CN113125262B - Method for quickly forecasting deformation of breakable calcareous sand in loading process - Google Patents
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Abstract
The invention relates to a method for quickly forecasting deformation of breakable calcareous sand in a loading process, which is used for decoupling stress ratio and body strain calculation, and can be used for independently calculating the change of the stress ratio along with shear strain and then calculating the change of the body strain along with the shear strain without interactively calculating the stress ratio and the body strain increment in each increment step, thereby simplifying the calculation process and improving the calculation efficiency.
Description
Technical Field
The invention belongs to the field of geotechnical engineering research, and particularly relates to a rapid prediction method for deformation of breakable calcareous sand in a loading process.
Background
The calcareous sand used for sea artificial hydraulic reclamation island construction has obvious particle crushing characteristic after being loaded. For the crushable calcareous sand, a large number of triaxial test researches are carried out to analyze the stress-strain characteristics of the crushable calcareous sand, but a method for quickly forecasting the deformation of the crushable calcareous sand in the crushing process under load is lacked at present, for example, when continuous compressive strain exists in the vertical direction, the changes of the stress and the volume deformation are decoupled and forecasted. In the existing method, the stress and the volume deformation need to be interactively calculated at each time step, the calculation of the stress and the volume deformation is not decoupled, and the calculation program is more complex.
Disclosure of Invention
The invention provides a rapid forecasting method for deformation of breakable calcareous sand in a loading process, which aims to decouple and forecast the stress and the volume deformation of the breakable calcareous sand in the loading process and carry out rapid forecasting on the deformation of the breakable calcareous sand in the loading process without interactively calculating the stress and the volume deformation of a soil sample at each time step.
The invention relates to some abbreviations and symbols, the following are notes:
σ 1 and σ' 1 : vertical stress, σ ', to which the aggregate of particles is subjected' 1 For effective stress, where σ 1 And σ' 1 Taking the same value;
σ 2 and σ 3 ，σ′ 2 And σ' 3 : horizontal stress, σ, to which the assembly of particles is subjected 2 And σ 3 Is perpendicular to the direction of' 2 And σ' 3 For effective stress, where σ 2 And σ' 2 Taking the same value, σ 3 And σ' 3 Taking the same value;
ε 1 、ε 2 and ε 3 : strain and respectively stress σ 1 、σ 2 And σ 3 The directions are the same;
p': the average effective stress of the steel is measured,
q: the shear stress q is set to a value of,
η: the stress ratio eta is such that,
ε v : bulk strain, epsilon v ＝ε 1 +ε 2 +ε 3 ；
ε s : the shear strain is generated by the shear strain,
t 0 ,t 1 ,t 2 ,…,t i ,…,t n : the recorded starting time in the loading process is t 0 The time recorded later is t from small to large 1 ,t 2 ,…,t i ,…,t n Where i is more than or equal to 1 and less than or equal to n, and n +1 is the number of recorded time points;
(ε v ) i : t th i Body strain epsilon corresponding to time v ；
(ε s ) i : t th i Shear strain epsilon corresponding to time s ；
(△ε v ) i : increase in bulk strain, (. DELTA.. epsilon.) v ) i ＝(ε v ) i -(ε v ) i-1 ；
(△ε s ) i : increase in shear strain, (. DELTA.. epsilon.) s ) i ＝(ε s ) i -(ε s ) i-1 ；
M: material parameters and equal to critical stress ratio;
k 2 : a parameter;
beta: parameter and
p a : atmospheric pressure;
p′ 0 : an initial average effective stress;
ξ 1 and xi 2 : calculating a beta parameter;
α: material parameter, α ═ C 2 +1)exp(-C 1 ε s )+(C 2 -1)exp(-ε s )+1；
C 1 And C 2 : calculating an alpha parameter;
the technical scheme of the invention is as follows: a method for quickly forecasting the deformation of breakable calcareous sand in a loading process comprises the following steps:
step 1: let the vertical stress of the aggregate of sand particles be sigma 1 Stress on the horizontal plane is respectively sigma 2 And σ 3 Where σ is 2 And σ 3 Is perpendicular to the direction of the particle assembly and the strain of the particle assembly is epsilon 1 、ε 2 And ε 3 Wherein strain ε 1 、ε 2 And ε 3 Respectively in the direction of the stress sigma 1 、σ 2 And σ 3 The directions are the same; defining average effective stress p', shear stress q, stress ratio eta, and bulk strain epsilon v And shear strain epsilon s ：
