# A discrete model for layered growth

Davide Renzi  Dipartimento di Ingegneria Civile, Informatica e delle
Tecnologie Aeronautiche, Roma Tre University, Via Vito Volterra 62, Roma,
00146, Italy  Sonia Marfia  Dipartimento di Ingegneria Civile, Informatica e
delle Tecnologie Aeronautiche, Roma Tre University, Via Vito Volterra 62,
Roma, 00146, Italy  Giuseppe Tomassetti  Dipartimento di Ingegneria
Industriale, Elettronica e Meccanica, Roma Tre University, Via Vito Volterra
62, Roma, 00146, Italy  Giuseppe Zurlo  School of Mathematical and Statistical
Sciences, University of Galway, University Road, Galway, Ireland

###### Abstract

In this work we present a discrete model that captures the fundamental
properties of additively manufactured solids in a minimal setting. The model
is based on simplified kinematics and allows for the onset of incompatible
deformations between discrete layers of an additively manufactured stack.
Thanks to the discrete nature of the model, we obtain an averaged formulation
of mechanical equilibrium for the growing stack, leading to closed-form
solutions that are both analytically simple and physically transparent. In
particular, we are able to explain the origin of residual stresses by the
accumulation of incompatible deformations between adjacent layers. At the same
time, we are able to formulate the technologically relevant inverse problem
that provides the deposition protocol required to produce a desired state of
internal stress in the manufactured stack. Another important aspect analyzed
in the work is the role played by an ideal “glue” between the layers, whose
presence is fundamental to prevent their sliding and whose mechanical behavior
can quantitatively influence the final stress distribution in the stack.
Although the model is an elementary approximation of additive manufacturing,
its simplicity makes it possible to highlight how the controls exerted during
deposition will have qualitative or quantitative effects on the final stress
state of the stack. This understanding is crucial in shedding light on the
complex mechanical behavior of additive manufactured solids.

##  1 Introduction

The modeling of growth phenomena has been a scientific challenge for many
years and continues to offer opportunities for development in various fields,
such as biomechanics, construction science, and materials science. Growth
refers to the change in mass of a body over time, and it is observed in many
contexts. In biology, growth is fundamental to describing many aspects of
life, from cell division to tumor growth in living beings. However, growth
processes also appear in many physical phenomena, where new material is added
to an evolving system, such as 3D printing, or where growth is associated with
phase transition phenomena, such as crystal growth [10, 24, 25].

Growth phenomena can be classified into two categories: volumetric growth [32,
23, 14, 30, 20, 21, 19] and surface growth [31, 11, 13, 33, 4, 41, 42, 34]. In
the former case, the addition or removal of matter takes places inside the
bulk; in the latter, material is added or removed at the boundary of the body.

Both volumetric and surface growth can result in residual stress. This is not
always an undesired outcome. In may biological contexts, residual stress plays
an important role: it helps maintaining the structural integrity and the
mechanical stability of a tissue; in blood vessels, residual stress
contributes to the maintenance of an optimal vessel diameter; residual stress
can influence how tissues respond to injury and their subsequent healing. In
many technological contexts residual stress is an important target, like for
example in the process of hot shrinking of metals, in reinforcing concrete
structures, or in prestressing glass layers to enhance their resistance to
fracture.

Apart of these (nowadays classical) applications that regard residual stress
as a beneficial feature, the last decade has seen a blooming of technologies
aimed at embedding target residual stress patterns into “shape lifting”
structures, an exciting avenue with an immense scenario of futuristic
applications, such as 4D printing technologies [ge2016], biomimetic tissues
and more.

The mathematical description of residual stress relies ultimately on the
concept of strain incompatibility, a measure of the level of “unfitness” of
sub-parts of the body to be arranged together without creating voids or
overlaps. In principle, this concept is purely geometrical, and then depending
on the stiffness of the various parts of the body, the same level of
incompatibility will ultimately result into different distributions of
residual stress.

Since the boundary of the body is directly accessible, it is intuitive that
surface growth offers more control over the accretion process, giving the
possibility of fine tuning, and hence program in advance the properties of the
resulting body. Illustrating examples of how the choice of the accretion
protocol affects the residual stresses may be found in [11] in the context of
masonry materials, and in [41] in the context of 3D printing.

The understanding of the mechanism behind the accumulation of residual stress
during surface growth was slowly achieved in stages over the span of the last
150 years. As early as 1883, G.H.Darwin (Charles’ son) acknowledged the
relevance of what he called “the historical element” in the horizontal thrust
of a mass of sand [15], accounting for the fact that the final distribution of
stress would depend on the way the sand had been layered in a container.

The first analytical treatment of the deposition-dependent stress distribution
in massive bodies is apparently traceable to E.I.Rashba [28], who pioneered a
prolific area of study of the residual stresses arising in surface growth
problems. Some years later, Brown and Goodman [12] recognize that the stress
state of a massive, self-gravitating body formed by layered accretion is not
the same as it would be if the body had been first fabricated, and then
endowed with mass. This is due to the fact that when a new layer is deposited,
it deforms the pre-existing material before hardening.

