The considered thermomechanical problem consists of the dynamic heat equation coupled with the static balance of linear momentum:

$$\begin{aligned} c(T)\, {\dot{T}} +\nabla \cdot \varvec{q}&= {\hat{r}} \end{aligned}$$

(1)

$$\begin{aligned} \nabla \cdot \varvec{\sigma }&= \varvec{0} \end{aligned}$$

(2)

with the primary variables temperature *T* and displacement \(\varvec{u}\). The magnitudes of displacements and rotations as arising from typical PBFAM processing conditions can be assumed as small, hence the problem is commonly modeled within the theory of geometrically linear continuum mechanics where the (engineering) strain tensor is defined by:

$$\begin{aligned} \varvec{\varepsilon } = \frac{1}{2}\left( \nabla \varvec{u} + (\nabla \varvec{u})^{T}\right) \end{aligned}$$

(3)

The coupling between the two Eqs. (1) and (2) is for now hidden in the structural material law \(\varvec{\sigma } = \varvec{\sigma }(\varvec{\varepsilon }(\varvec{u}), T)\) which will be discussed in detail in section “Mechanical constitutive law”. The heat flux \(\varvec{q}\) is specified by Fourier’s law of heat conduction,

$$\begin{aligned} \varvec{q} = -k(T) \nabla T. \end{aligned}$$

(4)

The material parameters appearing in the thermal problem, namely volumetric heat capacity *c* and heat conductivity *k*, will in general depend on the temperature and phase. Their modeling is discussed in detail in the authors’ publication [49] and is briefly reviewed in section “Temperature- and phase-dependent parameters”. The source term \({\hat{r}}\) is used to model the incident laser beam power based on [33].

### Remark

Technically, the heat equation (1), which is derived from energy conservation, can contain coupled mechanical terms. The present, geometrically linear problem formulation without these coupling terms results in a one-way coupling, i.e., the temperature field influences the structural field, but not vice versa. This assumption is ubiquitous in the macroscale PBFAM simulation literature [25,26,27, 45] and seems justified as strain-rate dependent heating effects are not relevant for a quasi-static solid mechanics problems. A model that considers the two-way coupled thermomechanical problem can be found in [48].

The initial boundary value problem is completed by initial conditions for the temperature field and Dirichlet and Neumann boundary conditions for the thermal and structural problem:

$$\begin{aligned} T&= T_0,\quad&\text {in } \Omega \text { for }t=0, \end{aligned}$$

(5)

$$\begin{aligned} T&= {\hat{T}},\quad&\text {on } \Gamma _T, \end{aligned}$$

(6)

$$\begin{aligned} \varvec{q}\cdot \varvec{n}&= {\hat{q}}, \quad&\text {on } \Gamma _{\varvec{q}}, \end{aligned}$$

(7)

$$\begin{aligned} \varvec{u}&= \hat{\varvec{u}}, \quad&\text {on } \Gamma _{\varvec{u}}, \end{aligned}$$

(8)

$$\begin{aligned} \varvec{\sigma }\cdot \varvec{n}&= \hat{\varvec{t}}, \quad&\text {on } \Gamma _{\varvec{\sigma }} \end{aligned}$$

(9)

where \(\Omega \) is the problem domain, \(\Gamma _T\) the Dirichlet and \(\Gamma _{\varvec{q}}\) the Neumann boundary of the thermal problem, \(\Gamma _{\varvec{u}}\) the Dirichlet and \(\Gamma _{\varvec{\sigma }}\) the Neumann boundary of the solid problem and quantities \(\hat{(\cdot )}\) are prescribed values on the respective boundaries. No initial conditions are required for the quasi-static balance of linear momentum (2).

In the present work an apparent capacity method accounts for the effects of latent heat. Essentially, this method modifies the heat capacity *c*, details on the derivation can again be found in [49].

### Remark

(Static vs. dynamic solid mechanics problem) The balance of linear momentum can either be treated as a static or dynamic problem. The PBFAM process can be assumed as quasi-static as no high accelerations takes place (in the solid phase) and inertia effects are thus negligible. Consequently, most of the literature focuses on the static structural problem [25, 36, 46, 52]. Note that history-dependent, potentially irreversible phenomena, which play an important role for the considered class of phase change processes, are still captured by means of history information in the thermomechanical constitutive equations.

