Introduction

Constitutive models research holds significant importance in engineering. Having a thorough understanding of constitutive models allows engineers to optimize designs, predict material responses, and improve the overall performance and durability of structures. It is worth emphasizing that, in real engineering environments, materials are not solely exposed to a stress field but are also frequently exposed to various physical fields, such as temperature1, chemical2,3, and radiation4. The interaction between materials and these diverse physical fields plays a crucial role in determining their performance and durability. By understanding how materials respond to these different types of stimuli, engineers can develop more robust and reliable designs. Consequently, extensive development has been carried out on constitutive models that account for the impact of multiple physical fields. However, when multiple physical fields coexist, the influence of each physical field on the mechanical properties of the materials will also change due to the interaction between the physical fields. Therefore, when considering the influence of multiple physical fields on the mechanical properties of the materials, it is necessary to consider not only the individual effects of each physical field on the materials but also the mutual influence of multiple physical fields. For example, during the loss of high-temperature deformation stability (Portevin-Le Chatelier) in AlMg alloys, the transition from negative to positive Strain rate sensitivity (SRS) is precisely the result of the mutual influence between the temperature field and the strain rate field (a more detailed discussion will be provided later in the text).

Numerous excellent research achievements have been made in the study of constitutive models under multiple physical fields. These models have demonstrated remarkable accuracy and simplicity, depending on the materials and physical fields involved. Due to space limitations, it is not possible to discuss each constitutive model in detail, but we can summarize the methods. Common methods for establishing constitutive models that consider various physical fields include empirical models, damage mechanics models, thermodynamic models, and artificial neural network models, etc. Empirical models focus on the total strain theory, making them simple and practical, e.g. well-developed constitutive model under strain rate and temperature fields Johnson-Cook model5, Zerilli-Armstrong model6, Preston-Tonks-Wallace model7, Rusinek-Klepaczko model8. Damage mechanics models elucidate the physical meaning of constitutive model parameters, which is beneficial for considering complex mechanical mechanisms and various physical field conditions, e.g. chemical damage9, hydro-chemical damage10, radiochemical damage11, freeze-thaw damage12, Seepage damage13. Thermodynamic models are relatively complex, but they have a rigorous theoretical foundation and a long history of development in modeling coupled field constitutions14,15. Artificial neural network models represent an important trend in current research on constitutive models16,17,18,19. While there are many constitutive models that consider multiple physical fields, few of them take into account the mutual influence of multiple physical fields. This paper focuses on thermodynamic constitutive models and attempts to depict the the mutual influence of multiple physical fields on elastic-plastic behavior of materials within the thermodynamic framework.

Taking some characteristic variable of a physical field as a parameter to modify the material coefficients in the constitutive equation is a common method in thermodynamic modeling of multiple physical fields. Chaboche believes that when describing the elastic-plastic behavior of materials under multiple physical field conditions, the characteristic variables of temperature and other physical fields can be regarded as “parameters”, and the thermodynamic forces can be functions of the temperature parameter, thus considering the temperature effect within the thermodynamic framework. Following this idea, it is possible to establish a thermodynamic constitutive model that considers the temperature effect without changing the traditional framework based on internal state variables. Currently, this approach idea is widely recognized. For example, Egner14 models the influence of temperature on the elastic-plastic mechanical properties of metal materials by defining the thermodynamic force evolution equation containing the temperature parameter. Cailletaud15 proposes two kinematic hardening state variables and defines them as functions of the temperature parameter to describe the temperature effect. Shakiba20 introduces a correction function into the damage variable evolution equation caused by water erosion using the temperature parameter to describe the temperature effect on porous materials such as soil or asphalt. Laloui21 defines a dissipative potential affected by temperature as a temperature parameter and describes the temperature effect on thermal deformation of soil based on irreversible thermodynamics. Under the condition of multiple physical fields, if the temperature effect, strain rate effect, or other physical field effects are fully considered, Chaboche’s approach can be extended by defining all parameters in the thermodynamic model as functions of temperature, strain rate, or other characteristic variables of physical fields. For example, Barrett22 defines the dynamic equation of thermodynamic flow as functions of temperature and strain rate parameter to describe the influence of temperature and strain rate on the plastic mechanical properties of P91 steel under fatigue loading. Darabi23 defines nonlinear effective stress as functions of temperature and strain rate to describe the influence of temperature and strain rate on the elastic-plastic deformation of asphalt materials. Arman24 provided a modified constitutive model for Shape memory polymers(SMP) based on the concept of internal state variables and rational thermodynamics is proposed in large deformation. Taking its basis from the nonlinear hyperelasticity and viscoelasticity under the influence of temperature rates and applied stretch ratios, the model can provide a more accurate prediction of SMPs response. Subrat25 presented a damage constitutive model to characterize the rheological behavior of Electro magneto viscoelastic (EMV) materials. In this model an alternative invariant-based damage function is introduced to incorporate the damage-induced stress softening. Malki26 presented a damage constitutive model to characterize thermoelastoviscoplastic under the influence of magnetism. In this model magnetism-dependent elasto-viscoplasticity and damage for magnetic materials are introduced. Currently, the traditional coupled-field thermodynamic methods can account for the influence of physical fields on the mechanical properties of materials, but rarely consider the mutual influence among multiple physical fields. Incorporating the cross-effects of physical fields is crucial for enhancing the accuracy of coupled-field constitutive models.

When the heat equation is solved independently of the mechanics, the found temperature field becomes a parameter field for mechanical computations where the temperature is known and influences the constitutive parameters. However, it is consensus within thermodynamics that temperature is a state variable. Both of these interpretations of temperature are reasonable. So, within the existing framework of thermodynamics, these two interpretations of temperature coexist in the formulas of constitutive equations and share the same symbol. This approach to understanding temperature seems somewhat unsatisfactory, even though it can be flexibly distinguished in front of experts, it always seems confusing for newcomers to thermodynamics. Because when two different people share the same name, it always brings about some inconvenient things. Distinguishing between the two interpretations of temperature is a positive step towards making the thermodynamic framework more rigorous. However, researchers familiar with the thermodynamic framework believe that such a drastic approach to quickly familiarize newcomers with the thermodynamic framework is not worthwhile. In fact, a precise understanding of temperature is important, for example in the definition of thermodynamic forces. If temperature is treated as a parameter, the direct result is that the evolution equation of the thermodynamic force of plastic hardening becomes a function of temperature in the thermodynamic framework, as shown in Eq. (1).

$$\begin{aligned} {\Lambda _{ij}} = \frac{{\partial \Psi }}{{\partial {\varsigma _{ij}}}} = {\Lambda _{ij}}\left( {{\varsigma _{ij}},T} \right) \end{aligned}$$
(1)

Where, \({\Lambda _{ij}}\) is the conjugate thermodynamic force of the internal state variable \({\varsigma _{ij}}\). When temperature T is defined in terms of state variables, the thermodynamic force is related to both the temperature T and the internal state variable \({\varsigma _{ij}}\). As is well known, conjugation in thermodynamics is defined as the thermodynamic force \({\Lambda _{ij}}\) being only related to its dual internal state variable \({\varsigma _{ij}}\) and independent of other state variables. If the thermodynamic force \({\Lambda _{ij}}\) is also related to the temperature T, the definition equation of the thermodynamic force \({\Lambda _{ij}} = {{\partial \Psi }/ {\partial {\varsigma _{ij}}}}\) will no longer hold. Notably, the “single value” property from conjugation is the base of orthogonality approach of Ziegler.