ε v ＝ε 1 +ε 2 +ε 3 (4)
Step 2: calculating the stress ratio:
the stress ratio eta is calculated by the following formula:
where M is a material parameter and is equal to the critical stress ratio, k 2 Is a parameter, β is a parameter;
in the above formula p a Is atmospheric pressure, p' 0 Is the initial mean effective stress, ξ 1 And xi 2 Is a parameter;
thus resulting from shear strain epsilon s The stress ratio eta can be directly calculated;
and step 3: calculating the volume deformation:
let the start time recorded during loading be t 0 The time recorded later is t from small to large 1 ,t 2 ,…,t i ,…,t n Where 1 ≦ i ≦ n, n +1 is the number of recorded time points, let t i Body strain epsilon corresponding to time v Is (epsilon) v ) i Let t be i Shear strain epsilon corresponding to time s Is (epsilon) s ) i Let the shear strain increment (Delta epsilon) generated by the difference between two adjacent moments s ) i Equal, define the bulk strain increment ([ Delta ] [ epsilon ] v ) i And increase in shear strain (. DELTA.. di-elect cons.) s ) i ：
(△ε v ) i ＝(ε v ) i -(ε v ) i-1 (8)
(△ε s ) i ＝(ε s ) i -(ε s ) i-1 (9)
Where the strain (epsilon) is cut at every moment s ) i And increase in shear strain (DELTA ε) s ) i For a known set value, the volume strain increment (Δ ε) is calculated as follows v ) i ：
In the above formula, α is a material parameter, and a relationship of α with a change in shear strain is defined as:
α＝-(C 2 +1)exp(-C 1 ε s )+(C 2 -1)exp(-ε s )+1 (11)
in the above formula C 1 And C 2 For the material parameter, each t can be calculated from the equation (10) i (i is more than or equal to 1 and less than or equal to n) corresponding to the moment v ) i By increasing these bulk strains by an amount (. DELTA.. di-elect cons.) v ) i When added together, the body strain (epsilon) at each moment is obtained v ) i I.e., (. epsilon.) v ) i ＝(ε v ) i-1 +(△ε v ) i And the initial t is known 0 Time-of-day bulk strain (. DELTA.. di-elect cons.) v ) i Is 0.
In the step 2, calculating to obtain a stress ratio eta when the stress is horizontally effective stress sigma' 2 And σ' 3 When the stress is constant and equal, the vertical effective stress sigma 'can be obtained' 1 And average effective and shear stresses:
in step 3, when the strain is horizontal 2 And ε 3 When the two strains are equal, the vertical strain epsilon can be obtained 1 And strain in horizontal direction epsilon 3 ：
The invention has the beneficial effects that: stress ratio and body strain are calculated and decoupled, stress ratio changing along with shear strain can be calculated independently, then body strain changing along with shear strain is calculated, and stress ratio and body strain increment do not need to be calculated in each increment step in an interactive mode, so that the calculation process is simplified, and the calculation efficiency is improved. In addition, the calculation formula of the ratio of the volume strain increment and the shear strain increment given by the formula (10) along with the stress ratio is more in line with the test curve of the breakable calcareous sand compared with the existing formula.
Drawings
FIG. 1 is a schematic view of an assembly of sand particles stressed vertically and horizontally;
FIG. 2 is a graph showing the variation of stress ratio and bulk strain with shear strain;
FIG. 3 is a plot of the ratio of bulk strain increase to shear strain increase versus stress ratio.
Fig. 1. assembly of sand particles.
Detailed Description
The invention will be further described with reference to the following detailed drawings in order to make the technical means, the novel features, the attainment objects and the effects of the invention apparent.
With the aggregate 1 of sand particles shown in FIG. 1, stress σ is applied in the vertical direction 1 Is stressed horizontally by a stress 2 And σ 3 ，σ 2 And σ 3 Is vertical. Carrying out a triaxial test with a confining pressure sigma 2 ＝σ 3 And a change curve of the stress ratio with the shear strain and a change curve of the body strain with the shear strain can be drawn. The confining pressures in the three-axis test of the breakable calcareous sand are taken as 100kPa, 200kPa, 300kPa, 500kPa, 700kPa and 900kPa, and the corresponding stress ratio curve and the corresponding body strain curve are shown in FIG. 2. FIG. 3 shows σ 2 ＝σ 3 The ratio of the increase in bulk strain to the increase in shear strain at 900kPa is plotted against the stress ratio.
