Advances in Civil Engineering

Volume 2019, Article ID 6016350, 11 pages

https://doi.org/10.1155/2019/6016350

## Enhancing Constitutive Models for Soils: Adding the Capability to Model Nonlinear Small Strain in Shear

Department of Civil Engineering, Aalto University, Espoo, Finland

Correspondence should be addressed to S. Seyedan; if.otlaa@nadeyes.davajdammahomdeyes

Received 30 June 2018; Accepted 30 January 2019; Published 28 April 2019

Academic Editor: Eric Lui

Copyright © 2019 S. Seyedan and W. T. Sołowski. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The deviatoric stress-deviatoric strain relationship in soils is highly nonlinear, especially in the small strain range. However, the constitutive models which aim to replicate the small strain nonlinearity are often complex and rarely used in geotechnical engineering practice. The goal of this study is to offer a simple way for updating the existing constitutive models, widely used in geotechnical practice, to take into account the small strain shear modulus changes. The study uses an existing small strain relationship to derive a yield surface. When the yield surface is introduced to an existing soil model, it enhances the model with the nonlinear deviatoric stress-deviatoric strain relationship in the small strain range. The paper also gives an example of such a model enhancement by combining the new yield surface with the Modified Cam Clay constitutive model. The validation simulations of the undrained triaxial tests on London Clay and Ham River sand with the upgraded constitutive models replicate the experiments clearly better than the base models, without any changes to existing model parameters and the core source code associated with the base model.

#### 1. Introduction

Soil shear stress-strain behaviour in small strain range is highly nonlinear and well documented in the literature of geotechnical engineering. Within the last 50 years, many researchers [1–7] have studied nonlinear shear stress-shear strain behaviour in different types of soils and proposed formulations describing the variation of shear modulus with strain in the small strain range [1, 8–12].

Despite the extensive investigations on the nonlinear shear stress-strain behaviour of soil, the most common soil constitutive models in engineering practice ignore this aspect of soil behaviour. They employ linear or simple nonlinear elasticity instead, which fails to describe the nonlinear shear stress-shear strain behaviour in the small strain range [13]. Furthermore, advanced constitutive models capable of describing nonlinear behaviour of soils in the small strain range are often complex and require special soil investigation and extra parameters beyond those related to small strain behaviour [10], which prevents engineers from applying them in routine practice.

Many advanced constitutive models, which simulate special aspects of soil behaviour, such as Barcelona basic model (BBM) [14] and constitutive models used for simulation of large strain in soils [15, 16], are based on commonly used soil constitutive models. Thus, they do not consider nonlinear shear stress-strain behaviour accurately. Therefore, there is a need for simple constitutive models capable of capturing realistic nonlinear shear stress-shear strain behaviour in the small strain range.

Benz [17] introduced a small strain overlay model which can replicate small strain behaviour of soil with only 2 extra parameters. Combining this overlay model with elastoplastic constitutive models enhances them with nonlinear shear stress-shear strain behaviour. However, the combination with the overlay small strain model requires alternation of some other aspects of elastoplastic constitutive models and may lead to thermodynamic inconsistency.

This paper updates the procedure for enhancing elastoplastic constitutive models with nonlinear shear stress-shear strain behaviour in the small strain range without changing other characteristics of the base models, first introduced in [18]. The presented algorithm preserves other features of the base constitutive model, including thermodynamic consistency. The proposed procedure can upgrade constitutive models with a constant and nonlinear shear modulus. Furthermore, implementation of the proposed algorithm typically does not require any changes of the code associated with the base model and therefore can be added to an existing code as a patch.

The paper first derives the constitutive equations for small strain shearing of a base model (i), shows the thermodynamic compatibility of that small strain shearing model (ii), gives incremental stress-strain equations for the small strain shearing and a procedure to combine the base constitutive model and small strain shearing (iii), shows the improvement in replication of laboratory tests on clayey and sandy soil by Modified Cam Clay and Mohr–Coulomb constitutive models enhanced with the proposed small strain shear nonlinear behaviour (iv), and finally discusses the required steps for enhancing other base constitutive models with the capability to model the small strain nonlinearity (v).