### Temperature- and phase-dependent parameters

This section briefly summarizes the modeling of the three different phases powder, melt and solid. For a more detailed motivation and derivation, the reader is referred to [49]. The commonly used liquid fraction *g* is introduced as

$$\begin{aligned} g(T) = {\left\{ \begin{array}{ll} 0, &{} T < T_s\\ \frac{T-T_s}{T_l-T_s}, &{}T_s \le T \le T_l\\ 1, &{}T > T_l \end{array}\right. } \end{aligned}$$

(10)

where \(T_s\) and \(T_l\) represent the solidus and liquidus temperature. The irreversibility of the powder-to-melt transition is captured via the consolidated fraction

$$\begin{aligned} r_c(t) = {\left\{ \begin{array}{ll} 1, &{} \text {if } r_c(0)=1 \text { (i.e., initially consolidated)}\\ \underset{{\tilde{t}}\le t}{\max }\, g(T({\tilde{t}})), &{} \text {if } r_c(0)=0 \text { (i.e., initially powder)}\\ \end{array}\right. } \end{aligned}$$

(11)

The resulting fractions of powder (*p*), melt (*m*) and solid (*s*) are computed as

$$\begin{aligned} r_p&= 1 - r_c, \end{aligned}$$

(12)

$$\begin{aligned} r_m&= g, \end{aligned}$$

(13)

$$\begin{aligned} r_s&= r_c -g. \end{aligned}$$

(14)

These phase fractions can be used to interpolate arbitrary material parameters:

$$\begin{aligned} f_\text {interp} = r_p(T) f_p(T) + r_m(T) f_m(T) + r_s(T) f_s(T), \end{aligned}$$

(15)

where \(f_\text {interp}\) is the interpolated parameter and \(f_p\), \(f_s\) and \(f_m\) are the single phase parameters. This technique is applied to the thermal conductivity *k* and the heat capacity *c*. For the mechanical material properties we refer to the next section.

### Mechanical constitutive law

#### Mathematical formulation

An iso-strain homogenization (also known as Voigt-type homogenization) assumes that the strain in all phases is identical. Accordingly, the stress of the mixture is given by a weighted sum of the individual contributions, a procedure that is in fact similar to the interpolation scheme (15):

$$\begin{aligned} \varvec{\sigma } = \sum _i r_i \varvec{\sigma }_i \quad \text {with} \quad i \in \lbrace p,m,s\rbrace . \end{aligned}$$

(16)

Based on the iso-strain assumption, the total kinematic strain (3) is equal for all phases. For each single phase \(i \in {p,m,s}\), it can be additively split according to

$$\begin{aligned} \varvec{\varepsilon }_i = \varvec{\varepsilon } = \varvec{\varepsilon }_{\sigma ,i} + \varvec{\varepsilon }_{p,i} + \varvec{\varepsilon }_{T,i} + \varvec{\varepsilon }_{\text {ref},i}, \end{aligned}$$

(17)

although not all terms will be utilized for each phase. The first term on the right-hand side of (17) is the elastic strain \(\varvec{\varepsilon }_{\sigma ,i}\) which induces a stress \(\varvec{\sigma }_i\) in each phase according to a linear hyper-elastic material

$$\begin{aligned} \varvec{\sigma }_i = \mathbbm {C}_i : \varvec{\varepsilon }_{\sigma ,i}, \end{aligned}$$

(18)

where \(\mathbbm {C}_i\) is the fourth-order constitutive tensor

$$\begin{aligned} \mathbbm {C}_i = \lambda _i\, \delta _{ab}\delta _{cd} + \mu _i (\delta _{ac}\delta _{bd} + \delta _{ad}\delta _{bc}), \quad \lambda _i = \frac{E_i\nu }{(1+\nu )(1-2\nu )},\quad \mu _i=\frac{E_i}{2(1+\nu )}.\qquad \end{aligned}$$

(19)

The artificial Young’s modulus in powder, \(E_p\), and melt, \(E_m\), will be chosen orders of magnitude below the physically consistent value of the solid, \(E_s\). The Poisson’s ratio is assumed to be the same in all phases.