In fact, taking temperature as parameter has achieved considering the influence of the temperature field, but it does not discuss in detail the rationality in the thermodynamic framework. This method results in temperature having a “dual” attribute of both a state variable and a parameter. The traditional parametric method results in the loss of duality between thermodynamic forces and the stress field under coupled-field conditions, thereby compromising the rigor of the thermodynamic framework. Establishing a thermodynamics-based coupled-field constitutive model while maintaining this duality is a key focus of current research.This article attempts to propose a new thermodynamic approach that considers the temperature to address the issue of temperature’s “dual” attribute.

Another issue that this paper wishes to address is the mutual influence of multiple physical fields, as shown in Fig. 1 under multi-field conditions, the mutual influence between physical fields can cause changes in the effects of each physical field on material mechanical properties. In practice, new parameters can be introduced to describe this mutual influence within Chaboche’s framework. However, it is important to consider whether these parameters can be based solely on empirical settings. This paper proposed a restriction matrix based on the Onsager reciprocal relations for the mutual influence between multiple physical fields, which gives a restriction on parameters settings.

Fig. 1
figure 1

Evolution of material mechanical properties under multi-physical field conditions.

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The model establishment is divided into two steps. In the first step, the concept of intrinsic variable of physical fields is proposed, the thermodynamic constitutive relations framework under multi-physical field is constructed, and a generalized thermodynamic constitutive model is established which is independent of material types and physical field types. The second step is degenerating the generalized thermodynamic constitutive model into specific constitutive models based on material properties and physical field characteristics. At the end of the paper, an application example is provided, which demonstrates the advantage of the thermodynamic constitutive on describing evolution of material mechanical properties under multi-physical field conditions that the positive-to-negative transition of SRS of AlMg alloys during the loss of high-temperature deformation stability within PLC regime.

Thermodynamic based constitutive model under multiple physical fields

The intrinsic variable of physical fields

As is well known, in the traditional irreversible thermodynamic framework with internal state variables, the thermodynamic state potential of the material is determined by both observable state variables and internal state variables, as shown in Eq. (2).

$$\begin{aligned} \Psi = H\left( {\varepsilon _{ij}^e,T,{\varsigma _{ij}}} \right) \end{aligned}$$
(2)

Where, \(\Psi\) is the thermodynamic state potential; \(\varepsilon _{ij}^e\) is elastic strain(observable state variable); T is temperature(observable state variable); \({\varsigma _{ij}}\) is internal state variable. Under the influence of multiple physical fields, theoretically, the thermodynamic state potential of a material depends not only on the state variables and internal state variables but also on certain characteristic variables of the physical fields (such as temperature, equivalent high-temperature holding time, etc.). Its expression is shown in Eq. (3).

$$\begin{aligned} {\Psi ^ * } = {\Psi ^ * }\left( {\varepsilon _{ij}^e,T,{\varsigma _{ij}},{a_{ij}}} \right) \end{aligned}$$
(3)

Where, \({\Psi ^ * }\) is the thermodynamic state potential of a material under the influence of multiple physical fields; \({a_{ij}}\) is the characteristic variable of the physical field. In this paper, the characteristic variable \({a_{ij}}\) is defined as the intrinsic variable of the physical fields. According to the definition of the thermodynamic state potential under multiple physical fields, the intrinsic variable is still one kind of state variable that describes the thermodynamic state potential. The intrinsic variable is an extension of the state variable, which incorporates the information of the physical field in multi-physical field conditions.

When further classifying the intrinsic variable, it should be classified as observable state variables. Physical field intrinsic variables are used to describe the characteristics of the physical field, and they are observable variables of the physical field, rather than internal state variables. Internal state variables are complementary to the strain state variables and their introduction allows for a clearer description of the characteristics of the strain field. Therefore, in terms of hierarchy, intrinsic variables correspond to the strain state variables. When classifying intrinsic variables as observable state variables, the classification of material state variables in multi-physics fields are shown in Table 1.

So far, the attributes of physical field intrinsic variables in this paper have been clarified. They belong only to observable state variables and have no other attributions.

Table 1 Classification of state variables of materials in coupled field.
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When the physical field discussed is a temperature field, temperature \({\tilde{T}}\) can be considered as an intrinsic variable of the temperature field. However, this intrinsic variable of temperature \({\tilde{T}}\) is different from the state variable T. The state variable T describes the changes in the temperature state of the material system during thermodynamic processes. The intrinsic variable \({\tilde{T}}\) describes the degree of change in material’s mechanism affected by the external temperature. The temperature state variable T describe the thermodynamic state in which the material is located. In fact, the intrinsic variable \({\tilde{T}}\) can be temperature, or other variable that can describe the influence of the temperature field on the stress field. However, the state variable T must be temperature.

The intrinsic variables of a physical field are representative variables that describe the characteristics of the field. However, the term “representative” is not precise, meaning that the representative variables of a physical field are not unique and depend on the observer. For example, in geotechnical materials, the flow of water in soil significantly affects the mechanical properties of the material. Therefore, the study often considers the flow field, with variables such as porosity, saturation, and permeating pressure used to describe its characteristics. While each variable can partially reflect the properties of the flow field, none can fully and perfectly do so. Therefore, when choosing an intrinsic variable as the representative variable of a physical field, it is usually necessary to consider omitting some secondary influencing factors. In Chapter 2 of the paper, the application of the constitutive model will provide a more detailed introduction to the selection of intrinsic variables for specific physical fields.

The general thermodynamic based constitutive model

According to the thermodynamic state potential under multiple physical fields in Eq. (3), it can be obtained:

$$\begin{aligned} {{\dot{\Psi }} ^ * } = \frac{{\partial {\Psi ^ * }}}{{\partial \varepsilon _{ij}^e}}{\dot{\varepsilon }} _{ij}^e + \frac{{\partial {\Psi ^ * }}}{{\partial T}}\dot{T} + \frac{{\partial {\Psi ^ * }}}{{\partial {\varsigma _{ij}}}}{{\dot{\varsigma }} _{ij}} + \frac{{\partial {\Psi ^ * }}}{{\partial {a_{ij}}}}{\dot{a}_{ij}} \end{aligned}$$
(4)

Substituting Eq. (4) into the first and second laws of thermodynamics yields (with detailed steps explained in Appendix A):

$$\begin{aligned} {\sigma } _{ij}= & \frac{{\partial {\Psi ^ * }}}{{\partial \varepsilon _{ij}^e}} \end{aligned}$$
(5)
$$\begin{aligned} s= & - \frac{{\partial {\Psi ^ * }}}{{\partial T}} \end{aligned}$$
(6)
$$\begin{aligned} & {\sigma } _{ij}{\dot{\varepsilon }} _{ij}^p - {\Lambda _{ij}}{{\dot{\varsigma }} _{ij}} - {A_{ij}}{\dot{a}_{ij}} \ge 0 \end{aligned}$$
(7)

Where, \({\sigma } _{ij}\) is the stress tensor; s is entropy; \({\Lambda _{ij}} = - \frac{{\partial {\Psi ^ * }}}{{\partial {\varsigma _{ij}}}}\) is the conjugate thermodynamic force of the internal state variable \({\varsigma _{ij}}\); \({A_{ij}} = - \frac{{\partial {\Psi ^ * }}}{{\partial {a_{ij}}}}\) is the conjugate thermodynamic force of the intrinsic variable \({a_{ij}}\).