#### 2. Enhancing Constitutive Models with Small Strain Shear Nonlinearity

This section describes derivation of a yield surface for describing small strain shearing behaviour which can be introduced into simple constitutive models, with no other change in the constitutive behaviour of the base model. The section starts with (1) brief review of the small strain models followed by (2) the derivation of the constitutive equations for small strain shearing yield surface, (3) the introduction of hyperplasticity formulations for small strain shearing, (4) the development of the incremental stress-strain equations during small strain shearing, and (5) the coupling between the base constitutive model and the small strain one.

This section will also show an example of enriching the constitutive model with small strain nonlinearity. The base model used in this example, the Modified Cam Clay model with constant shear modulus, is a very common model in geotechnical engineering practice. It is also a base for some advanced constitutive model (e.g. BBM) and can be formulated in thermodynamically consistent fashion [19]. The same procedure can upgrade constitutive models with isotropic hardening as well as those with constant yield surface, as long as their formulation uses a constant elastic shear modulus. Upgrading constitutive models with a nonconstant shear modulus requires modification to the given procedure and is beyond the scope of this paper. Nevertheless, the paper will discuss steps required for enhancing such constitutive models in the next section.

##### 2.1. Brief Review of the Hyperbolic Models for the Small Strain Behaviour in Shear

The limited selection of models discussed here aim to alter the shear modulus in small strain range in order to capture the significant nonlinearity of the shear strain-shear stress curve. First, Hardin and Drnevich [1] proposed one of the first hyperbolic formulations:where is the secant shear modulus, represents the elastic shear modulus of soil before shearing, corresponds to the shear strain, and is the reference shear strain which is equal to the maximum shear stress divided to the elastic shear modules (). This formulation requires calibration of which is often difficult as the maximum shear stress changes with mean effective stress level [10]. In order to allow a stress independent calibration, Darendeli [8] suggested a slightly altered formulation for sand:where is a curvature parameter and is the reference shear strain parameter at which the shear modulus reduces to 50% of value.

To address both clayey and sandy soils, Correia et al. [9] and dos Santos et al. [11, 12] introduced a formulation to predict secant shear modulus in the small strain range for both clayey and sandy soils:where is the reference shear strain and equal to shear strain in which shear modulus reduces to 70% of its maximum value. This equation is only applicable in the small strain region and can replicate the nonlinear behaviour of both clayey and sandy soils. Due to that comprehensiveness, equation (3) is selected to be introduced into constitutive models.

##### 2.2. Derivation of Constitutive Equations

Shearing of soils may cause both recoverable (elastic) and irrecoverable (plastic) deformations. Therefore, its modelling requires, for example, an elastoplastic formulation. Such an elastoplastic formulation needs definitions of a hardening law, a yield surface, and a plastic potential function.

Equations (1)–(3), and similar expressions, provide nonlinear relation between shear stress and total shear strain:where is the deviatoric shear stress, is the deviatoric shear strain, and is the secant shear modulus which is a function of strain. Elastic and plastic parts are the two composing parts of , although only the elastic strain leads to stress increment:where is the elastic part of deviatoric strain and is the secant elastic shear modulus during small strain shearing. This secant elastic modulus controls soil behaviour when subjected to very small shear strain at moderate rate. Experiments, based on the measurements of the shear wave velocity with bender elements, show that in the beginning of shearing has its maximum (initial) value which changes with the density and the stress state of soil [20, 21]. During shearing, the value of the secant elastic shear modulus decreases. Combining equation (4) and (5) leads to the calculation of plastic strain, based on deviatoric shear stress level:where is the plastic part of deviatoric strain. Equation (6) also provides a way to find shear stress based on the amount of plastic strain. This equation can also lead to the hardening law for small strain if and of small strain shearing are specified.

