A method and system for predicting shear strength of unsaturated soil

Abstract

The present invention discloses a method and system for predicting the shear strength of unsaturated soil, and relates to the technical field of calculating the shear strength of unsaturated soil, and comprises the following steps: obtaining multiple hydraulic paths of any point on the soil water retention curve, obtaining multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of the point in its different hydraulic histories based on the multiple hydraulic paths, wherein the hydraulic paths include moisture absorption paths and dehumidification paths, and the hydraulic history is the repeated change process of dehumidification and moisture absorption of the point from the initial state to the research state; inputting multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of any point in the soil water retention curve in its different hydraulic histories and the saturated shear strength parameters of the test soil into the modified shear strength formula, obtaining the corresponding shear strength, and completing the prediction of the shear strength of unsaturated soil. The present invention can significantly improve the accuracy of the calculation of the shear strength of unsaturated soil, and is conducive to the popularization and application of the unsaturated soil strength calculation in actual engineering.

Description

Technical Field

[0001] The invention relates to the technical field of unsaturated soil shear strength calculation, and in particular to a method and system for predicting unsaturated soil shear strength. Background Art

[0002] Accurately evaluating soil strength is an important basis for solving geotechnical engineering problems. First, due to the existence of hydraulic hysteresis effect (including capillary hysteresis effect and residual air content effect), soil with the same water content may be in different hydraulic paths, resulting in different shear strength of the soil, which is called hydraulic path dependence of shear strength. Secondly, in the natural environment, hydraulic environmental changes such as intermittent rainfall and groundwater level fluctuations will cause the soil to experience different hydraulic histories.

[0003] The shear strength characteristics of soil in the dehumidification process and the moisture absorption process are different. However, the dehumidification curve is usually used for calculation without considering the hydraulic path dependence, which will obviously overestimate the strength of the soil. Therefore, ignoring the strength analysis of the hydraulic state will lead to unsafe calculation results, which may cause serious engineering safety problems. Therefore, considering the hydraulic state of soil (hydraulic path and hydraulic history) is closely related to the influence on the shear strength of unsaturated soil, which is of great significance to improve the prediction accuracy of soil mechanical response.

[0004] Unlike saturated soils, the shear strength of unsaturated soils involves two independent stress state variables, namely, the net normal stress Matrix suction ,in, is the normal stress, u a is the pore pressure, u w is the pore water pressure. Scholars have proposed many equations that use soil-related parameters (such as soil water retention curve SWRC) to predict the shear strength of unsaturated soil under different water content conditions. The existing shear strength equations can be divided into fitting equations and prediction equations according to the nature of the equations. The prediction equation uses the shear strength parameters of saturated soil or parameters related to the behavior of unsaturated soil (such as SWRC and shear strength failure envelope, etc.) to predict the shear strength. In comparison, the predictive shear strength equation is more suitable for unsaturated shear strength calculation.

[0005] According to the dual stress variable method proposed by Fredlund and Morgenstern, in the analysis of the mechanical behavior of unsaturated soil, the net normal stress Matrix suction The effects of are treated independently, and the macroscopic mechanical behavior is described by independent stress state variables and conjugate material properties, which are expressed in a bilinear form. Based on this, the shear strength of unsaturated soil can be described as:

[0006] (1);

[0007] in, is the shear strength, As an additional internal friction angle to describe the contribution of matrix suction to the shear strength, is the effective cohesion of saturated soil, is the effective internal friction angle when the soil is saturated. belongs to the forces on the soil particle skeleton at the macroscopic scale, while It belongs to the capillary force caused by surface tension at the pore scale. Using matrix suction and net normal stress as independent stress state variables is a mixed scale and does not conform to the physical meaning of continuous medium mechanics of multiphase systems. Its non-uniqueness in different saturations also limits the practical use of the Fredlund dual stress variable method.

[0008] According to the effective stress principle proposed by Bishop, the shear strength of unsaturated soil can be established by the effective stress defined by formula (2):

[0009] (2).

