanical characteristics of the composite material.

[0003] At present, for the stress-strain relationship evaluation method of the ceramic matrix composite, scholars at home and abroad have developed some methods based on the micromechanics, continuous medium damage mechanics or elastoplasticity theory framework, but mainly focus on the nonlinear stress-strain relationship evaluation method of the composite material under static tension. Therefore, it is of great significance to propose a ceramic matrix composite nonlinear stress-strain relationship evaluation method which can consider the evolution process of the stress-strain hysteresis loop under cyclic loading. SUMMARY

[0004] The purpose of the application is to propose a ceramic matrix composite nonlinear stress-strain relationship evaluation method under the condition of cyclic loading, the method is to improve the traditional Armstrong-Frederick model, so that the improved model can be applied in the field of anisotropic ceramic matrix composite, and can describe the mechanical nonlinear characteristics and ratchet effect of different material principal axes of the composite material; the method can use less test cost to construct a relatively simple and easy-to-use ceramic matrix composite stress-strain relationship model.

[0005] In order to achieve the above purpose, the application adopts the following technical scheme:

[0006] Step 1. Perform monotonic tension test and tension-tension cyclic loading and unloading test of the ceramic matrix composite, and establish monotonic curves and loading and unloading curves in the 0° direction (along the fiber direction), the 90° direction (perpendicular to the fiber direction) and the shear direction.

[0007] Step 2. The anisotropy difference in the inelastic deformation process of each material principal direction (along the fiber direction, perpendicular to the fiber direction and shear direction) and the accumulation of ratcheting strain in the fiber direction when the ceramic matrix composite is under load in the plane stress state can be characterized by the kinematic hardening model.

[0008] When considering the kinematic hardening form, the yield function is generally expressed as:

[0009]

[0010] where σ i , i = 1, 2, 12 are stress components defined in the principal coordinate system of the ceramic matrix composite; α i , i = 1, 2, 12 are back stress components, representing the size of the movement of the yield surface center in the loading process; a2, a 12 are parameters reflecting the anisotropy of the mechanical characteristics of the ceramic matrix composite, and the size of the parameters represents the degree of stress contribution of each direction of the ceramic matrix material to yield, which needs to be determined through monotonic tensile test data along the fiber direction, perpendicular to the fiber direction and shear direction; is the yield stress used to determine whether to yield.

[0011] The evolution equation of the increment of the anisotropic back stress component is proposed as:

[0012]

[0013] where dα is the increment of the back stress tensor, dε p is the increment of the plastic strain tensor, is the equivalent plastic strain, c i , r i i = 1, 2, 3 are fitting parameters used to modify the anisotropic evolution law of the increment of the back stress component in each principal direction.

[0014] Step 3. The plastic strain can be obtained by the relevant plastic flow rule;

[0015]

[0016] where dε p is the increment of the plastic strain tensor, and dλ is the proportional coefficient.

[0017] Step 4. It can be found from the curve obtained from the cyclic loading and unloading test that the secant modulus of the hysteresis loop of the ceramic matrix composite decreases with the increase of the loading and unloading stress amplitude, so the damage variable of the material principal direction is defined as:

[0018]

[0019] Wherein, E1', E'2, G1'2 are initial modulus of ceramic matrix composite in each main direction; E1, E2, G 12 The secant modulus of the hysteresis loop varies with the stress level.

[0020] Step 5. Elastic strain and stress relationship

[0021] Delta sigma equals S (d) colon delta epsilon e

[0022]

[0023] Wherein S (d) is the damage stiffness matrix of the composite material, v1'2, v' 21 The Poisson's ratio of the material.

[0024] Step 6. Select the numerical simulation method based on the increment theory: the fully implicit backtracking algorithm; first, the algorithm assumes that the strain state at the beginning of the current increment step is all in the elastic state, an elastic trial stress is updated, and the plastic strain, back stress, equivalent plastic strain of the current increment step are the same as those at the end of the previous increment step; then, the current value is brought into the yield equation, if the calculation result satisfies the error value, then the current increment step is considered as elastic loading, the stress at the end of the increment step is the elastic trial stress, and the plastic strain, back stress, equivalent plastic strain remain unchanged; if the calculation result does not satisfy the yield equation, the trial stress needs to be corrected, and the plastic strain, back stress, equivalent plastic strain of the current increment step are updated until the result satisfies the yield function.

[0025] The above algorithm can be written as the following equation group, and the nonlinear equation group can be solved by Newton-Raphson method.

[0026]

[0027]

[0028]

[0029]

[0030]

[0031] In the formula, The elastic trial stress, i, j=1, 2, 12.

[0032] The ceramic matrix composite nonlinear stress-strain relationship evaluation method provided by the application has the following advantages:

[0033] 1. The evolution equation of anisotropic back stress increment proposed in the present application can simulate the monotonic loading curve and cyclic stress-strain curve of anisotropic ceramic matrix composite in the 1 direction (fiber direction), 2 direction (perpendicular to the fiber direction) and shear direction, and these curves in different directions often have large differences in morphology.

[0034] 2. The nonlinear stress-strain relationship evaluation method proposed in the present application introduces back stress to simulate the ratcheting effect of the hysteresis loop of ceramic matrix composite under uniaxial cyclic loading, and the model is simple and easy to use, and the required test amount is small, saving the cost. BRIEF DESCRIPTION OF DRAWINGS

[0035] Figure 1 is a 0° specimen monotonic tensile curve;

[0036] Figure 2 is a 90° specimen monotonic tensile curve;

[0037] Figure 3 is a 0° specimen shear curve;

[0038] Figure 4 is a 0° specimen cyclic loading and unloading curve;

[0039] Figure 5 is a schematic diagram of stress components of ceramic matrix composite;

[0040] Figure 6 is a flowchart of a fully implicit back-mapping algorithm; DETAILED DESCRIPTION

[0041] The specific embodiments of the present application are described in conjunction with the accompanying drawings.