The secant shear modulus depends on the type of soil and the base constitutive model. Several formulations are available for defining . These equations describe different soils, vary in accuracy, and depend on different variables and parameters. Although it is theoretically possible to use almost any formulation for the secant shear modulus , the choice affects the complexity of the hardening law and may help to keep the base model intact. This paper uses equation (3) for describing nonlinear small strain behaviour of soil due to its simplicity and applicability for both clayey and sandy soils. Equation (3) may be recast into the deviatoric stress space:where is the reference deviatoric shear strain and is equal to deviatoric shear strain in which shear modulus reduces to 70% of its maximum value.

The base constitutive model in this study uses a constant elastic shear modulus () and ignores variation of elastic shear modulus during shearing. Taking a different elastic shear modulus for small strain shearing formulation (i.e., assuming ) is a more realistic approach but it would lead to deviation from behaviour of the base model and is against the main goal of this section. Hence, the study assumes the constant of base model for small strain shearing (). This constant value cannot represent variation of shear modulus during small strain shearing. Inevitably, it leads to erroneous calculation of elastic strain and the equation (5) changes towhere stands for error caused by constant . Combining this equation with equation (4) results in recalculation of equation (6) as

This paper introduces an internal variable aswhere is a scalar variable. This variable is negative when has higher value than and is similar to kinematic internal variable in Houlsby and Puzrin hyperplasticity framework [22], which will be investigated in next subsection. The internal variable replaces in calculation of hardening law to include errors caused by constant .

After defining and for small strain shearing, finding an equation relating to leads to calculation of the hardening law. Combining equations (4), (7), and (9) provides a function for based on :where is equal to and is . Inverse of (11) defines based on :where the value of is negative and is between 0 and .

Small strain hardening law is calculated after combining equations (4), (7), and (12):where is the shear hardening parameter. The added −1 in this equation is an extra change of the variable and helps to simplify the hyperplastic derivation of the model. Equation (13) is independent from the volumetric plastic strain. Therefore, the Modified Cam Clay yield surface remains unchanged during small strain shearing leading to preservation of the elastoplastic behaviour of the base model.

The small strain yield function separates the elastic and the elastoplastic regions. This function should remain 0 during small strain shearing and generate negative value beyond it. Defining small strain yield surface function assatisfies these criteria. Plastic potential function defines direction of plastic strain during elastoplastic behaviour. As this study assumes associated flow rule, the plastic potential function is the same as the yield surface.

##### 2.3. Hyperplastic Formulation of the Small Strain Model

This subsection employs Houlsby and Puzrin [22] hyperplastic framework to derive the small strain model. The hyperplastic derivation provides an alternative way to look at the model enhancement. It also ensures thermodynamic consistency, as the first and second laws of thermodynamics are automatically enforced in the hyperplastic framework.

The following derivation utilizes a procedure similar to derivation of Modified Cam Clay model with constant [22]. It first introduces the Gibbs free energy function for small strain model. Then, it defines the dissipation function and uses it for finding the yield equation. Finally, Ziegler’s orthogonality condition leads to recalculation of the shear small strain yield surface given in the previous subsection (). This procedure assumes that soil thermodynamic state is dependent on the internal kinematic variable () as well as the state variables (*p*, *q*). The definition of is the same as in the last subsection (equation (10)).