[0010] Bishop will It is defined as a parameter related to saturation, with a value between 0 and 1. =1, and 0 when completely dry. The Bishop effective stress formula is simple in form and can be reduced to the Terzaghi effective stress principle when fully saturated. Substitute formula (2) into the Mohr-Coulomb failure criterion:

[0011] (3).

[0012] By describing the established saturated soil mechanics framework, the unsaturated soil shear strength expression under the modified effective stress framework can be obtained:

[0013] (4).

[0014] Scholars have proved through research that effective stress parameters It is not only a function of the saturation of unsaturated soil, but also related to factors such as the hydraulic path and hydraulic history of the soil. There is no unique definiteness between the value and saturation, which makes the Bishop method have certain limitations in research and practical applications.

[0015] Effective stress parameters The characteristics affected by hydraulic history can be described by the concept of hysteresis. Hydraulic hysteresis is common in porous media and has a non-negligible influence on the shear strength of soil. At present, most researchers only consider using one unsaturated soil shear strength parameter and ignore the hydraulic path dependence of the soil. According to the suction stress concept proposed by Lu, the shear strength of unsaturated soil is generated by the friction between soil particles, while the Terzaghi effective stress theory defines the force transmitted along the skeleton, which is transmitted through the suction stress. is used to describe the contribution of matrix suction to the effective stress.

[0016] Wei proposed the concept of intergranular stress in 2014. By introducing the soil water retention curve, the shear strength of unsaturated soil under different hydraulic paths can be characterized. When considering physical and chemical effects, the repulsive force generated by the double layer effect can be characterized by the Donnan osmotic pressure. For low plasticity soils, the double layer effect is not obvious, and the Donnan osmotic pressure can be ignored. The shear strength formula based on intergranular stress is shown in Equation (5).

[0017] (5);

[0018] in, is the current volumetric water content of the soil, is the volumetric water content at saturation under this path, is the matrix suction. According to formula (5), only one of the dehumidification path or the moisture absorption path in the soil water retention curve is considered to calculate the soil shear strength.

[0019] In summary, the traditional unsaturated soil shear strength prediction model can reflect the response law of soil shear strength under changes in water content to a certain extent, but the traditional unsaturated soil shear strength prediction model does not consider the influence of soil hydraulic path and hydraulic history on shear strength. Although the shear strength model under the intergranular stress framework can consider the influence of the current hydraulic path of the soil on the shear strength, it does not consider the influence of soil-water history on the shear strength. Therefore, the existing unsaturated soil shear strength prediction model has poor prediction accuracy when predicting shear strength. Summary of the invention

[0020] The present invention provides a method and system for predicting the shear strength of unsaturated soil, which solves the problem that the existing unsaturated soil shear strength prediction model does not consider the influence of soil water history on shear strength when predicting shear strength, resulting in poor prediction accuracy.

[0021] In a first aspect, the present invention provides a method for predicting the shear strength of unsaturated soil, comprising the following steps:

[0022] Obtain the saturated shear strength parameters and soil water retention curve of the test soil;

[0023] Obtain multiple hydraulic paths of any point on the soil water retention curve, and obtain multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of the point in its different hydraulic histories based on the multiple hydraulic paths, wherein the hydraulic path includes a moisture absorption path and a dehumidification path, and the hydraulic history is the repeated change process of moisture absorption and dehumidification of the point from the initial state to the study state;

[0024] The intergranular stress expression is modified based on the integral of multiple groups of dehumidification path curves and moisture absorption path curves with different hydraulic histories, and the modified intergranular stress expression is input into the effective stress formula to obtain the modified shear strength formula;

[0025] The integral of multiple groups of moisture absorption path curves, the integral of dehumidification path curves and the saturated shear strength parameters of the test soil at any point in the soil water retention curve in its different hydraulic history are input into the modified shear strength formula to obtain the corresponding shear strength.

[0026] Preferably, obtaining the saturated shear strength parameters of the test soil comprises the following steps:

[0027] Prepare the test soil and saturate it with vacuum to obtain a fully saturated test soil;

[0028] A quadruple direct shear test is performed on the fully saturated test soil to obtain saturated shear strength parameters, which include cohesion and internal friction angle.