The invention relates to some abbreviations and symbols, the following are notes:
σ 1 and σ' 1 : vertical stress, σ ', to which the aggregate of particles is subjected' 1 For effective stress, where σ 1 And σ' 1 Take the same value
σ 2 And σ 3 ，σ′ 2 And σ' 3 : horizontal stress, σ, to which the assembly of particles is subjected 2 And σ 3 Is perpendicular to the direction of' 2 And σ' 3 For effective stress, where σ 2 And σ' 2 Taking the same value, σ 3 And σ' 3 Take the same value
ε 1 、ε 2 And ε 3 : strain and respectively stress sigma 1 、σ 2 And σ 3 Same direction
p': the average effective stress of the steel is measured,
q: the shear stress q is a stress that is applied,
eta: the stress ratio eta is such that,
ε v : bulk strain, epsilon v ＝ε 1 +ε 2 +ε 3
ε s : the shear strain is set to a value that is,
t 0 ,t 1 ,t 2 ,…,t i ,…,t n : the recorded starting time in the loading process is t 0 The time recorded later is t from small to large 1 ,t 2 ,…,t i ,…,t n Where 1 ≦ i ≦ n, n +1 is the number of recorded time points
(ε v ) i : t th i Body strain epsilon corresponding to time v
(ε s ) i : t th i Shear strain epsilon corresponding to time s
(△ε v ) i : increase in bulk strain, (. DELTA.. epsilon.) v ) i ＝(ε v ) i -(ε v ) i-1
(△ε s ) i : increase in shear strain, (. DELTA.. epsilon.) s ) i ＝(ε s ) i -(ε s ) i-1
M: material parameter equal to critical stress ratio
k 2 : parameter(s)
Beta: parameter and
p a : atmospheric pressure
p′ 0 : initial mean effective stress
ξ 1 And xi 2 : calculating beta parameter
α: material parameter,. alpha. - (C) 2 +1)exp(-C 1 ε s )+(C 2 -1)exp(-ε s )+1
C 1 And C 2 : calculating alpha parameter
The technical scheme of the invention is as follows: a method for quickly forecasting the deformation of breakable calcareous sand in a loading process comprises the following steps:
step 1: let the vertical stress of sand particle aggregate be sigma 1 Stress on the horizontal plane is respectively σ 2 And σ 3 Where σ is 2 And σ 3 Is perpendicular to the direction of the particle assembly and the strain of the particle assembly is epsilon 1 、ε 2 And ε 3 In which strain epsilon 1 、ε 2 And ε 3 Respectively in the direction of the stress sigma 1 、σ 2 And σ 3 The directions are the same; defining average effective stress p', shear stress q, stress ratio eta, and bulk strain epsilon v And shear strain epsilon s ：
ε v ＝ε 1 +ε 2 +ε 3 (4)
Step 2: calculating the stress ratio:
the stress ratio eta is calculated by the following formula:
where M is a material parameter and is equal to the critical stress ratio, k 2 As parameters, β is:
in the above formula p a Is atmospheric pressure, p' 0 Is the initial mean effective stress, ξ 1 And xi 2 Is a parameter such that the shear strain ε s The stress ratio eta can be directly calculated;
in this example, the horizontal effective stress σ' 2 And σ' 3 Constant and equal, so that the vertical effective stress sigma 'can be obtained' 1 And average effective and shear stresses:
and 3, step 3: calculating the volume deformation:
let the start time recorded during loading be t 0 The time recorded later is t from small to large 1 ,t 2 ,…,t i ,…,t n Where 1 ≦ i ≦ n, n +1 is the number of recorded time points, let t i Body strain epsilon corresponding to time v Is (epsilon) v ) i Let a t i Shear strain epsilon corresponding to time s Is (epsilon) s ) i Let the shear strain increment (Delta epsilon) generated by the difference between two adjacent moments s ) i Equal, defining the bulk strain increment (. DELTA.. di-elect cons.) v ) i And increase in shear strain (DELTA ε) s ) i ：
(△ε v ) i ＝(ε v ) i -(ε v ) i-1 (8)
(△ε s ) i ＝(ε s ) i -(ε s ) i-1 (9)
Where the strain (epsilon) is cut at every moment s ) i And increase in shear strain (DELTA ε) s ) i For a known set value, the bulk strain delta (. DELTA.. di-elect cons.) is calculated as follows v ) i ：
In the above formula, α is a material parameter, and a relationship of α with a change in shear strain is defined as:
α＝-(C 2 +1)exp(-C 1 ε s )+(C 2 -1)exp(-ε s )+1 (11)
in the above formula C 1 And C 2 For the material parameter, each t can be calculated from the equation (10) i (i is more than or equal to 1 and less than or equal to n) corresponding to the moment v ) i By increasing these bulk strains by an amount (. DELTA.. di-elect cons.) v ) i When added together, the body strain (epsilon) at each moment is obtained v ) i I.e., (. epsilon.) v ) i ＝(ε v ) i-1 +(△ε v ) i And the initial t is known 0 Time body strain (Delta epsilon) v ) i Is 0;
k 2 is a parameter, C 1 And C 2 It is calculated in the following manner,
C 2 ＝-b 1 (p′ 0 /p a )+b 2 ，a 1 、a 2 、b 1 、b 2 are all parameters.