The most recent developments in this vein have elucidated the geometric
aspects of “non-Euclidean” surface growth [37, 36] and have established the
non-locality, both in space and time, between the controls that are exerted on
the growth surface and the ensuing state of incompatibility that is nailed in
the body at the end of the process [41]. What emerges from these studies is
that strain incompatibility can not be controlled locally, in the sense that
even assuming that the “gluing” takes place immediately after deposition, the
local value of strain incompatibility depends not only on the pre-strain
imposed on the new block prior to deposition, but also on the elastic
adjustment of the underlying body, which is required to reattain equilibrium,
and thus implicitly, also on the whole accretion history.

Three-dimensional problem involving incompatibility are often challenging to
solve analytically, unless symmetry assumptions are made [16, 17, 33, 38].
However, one should be careful when making such assumptions, as symmetric
solutions may be unstable during growth [5, 6]. One might consider simplifying
the problem by reducing the number of dimensions to one. However, doing so
would eliminate the issue of incompatibility altogether, which is of course
not an option. As a solution [42], suggests retaining a trace of higher-
dimensionality in the problem, through a 1.5-dimensional model.

In the spirit of [42] in the present paper we propose an approach where the
growth of a body _occurs primarily in one dimension_ but the impact of the
other dimensions is considered through the integration of _extra state
variables that account for stress and strain in the perpendicular directions_.
Differently from previous studies, the problem is discrete _both in space and
time_. Specifically, the growing body is a stack elastic blocks glued
sequentially, at discrete times, on top of another. It is quite intuitive that
when two elastic blocks are pre-strained, and then glued, residual stresses
may arise depending on the loads that were applied to the blocks prior to
attachment, and on the type of glue that was used in the process. A fast glue
would freeze the state of the bonding interface at the very instant of
adhesion, some other glues may allow for a partial or a total relaxation of
stress before activating the bond. These intuitive considerations disclose the
possibility to play with the order and manner in which an elastic body are
assembled by the juxtaposition of elementary “blocks”, in order to create
desired distributions of residual stress in the resulting patch-worked body.
This is the main idea explored in this paper.

We introduce our setup in Section 2, where we formulate the equilibrium
problem for a weightless stack of $j$ blocks. Our key modeling assumptions are
that each block undergoes a shear-less homogeneous strain, and that the
difference between the horizontal strains of block $j+1$ and block $j$ be a
prescribed _incompatibility_ $\delta_{j}$. Since the average horizontal stress
in each block vanishes the problem consists of oneglobal equilibrium
condition, $j-1$ compatibility equations, and $j$ constitutive equations, for
a total of $2j$ unknowns representing the horizontal stress and horizontal
strain of the blocks. Of course this way to account for equilibrium in the
horizontal direction is a crude approximation. In a real system it is
inhomogeneous strain will occur, and therefore it should be understood that
our analysis is only a first order approximation of a more refined model, to
be developed elsewhere.

For this problem, we work out an analytical solution which shows that in the
absence of external forces the horizontal stresses vanish only in the special
case when the incompatibility vanishes as well. We then take into
consideration external forces by extending the formulation to the case when
each block is topped by a lumped mass, $b_{i}$. In this case, the vertical
stresses must be taken into account, and can be computed by means of $j$
equilibrium conditions.

In Section 3, we shift our focus to the most fascinating issues in surface
growth, specifically those where the incompatibilities $\delta_{i}$, are not
predefined, but instead dictated by the accretion rule. Starting from a stack
of $j$ weighting blocks in equilibrium with prescribed body forces $b_{i}$,
$i=1,\ldots,j$ and incompatibilities $\delta_{i}$, $i=1,\ldots,j-1$, we
position a new pre-stressed block atop the stack with a layer of glue between
the two, and we subsequently release the block letting the whole system attain
equilibrium. The resulting stack will have $j+1$ block, where the
incompatibility $\delta_{j}$ between block $j$ and block $j+1$ is determined
by the character of the glue used for bonding. We consider two extreme cases:
a “fast” glue and a “slow” glue.

We remark that the new block perturbs the equilibrium state of the underlying
stack only _after_ has been released. If the glue is fast, it locks the
incompatibility $\delta_{j}$ _before_ the new block is released, when the
strain of the new block is determined by its pre-stress, and the strain of the
underlying stack is the same as in the previous accretion step. If the glue is
very slow, on the other hand, the incompatibility $\delta_{j}$ locks in on
only _after_ the block has been released. We assume that in the time span
between the release of the new block and the solidification of the glue, the
new block and the stack interact with each other with a _frictionless_
contact. As a result, when equilibrium is attained, the strain of the new
block is determined by its own weight and not by its pre-stressed state;
likewise, the strain of the underlying stack will change due to the additional
weight of the new element. The strain difference between the last two blocks
of the new stack determines the incompatibility after the glue solidifies.