[0029] Preferably, the soil water retention curve is obtained through a hydraulic hysteresis model.

[0030] Preferably, the intergranular stress expression is corrected by using multiple sets of dehumidification path curve integrals and moisture absorption path curve integrals based on different hydraulic histories, as shown in the following formula:

[0031] ;

[0032] In the formula, To correct the intergranular stress, is the normal stress, u a is the pore pressure, is the current volumetric water content of the soil, is the matrix suction, is the integral of the soil dehumidification path curve, is the integral of the soil moisture absorption path curve, is the volumetric water content at saturation under this path, It is the volumetric water content of the soil when the desorption path is converted.

[0033] Preferably, the modified intergranular stress expression is input into the effective stress formula to obtain a modified shear strength formula, which is specifically as follows:

[0034] ;

[0035] In the formula, is the shear strength, is the effective cohesion of saturated soil, is the effective internal friction angle when the soil is saturated.

[0036] In a second aspect, the present invention provides a system for predicting the shear strength of unsaturated soil, comprising:

[0037] The first acquisition module is used to obtain the saturated shear strength parameters and soil water retention curve of the test soil;

[0038] The second acquisition module is used to obtain multiple hydraulic paths of any point on the soil water retention curve, and obtain multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of the point in its different hydraulic histories based on the multiple hydraulic paths, wherein the hydraulic path includes a moisture absorption path and a dehumidification path, and the hydraulic history is a repeated change process of moisture absorption and dehumidification of the point from the initial state to the research state;

[0039] A construction module is used to correct the intergranular stress expression based on multiple sets of dehumidification path curve integrals and moisture absorption path curve integrals with different hydraulic histories, and input the corrected intergranular stress expression into the effective stress formula to obtain a corrected shear strength formula;

[0040] The prediction module is used to input the integral of multiple groups of moisture absorption path curves, the integral of dehumidification path curves and the saturated shear strength parameters of the test soil at any point in the soil water retention curve in its different hydraulic histories into the modified shear strength formula to obtain the corresponding shear strength.

[0041] In a third aspect, the present invention provides a computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor implements the above-mentioned method for predicting the shear strength of unsaturated soil when executing the program.

[0042] In a fourth aspect, the present invention provides a computer-readable storage medium, wherein the storage medium stores a computer program, and when the computer program is executed by a processor, the above-mentioned method for predicting the shear strength of unsaturated soil is implemented.

[0043] Compared with the prior art, the present invention has the following beneficial effects:

[0044] The present invention first obtains multiple hydraulic paths at any point on the soil water retention curve, and obtains the curve integrals of multiple groups of moisture absorption paths and dehumidification paths at this point in its hydraulic history. Compared with the prior art that only considers the parameters of a single hydraulic path, the present invention also obtains the relevant parameters in the hydraulic history of different hydraulic paths. The intergranular stress expression is corrected by integrating the curves of multiple dehumidification paths and moisture absorption paths with different hydraulic histories, and then the corrected intergranular stress expression is input into the effective stress formula to obtain the corrected shear strength formula. This formula not only takes into account the influence of hydraulic path dependence on the shear strength of the soil, but also takes into account the influence of hydraulic history on the shear strength of the soil. The present invention can accurately predict the shear strength of the soil under any hydraulic state when the shear strength parameters of saturated soil and the hysteresis model parameters are known, which is conducive to the promotion and application of unsaturated soil strength calculation in actual engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0045] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0046] Figure 1 It is a schematic diagram of the unsaturated soil shear strength prediction process of the present invention;

[0047] Figure 2 It is a schematic diagram of the repeated change process of dehumidification and moisture absorption in different hydraulic paths of the present invention;

[0048] Figure 3 It is a schematic diagram of the calculation method of different hydraulic paths of the present invention. DETAILED DESCRIPTION

[0049] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0050] Based on the shortcomings of the existing models, the main purpose of this invention is to propose a method for predicting the shear strength of unsaturated soil. Specifically, a modified intergranular stress model is proposed. This method makes up for the problem that the intergranular stress model cannot consider the hydraulic history of the soil, and improves the accuracy of the intergranular stress model. Figure 1 , specifically including the following steps:

[0051] Step 1: Obtain the shear strength parameters of the test soil under fully saturated state and the soil water retention curve.

