1. Introduction
Cement and emulsified asphalt mortar (CA mortar) is an inorganic–organic composite material, which mainly consists of cement, emulsified asphalt, sand, and admixtures and is considered as a key material used in the construction of high-speed railway and road pavement [
1,
2,
3,
4,
5,
6]. In China, CA mortar is employed as the cushion layer and casted between the ballastless track slab and the concrete bed plate for providing leveling, load bearing and transmitting, vibration absorption, segregation and geometrical adjustment in CRTS (China Railway Track System) type I and II track structure [
7,
8].
To date, much effort has been directed towards the exploration of relevant properties for CA mortar due to its crucial role in safety and life control for railway track structures. Generally, three major categories can be identified in present research topics referring to CA mortar: (a) physical and chemical properties of fresh CA mortar [
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19], which mainly incorporate the properties of rheology, water absorption and expansibility, seepage, and expansibility; (b) mechanical properties of hardened CA mortar [
8,
20,
21,
22,
23,
24,
25,
26,
27,
28,
29,
30,
31,
32,
33,
34,
35,
36], where particular attention is chiefly paid to the issues of critical compressive strength, Young’s modulus, strain–stress relationship, strain rate effect, and temperature sensitivity; (c) investigations of the durability properties of hardened CA mortar [
37,
38,
39,
40,
41,
42,
43,
44,
45,
46,
47], which primarily aim at the behaviors of fatigue properties, creep and stress relaxation, freezing resistance, and water erosion.
Among foregoing categories of research topics, the mechanical behavior of CA mortar is recognized as paramount. In view of CA mortar normally being under the coupling effect of varied loading frequencies and temperatures during the service condition, the mechanical properties have been usually examined incorporated with strain rate effects and temperature sensitivity. Such approaches have been applied successfully to deal with relative topics, and a variety of works can be traced [
8,
21,
23,
24,
25,
26,
28,
29,
30,
31,
32,
33,
45]. In detail, several conclusions are can be drawn according to those researches: (1) the compressive strength, compressive strain, and the elastic modulus are much more sensitive than traditional Portland cement mortar and concrete to strain rate effect [
28]; (2) the peak strength, discrete dynamic Young’s modulus, and specific energy absorption increase with strain rate [
8,
21,
23,
25,
26,
31]; (3) the compressive strength and elastic modulus increase monotonically with confining pressure [
24]; (4) increasing temperature results in the decrease of mechanical properties such as resilient modulus, compressive strength, and flexural strength, the temperature sensitivity and loading rate dependence for mechanical properties with higher asphalt-to-cement content ratio (A/C) are greater than those with lower A/C [
22,
25,
32,
33,
45]; (5) the storage modulus gradually decreases with increasing temperature and decreases with the increase of A/C and water content [
29].
As a fundamental problem with respect to mechanical behaviors, the constitutive relationship of the material is regarded as the basis of the research on the performance of structural analysis. Up until now, a variety of empirical or theoretical approaches have been successfully developed and applied in modelling such behaviors for different materials [
8,
21,
22,
23,
24,
25,
35,
46,
48]. However, relevant studies for CA mortar are limited at present. From a summary standpoint, present research can be classified into two categories: (a) Empirical or semi-empirical models, where the constitutive relationship are accounted for based on the observation of the experimental results; for example, Wang et al. [
24] developed a modified Domaschuk model to characterize the full deviatoric stress–strain relationship of CA mortar in both the confined and uniaxial cases based on the experimental results of a triaxial compressive test. Further, by coupling of the effects of loading rate and temperature sensitivity, Wang et al. [
25] proposed a stress–strain model for CA mortar by using a modified Guo’s model. Although such empirical models may have the advantage of simplifying relative design/analysis process towards certain problems, the formulated equations can be only used for representing a specific observed phenomenon. (b) Theoretical analysis models mainly based on adopting the statistical damage approach [
8,
21,
23,
35]; the basic idea of this approach is to use mesoscopic elements for delineating the damaging process and failure characteristics of the macroscopic behavior of materials caused by the initiation, growth, and coalescence of micro cracks during the loading. Normally, the introduction of suitable probability density functions for the failure behavior of the mesoscopic elements plays an essential role. In terms of failure behavior, two types of formalisms for the strength criterion are used to govern the fracture of such elements, viz., the stress space or strain space. For instance, Fu et al. [
35] proposed a statistical damage constitutive model by adopting the Weibull distribution function and incorporating the Mises strength criterion, to describe the stress–strain relationships for CA mortar under monotonic uniaxial compression. Later, based on the Kelvin model, Fu et al. [
23] developed a model by employing the same distribution function to the strength criterion in terms of fracture strain. This model is able to describe the stress– strain relationship of CA mortar with the strain rate effect. Similar attempts could be also found in his work [
8] on characterizing the stress–strain relationship of CA mortar under dynamic compressive loading.