Concerning the two modes of accretion, we work out explicit formulas for the
incompatibility $\delta_{j}$. An interesting finding is that for the fast glue
the incompatibility $\delta_{j}$ is determined by the pre-strain (and hence
the pre-stress) in block $j+1$ and block $j$, as well as on the weights of the
blocks. For the case of slow glue, instead, the incompatibility $\delta_{j}$
is determined solely by the weight $b_{j}$ of the $j-th$ element, and is
unaffected, as anticipated, by the pre-stress.

Having established these general results, we consider in Section 4 some
specific examples. In the first example we consider the growth of a stack up
to the addition of 20 blocks, with constant horizontal tensile pre-stress, and
we observe how the stress in the first block evolves as news blocks are added.
What we observe is that the tensile pre-stress of the newly added blocks
produces a contraction in the blocks beneath, and in particular, in the first
block. However, as more and more blocks are added, the weight of the entire
stack on the first block compensates the above-mentioned horizontal
contraction, and as a result the horizontal strain of the first block becomes
eventually positive. We also determine the strain profile in the entire stack,
which shows that, while a constant incompatibility would result in a linear
horizontal stress, a constant pre-stress produces a horizontal stress profile
of “parabolic” type. In the second part of Section 4 we repeat the
calculations for the case of slow glue, and we verify that the “slow glue”
protocol results into lower final stressed.

We finally analyze the residual stresses that result from the removal of the
weights, considering accretion protocols that differ not only for the type of
glue used, but also for the weights of the blocks. If the weights $b_{i}$ are
increasing, the profiles resulting from the “fast” and “slow” glue protocols
are similar in shape, but differ by approximately an order of magnitude, with
the residual stress being much larger in the case of “fast glue”. If the
weights are decreasing, on the other hand, the two profile have opposite
convexities. Finally, if all weights are the same, the residual stress in the
“slow glue” case is linear (since in this case the generated incompatibilities
$\delta_{i}$ are all equal).

Section 5 deals with an even more intriguing problem: choosing a deposition
strategy in order to attain a target residual stress, the so-called “inverse
problem”. As a first test, we start from the residual stress resulting from an
analysis performed in Section 4 on the case when all weights are equal, and
the pre-stress vanishes. We indeed verify that the original pre-stresses are
recovered. As a further test, we consider the case of a sinusoidal residual
stress, and we determine the pre-strains needed to obtain it. We then extend
the model to incorporate Tresca’s resistance criterion. We then repeat the
treatment for the case of slow glue. Additional considerations and insights
are contained in the concluding section.

##  2 Equilibrium of an incompatible stack

###  2.1 The notion of incompatibility

Figure 1:  $a)$ reference (dashed) and strained (solid) block. $b)$ the block
is assumed to deform homogeneously under the action of horizontal and vertical
stresses $\sigma_{x},\sigma_{y}$. $c)$ to simplify the notation, a grey block
will denote the union of a lumped mass (of weight $b_{i}$) on top of a
massless elastic block.

To set the stage, we consider a planar system consisting of a stack of blocks
of linear elastic material, which deform homogeneously and without shear
strains.

We denote by $\epsilon_{x}^{i}$ and $\epsilon_{y}^{i}$ the horizontal and
vertical strain of the $i$-th block. The corresponding stresses on the
horizontal and vertical directions are $\sigma_{x}^{i}$ and $\sigma_{y}^{i}$,
as shown in Fig.1. The “elastic state” of the system is described by $2n$
_strain components_ and $2n$ _stress components_ , respectively,

$\mathcal{E}=\\{(\epsilon_{x}^{i},\epsilon_{y}^{i}),i=1,\ldots
n\\},\quad\text{and}\quad\Sigma=\\{(\sigma_{x}^{i},\sigma_{y}^{i}),i=1,\ldots,n\\}.$
(1)

We assume that the stress and the strain components are related by the
_costitutive equations_ :

$\begin{aligned} \epsilon_{x}^{i}=(\sigma_{x}^{i}-\nu\sigma_{y}^{i})/E\\\
\epsilon_{y}^{i}=(\sigma_{y}^{i}-\nu\sigma_{x}^{i})/E\end{aligned}\qquad
i=1,...,n,$ (2)

where $E$ is the 2D _Young modulus_ (force/length) and $\nu$ is the _Poisson
ratio_ (dimensionless). As a key ingredient in our model, we also assume that
the stack comes with _prescribed incompatibility_

$\Delta=\\{\delta_{i},i=1,\ldots,n-1\\},$ (3)

and we impose the _incompatibility constraints_ :

$\epsilon_{x}^{i+1}-\epsilon_{x}^{i}=\delta_{i}\qquad i=1,...,n-1,$ (4)

These incompatibilities are maintained in the by a “glue”, whose nature will
be detailed in the sequel. We shall see that, consistent with physical
intuition, the enforcement of a non-trivial incompatibility $\Delta\neq 0$
induces a non-homogeneous strain state $\mathcal{E}$, as sketched in Fig. 2
below, and, as a result, a non-vanishing residual stress $\Sigma$.

Figure 2: Comparison between a compatible weightless stack, and an
incompatible weightless stack with mismatches $\delta_{i}$ between the blocks
$i$ and $i+1$.