It assumes that there exists a potential function \({F^*}\) that is an n-th order convex function of thermodynamic flows \({\dot{\varepsilon }} _{ij}^p\), \({{\dot{\varsigma }} _{ij}}\), \({\dot{a}_{ij}}\). When \(n > 1\), the mechanical response of the material in relation to the rate (time) can be considered27, and we have:

$$\begin{aligned} {F^*} = \frac{{\partial {F^*}}}{{\partial {\dot{\varepsilon }} _{ij}^p}}{\dot{\varepsilon }} _{ij}^p + \frac{{\partial {F^*}}}{{\partial {{{\dot{\varsigma }} }_{ij}}}}{{\dot{\varsigma }} _{ij}} + \frac{{\partial {F^*}}}{{\partial {{\dot{a}}_{ij}}}}{\dot{a}_{ij}} \ge 0 \end{aligned}$$
(8)

By comparing Eqs. (7) and (8), it can be concluded that

$$\begin{aligned} {\sigma } _{ij}= & \frac{{\partial {F^*}}}{{\partial {\dot{\varepsilon }} _{ij}^p}} \end{aligned}$$
(9)
$$\begin{aligned} {\Lambda _{ij}}= & \frac{{\partial {F^*}}}{{\partial {{{\dot{\varsigma }} }_{ij}}}} \end{aligned}$$
(10)
$$\begin{aligned} {A_{ij}}= & \frac{{\partial {F^*}}}{{\partial {{\dot{a}}_{ij}}}} \end{aligned}$$
(11)

By applying the Legendre transform, the potential function F can be obtained, which satisfies Eqs. (12) to (14). For more details about the Legendre transform, please refer to the appendix of Houlsby26.

$$\begin{aligned} {\dot{\varepsilon }} _{ij}^p= & \lambda \frac{{\partial F}}{{\partial {\sigma } _{ij}}} \end{aligned}$$
(12)
$$\begin{aligned} {{\dot{\varsigma }} _{ij}}= & \lambda \frac{{\partial F}}{{\partial {\Lambda _{ij}}}} \end{aligned}$$
(13)
$$\begin{aligned} {\dot{a}_{ij}}= & \lambda \frac{{\partial F}}{{\partial {A_{ij}}}} \end{aligned}$$
(14)

Equations (12) to (14) adopt the same orthogonality approach as Ziegler and Collins (refer to Chapter 15 of Ziegler28 or Eq. 2.15 of Collins29. Equations (12) to (14) are a more general form than the orthogonality postulate proposed by Drucker. We do not argue whether Ziegler’s assumption is correct or not, as it is often considered a stronger condition than the second law of thermodynamics and applies to a wide range of materials.

In thermodynamic processes, orthogonality can be represented in general by Eq. (15).

$$\begin{aligned} {j_i} = \sum \limits _{f = 1}^n {{L_{if}}} {X_f} \end{aligned}$$
(15)

Where, \({j_i}\) is thermodynamic flow; \({X_f}\) is a generalized thermodynamic force; \({L_{if}}\) is the phenomenon coefficient. When taking \({j_i} = {\dot{\varepsilon }} _{ij}^p\), Eq. (12) can be express as Eq. (16) in general thermodynamic form.

$$\begin{aligned} {\dot{\varepsilon }} _{ij}^p = \lambda \frac{{\partial F}}{{\partial {\sigma } _{ij}}} = \sum \limits _{f = 1}^n {{L_{if}}{X_f}} = {L_{\varvec{\sigma } \varvec{\sigma } }}{\sigma } _{ij} + {L_{\varvec{\sigma } \Lambda }}{\Lambda _{ij}} + {L_{\varvec{\sigma } A}}{A_{ij}} \end{aligned}$$
(16)

In Eq. (16), \({L_{\varvec{\sigma } \Lambda }}\) describes the influence of plastic mechanisms controlled by internal state variables on the plastic strain thermodynamic flow; \({L_{\varvec{\sigma } A}}\) describes the influence of plastic mechanisms controlled by intrinsic state variables on the plastic strain thermodynamic flow. If internal state variables are not considered and the influence of physical fields on materials is also ignored, we will have \({L_{\varvec{\sigma } \Lambda }} = 0\); \({L_{\varvec{\sigma } A}} = 0\). With \({L_{\varvec{\sigma } \Lambda }} = 0\); \({L_{\varvec{\sigma } A}} = 0\), Eq. (16) will be degenerated into the orthogonality rule of traditional continuum mechanics. If the physical field is considered, we will have \({L_{\varvec{\sigma } A}} \ne 0\).

It is worth noting that the Onsager reciprocal relations require the influences of thermodynamic forces to be mutually reciprocal, meaning that \({L_{if}} = {L_{fi}}\). Therefore, in the analysis of multiple physical fields, not only the values of \({L_{\varvec{\sigma } A}}\) need to be considered, but also the coordination of the influences of thermodynamic forces. Hence, Eq. (17) is derived.

$$\begin{aligned} \left( {\begin{array}{ll} {{\dot{\varepsilon }} _{ij}^p}\\ {{{{\dot{\varsigma }} }_{ij}}}\\ {{{\dot{a}}_{ij}}} \end{array}} \right) = \left( {\begin{array}{lll} {{L_{\varvec{\sigma } \varvec{\sigma } }}}& {{L_{\varvec{\sigma } \Lambda }}}& {{L_{\varvec{\sigma } A}}}\\ {{L_{\Lambda \varvec{\sigma } }}}& {{L_{\Lambda \Lambda }}}& {{L_{\Lambda A}}}\\ {{L_{A\varvec{\sigma } }}}& {{L_{A\Lambda }}}& {{L_{AA}}} \end{array}} \right) \left( {\begin{array}{ll} {{\sigma } _{ij}}\\ {{\Lambda _{ij}}}\\ {{A_{ij}}} \end{array}} \right) \end{aligned}$$
(17)

Where, \({L_{\varvec{\sigma } \Lambda }} = {L_{\Lambda \varvec{\sigma } }}\); \({L_{\varvec{\sigma } A}} = {L_{A\varvec{\sigma } }}\); \({L_{\Lambda A}} = {L_{A\Lambda }}\). In Eq. (17), \({L_{A\varvec{\sigma } }}\) and \({L_{A\Lambda }}\) represent the reciprocal effects between the physical field and stress field. Chaboche’s model considers temperature as a parameter to modify the hardening (softening) function. However, this method only takes into account the influence of the physical field on the thermodynamic forces (i.e., only \({L_{\varvec{\sigma } A}}\); \({L_{\Lambda A}}\), and without \({L_{A\varvec{\sigma } }}\); \({L_{A\Lambda }}\) ). It does not consider the reciprocal effects of the stress field on the physical field and the changing of the physical field’s influence on material’s non-linear dissipation. This distinction is important and sets apart the method proposed in this paper from Chaboche’s method. In this paper, the matrix in Eq. (17) is referred to as the multiple physical fields constraint matrix. Additionally, when multiple physical fields are present, Eq. (18) is expressed as follows:

$$\begin{aligned} \left( {\begin{array}{ll} {{\dot{\varepsilon }} _{ij}^p}\\ {{{{\dot{\varsigma }} }_{ij}}}\\ {{{\dot{a}}_{1ij}}}\\ {{{\dot{a}}_{2ij}}} \end{array}} \right) = \left( {\begin{array}{llll} {{L_{\varvec{\sigma } \varvec{\sigma } }}}& {{L_{\varvec{\sigma } \Lambda }}}& {{L_{\varvec{\sigma } {A_1}}}}& {{L_{\varvec{\sigma } {A_2}}}}\\ {{L_{\Lambda \varvec{\sigma } }}}& {{L_{\Lambda \Lambda }}}& {{L_{\Lambda {A_1}}}}& {{L_{\Lambda {A_2}}}}\\ {{L_{{A_1}\varvec{\sigma } }}}& {{L_{{A_1}\Lambda }}}& {{L_{{A_1}{A_1}}}}& {{L_{{A_1}{A_2}}}}\\ {{L_{{A_2}\varvec{\sigma } }}}& {{L_{{A_2}\Lambda }}}& {{L_{{A_2}{A_1}}}}& {{L_{{A_2}{A_2}}}} \end{array}} \right) \left( {\begin{array}{ll} {{\sigma } _{ij}}\\ {{\Lambda _{ij}}}\\ {{A_{1ij}}}\\ {{A_{2ij}}} \end{array}} \right) \end{aligned}$$
(18)