The remaining terms in (17) are inelastic contributions which are considered in more detail in the following. The plastic strains \(\varvec{\varepsilon }_{p,i}\), which are only relevant in the solid phase, could be calculated with standard approaches, e.g., an incremental problem formulation in combination with a return mapping algorithm. For simplicity, however, plastic strains will not be considered in the numerical examples in this work (\(\varvec{\varepsilon }_p = 0\)). The strains due to thermal expansion \(\varvec{\varepsilon }_T\) are assumed equal in all phases and read

$$\begin{aligned} \varvec{\varepsilon }_{T,i} = \varvec{\varepsilon }_{T} = \varvec{I}\int _{T_\text {ref}}^T\alpha _{T}\,\mathrm {d}T = \alpha _{T}(T-T_\text {ref}) \varvec{I}, \end{aligned}$$

(20)

where \(\alpha _{T}\) is the (constant) coefficient of thermal expansion and \(T_\text {ref}\) is a reference temperature. Finally, the following reference strain \(\varvec{\varepsilon }_{\text {ref},s} =: \varvec{\varepsilon }_{\text {ref}}\), which is only relevant for the solid phase, i.e., \(\varvec{\varepsilon }_{\text {ref},p} = \varvec{\varepsilon }_{\text {ref},m} = 0\) , is proposed in rate form:

$$\begin{aligned} \varvec{\varepsilon }_\text {ref} = \frac{1}{r_s}\hat{\varvec{\varepsilon }}_\text {ref}, \quad \text {with} \quad \dot{\hat{\varvec{\varepsilon }}}_\text {ref} = {\left\{ \begin{array}{ll} (\varvec{\varepsilon }- \varvec{\varepsilon }_p -\varvec{\varepsilon }_T)\, {\dot{r}}_s, &{}\text {if } {\dot{r}}_s > 0\\ {\hat{\varvec{\varepsilon }}}_\text {ref}\, \frac{{\dot{r}}_s}{r_s}, &{}\text {if } {\dot{r}}_s < 0\\ 0, &{} \text {otherwise} \end{array}\right. }, \quad \hat{\varvec{\varepsilon }}_\text {ref}(0) = 0, \end{aligned}$$

(21)

where \(\hat{\varvec{\varepsilon }}_\text {ref}\) represents an accumulation of reference strain contributions weighted by solid fractions, which is used as an intermediate variable. The first case in (21) refers to a solidifying material point (\({\dot{r}}_s > 0\)) and is motivated by physics as discussed in the next section, while the second case, for a melting material point (\({\dot{r}}_s < 0\)), ensures that the reference strain \(\varvec{\varepsilon }_\text {ref}\) does not change during melting, i.e., \({\dot{\varvec{\varepsilon }}}_\text {ref} = \frac{1}{r_s} \dot{{\hat{\varvec{\varepsilon }}}}_\text {ref} - \frac{1}{r_s^2}{\dot{r}}_s{\hat{\varvec{\varepsilon }}}_\text {ref} = 0\) given \({\dot{r}}_s < 0\). This case is necessary for a consistent notation but, as we will see later, can be circumvented in practice. Note, how the rate formulation in (21) causes a continuous change in the stress over the phase change interval \([T_s; T_l]\), which is beneficial for a numerical solution, in contrast to existing approaches with an instantaneous reset of stresses at melting temperature.

For completeness, all introduced strain contributions can be inserted into (16), which after some rearrangement yields the following total stress of the phase mixture:

$$\begin{aligned} \varvec{\sigma }&= (r_p\mathbbm {C}_p + r_m\mathbbm {C}_m + r_s\mathbbm {C}_s) : (\varvec{\varepsilon }- \alpha _T(T-T_\text {ref})\varvec{I}) - r_s\mathbbm {C}_s:\varvec{\varepsilon }_\text {ref}\end{aligned}$$

(22)

The pre-factor of the first term in (22) is equivalent to an average of the single phase material parameters weighted with the phase fractions \(r_i\) similar to (15).