In this work we will focus attention both on weightless blocks, that according
to Fig.1a,b will be described by unfilled blocks, and on heavy blocks, that
will be filled in grey instead. In order to simplify the description of the
state of deformation of the blocks under gravity, each block will be conceived
as if it is formed by a heavy lumped mass, denoted in black in Fig.1c, topping
a massless block through a frictionless interface. This way we will be
legitimated to keep describing the deformation of each block as homogeneous,
whereas clearly in reality a heavy block with distributed mass will deform
inhomogeneously.

The constitutive equations (2) and the incompatibility constraints (4) are
complemented by equilibrium equations, that we introduce in the next Section.

###  2.2 Weightless blocks

We first consider the problem of equilibrium of a weightless stack, as the one
represented in Fig.2. If the incompatibilities are prescribed and there are no
external loads acting on the stack, to determine the state of stress we
complement (4) with the following $n+1$ equilibrium equations:

$\left\\{\begin{array}[]{lll}{\displaystyle\sum_{i=1}^{{n}}\sigma_{x}^{i}=0},\\\\[10.00002pt]
{\sigma_{y}^{i}=0\qquad i=1,...,{n}}.\end{array}\right.$ (5)

The first of (5) tells us that the stress average for the entire stack
vanishes along the horizontal direction, whereas the remaining equations
impose stress equilibrium of the individual blocks along the vertical
direction. The averaged formulation of equilibrium (5) in the context of
accreting media appeared for the first time in a continuous setting in the
pioneering work of Palmov [1], in the study of solidification. The discrete
form presented above can be derived from the principle of virtual work. Since
the incompatibilities are fixed, by (4), every virtual variation

$\widetilde{\mathcal{E}}=\\{(\widetilde{\epsilon}^{i}_{x},\widetilde{\epsilon}^{i}_{y}),i=1,\ldots,n\\}$
(6)

of the strain state must satisfy

$\widetilde{\epsilon}_{x}^{i+1}-\widetilde{\epsilon}_{x}^{i}=0,$ (7)

which imply that

$\widetilde{\epsilon}_{x}^{i}=\widetilde{\epsilon}_{x},$ (8)

with $\widetilde{\epsilon}_{x}$ an arbitrary constant. Thus, the internal
virtual work associated to a virtual variation of the strains is

$W[\widetilde{\mathcal{E}}]=\sum_{i=1}^{n}\sigma^{i}_{x}\widetilde{\epsilon}^{i}_{x}+\sum_{i=1}^{n}\sigma^{i}_{y}\widetilde{\epsilon}^{i}_{y}=\Big{(}\sum_{i=1}^{n}\sigma^{i}_{x}\Big{)}\widetilde{\epsilon}+\sum_{i=1}^{n}\sigma^{i}_{y}\widetilde{\epsilon}^{i}_{y}.$
(9)

At equilibrium, in the absence of applied loads, the internal work must vanish
for every virtual variation of the horizontal strains, whence $\eqref{2}$,
since $\widetilde{\epsilon}_{x}$ and
$\\{\widetilde{\epsilon}_{y}^{i},i=1,\ldots,n\\}$ are arbitrary.

The linear system (2)–(5) is well-posed: it is comprised of linearly
independent equations whose number equals that of the unknowns. The unique
solution is:

$\sigma_{x}^{i}=E\left(\sum_{k=1}^{i-1}\delta_{k}-\frac{1}{{n}}\sum_{a=1}^{n}\sum_{k=1}^{a-1}\delta_{k}\right)\qquad
i=1,...,{n},$ (10)

where we use the convention that sums over an empty set of indices vanish,
i.e., $\sum_{k=1}^{0}\delta_{k}=0$. The solution (10) confirms that the
quantities $\delta_{i}$ are source of stress, even in the absence of loading.
This motivates our interpretation of $\delta_{i}$ as a lumped version of
strain incompatibility.

For illustrative purposes, sample distributions of horizontal stress obeying
(11) for ${n}=20$ are represented in Fig.3. In the special situation when the
incompatibility is constant ($\delta_{i}=\delta$), the stress is (see Fig.3a):

$\sigma_{x}^{i}=\left(i-\frac{{n}+1}{2}\right)E\delta\qquad i=1,...,{n}.$ (11)

Note that the blocks in the upper half are in tension along the horizontal
direction; conversely, the blocks in the lower half of the stack are in
compression.

Figure 3:  Renormalized (stress/Young modulus) residual stress in a stack with
$n=20$ blocks. $a)$ constant incompatibilities $\delta_{i}=1$; $b)$ linear
incompatibilities; $c)$ singular incompatibility.

Since equilibrium in the horizontal direction is imposed _globally_ , our
model does not display a localized residual stress in the proximity of an
isolated disarrangement, as one would expect.