[0052] The sample is prepared and vacuum saturated to obtain a fully saturated test soil. A quadruple direct shear test is performed on the fully saturated test soil to obtain the shear strength parameters of the saturated soil.

[0053] The direct shear test is an earlier method used to determine the shear strength of soil. The test is simple to operate and can directly measure the shear strength of soil on the shear plane. It is widely used in geotechnical engineering.

[0054] In this embodiment, the test uses a computer-controlled strain-type quadruple direct shear apparatus, which consists of a shear box, a sensor, a shear transmission system, and a collection instrument system. The diameter of the direct shear ring is 61.8 mm and the height is 20 mm. In order to simulate the actual natural situation of rainfall-induced landslides, the undrained quick shear test is carried out at a rate of 1.2 mm / min and a direct pressure of 50 kPa, 100 kPa, 200 kPa, and 300 kPa. During the test, the room temperature is controlled at about 25 ° C to ensure that the moisture content of the sample does not change before and after shearing.

[0055] Place the test soil with a height of 20mm between the upper shear box and the lower shear box. A permeable stone and filter paper should be placed on the top and bottom surfaces of the test soil sample for drainage. During the test, first apply the specified vertical pressure, shear velocity, horizontal thrust and shear displacement to the test soil. Apply the specified normal stress to increase the shear stress on the shear surface until the end of the test, and then you can get the displacement and shear stress relationship curve, and finally get the shear strength parameters of the saturated soil.

[0056] Through the ISVH-TRAP (Internal State Variables hysteresis model-airentrapment) hydraulic hysteresis model, which takes into account the influence of capillary hysteresis effect and residual air content effect on the dry-wet cycle of unsaturated soil, a Fortran numerical analysis program was developed based on the hydraulic hysteresis model. Under the conditions of complex hydraulic environmental changes, the water retention curve of soil under any hydraulic state can be obtained through this program.

[0057] Step 2: Obtain multiple hydraulic paths for any point on the soil water retention curve, and based on the multiple hydraulic paths, obtain multiple sets of moisture absorption path curve integrals and dehumidification path curve integrals for the point in its different hydraulic histories.

[0058] The hydraulic path includes the moisture absorption path and the dehumidification path. The hydraulic history is the repeated change process of moisture absorption and dehumidification of the point from the initial state to the studied state, that is, the repeated change process of moisture absorption and dehumidification of the point from the beginning of its formation to the stop of its study. Figure 2The hydraulic path at point a includes a dehumidification path, a hygroscopic path, a dehumidification path and a hygroscopic path.

[0059] Step 3: Construct the modified shear strength formula.

[0060] Due to the influence of hydraulic path on the mechanical behavior of unsaturated soil, the shear strength formula under the traditional effective stress framework cannot accurately describe the shear strength of unsaturated soil.

[0061] The concept of average intergranular stress can describe the influence of physical and chemical effects on the behavior of unsaturated soil. Intergranular stress refers to the unified description of capillary force and physical and chemical forces in soil particles, which can better describe the deformation and strength of soil. The intergranular stress expression of unsaturated soil proposed by Wei is:

[0062] (6);

[0063] in, is the suction stress, is the volumetric water content, is the matrix suction, It is a generalized osmotic pressure, which is a macroscopic expression of the microscopic physical and chemical forces between particles inside the soil. It can be explained by the difference between the measured pore water pressure and the true pore pressure in the soil.

[0064] Based on the fact that formula (6) only considers the influence of hydraulic path on shear strength, the present invention adds the influence of hydraulic history on shear strength. Generally speaking, when physical and chemical effects are not considered, the modified intergranular stress expression is:

[0065] (7);

[0066] in, is the integral of the dehumidification path curve, is the integral of the moisture absorption path curve, the upper limit of the integral is the volumetric water content at saturation under this path, is the volumetric water content of the soil when the dehumidification path is converted, is the current volumetric water content of the soil.