Gibbs free energy is one of the possible options for defining the soil energy equation. This function depends on state and internal kinematic variables and allows for the calculation of the strains and the generalized stress:where is the volumetric strain and is the generalized stress conjugate of deviatoric internal variable. The study assumes the Gibbs free energy function as follows:where is the Modified Cam Clay hardening parameter and is a function of defined as

The first two terms in equation (18) relate to the elastic behaviour of small strain model, whereas the third term adds the deviatoric internal variable, ensuring correct calculation of the deviatoric strain. The remaining part of the function introduces the hardening behaviour during the small strain shearing. The Gibbs free energy function and equations (15)–(17) provide , , and as follows:

The dissipation function is another function required in hyperplasticity. This function must be nonnegative, homogenous of first order, and depend on the changes of internal variable . The dissipation function allows for calculation ofwhere is a dissipative generalized stress, conjugate to the deviatoric internal variable change. The study chooses dissipation function aswhich satisfies requirements of the dissipation function and leads to the calculation of as

In the hyperplastic framework, the Lagrange transformation of the dissipation function leads to calculation of the yield function. This yield function has to be a function of and is found as

Ziegler’s orthogonality condition states that . Therefore, the hyperplastic yield surface of equation (26) becomes

Combining equations (22) and (27) leads to the shear small strain yield surface definitionwhich is the same as small strain yield surface of equation (14). This shows that the small strain shearing model is well defined within the hyperplastic framework. It also confirms that the proposed model satisfies the first and second laws of thermodynamics. This conservative behaviour is inherited from the base model since the small strain shearing uses the elastic law of the base model.

##### 2.4. Incremental Stress-Strain Equations

This subsection focuses on derivation of incremental stress-strain equations in the small strain range. It first recasts the derived yield surface into the general stress space. Next, it introduces a new internal variable to compensate using the elastic matrix of the Modified Cam Clay model instead of the small strain shearing tangent matrix. Finally, it uses the consistency condition to calculate plastic multiplier and the increment of stress. Combining these derived equations with the base model allows for replication of the small strain nonlinearity during shearing.

The enhancement of the Modified Cam Clay model with the small strain shear nonlinearity requires the small strain model to be formulated in the general stress space. Recasting the yield surface into the general stress space leads towhere , , , , , and are components of the stress tensor (). The differential of this yield surface isand it should remain equal to 0 during small strain shearing in order to satisfy the Prager consistency condition. In this equation, is the change of hardening parameter andwhere is change of internal soil variable. Equation (11) allows to calculate :where is the nonnegative deviatoric strain increment. This equation becomes undefined when is equal to and is thus only valid for below . Equation (32) also leads to a negative value for increment when the deviatoric strain increases and results in positive value for if the deviatoric strain decreases.

The stress increment in equation (30) is associated with change of elastic strain:where is the tangent elastic matrix for the small strain shearing and is the increment of the elastic strain. The elastic matrix of the base model is dependent on constant and should not be the same as which is constructed with . Nevertheless, to keep elastoplastic behaviour of the combined model same as the base model, the paper assumes that . Therefore, equation (33) changes towhere represents error caused by using as elastic matrix for small strain shearing. This error is similar to the error calculated in equation (8). Similarly, the new internal variable for soil iswhere is a tensorial variable similar to plastic strain. This variable replaces plastic strain in the incremental stress-strain equations and leads to the correct calculations of strain change:

The formulation uses an associated flow rule. Therefore, the increments of are computed aswhere is the scalar plastic multiplier. Combining (34), (36), and (37) gives the stress increment as

Introducing (31), (32), and (38) into (30) results in calculation of the plastic multiplier:

Calculation of the plastic multiplier allows for calculation of when the stress state satisfies the yield equation of small strain shearing.

Figure 1 shows how the yield surface evolve during small strain shearing. Before shearing starts, the small strain yield surface is a horizontal line located on the deviatoric strain axis. Increases of deviatoric strain result in negative values for . These negative result in calculation of positive values for . The change of stress , which is calculated using equations (38) and (39), has a deviatoric part () equal to . Therefore, the small strain yield surface moves upward during the small strain shearing. The movement of the small strain yield surface does not produce any volumetric plastic strain. Therefore, the volumetric part of strain change () results in the mean stress change according to the elastic law of the base model. Furthermore, application of a strain increment consisting only from the volumetric strain leads to no changes in the small strain yield surface.