To see this, consider for instance the stress profile in Fig. 3c, which is
obtained for a stack of $n=20$ blocks with $\delta_{10}=\delta>0$ and
$\delta_{i}=0$. The horizontal stress is

$\sigma_{x}^{i}=\left\\{\begin{array}[]{lll}-E\delta/2&i<10\\\ E\delta/2&i\geq
11.\end{array}\right.$ (12)

The stress resultant in the horizontal direction would sum to zero, but the
whole stack beneath and above the interface between the blocks $10$ and $11$
has uniform stress, whereas in a real system one would expect this stress to
decay moving towards the boundaries. This limitation of the model could
clearly be relieved in multiple ways at the discrete level, for example by
incorporating concentrated shear springs between all blocks, but this would
significantly complicate the model and therefore would affect its
transparency.

###  2.3 Heavy blocks

We now consider the case when, along with incompatibilities, an external
loading is applied to the stack, which we ascribe to gravity. As a modelling
assumption, we consider that at the interface between every pair of blocks
there is a lumped mass, as shown in Figure 4.

The weights of the lumped masses, renormalized by the bock width, are denoted
by the symbol $b_{i}$. These have the same dimensions of the 2D Young modulus
$E$ and the 2D stress components $\sigma_{x/y}^{i}$, as a result of having
dimensions of force per unit of length. As a remark, the $b_{i}$ should not be
confused with 2D body forces (force/area), yet we continue to use the same
symbol because of how these are physically related to weight.

As depicted in Fig.1c, heavy blocks made of a lumped mass topping a massless
elastic block will be filled in gray, whereas massless blocks are unfilled.
Note that as a further simplification in our model, thinking of the weight as
concentrated supports the hypothesis that the heavy blocks undergo homogeneous
strain.

Gravity is applied to the stack while the incompatibilities between the blocks
are kept frozen to their prescribed values. Therefore, during the application
of the loading, the blocks can not slide relative to each other, but they
deform laterally as a whole. The equilibrium equations (10) are now replaced
by:

$\left\\{\begin{array}[]{lll}{\displaystyle\sum_{i=1}^{n}\sigma_{x}^{i}=0},\\\\[15.00002pt]
{\displaystyle\sigma_{y}^{i+1}-\sigma_{y}^{i}=b^{i}\qquad
i=1,...,n-1},\\\\[10.00002pt]
{\displaystyle\sigma_{y}^{n}=-b^{n}}.\end{array}\right.$ (13)

Figure 4: a) Equilibrium configuration of a stack with $b_{i}=0$ and
$\delta_{i}=0$. b) Stack in equilibrium with $b_{i}>0$ and $\delta_{i}=0$. c)
Stack in equilibrium with $b_{i}=0$ and $\delta_{i}\neq 0$. d) Combined effect
of the weights and incompatibilities. In $b)$ and $c)$ the black line denote a
smooth rigid foundation, that allows horizontal sliding but holds the
structure vertically.

The equilibrium equations determine the stresses along the vertical direction:

$\sigma_{y}^{i}=-\sum_{s=i}^{n}b_{s}\qquad i=1,...,n.$ (14)

The stresses in the horizontal direction are instead obtained by solving the
first of (13), together with the constitutive equations (2) and the
incompatibility constraints (4). The solution is (the details of the
derivation may be found in Appendix 7.1):

$\sigma_{x}^{i}=\sum_{k=1}^{i-1}(E\delta_{k}+\nu
b_{k})-\frac{1}{{n}}\sum_{k=1}^{{n}-1}(({n}-k)(E\delta_{k}+\nu b_{k}))\qquad
i=1,...,{n}.$ (15)

The mechanical effect of the weights $b_{i}$ and of the incompatibilities
$\delta_{i}$ is illustrated in the cartoon in Fig.4. Due to the linearity of
the problem, the final shape represented in $d)$ may be seen as the result of
superposing $a)$, wherein $\delta_{i}=0$ and $b_{i}\neq 0$, and $c)$, where
instead $\delta_{i}\neq 0$ and $b_{i}=0$.

It is worth noting that the weight $b_{n}$ of the last block _does not affect
the horizontal stress._ Similarly to what we did to obtain (11), we can
consider the special case when the incompatibilities and the weights are
spatially uniform: $\delta_{i}=\delta$, $b_{i}=b$. In this case, the
horizontal stress is an affine function of $i$:

$\sigma_{x}^{i}=\left(i-\frac{{n}+1}{2}\right)(E\delta+\nu b)\qquad
i=1,...,{n},$ (16)

and the profile of the horizontal residual stress is qualitatively the same as
in Figure 3. The horizontal strain is

$\varepsilon_{x}^{i}=\left(i-\frac{n+1}{2}\right)\delta+\nu\frac{n+1}{2}\frac{b}{E}.$
(17)

We notice from (17) that the effect of a constant weight is to shift the
average value of the horizontal strain from zero. The scaling factor $(n+1)/2$
is due to the fact that the mechanical work performed along the loading path
by the weight on top of the $i$-th block is increasing with $i$, due to a sort
of “telescopic effect”. Hence, the weights do not perform the same work.

A remarkable feature of the general solution (15) is that the dependence of
the stress $\sigma_{x}^{i}$ on the incompatibilities $\delta_{i}$ and on the
weights $b_{i}$ is functionally similar (apart from the different coefficients
$E$ and $\nu$). Albeit this model is extremely simplified, it therefore
retains a feature that is present at the level of the three-dimensional theory
of elasticity, where incompatible strains and body forces play a similar role
as sources of residual stress fields [3].