### Remark

(Modeling assumptions) One of the main assumptions underlying the present and most existing thermo-mechanical PBFAM models is that mechanical stresses in the (open-surface) powder and melt phase domains are negligible. This behavior is approximated by applying a simple elastic constitutive law to these phases, with stiffness parameters that are considerably lower as compared to the solid phase, i.e., \(E_p , E_m \ll E s\) . In practice, this approximation turns out to result in moderate, i.e., limited, strains, since the deformation of these powder and melt domains is mostly kinematically controlled by the motion of the significantly stiffer solid phase domains, thus yielding only small stress contributions as desired. Moreover, as compared to approaches exactly satisfying the zero-stress assumption in powder and melt, no additional means are required for tracking and discretization of sharp interfaces inside elements. Note, the assumption that thermal strains exist also in the powder and melt phase, and are equal to thermal strains in the solid phase, has been made for simplicity here. This assumption is neither necessary nor has it a significant influence on the resulting residual stresses due to the low stiffness of these phases and the definition of reference strains (21), which ensures that newly created solid material is stress-free. Further, we assume that from the beginning powder already has the volume and density it would have after consolidation, which has to be accounted for by defining correspondingly decreased layer thicknesses. Modeling solidification shrinkage, i.e., a density increase when powder consolidates, is deemed unnecessary due to the free surface at the top of the currently processed power layer, which allows for (approximately) stress-free consolidation and shrinkage in thickness direction when the powder melts.

The irreversibility of phase change and the reference strain (21) make the material behavior non-conservative. While the proposed approach is very general and can be combined with arbitrary (standard) solid material laws, e.g., elasto-plasticity, we purposefully restrict ourselves to purely elastic solid material behavior in the studied examples. In this scenario, the proposed reference strain term is the only non-conservative contribution to the overall material model, underlining that the creation of a stress-free state at solidification start is the most significant non-conservative aspect of the overall thermo-mechanical problem.

In the simple case of purely elastic material behavior (i.e, stresses do not exceed the yield stress) a heating to a maximum temperature below melting temperature and subsequent cooling to the initial temperature would not lead to any residual stress. Only when the melting temperature is exceeded, the reference strain term will cause an (approximately) stress-free configuration at solidification start, and thus, a residual stress will remain after cooling to the initial temperature. Therefore, we identify the reference strain contribution as the minimal necessary effect for residual stress prediction in such a simplified model.

#### Physical motivation and discussion

The reference strain (21) and stress (22) form a simple yet consistent solid constitutive law. In the following, we want to discuss some properties of the material law and their physical motivation by means of analytically tractable cases. For simplicity, the plastic strain terms are neglected from now on.

Note, that the reference strains \(\varvec{\varepsilon }_\text {ref}\) in (21) only change when the solid phase fraction increases according to \({\dot{r}}_s > 0\), i.e., for temperatures \(T \in [T_s ; T_ l ]\) in the phase change interval and negative temperature rates \({\dot{T}} < 0\). An elastic constitutive law with low stiffness values (i.e., \(E_p , E_m \ll E s\)) as applied to powder and melt leads to small stresses yet considerable total strains in these phases. In this context, the reference strains according to (21) ensure that these strains do not translate into stresses during solidification. For the special case that kinematic \(\varvec{\varepsilon }\) and thermal strains \(\varvec{\varepsilon }_T\) (as well as plastic strains \(\varvec{\varepsilon }_p\)) are constant during solidification, which approximately holds if the phase change interval \(T_l - T_s\) is sufficiently small, it can easily be verified from (17) and (21) that the elastic strain, and thus due to (18) the resulting stresses, in the evolving solid phase vanish. This fact corresponds to the physical intuition that newly formed solid should lose all history information and exhibit a new stress-free configuration when solidification starts.

From a stress homogenization point of view, the model assumptions underlying the reference strain formulation state that for each solidifying fraction of material, the reference strain contribution effectively creates its new, stress-free reference configuration, from which strains are calculated. This is illustrated in Fig. 1.

Two more involved examples, which illustrate the evolution of the reference strain over multiple repeated melt and solidification cycles, can be found in Appendix A.