In this example, when the strain epsilon is horizontal 2 And epsilon 3 When the two strains are equal, the vertical strain epsilon can be obtained 1 And strain in horizontal direction epsilon 3 ：
The values of the parameters are as follows:
TABLE 1 parameter values
The predicted shear stress curve and body strain curve are shown in fig. 2, and the predicted change curve of the ratio of the increment of body strain to the increment of shear strain along with the stress ratio is shown in fig. 3. In fig. 3, the prediction curve given by the conventional Cam-clay model is also given, and it can be seen that the conventional Cam-clay model does not describe the initial bending segment and the right-side hooking segment, and the prediction method given herein is more consistent with the actual experiment.
Claims (6)
1. A method for quickly forecasting the deformation of breakable calcareous sand in a loading process is characterized in that: which comprises the following steps:
step 1: let the vertical stress of the aggregate of sand particles be sigma 1 Stress on the horizontal plane is respectively sigma 2 And σ 3 Where σ is 2 And σ 3 Is perpendicular to the direction of (A), and the horizontal effective stress of the particle aggregate is σ' 2 And σ' 3 The vertical effective stress is sigma' 1 (ii) a Strain of the particle aggregate is epsilon 1 、ε 2 And epsilon 3 In which strain epsilon 1 、ε 2 And epsilon 3 Respectively in the direction of the stress sigma 1 、σ 2 And σ 3 The directions are the same; defining average effective stress p', shear stress q, stress ratio eta, and bulk strain epsilon v And shear strain epsilon s ：
ε v ＝ε 1 +ε 2 +ε 3 (4)
Step 2: calculating the stress ratio:
the stress ratio eta is calculated by the following formula:
in the above formula, M is a materialMaterial parameter equal to critical stress ratio, k 2 Is a parameter, β is a parameter;
thus resulting from shear strain epsilon s The stress ratio eta can be directly calculated;
and step 3: calculating the volume deformation:
let the start time recorded during loading be t 0 The time recorded later is t from small to large 1 ,t 2 ,…,t i ,…,t n Where 1 ≦ i ≦ n, n +1 is the number of recorded time points, let t i Body strain epsilon corresponding to time v Is (epsilon) v ) i Let a t i Shear strain epsilon corresponding to time s Is (epsilon) s ) i Let the shear strain increment (Delta epsilon) generated by two adjacent time differences s ) i Equal, define the bulk strain increment ([ Delta ] [ epsilon ] v ) i And increase in shear strain (. DELTA.. di-elect cons.) s ) i ：
(△ε v ) i ＝(ε v ) i -(ε v ) i-1 (8)
(△ε s ) i ＝(ε s ) i -(ε s ) i-1 (9)
Where the strain (epsilon) is cut at every moment s ) i And increase in shear strain (DELTA ε) s ) i For a known set value, the volume strain increment (Δ ε) is calculated as follows v ) i ：
In the above formula, α is a material parameter, and each t can be calculated from the formula (10) i Increment of body strain (delta epsilon) corresponding to time v ) i By increasing these bulk strains by an amount (. DELTA.. di-elect cons.) v ) i When added together, the body strain (epsilon) at each moment is obtained v ) i I.e., (. epsilon.) v ) i ＝(ε v ) i-1 +(△ε v ) i And the initial t is known 0 Time body strain (Delta epsilon) v ) i Is 0.
2. The method for rapidly forecasting the deformation of the breakable calcareous sand in the loading process according to claim 1, characterized in that: in the step 2, the step of the method is carried out,
in the above formula p a Is atmospheric pressure, p' 0 Is the initial mean effective stress, ξ 1 And xi 2 Is a parameter.
3. The method for rapidly forecasting the deformation of the breakable calcareous sand in the loading process according to claim 1, characterized in that: in step 3, the relationship of α with the change in shear strain is defined as α ═ C 2 +1)exp(-C 1 ε s )+(C 2 -1)exp(-ε s )+1，C 1 And C 2 Is a material parameter.
4. The method for rapidly forecasting the deformation of the breakable calcareous sand in the loading process according to claim 1, characterized in that: in step 2, calculating to obtain a stress ratio eta, and calculating to obtain an effective stress sigma 'in the horizontal direction' 2 And σ' 3 When the stress is constant and equal, the vertical effective stress sigma 'can be obtained' 1 And average effective and shear stresses:
5. the method for rapidly forecasting the deformation of the breakable calcareous sand in the loading process according to claim 1, characterized in that: in step 3, when the strain is horizontal 2 And ε 3 When the two strains are equal, the vertical strain epsilon can be obtained 1 And waterStrain in the horizontal direction epsilon 3 ：
6. The method for rapidly forecasting the deformation of the breakable calcareous sand in the loading process according to claim 3, characterized in that: c 1 And C 2 It is calculated in the following manner,
C 2 ＝-b 1 (p′ 0 /p a )+b 2 ，a 1 、a 2 、b 1 、b 2 are all parameters.
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