Another interesting feature of the general solution (15) is that that
horizontal stresses would vanish inside each block ($\sigma_{x}^{i}=0$) if and
only if

$\delta_{i}=-\frac{\nu}{E}b_{i}.$ (18)

Equation (18) also unveils that the incompatibilities $\delta_{i}$, if these
can be externally controlled, could be used to anneal the horizontal stress in
a heavy stack, through a sort of remodeling process.

If, instead, the incompatibilities can be nailed and maintained in the stack,
one could exploit (18) to _compensate_ for the effect of the concentrated
weights $b_{i}$, for example, to annihilate residual stresses in the deformed
stack. Therefore, the loaded column would have zero horizontal stresses in all
blocks, whereas of course the unloaded column (where weights are removed)
would be residually stressed.

##  3 Growth of an heavy stack: the direct problem

While until now the incompatibilities were prescribed, in this section we
establish how their distribution arises during a construction process of
layered deposition, where a hypothetical printing devices deposits the blocks
one by one, through a sequence of steps, with the incompatibilities being
locked at the end of each step by a layer of glue between adjacent blocks. We
call this the _direct problem_ , as opposed to the inverse problem [41, 34],
which consists in determining the incompatibilities that are required to
produce a desired distribution of residual stress.

We describe the construction process incrementally. Making reference to the
cartoon of Fig.5, the process proceeds as follows. At the first step of the
construction process, the first block $j=1$ is laid on a frictionless support.
For $b^{1}$ the weight of the block, the stress and strain state at the end of
the first step are

$\Sigma^{1}=\\{(\sigma_{x}^{1,1},\sigma_{y}^{1,1})\\}=\\{(0,-b^{1})\\}.$ (19)

The corresponding strains are

$\mathcal{E}^{1}=\\{(\epsilon_{x}^{1,1},\epsilon_{y}^{1,1})\\},$ (20)

with

$\epsilon_{y}^{1,1}=-\frac{b^{1}}{E},\qquad\epsilon_{x}^{1,1}=-\nu\epsilon_{y}^{1,1}.$
(21)

Now, suppose that we are at the end of the $j^{\text{th}}$ step (see Fig.
5.a). The stack is composed of $j$ blocks, the glue that bonds all the blocks
is solidified, the incompatibilities are:

$\Delta^{j}=(\delta_{i},i=1,...,j-1).$

Note that at the end of step $j=1$ there is no incompatibility. The
incompatibilities are _locked_ , and the stack is in equilibrium under the
action of the weights $b_{i}$, $i=1,\ldots j$. In this configuration,
equilibrium is described by a $j$-uple of horizontal and vertical stresses

$\Sigma^{j}=\\{(\sigma_{x}^{i,j},\sigma_{y}^{i,j}),i=1,...,j\\}$

and by a $j$-uple of horizontal and vertical strains

$\mathcal{E}^{j}=\\{(\epsilon_{x}^{i,j},\epsilon_{y}^{i,j}),i=1,\ldots,j\\}$

which solve the equilibrium problem discussed in the previous section, namely:

$(P_{j})\left\\{\begin{array}[]{lll}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sum_{i=1}^{j}\sigma_{x}^{i,j}=0}\\\\[15.00002pt]
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{y}^{i+1,j}-\sigma_{y}^{i,j}=b^{i}\qquad
i=1,...,j-1}\\\\[10.00002pt]
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{y}^{j,j}=-b^{j}}\\\\[10.00002pt]
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\epsilon_{x/y}^{i,j}=\frac{\displaystyle\sigma_{x/y}^{i,j}-\nu\sigma_{y/x}^{i,j}}{\displaystyle
E}\qquad i=1,...,j}\\\\[10.00002pt]
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\epsilon_{x}^{i+1,j}-\epsilon_{x}^{i,j}=\delta_{i}\qquad
i=1,...,j-1}\end{array}\right.$ (22)

The $(j+1)^{\rm th}$ construction step takes place in two stages. In the first
stage, the printing device deposits the $(j+1)^{\text{th}}$ block on top of
the stack, keeping the block under a self-balanced couple of “deposition
surface stress”:

$\mathring{\sigma}_{x}^{j+1}:=\sigma_{x}^{j+1,j}.$ (23)

In addition, the printing device maintains the $(j+1)^{\text{th}}$ block in
equilibrium, by supplying a vertical force

$\mathring{\sigma}_{y}^{j+1}=-b_{j+1},$ (24)

at the bottom face of the block, see Fig.5b, which equilibrates the weight
$b_{j+1}$.

In writing (23), we take inspiration from previous work on continuous surface
growth [41, 34, 42], where the surface (i.e. tangential) components of the
stress are controlled at the growth surface, besides the usual tractions or
displacements, ordinarily controlled in non-growing bodies.