\({a_{1ij}}\) and \({a_{2ij}}\) are two kinds of intrinsic variables. \({L_{{A_1}{A_2}}} = {L_{{A_2}{A_1}}}\) represents the mutual influence between these two physical fields. In previous multi-physics field constitutive modeling, such as the modeling of shape memory alloys, the influence of field 1 on the stress field was only described by internal state variables, while the influence of field 1 on field 2 and the influence of the stress field on field 2 were ignored. However, when the influence of field 2 on the stress field is described by introducing internal state variables, the final description of the multi-physics field becomes inaccurate due to the neglect of the influence of field 2 by field 1 and the stress field. This difference is an important distinction between the method presented in this paper and the traditional thermodynamic multi-physics field modeling method.

Assuming that the potential function is the same as the yield surface and using the associated flow rule, the abstract form of the yield condition can be represented by Eq. (19). In this discussion, we temporarily exclude the variations of special yield functions caused by material properties (such as volumetric yield of soils, shear dilatancy, etc.) and the non-associated flow rules resulting from them. The specific expression of the yield condition will be discussed in detail in Chapter 3.

$$\begin{aligned} f = F\left( {{\sigma } _{ij}\left( {\varepsilon _{ij}^p} \right) ,{\Lambda _{ij}}\left( {{\varsigma _{ij}}} \right) ,{A_{ij}}\left( {{a_{ij}}} \right) } \right) = 0 \end{aligned}$$
(19)

By substituting Eqs. 13 and 14 into the differential form of the yield condition Eq. 19, we obtain Eq. 20.

$$\begin{aligned} df = \frac{{\partial f}}{{\partial \mathbf{{\varvec{\varvec{\sigma }}}}}}:d\mathbf{{\varvec{\varvec{\sigma }}}} + \frac{{\partial f}}{{\partial {\Lambda _{ij}}}}d{\Lambda _{ij}} + \frac{{\partial f}}{{\partial {A_{ij}}}}d{A_{ij}}\mathrm{{ = }}\frac{{\partial f}}{{\partial \mathbf{{\varvec{\varvec{\sigma }} }}}}:d\mathbf{{\varvec{\varvec{\sigma }} }} + \frac{{\partial f}}{{\partial {\Lambda _{ij}}}}\frac{{\partial {\Lambda _{ij}}}}{{\partial {\varsigma _{ij}}}}\frac{{\partial F}}{{\partial {\Lambda _{ij}}}}\lambda + \frac{{\partial f}}{{\partial {A_{ij}}}}\frac{{\partial {A_{ij}}}}{{\partial {a_{ij}}}}\frac{{\partial F}}{{\partial {A_{ij}}}}\lambda \mathrm{{ = }}0 \end{aligned}$$
(20)

Let

$$\begin{aligned} \beta = \frac{1}{{\frac{{\partial f}}{{\partial {\Lambda _{ij}}}}\frac{{\partial {\Lambda _{ij}}}}{{\partial {\varsigma _{ij}}}}\frac{{\partial F}}{{\partial {\Lambda _{ij}}}} + \frac{{\partial f}}{{\partial {A_{ij}}}}\frac{{\partial {A_{ij}}}}{{\partial {a_{ij}}}}\frac{{\partial F}}{{\partial {A_{ij}}}}}} \end{aligned}$$
(21)

Form Eq. (20), we obtain Eq. (22).

$$\begin{aligned} \lambda = - \beta \frac{{\partial f}}{{\partial \mathbf{{\varvec{\sigma } }}}}:d\mathbf{{\varvec{\sigma } }} \end{aligned}$$
(22)

Substituting Eq. (22) into Eq. (12) yields

$$\begin{aligned} d\varepsilon _{ij}^p = - \beta \frac{{\partial f}}{{\partial \mathbf{{\varvec{\sigma } }}}}:d\mathbf{{\varvec{\sigma } }}\frac{{\partial F}}{{\partial {\sigma } _{ij}}} \end{aligned}$$
(23)

Assuming that elasticity and plasticity are uncoupled, we can obtain Eq. (24).

$$\begin{aligned} d{\varvec{\varepsilon }} = d{{\varvec{\varepsilon }}^e} + d{{\varvec{\varepsilon }}^p} \end{aligned}$$
(24)

Substituting Eq. (23) into Eq. (24) yields

$$\begin{aligned} d{\varvec{\varepsilon }} = d{{\varvec{\varepsilon }}^e} + d{{\varvec{\varepsilon }}^p} = \left( {{{\left( {\frac{{{\partial ^2}{\Psi ^*}}}{{\partial {{\varvec{\varepsilon }}^e}^2}}} \right) }^{ - 1}} - \beta \frac{{\partial F}}{{\partial {{\varvec{\sigma } }}}}\frac{{\partial f}}{{\partial {{\varvec{\sigma } }}}}} \right) :d{{\varvec{\sigma } }} \end{aligned}$$
(25)

Let \(\bar{\delta }\) represent the flexibility tensor and \(\hat{\delta }\) represent the stiffness tensor.

$$\begin{aligned} \hat{\delta } = {\hat{\delta } ^{ - 1}} = {\left( {\frac{{{\partial ^2}{\Psi ^*}}}{{\partial {{\varvec{\varepsilon }}^e}^2}}} \right) ^{ - 1}} \end{aligned}$$
(26)

Form Eq. (25), we obtain Eq. (27).

$$\begin{aligned} d{\varvec{\varepsilon }} = \left( {\hat{\delta } - \beta \frac{{\partial F}}{{\partial \mathbf{{\varvec{\sigma } }}}}\frac{{\partial f}}{{\partial \mathbf{{\varvec{\sigma } }}}}} \right) :d\mathbf{{\varvec{\sigma } }} \end{aligned}$$
(27)

Let

$$\begin{aligned} {\mathbf{{Q}}_\mathbf{{1}}}= & \frac{{\partial F}}{{\partial \mathbf{{\varvec{\sigma } }}}} \end{aligned}$$
(28)
$$\begin{aligned} {\mathbf{{Q}}_\mathbf{{2}}}= & \frac{{\partial f}}{{\partial \mathbf{{\varvec{\sigma } }}}} \end{aligned}$$
(29)

Form Eq. (27), we obtain Eq. (30).

$$\begin{aligned} d{\varvec{\varepsilon }} = \left( {\hat{\delta } - \beta {\mathbf{{Q}}_\mathbf{{1}}}{\mathbf{{Q}}_\mathbf{{2}}}} \right) :d{\varvec{\sigma }} \end{aligned}$$
(30)

Form Eq. (30), we obtain Eq. (31). Please refer to Appendix B for a detailed derivation.

$$\begin{aligned} d{\varvec{\sigma } } = \hat{\delta }:\left[ {d{\varvec{\varepsilon }} - \beta {\mathbf{{Q}}_\mathbf{{1}}}{{\left( {1 + \beta {\mathbf{{Q}}_\mathbf{{2}}}: \hat{\delta }:{\mathbf{{Q}}_\mathbf{{1}}}} \right) }^{ - 1}}{\mathbf{{Q}}_\mathbf{{2}}}: \hat{\delta }:d{\varvec{\varepsilon }}} \right] \end{aligned}$$
(31)

Equation (31) is a general model that does not specify the properties of the material or the type of physical field. Once the material and physical field are determined, Eq. (31) can be degraded into a specific thermomechanical constitutive model. Chapter 3 will discuss the degradation of the general model in detail.