[0067] Combined with Bishop's effective stress formula The modified shear strength formula under the intergranular stress framework can be obtained by using the Mohr-Coulomb criterion:

[0068] (8);

[0069] in, is the volumetric moisture content of soil, is the matrix suction. In the direct shear test, when the normal pressure When applied directly to the specimen , when the pore pressure is at atmospheric state, u a =0.

[0070] Since the calculation method takes into account the influence of hydraulic state (including hydraulic path and hydraulic history of soil), the accuracy of hydraulic hysteresis model will directly affect the calculation effect of modified intergranular stress model. In the present invention, ISVH-TRAP model is adopted, which can not only consider the capillary hysteresis effect, but also the influence of residual air content, and can accurately predict the hydraulic path of soil, which is beneficial to the accuracy of shear strength calculation.

[0071] Step 4: Based on the known saturated soil strength parameters and soil hydraulic path, the shear strength of soil with different hydraulic paths and different hydraulic histories can be accurately predicted by equation (8). Figure 3 As shown. For example, point C in a certain hydraulic state can experience two different hydraulic paths:

[0072] (1) In the fully saturated state, A dehumidifies to B and finally absorbs moisture to C.

[0073] (2) Near saturation state (Residual air content effect) Dehumidification to B.

[0074] The calculation method of intergranular stress in the first hydraulic path is: ABC :

[0075] .

[0076] The calculation method of intergranular stress in the second hydraulic path is: A'-C :

[0077] .

[0078] It can be seen that even if the soil is in the same hydraulic state, the influence of different hydraulic histories on its shear strength cannot be ignored. The weakening law of the shear strength of the test soil is obtained by calculating multiple points.

[0079] Based on the same concept, the present invention also provides a system for predicting the shear strength of unsaturated soil, comprising a first acquisition module, a second acquisition module, a construction module and a prediction module.

[0080] The first acquisition module is used to obtain the saturated shear strength parameters of the test soil and the soil water retention curve.

[0081] The second acquisition module is used to obtain multiple hydraulic paths of any point on the soil water retention curve, and based on the multiple hydraulic paths, obtain multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of the point in its different hydraulic histories, wherein the hydraulic path includes a moisture absorption path and a dehumidification path, and the hydraulic history is the repeated change process of dehumidification and moisture absorption of the point from the initial state to the study state.

[0082] The construction module is used to correct the intergranular stress expression based on multiple sets of dehumidification path curve integrals and moisture absorption path curve integrals with different hydraulic histories, and the corrected intergranular stress expression is input into the effective stress formula to obtain the corrected shear strength formula.

[0083] The prediction module is used to input the integral of multiple groups of moisture absorption path curves, the integral of dehumidification path curves and the saturated shear strength parameters of the test soil at any point in the soil water retention curve in its different hydraulic histories into the modified shear strength formula to obtain the corresponding shear strength.

[0084] The present invention also provides a computer device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, the above-mentioned method for predicting the shear strength of unsaturated soil is implemented.

[0085] The present invention also provides a computer-readable storage medium, wherein the storage medium stores a computer program, and when the computer program is executed by a processor, the above-mentioned method for predicting the shear strength of unsaturated soil is implemented.

[0086] Although the preferred embodiments of the present invention have been described, those skilled in the art may make other changes and modifications to these embodiments once they have learned the basic creative concept. Therefore, the appended claims are intended to be interpreted as including the preferred embodiments and all changes and modifications that fall within the scope of the present invention.

[0087] Obviously, those skilled in the art may make various changes and modifications to the present invention without departing from the spirit and scope of the present invention. Thus, if these modifications and variations of the present invention fall within the scope of the claims of the present invention and their equivalents, the present invention is also intended to include these modifications and variations.