Figure 5:  a) A block $j+1$ is prepared, in order to be deposited on top of an
equilibrated stack of $j$ block. b) Immediately prior to deposition, the
“printer” pre-stresses the block through a self-balanced couple of horizontal
forces of magnitude $\mathring{\sigma}_{x}^{j+1}$, and balances the weight of
the block by exerting a vertical force $\mathring{\sigma}_{y}^{j+1}$ on the
the bottom of the block. c) When the block is dropped, the new stack made of
$j+1$ blocks reattains equilibrium.

In this stage, the deposition strains of the upper block are:

$\mathring{\epsilon}_{x}^{j+1}=\frac{\mathring{\sigma}_{x}^{j+1}+\nu
b_{j+1}}{E},\qquad\mathring{\epsilon}_{y}\\\
^{j+1}=-\frac{b_{j+1}+\nu\mathring{\sigma}_{x}^{j+1}}{E}.$ (25)

We suppose that the glue begins its solidification process at the end of the
first stage, as soon as the block $j+1$ is brought to contact with the stack.

In the second stage of step $j+1$, the printing devices releases the block,
and the construction process pauses until the glue has solidified. At the end
of the solidification process, the stack has attained a new equilibrium
configuration (see Fig.5c), described by a a new stress state

$\Sigma^{j+1}=\\{(\sigma_{x}^{i,j+1},\sigma_{y}^{i,j+1}),i=1,...,j+1\\},$ (26)

and by a $j$-uple of horizontal and vertical strains

$\mathcal{E}^{j+1}=\\{(\epsilon_{x}^{i,j+1},\epsilon_{y}^{i,j+1}),i=1,\ldots,j+1\\}.$
(27)

As we shall see below, the new stress and strain distributions depend on the
nature of the glue between block $j$ and block $j+1$. Whatever, these
distributions be, one must expect that, in general, the stress
$\sigma_{x}^{j+1,j+1}$ in the upper block after the attainment of equilibrium
be different from the pre-deposition stresses $\mathring{\sigma}_{x}^{j+1}$.
This, as we will illustrate, has an important quantitative effect in the
distribution of incompatibilities resulting from the deposition process.

The stress and strain states $\Sigma^{j+1}$ and $\mathcal{E}^{j+1}$, after the
printing device is removed, and after the glue has solidified, depends on the
solidification speed of the glue. We consider two extreme cases, which we
refer to as “fast glue”, which “freezes” the incompatibility at the very
instant of deposition, and “slow glue”, which allows the last block to slide
on top of the second-to-last until the horizontal stress in the last block
vanishes.

A point we would like to stress is that, even if at the current stage we have
not yet described how the incompatibility between the last and second to last
blocks is calculated, we will assume that incompatibilities between all pairs
of block do not evolve during the process. In other words, their value is
frozen at the value determined at deposition, and the latter will depend
crucially on the mechanical behavior of the “glue”.

###  3.1 Fast glue

The “fast glue” solidifies at the precise instant when block $j+1$ is brought
into contact with the stack, that is, when the printing device is still
holding the block, i.e. the weight $b_{j+1}$ is not trasmitted to the stack
yet. Prior to deposition, the equilibrium of the stack is not perturbed, hence
the stresses and strains are those resulting from Problem $(P_{j})$; moreover,
the strain state of block $j+1$ is given by (25). Thus, the incompatibility
between block $j$ and block $j+1$ calculates as:

$\delta_{j}=\mathring{\epsilon}_{x}^{j+1}-\epsilon_{x}^{j,j}=-\frac{b_{j+1}+\nu\mathring{\sigma}_{x}^{j+1}}{E}-\epsilon_{x}^{j,j}.$
(28)

We remark that $\delta_{j}$ depends not only on the pre-stress
$\mathring{\sigma}_{x}^{j+1}$ imparted on the new block by the printing
device, but also, through Problem $(P_{j})$, on the incompatibilities
$(\delta_{1},\ldots,\delta_{j-1})$, _which keep memory of the history of the
construction process_.

In Fig.5 we can see the cartoon of the process. At deposition of the
$(j+1)-$th block, when the latter is still supported by the printing machine,
the fast glue perfects the bond, therefore locking the incompatibility at the
value (28). At this point, the printing device releases the block, and this in
turn imparts both a vertical action and an horizontal action on the underlying
stack, which at this instant is yet unbalanced. Thus, for the case of the fast
glue, the new stress and strain distributions (26) and (27) are the solution
of Problem $(P_{j+1})$, where the incompatibility distribution

$\Delta^{j+1}=(\delta_{1},\ldots,\delta_{j-1},\delta_{j}),$ (29)

with $\delta_{1},\ldots,\delta_{n}$ the same as in the previous step and
$\delta_{n}$ given by (28). The process is then iterated with the deposition
of new blocks.