Discussion

Assuming that the damage field is a special type of physical field, Eq. (31) will degenerate into the quasi damage model. So, what are the differences between Eq. (31) and traditional damage mechanics? This section will discuss it.

In damage mechanics, the basic thermodynamic flows are given by Eqs. (32)–(34)30.

$$\begin{aligned} {\dot{\varepsilon }} _{ij}^p= & \frac{3}{2}\frac{{{\tilde{\sigma }} _{ij}^D}}{{{{\left( {{\tilde{\sigma }} _{ij}^D} \right) }_{eq}}\left( {1 - D} \right) }}{\dot{\lambda }} \end{aligned}$$
(32)
$$\begin{aligned} \dot{r}= & {\dot{\lambda }}\end{aligned}$$
(33)
$$\begin{aligned} \dot{D}= & \frac{{Y{H_{\left( {p - pD} \right) }}}}{{S\left( {1 - D} \right) }}{\dot{\lambda }} \end{aligned}$$
(34)

Where, \(\varvec{\sigma } _{ij}^D = {\sigma } _{ij} - {1/3}{\varvec{\sigma } _{kk}}{\delta _{ij}}\); \({{\tilde{\sigma }}} _{ij} = {{{\sigma } _{ij}}/ {\left( {1 - D} \right) }}\); r is cumulative plastic strain; D is damage variable; \(Y = {1/2}{E_{ijkl}}\varepsilon _{ij}^e\varepsilon _{kl}^e\); \({H_{\left( {p - pD} \right) }}\) is step function; S is damage parameter.

\({\dot{\varepsilon }} _{ij}^p\) is related to stress and damage state variables in Eqs. (32) and (34), as shown in Eq. (35). \(\dot{D}\) is related to damage thermodynamic force Y and plastic strain \(\varepsilon _{ij}^p\) ( In fact, \(\dot{D}\) is Related to Y and p. However, \(p = {\left( {{2/ 3}{\dot{\varepsilon }} _{ij}^p{\dot{\varepsilon }} _{ij}^p} \right) ^{{1/2}}}\) ), as shown in Eq. (36).

$$\begin{aligned} {\dot{\varepsilon }} _{ij}^p= & {\dot{\lambda }} {g_1}\left( {{\sigma } _{ij},D} \right) = {\dot{\lambda }} \frac{{\partial F}}{{\partial {\sigma } _{ij}}} \end{aligned}$$
(35)
$$\begin{aligned} \dot{D}= & {\dot{\lambda }} {g_2}\left( {Y,\varepsilon _{ij}^p} \right) = {\dot{\lambda }} \frac{{\partial F}}{{\partial Y}} \end{aligned}$$
(36)

Although damage mechanics can describe the cross-effects of thermodynamic processes, such as defining the strain thermodynamic flow as a function of stress and damage state variables, or the damage thermodynamic flow as a function of the thermodynamic force and plastic strain state variables, it does not satisfy the strict equality of cross-effects required by Onsager’s reciprocal relations. This is because Onsager’s reciprocal relations require the cross-effects of thermodynamic processes to be realized through thermodynamic forces, rather than through state variables. However, in damage mechanics, the cross-effects are realized through state variables, as shown in Fig. 2.

Fig. 2
figure 2

The difference between damage mechanics and thermodynamics in implementing the mutual interaction of physical fields.The blue represents traditional damage mechanics, while green represents the thermodynamic constitutive model proposed in this paper.

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Currently, there are numerous studies on coupled field constitutive models based on damage mechanics. These models define damage variables for physical fields and modify damage mechanics. However, it has been observed that the damage variables of physical fields still have shortcomings in obeying the Onsager reciprocal relation, similar to traditional damage mechanics. The Onsager reciprocal relation requires that cross effects be expressed through thermodynamic forces, with both cross effects having an equal influence on each other. This necessitates the symmetric nature of the multiple physical fields constraint matrix. However, this aspect has not received sufficient attention in multi-field constitutive models based on damage mechanics. Neglecting the mutual influence of physical fields in damage mechanics does not cause the model to collapse under single-coupled field conditions. However, it leads to unreasonable results under multiple physical field conditions. In Chapter 3, we will describe in detail the unreasonable results that occur when the mutual influence of multiple physical fields is ignored. We will also apply the thermodynamic generalized model proposed in this paper to verify its effectiveness.

Degenerate applications and validation of generalized thermodynamic constitutive model under multiple physical fields

In alloy materials, the stability of deformation can be lost when the strain exceeds a critical level within a specific range of strain rate and temperature. This instability of deformation is evident through a serrated stress-strain curve observed during tensile testing30. The range of strain rate and temperature that causes this instability is referred to as the Dynamic strain aging (DSA) regime. This phenomenon, characterized by the loss of deformation stability, is known as the Portevin-le chatelier (PLC) effect. At low temperatures, the material exhibits a negative Strain Rate Sensitivity (SRS), meaning that the plastic flow stress decreases with increasing strain rate within the PLC regime. However, at high temperatures, the SRS transitions from negative to positive, indicating that the plastic flow stress increases with increasing strain rate within the PLC regime.

In traditional thermal damage constitutive models, it is necessary to define “destructive” thermal damage and “healing” thermal damage to achieve positive and negative changes in SRS. While the concept of “healing” thermal damage can mathematically describe changes in material mechanical properties, there is still controversy surrounding its concept. Lemaitre explicitly states that damage is irreversible in damage mechanics. However, the transition from positive to negative represents the reverse change of damage. The sudden change in the nature of damage variables caused by “healing” damage lacks a physical mechanism. The thermodynamic constitutive model proposed in this paper can effectively describe positive and negative changes in SRS without the need for defining “healing” damage. Within the thermodynamic framework, the mechanism behind these changes lies in the interaction between multiple physical fields, specifically the interaction between the temperature field and the strain rate field.

Intrinsic variables of the temperature-strain rate physical field

Temperature field

Due to dynamic diffusion, solute atoms tend to temporarily arrest at obstacles, creating additional barriers to dislocation motion. This micro-mechanism plays a significant role in the occurrence of the PLC effect. The temperature field influences the efficiency of these additional barriers at different temperatures, which is the micro-mechanism for the PLC temperature effect. Metrics such as mean center, weighted mean center, and median center can effectively measure the distribution of these barriers. These metrics can be used as intrinsic variables to quantify clustering efficiency and represent the PLC temperature effect. However, accurately measuring these microphysical quantities still poses challenges at the microscale. One approach to overcome this challenge is to choose macroscopic variables that impact microphysical quantities and indirectly measure the efficiency of solute atom arresting. Temperature, as the most fundamental and simple macroscopic variable in the temperature field, affects the efficiency of solute atom clustering, which in turn affects the PLC effect in metals. In this study, temperature was selected as the indirect intrinsic variable. As experimental observation methods advance, direct microphysical intrinsic variables of the temperature field can be chosen.