Claims

1. A method for predicting the shear strength of unsaturated soil, characterized in that: The following steps are involved: Obtain the saturated shear strength parameters and soil water retention curve of the test soil; Obtain multiple hydraulic paths of any point on the soil water retention curve, and obtain multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of the point in its different hydraulic histories based on the multiple hydraulic paths, wherein the hydraulic path includes a moisture absorption path and a dehumidification path, and the hydraulic history is the repeated change process of moisture absorption and dehumidification of the point from the initial state to the study state; The intergranular stress expression is modified based on the integral of multiple groups of dehumidification path curves and moisture absorption path curves with different hydraulic histories, and the modified intergranular stress expression is input into the effective stress formula to obtain the modified shear strength formula; Input the integral of multiple groups of moisture absorption path curves, the integral of dehumidification path curves and the saturated shear strength parameters of the test soil at any point in the soil water retention curve in its different hydraulic history into the modified shear strength formula to obtain the corresponding shear strength; The intergranular stress expression is corrected by integrating multiple groups of dehumidification path curves and hygroscopic path curves based on different hydraulic histories, as shown in the following formula: In the formula, σ" is the modified intergranular stress, σ is the normal stress, and u a is the pore pressure, θ is the current volumetric water content of the soil, ψ is the matrix suction, is the integral of the soil dehumidification path curve, is the integral of the soil moisture absorption path curve, θ s is the volumetric water content at saturation under this path, It is the volumetric water content of the soil when the desorption path is converted.

2. A method for predicting the shear strength of unsaturated soil according to claim 1, characterized in that: The method of obtaining the saturated shear strength parameter of the test soil comprises the following steps: Prepare the test soil and saturate it with vacuum to obtain a fully saturated test soil; A quadruple direct shear test is performed on the fully saturated test soil to obtain saturated shear strength parameters, which include cohesion and internal friction angle.

3. The method for predicting the shear strength of unsaturated soil according to claim 1, characterized in that: The soil water retention curve is obtained through the hydraulic hysteresis model.

4. The method for predicting the shear strength of unsaturated soil according to claim 1, characterized in that: The modified intergranular stress expression is input into the effective stress formula to obtain the modified shear strength formula, which is specifically shown as follows: In the formula, τ f is the shear strength, c′ is the effective cohesion when the soil is saturated, and φ′ is the effective internal friction angle when the soil is saturated.

5. A system for predicting the shear strength of unsaturated soil, characterized in that: include: The first acquisition module is used to obtain the saturated shear strength parameters and soil water retention curve of the test soil; The second acquisition module is used to obtain multiple hydraulic paths of any point on the soil water retention curve, and obtain multiple groups of moisture absorption path curve integrals and dehumidification path curve integrals of the point in its different hydraulic histories based on the multiple hydraulic paths, wherein the hydraulic path includes a moisture absorption path and a dehumidification path, and the hydraulic history is a repeated change process of moisture absorption and dehumidification of the point from the initial state to the research state; A construction module is used to correct the intergranular stress expression based on multiple sets of dehumidification path curve integrals and moisture absorption path curve integrals with different hydraulic histories, and input the corrected intergranular stress expression into the effective stress formula to obtain a corrected shear strength formula; A prediction module is used to input the integral of multiple groups of moisture absorption path curves, the integral of dehumidification path curves and the saturated shear strength parameters of the test soil at any point in the soil water retention curve in its different hydraulic histories into the modified shear strength formula to obtain the corresponding shear strength; The intergranular stress expression is corrected by integrating multiple groups of dehumidification path curves and hygroscopic path curves based on different hydraulic histories, as shown in the following formula: In the formula, σ″ is the modified intergranular stress, σ is the normal stress, and u a is the pore pressure, θ is the current volumetric water content of the soil, ψ is the matrix suction, is the integral of the soil dehumidification path curve, is the integral of the soil moisture absorption path curve, θ s is the volumetric water content at saturation under this path, It is the volumetric water content of the soil when the desorption path is converted.

6. A computer device, characterized in that: The method comprises a memory, a processor and a computer program stored in the memory and executable on the processor. When the processor executes the program, the method for predicting the shear strength of unsaturated soil described in any one of claims 1 to 4 is implemented.

7. A computer-readable storage medium, characterized in that: The storage medium stores a computer program, and when the computer program is executed by a processor, the method for predicting the shear strength of unsaturated soil according to any one of claims 1 to 4 is implemented.

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