However, even above-mentioned models are capable of addressing and modelling the nonlinearity emerged in stress-strain relationship of CA mortar, they are in deficiency of describing relative stochastic properties (e.g., the variation of strain–stress relationships induced by loading for a group of identical specimens), which limit the application of CA mortar in the engineering field inevitably. Usually, nonlinearity and randomness are two essential characteristics of materials’ mechanical behaviors and could be easily observed in experiments [
47,
49,
50,
51,
52,
53,
54]. For heterogeneous material subjected to external load, internal pre-existing flaws (initial damage) will develop and further result in the appearance of nonlinearity in the stress–strain relationship. Meanwhile, due to the randomly distributed properties of the components for heterogeneous material, the initial damage and subsequent damage evolution process are also endowed with random characteristics, which finally cause the emergence of the variations in strength and the constitutive relationship. Hence, the coupling effects of the nonlinearity and randomness for certain materials will cause the fluctuations in nonlinearity and variability of relative structures’ behaviors, which are directly related to the safety of structures. In fact, the characteristic of such randomness for other materials (e.g., concrete, rock) has already attracted the attention of several intensified studies by researchers for years, and a variety of experimental investigations and stochastic constitutive models have been conducted and developed [
47,
49,
50,
51,
52,
53,
54]. With the aid of those stochastic constitutive models, the randomness emerging in a material’s mechanical behaviors can be addressed, which enables us to gain comprehensive information to characterize relative structural performances.
Precisely, CA mortar is a heterogeneous material that inevitably contains randomly distributed pre-existing flaws with assorted mechanical behaviors. During the loading phase, its stochastic response can therefore be assumed as the coactions of a number of factors, including ingredient proportions, flaw distribution, grain size, and bonding capacity. Therefore, the success of a mechanics-based design of relevant structures depends heavily on the development of appropriate constitutive models of CA mortar capable of accounting for stochastic mechanical behaviors. Unfortunately, up to now, systematic experimental data and suitable constitutive models for delineation of the stress–strain response for CA mortar incorporating the stochastic responses are still lacking, which restricts the development and verification of stochastic constitutive models for CA mortar in relevant structural analyses and safety control.
Therefore, based on the above assertions, the main objective of the current research is to experimentally investigate both the mean and variation of the mechanical properties for CA mortar under uniaxial compressive loading and further develop an analytical model which is able to effectively predict the stochastic constitutive relationship. With this objective, the remainder of this work is organized as follows. In
Section 2, the materials and experimental methods in this work are firstly introduced; then the experimental results are investigated by examining the representative mechanical properties, especially the mean and the standard variation (STD) of the stress–strain curves. In
Section 3, after a brief recall of a statistical damage model named fiber bundle-plastic chain model (BCM), an analytical model for predicting the stochastic constitutive relationship of CA mortar is developed and the effectiveness of the proposed model is also verified against experimental results and one existing model. It is then followed by
Section 4, where the transition process for CA mortar under compression and the comparisons of the constitutive relationships among CA mortar investigated in this work, CRTS-I type CA mortar and typical concrete, are analyzed and discussed. The conclusions of this work are finally given in
Section 5.