Strain rate field

Solute atom arresting requires time to form an additional barrier. However, the available time for solute atom arresting is directly influenced by the strain rate. Ultimately, the strain rate affects the PLC effect. Therefore, the strain rate was chosen as the macroscopic indirect intrinsic variable to represent the PLC strain rate effect. As experimental observation methods advance, direct microphysical intrinsic variables of the strain rate field can be selected, similar to the temperature field. Additionally, we will not discuss the anisotropic additional barrier caused by solute atom arresting. Instead, we will use the equivalent strain rate \(\mu \mathrm{{ = }} - {\log _{10}}\left( {{\varvec{{\dot{\varepsilon }} }}} \right)\) as the eigenvariable of the strain rate field. The equivalent strain rate only reflects the magnitude of the strain rate, regardless of direction. This simplification is not because alloy materials are insensitive to the anisotropy of strain rate, but because the focus of this paper is on the interaction between the temperature field and strain rate field. A comprehensive discussion of strain rate anisotropy would divert from the main topic. It is worth noting that strain rate anisotropy is a complex and independent issue that is expected to be addressed in future research.

Thermodynamic forces under the temperature-strain rate physical field

Thermodynamic forces of stress field

The plastic deformation of materials typically involves multiple physical mechanisms that correspond to different thermodynamic force functions. To simplify the thermodynamic constitutive equations, this paper focuses on representative isotropic hardening plastic mechanisms. Equation (37) shows the typical representation form of the isotropic hardening thermodynamic potential function in Lemaitre’s damage mechanics30.

$$\begin{aligned} {R^G} = R_{\max }^G\left[ {1 - exp\left( { - {b^G}{r^G}} \right) } \right] \end{aligned}$$
(37)

Where, \({R^G}\) is the isotropic hardening thermodynamic force; \({r^G}\mathrm{{ = }}{\left( {{2/ {3e_{ij}^pe_{ij}^p}}} \right) ^{{1/2}}}\) is the conjugate internal state variable of \({R^G}\); \(e_{ij}^p\) is the plastic strain tensor; \(R_{\max }^G\) equivalent extremum of isotropic hardening; \({b^G}\) rate coefficient of isotropic hardening.

Thermodynamic forces of temperature field

The temperature field’s thermodynamic forces \({\tilde{Y}}\) are given by Eq. (38). The coefficient \(\xi\) represents the influence of temperature on the thermodynamic force \({\tilde{Y}}\). Equation (38) establishes a mapping relationship between the thermodynamic force \({\tilde{Y}}\) and the intrinsic variable \({\tilde{T}}\) using a simple linear function. While it is possible to assume a polynomial or exponential function in Eq. (38) to enhance simulation accuracy, the focus of this section is not on optimizing the functional form of the thermodynamic force. Instead, it aims to introduce the degeneration process of generalized thermodynamic constitutive models, and so using a simple linear function as an example.

$$\begin{aligned} {\tilde{Y}} = \xi \left( {{\tilde{T}} - {{{\tilde{T}}}_c}} \right) \end{aligned}$$
(38)

Where, \({{\tilde{T}}_c}\) is reference room temperature.

Thermodynamic forces of strain rate field

The thermodynamic forces of strain rate field \({\tilde{W}}\) are given by Eq. (39).

$$\begin{aligned} {\tilde{W}} = \zeta \left( {{\tilde{\mu }} - {{{\tilde{\mu }} }_c}} \right) \end{aligned}$$
(39)

Where, \({{\tilde{\mu }} _c}\) is reference strain rate; The coefficient \(\zeta\) represents the influence of strain rate on the thermodynamic force \({\tilde{W}}\). In Eq. (39), the thermodynamic force \({\tilde{W}}\) is also represented by a simple linear function.

Thermodynamic state potential under the temperature-strain rate physical field

According to Eqs. (37)–(39) and the “State Kinetic Coupling theory”31, we can obtain the thermodynamic state potential under the temperature-strain rate physical field, as shown in Eq. (40).

$$\begin{aligned} {\Psi ^*}\mathrm{{ = }}\frac{1}{2} \mathbf{{D:}}{{\varvec{\varepsilon }}^\mathbf{{e}}} \mathbf{{:}}{{\varvec{\varepsilon }}^\mathbf{{e}}} + R_{\max }^G\left[ {{r^G} + \frac{1}{{{b^G}}}exp\left( { - {b^G}{r^G}} \right) } \right] + \frac{1}{2}\xi {{\tilde{T}}^2} - \xi {{\tilde{T}}_c}{\tilde{T}} + \frac{1}{2}\zeta {{\tilde{\mu }} ^2} - \zeta {{\tilde{\mu }} _c}{\tilde{\mu }} \end{aligned}$$
(40)

Where, \({\Psi ^*}\) is the thermodynamic state potential under the temperature-strain rate physical field; \(\mathbf{{D}}\) is elastic stiffness matrix of materials at room temperature; \({{\varvec{\varepsilon }}^\mathbf{{e}}}\) is elastic strain.

Yield criterion under the temperature-strain rate physical field

The yield criterion under the temperature-strain rate physical field is given as Eq. (41), and a detailed discussion of the yield criterion will follow.

$$\begin{aligned} f = \sqrt{3{J_2}} + \left[ {k\sin \left( {\theta {r^G}} \right) {\tilde{Y}} +\bar{k} + 1} \right] {R^G} + \left( {{\tilde{Y}} + {\tilde{W}} + {\tilde{m}}{\tilde{Y}}{\tilde{W}}} \right) \frac{{{r^G}}}{{r_{Max}^G}} = 0\mathrm{ } \end{aligned}$$
(41)

Where, k is the amplitude parameter; \(\bar{k}\) is the reference amplitude parameter, which can be directly obtained from experimental data when the reference state is determined; \(\theta\) is the vibration frequency parameter; \(r_{Max}^G\) is the stress at the end of the plastic stage. The end of the plastic stage can be understood as the end of plastic deformation, or the beginning of fracture, or in models considering only isotropic hardening, the point where the flow stress growth becomes slow. All the above parameters will be discussed in detail below.

By applying the associated flow, the dissipation potential can be given as follows.

$$\begin{aligned} {\tilde{f}} = {\tilde{F}} \end{aligned}$$
(42)

Based on dissipation potential Eq. (42), the multiple physical fields constraint matrix can be given as follows.

$$\begin{aligned} \left( {\begin{array}{ll} {{\dot{\varepsilon }} _{ij}^p}\\ {{{\dot{r}}^G}}\\ {\dot{\tilde{T}}}\\ {{\dot{\tilde{\mu }}}} \end{array}} \right) = {\dot{\lambda }} \left( {\begin{array}{llll} {\frac{{\sqrt{3} }}{{2\sqrt{{J_2}} }}}& 0& 0& 0\\ 0& {\frac{{\bar{k} + 1}}{{{R^G}}}}& {k\sin \left( {\theta {r^G}} \right) }& 0\\ 0& {k\sin \left( {\theta {r^G}} \right) }& {\frac{1}{{{\tilde{Y}}}}\frac{{{r^G}}}{{r_{Max}^G}}}& {{\tilde{m}}\frac{{{r^G}}}{{r_{Max}^G}}}\\ 0& 0& {{\tilde{m}}\frac{{{r^G}}}{{r_{Max}^G}}}& {\frac{1}{{{\tilde{W}}}}\frac{{{r^G}}}{{r_{Max}^G}}} \end{array}} \right) \left( {\begin{array}{ll} {{s_{ij}}}\\ {{R^G}}\\ {{\tilde{Y}}}\\ {{\tilde{W}}} \end{array}} \right) \end{aligned}$$
(43)

Where, \({s_{ij}}\) is stress deviation. According to Eq. (43), it can be observed that the thermodynamic flow \(\dot{\tilde{T}}\), \({\dot{\mu }}\) are influenced by both their own thermodynamic force \({\tilde{Y}}\), \({\tilde{W}}\) and the thermodynamic force of the stress field \({R^G}\). Additionally, the thermodynamic flow of the stress field \({\dot{r}^G}\), according to the Onsager reciprocal relation, is also influenced by the thermodynamic force of physical fields \({\tilde{Y}}\), \({\tilde{W}}\).