[0042] Step 1. Perform monotonic tensile test and tensile-tensile cyclic loading and unloading test of ceramic matrix composite, and establish monotonic curve and loading and unloading curve in 0° direction (along fiber direction), 90° direction (perpendicular to fiber direction) and shear direction, each curve is shown in Figures 1 to 4 .

[0043] Step 2. In the plane stress state, for ceramic matrix composite under load, the anisotropic difference exhibited in the inelastic deformation process of each material main direction (along the fiber direction, perpendicular to the fiber direction and shear direction) and the accumulation of ratcheting strain in the fiber direction under cyclic loading can be characterized by the kinematic hardening model.

[0044] When considering the kinematic hardening form, the yield function is generally expressed as:

[0045]

[0046] where σ ii = 1, 2, 12 are stress components defined in the material principal coordinate system, see Figure 5 ; a i i = 1, 2, 12 are back stress components, representing the magnitude of the movement of the center of the yield surface during loading; a2, a 12 are parameters of the mechanical anisotropy of the composite material, the magnitude of the parameters indicates the degree of the contribution of the stress in each direction to the yielding, which needs to be determined by monotonic tensile test data along the fiber direction, perpendicular to the fiber direction and the shear direction; is the yield stress used to determine whether yielding occurs.

[0047] The evolution equation of the increment of the anisotropic back stress components is given as:

[0048]

[0049] where dα is the increment of the back stress tensor, dε p is the increment of the plastic strain tensor, is the equivalent plastic strain, c i , r i i = 1, 2, 3 are fitting parameters used to modify the anisotropic evolution of the increment of each component of the back stress.

[0050] Step 3. The plastic strain can be obtained by the relevant plastic flow rule;

[0051]

[0052] According to the plastic theory, the increment of the plastic work per unit volume is expressed as,

[0053]

[0054] Substituting the relevant plastic flow rule into the above equation,

[0055]

[0056] where dλ is the proportional coefficient, is the equivalent stress.

[0057] Step 4. The curve obtained from the cyclic loading and unloading test can find that the secant modulus of the hysteresis loop of the ceramic matrix composite decreases with the increase of the loading and unloading stress amplitude, so the damage variable of the material principal direction is defined as:

[0058]

[0059] where E1', E'2, G1'2 are the initial moduli of each principal direction of the ceramic matrix composite; E1, E2, G 12 are the secant moduli of the hysteresis loop that change with the stress level.

[0060] Step 5. Elastic strain and stress relationship;

[0061] Δσ = S(d): Δε e

[0062]

[0063] where S(d) is the damage stiffness matrix of the composite, v1'2, v' 21 are the Poisson's ratios of the materials.

[0064] Step 6. As shown in Figure 6 the numerical simulation method based on the incremental theory: fully implicit return mapping algorithm is selected; first, the algorithm assumes that the strain state at the beginning of the current increment step is all in the elastic state, an elastic trial stress is updated, and the plastic strain, back stress, equivalent plastic strain of the current increment step are the same as those at the end of the last increment step; then, the current value is brought into the yield equation, if the calculation result satisfies the error value, then the current increment step is considered as elastic loading, the stress at the end of the increment step is the elastic trial stress, and the plastic strain, back stress, equivalent plastic strain remain unchanged; if the calculation result does not satisfy the yield equation, the assumed trial stress needs to be corrected, and the plastic strain, back stress, equivalent plastic strain of the current increment step are updated, until the result satisfies the yield function.

[0065] The above algorithm can be written as the following equation group, and the nonlinear equation group can be solved by Newton-Raphson method.

[0066]

[0067]

[0068]

[0069]

[0070]

[0071] In the formula, is the elastic trial stress, i, j = 1, 2, 12.

Claims

1. A method for evaluating the nonlinear stress-strain relationship of ceramic matrix composites, characterized in that: Under plane stress, the anisotropy differences exhibited by ceramic matrix composites in the inelastic deformation process of each principal direction of the material and the accumulation of ratchet strain in the fiber direction under cyclic loading are characterized by a kinematic hardening model. When considering the kinematic hardening form, the yield function is generally expressed as: ； in, , The parameter is used to reflect the anisotropy of the mechanical characteristics of ceramic matrix composites. The magnitude of the parameter represents the degree of contribution of stress in each principal direction of the material to the yield. It needs to be determined by monotonic tensile test data along the fiber direction, perpendicular to the fiber direction and shear direction. These are the stress components defined in the material's principal coordinate system. The back stress component represents the magnitude of the displacement of the center of the yield surface during loading. The yield stress is used to determine whether yielding has occurred. The evolution equation for the increment of the anisotropic back stress component is as follows: ； in, The increment of the back stress tensor For the increment of the plastic strain tensor, For equivalent plastic strain, , The fitting parameters are used to correct the anisotropic evolution of the increments of each component of the back stress. The improved yield function and the evolution equation of the anisotropic back stress increment are used to fit the different stress-strain curves of each principal direction of the composite material and the ratchet strain accumulation of the hysteresis loop, so as to realize the modeling of the nonlinear stress-strain relationship of ceramic matrix composites.

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