As is known, the key to the incremental form of the plastic constitutive equation is the \(d\varepsilon _{ij}^p = {\dot{\lambda }} \frac{{\partial F}}{{\partial {{\varvec{\sigma } }}}}\). \(\frac{{\partial F}}{{\partial {{\varvec{\sigma } }}}}\) determines the direction of \(d\varepsilon _{ij}^p\), and \(\dot{\lambda }\) determines the size of \(d\varepsilon _{ij}^p\). Under the temperature-strain rate physical field, \({\dot{\lambda }}\) not only depends on \({\dot{r}^G}\) but also on \(\dot{\tilde{T}}\), \({\dot{\mu }}\). \(\dot{\tilde{T}}\) and \({\dot{\mu }}\) depends on \({\tilde{f}}\) and \(\tilde{F}\). The appearance of \(\dot{\tilde{T}}\), \({\dot{\tilde{\mu }}}\), \(\tilde{Y}\), \({\tilde{W}}\) provide a more finely description of the plastic mechanism in multi-physical field situations. In this paper, we not only consider the influence of multiple physical fields on the mechanical properties of materials, but also emphasize that the introduction of thermodynamic forces cannot be completely subjective and should follow the Onsager reciprocity relations to satisfy the requirements of describing the mutual influence between various physical fields.

With thermodynamic forces Eqs. (37)–(39), state potential Eq. (40), yield criterion Eq. (41), dissipation potential Eq. (42) and multiple physical fields constraint matrix Eq. (43), the following physical mechanisms can be described:

As the temperature increases, stably additional barriers produced by solute atoms temporarily arresting will decrease. This leads to a decrease in the strengthening ability, as represented by the term in Eq. (38) , which \({\tilde{Y}}\) decreases as the temperature increases.

When the strain rate increases, it becomes more difficult for solute atoms that are clustered around dislocations to “escape” the temporarily arresting. This phenomenon is manifested macroscopically as a slight increase in plastic strength with a higher strain rate. In Eq. (39), \({\tilde{W}}\) increases with a rise in temperature.

Although strain rate can increase the Stability of solute atoms temporarily arresting (SSATA), it does not guarantee improved the SSATA when the material’s fluidity is enhanced by exceeding a certain temperature threshold. When the material’s fluidity is enhanced, solute atoms are more likely to cross dislocations and “escape” temporary arrest, resulting in reduced SSATA. In fact, as the temperature rises, the ability of the strain rate to enhance the SSATA decreases. These microscale mechanisms are observed on a macroscopic level as the coupling effect of temperature and strain rate fields. The parameter \({\tilde{m}}\) in Eq. (43) is used to describe the influence of the coupling effect of temperature and strain rate fields on the yielding characteristics of materials. Traditional damage mechanics do not consider the mutual influence of these coupled fields or typically use damage functions with opposing properties to describe the effect of strain rate fields on flow stress at different temperatures. However, these strain rate damage evolution equations, which are completely opposite, lack a reasonable microscopic physical explanation. This paper reveals that the essence of the influence of the strain rate field on materials lies in the coupling effect of temperature and strain rate fields.

Thermodynamic constitutive model under the temperature-strain rate physical field

Bring thermodynamic forces Eqs. (37)–(39), state potential Eq. (40), yield criterion Eq. (41), dissipation potential Eq. (42) and multiple physical fields constraint matrix Eq. (43) into general model Eq. (31), we have

$$\begin{aligned} d{\varvec{\sigma } } = \hat{\delta }:\left[ {d{\varvec{\varepsilon }} - {\beta _1}{\mathbf{{H}}_\mathbf{{1}}}{{\left( {1 + {\beta _1}{\mathbf{{H}}_\mathbf{{2}}}: \hat{\delta }:{\mathbf{{H}}_\mathbf{{1}}}} \right) }^{ - 1}}{\mathbf{{H}}_\mathbf{{2}}}:\hat{\delta }:d{\varvec{\varepsilon }}} \right] \end{aligned}$$
(44)

Where,

$$\begin{aligned} {\mathbf{{H}}_\mathbf{{1}}}= & {\mathbf{{H}}_\mathbf{{2}}} = \frac{{\partial F}}{{\partial \mathbf{{\varvec{\sigma } }}}} = \frac{{\sqrt{3} \mathbf{{s}}}}{{2\sqrt{{J_2}} }} \end{aligned}$$
(45)
$$\begin{aligned} {\beta _1}= & \frac{1}{{\frac{{\partial f}}{{\partial {R^G}}}\frac{{\partial {R^G}}}{{\partial {r^G}}}\frac{{\partial F}}{{\partial {R^G}}} + \frac{{\partial f}}{{\partial {\tilde{Y}}}}\frac{{\partial {\tilde{Y}}}}{{\partial {\tilde{T}}}}\frac{{\partial F}}{{\partial {\tilde{Y}}}} + \frac{{\partial f}}{{\partial {\tilde{W}}}}\frac{{\partial {\tilde{W}}}}{{\partial {\tilde{\mu }} }}\frac{{\partial F}}{{\partial {\tilde{W}}}}}}\mathrm{{ = }}\frac{1}{{{u_\mathbf{{1}}}R_{\max }^G{b^G}{e^{ - {b^G}{r^G}}} + {u_2}\xi + {u_3}\zeta }} \end{aligned}$$
(46)
$$\begin{aligned} {u_1}= & \frac{{\partial f}}{{\partial {R^G}}}\frac{{\partial F}}{{\partial {R^G}}} = {\left[ {\bar{k} + 1 + k\sin \left( {\theta {r^G}} \right) {\tilde{Y}}} \right] ^2} \end{aligned}$$
(47)
$$\begin{aligned} {u_2}= & \frac{{\partial f}}{{\partial {\tilde{Y}}}}\frac{{\partial F}}{{\partial {\tilde{Y}}}} = {\left[ {k\sin \left( {\theta {r^G}} \right) {R^G} + \left( {1 + {\tilde{m}}{\tilde{W}}} \right) \frac{{{r^G}}}{{r_{Max}^G}}} \right] ^2} \end{aligned}$$
(48)
$$\begin{aligned} {u_3}= & \frac{{\partial f}}{{\partial {\tilde{W}}}}\frac{{\partial F}}{{\partial {\tilde{W}}}} = {\left[ {\left( {{\tilde{m}}{\tilde{Y}} + 1} \right) \frac{{{r^G}}}{{r_{Max}^G}}} \right] ^2} \end{aligned}$$
(49)

\({\beta _1}\) is an important parameter determined by the thermodynamic flow and thermodynamic force under multiple physical field conditions. Compared to traditional damage mechanics, the thermodynamic constitutive model in this paper not only considers the influence of physical fields on the plastic mechanical properties of materials but also the mutual influence of physical fields by the parameter \({\tilde{m}}\) in the multiple physical fields constraint matrix.

Model validation

Xu31 conducted high-temperature tensile tests on AA5182-O AlMg alloy at various strain rates, and discuss the PLC effect and the SRS transitions. In his tests the alloy primarily consists of Mg (4.55 wt.%), along with Si, Fe, Cu, Mn, Cr, Zn, and Ti (0.11, 0.21, 0.0215, 0.27, 0.0033, 0.005 and 0.01 wt.%), respectively. The samples had a dumbbell-shaped design with dimensions of 12\(\times\)1.2\(\times\)60 mm. The temperatures tested were 298 K, 325 K, and 418 K, while the strain rates ranged from 10-3s-1 to 10-1s-1, as depicted in Fig. 3.

Fig. 3
figure 3

Stress-strain curves of AlMg alloy at different temperatures and strain rates31. The plastic flow stress of AA5182-O AlMg alloy decreases with increasing strain rate at low temperatures but increases at high temperatures.

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Model parameter values

We simulated the experimental data of Xu using the constitutive model given by Eq. (44). Compared to the continuum mechanics model, this model adds 6 parameters \(\xi\), \(\zeta\), \({\tilde{m}}\), k, \(\bar{k}\), \(\theta\). These parameters are used to describe two types of physical fields and their interaction. The types, physical meanings, and values of these parameters are shown in Table 2. The parameter identification method is in Appendix C.

Table 2 Types, physical meanings, and values of the model parameters.
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Compared to empirical models, thermodynamic models offer clearer physical interpretations for parameters when describing processes involving multiple physical fields. Understanding the physical meanings of these parameters can help us establish connections between material deformation mechanisms, material parameters, and constitutive models. When replacing a material, the parameters’ adjustment can be determined based on the material’s mechanical characteristics, other than just a mathematical fit. In cases where a material experiences a single physical field, regression analysis may be able to identify suitable parameters and adjust them to achieve satisfactory simulation results. However, in complex multi-physical field scenarios, regression analysis that disregards the material deformation physical mechanism often fails to produce satisfactory results, even with complex random parameter adjustments.

Simulation results and discussion

The simulation results of the model are as follows:

Fig. 4
figure 4

The simulation results of the model at different temperatures and strain rates. The test results are represented by black lines, while the simulation results are depicted using colored lines. The property that plastic flow stress of AA5182-O AlMg alloy decreases with increasing strain rate at low temperatures but increases at high temperatures has been modeled.

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The validation results show that the model has good predictive accuracy. Within a strain range of 0.15%, the maximum error in strength prediction is 4.9%, corresponding to experimental conditions of 297 K, strain rate \(1 \times {10^{ - 3}}/s\). the property that plastic flow stress of AA5182-O AlMg alloy decreases with increasing strain rate at low temperatures but increases at high temperatures has been modeled.

Figure 4a, b and c illustrate the negative steady-state strain rate sensitivity (SRS) of the alloy at low temperatures. This means that the plastic flow stress decreases as the strain rate increases within the PLC regime. On the other hand, Fig. 4g, h and i depict the positive SRS of the alloy at high temperatures. This indicates that the plastic flow stress increases with increasing strain rate within the PLC regime. Additionally, Fig. 4a, d , g demonstrate that the simulated curves capture the basic characteristics of stress amplitude decreasing with temperature increase in the PLC regime. The same features can be seen in Fig. 4b, e, h and c, f, i.

However, the simulated curves still do not perfectly match the experimental curves. For example, at 325K a strain rate of \(1 \times {10^{ - 3}}/s\) stress oscillations occurs. Similarly, at 298K and a strain rate of \(1 \times {10^{ - 3}}/s\), the stress oscillations increase. The main reason for these discrepancies is that this study aimed to simplify the model and reduce the complexity of the thermodynamic forces and their evolution equations. Improving the simulation accuracy can be achieved by assuming that the thermodynamic force evolution equations in formulas (38) and (39) are polynomial functions.

The parameters in the model have clear physical meanings within the thermodynamic framework, which makes it convenient for applying the model. When applying the model to new materials, parameters can be adjusted specifically based on the characteristics of the material, rather than relying solely on regression analysis. For instance, if the SRS characteristics of a certain material are sensitive, only the temperature field and strain rate field coupling parameters \({\tilde{m}}\) need to be adjusted.

Conclusion

In multiple physical fields, the mutual influence among these fields can significantly impact material elastoplasticity. This paper introduces a thermodynamic-based constitutive model that incorporates the mutual influence of multiple physical fields. Rather than treating physical field characteristics as adjustable “parameters” affecting material coefficients, it employs a thermodynamic dissipation potential derived from the Onsager reciprocity relations, accounting for the mutual influence of multiple physical fields. This dissipation potential ensures that the thermodynamic flow in the stress field is influenced by both newly constructed coupling field thermodynamic forces and traditional stress fields thermodynamic forces. Not only does this method capture the interactions among multiple physical fields, but it also addresses the issue of broken thermodynamic duality in parameter methods, thereby enhancing the rigor of thermodynamic modeling techniques.

Verification examples in the paper demonstrate that leveraging a thermodynamic framework enables the model to elucidate the behavior of plastic flow stress in AA5182-O AlMg alloy. Specifically, at low temperatures, plastic flow stress decreases with increasing strain rate, while at high temperatures, it increases with rising strain rate. Traditional constitutive models, such as damage models, fail to capture this nuance, as they only account for the decrease in plastic flow stress with increasing strain rate under low-temperature conditions. The fundamental aspect of this unique strain rate effect stems from the mutual influences between the temperature field and the strain rate field. Incorporating these the mutual influences of multiple physical fields offers valuable insights into complex mechanical behaviors in constitutive models.

The organization of this paper for the constitutional model is as follows: initially, a generalized thermodynamic model with broad applicability is developed to suit any material and physical fields. Subsequently, this generalized model is degenerated and applied to specific materials, such as AA5182-O AlMg alloy. This approach of degenerating a generalized model establishes a flexible framework suitable for various materials and coupling physical fields. In cases where materials differ from AA5182-O AlMg alloy in engineering practice, additional thermodynamic forces can be incorporated to describe their unique mechanical properties. Consequently, this approach of degenerating enables the creation of coupled-field thermodynamic constitutive models for new materials without altering the model framework, thereby enhancing its potential for widespread application.

The development of this paper model is constrained by the Onsager reciprocal relations. Consequently, the thermodynamic dissipation potentials usually become complex. In engineering practice, severe nonlinear changes in the mutual influence among multiple physical fields can render existing thermodynamic models too convoluted for widespread application. AI32,33,34,35,36,37,38,39 can be applied to construct thermodynamic forces for coupled fields, thereby reducing model complexity under nonlinear conditions. Traditional methods typically involve initially establishing these thermodynamic forces and subsequently determining unknown parameters within the force functions using experimental data. This approach is more suitable when the thermodynamic force in the coupled field is linear. However, when the thermodynamic forces evolve in a complex and nonlinear, conventional methods become less effective. By leveraging AI with increased experimental data sets, a nonlinear mapping between thermodynamic forces and dual state variables can be established, thus replacing traditional nonlinear force functions of coupled fields. This approach enhances simulation accuracy while reducing model complexity, thereby facilitating broader adoption.