The direct problem, as formulated above, admits a closed-form solution. The
details of its derivation may be found in Appendix 7.2. In terms of deposition
strains, the incompatibility $\delta_{i}$ is:

$\delta_{i}=\mathring{\epsilon}_{x}^{i+1}-\frac{i-1}{i}\mathring{\epsilon}_{x}^{i}-\frac{\nu
b_{i}}{E}.$ (30)

Using (25), we can rewrite (30) as:

$\delta_{i}=\frac{\mathring{\sigma}_{x}^{i+1}+\nu
b_{i+1}}{E}-\frac{i-1}{i}\frac{\mathring{\sigma}_{x}^{i}+\nu
b_{i}}{E}-\frac{\nu b_{i}}{E}.$ (31)

The finding that the incompatibility $\delta_{i}$ depends only on the last two
elements $\mathring{\sigma}_{x}^{i},\mathring{\sigma}_{x}^{i+1}$ of the
deposition history
$(\mathring{\sigma}_{x}^{2},\ldots,\mathring{\sigma}_{x}^{i},\mathring{\sigma}_{x}^{i+1})$,
was to be expected. Indeed, even in the continuous case the incompatibility
depends on the spatial derivative (the divergence) of the pre-deposition
surface stress [2, 41, 42]. Were this continuous problem to be discretized to
find a numerical solution, the approximation of the divergence a the boundary
would involve the values of the discretized deposition stress at the two grid
points closest to the boundary, in accordance with (31).

Once the incompatibilities are known, the horizontal stresses can be
calculated from (15), an equation that we repeat here for $n=j$:

$\sigma_{x}^{i,j}=\sum_{k=1}^{i-1}\left(E\delta_{k}+\nu
b_{k}\right)-\frac{1}{j}\sum_{k=1}^{j-1}\left(\left(j-k\right)\left(E\delta_{k}+\nu
b_{k}\right)\right).$ (32)

###  3.2 Slow glue

If the glue perfects the bond between the last and second to last blocks very
slowly, we may think that the interface between the $j+1$-th and the $j$-th
blocks is frictionless since the instant of deposition to the instant when the
whole stack finally reaches equilibrium. Since the glue cannot support shear
stresses prior to solidification, which occurs more slowly than the attainment
of equilibrium, the horizontal stress of block $j+1$ needs to vanish,

$\sigma_{x}^{j+1,j+1}=0.$ (33)

The system of equations that govern the elastic state in the stack is

$(\bar{P}_{j+1})\left\\{\begin{array}[]{lll}\displaystyle\sum_{i=1}^{j}\sigma_{x}^{i,j+1}=0,\\\\[10.00002pt]
\displaystyle\sigma_{y}^{i+1,j}-\sigma_{y}^{i,j}=b^{i},\qquad
i=1,...,j,\\\\[10.00002pt]
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\sigma_{y}^{j+1,j+1}=-b^{j+1}},\\\\[10.00002pt]
\epsilon_{x/y}^{i,j+1}=\frac{(\sigma_{x/y}^{i,j+1}-\nu\sigma_{y/x}^{i,j+1})}{E},\qquad
i=1,...,j+1,\\\\[10.00002pt]
{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\displaystyle\epsilon_{x}^{i+1,j+1}-\epsilon_{x}^{i,j+1}=\delta_{i},\qquad
i=1,...,j-1},\\\\[10.00002pt]
\displaystyle\sigma_{x}^{j+1,j+1}=0.\end{array}\right.$ (34)

When the glue perfects the bond at the attainment of equilibrium, the
difference between the horizontal strains of the $j+1$-th and the $j$-th block
remains frozen to its value at equilibrium, and the incompatibility
$\delta_{j}$ is therefore set to:

$\delta_{j}=\epsilon_{x}^{j+1,j+1}-\epsilon_{x}^{j,j+1}.$ (35)

Once the incompatibility is locked, the printing devices starts the deposition
of the next block, and the procedure is iterated until $j\leq n$.

In summary, the process proceeds as follows (following again the cartoon of
Fig.5): prior to deposition, the stack of $j$ blocks is in equilibrium, and
the block $j+1$ is made available geometrically; the block is then
prestretched and maintained in equilibrium by the printing device; at
deposition, however, differently than the case of fast glue, the horizontal
stress in the $j+1$ block drops to zero, and therefore the underlying stack
deforms to reach a new equilibrium only due to the weight of the last block.
Finally, the glue dries and the incompatibility is frozen at the value defined
by (35).

It is immediate to check (details in Appendix 7.3) that the solution to the
equilibrium condition (33) leads to

$\delta_{i}=-\frac{\nu}{E}b_{i}.$ (36)

Equation (36), which coincides with (18), confirms that in the slow glue
protocol, the incompatibility is determined by the local value of $b_{i}$.
This was to be expected: the slow glue solidifies only _after_ equilibrium has
been attained, meaning that prior to equilibrium the glue is frictionless,
which is the same assumption at the basis of (18), where the variables
$\delta_{i}$ were permitted to evolve freely upon attainment of equilibrium.

We conclude this section by observing that (15) with (36) imply that in the
slow glue protocol, the horizontal stresses $\sigma_{x}^{i,j}$ vanishes
everywhere in the stack during the construction process. Note however that,
since the incompatibilities $\delta_{i}$ remain frozen, the removal of the
loads upon completion of the construction results, in general, into a non-
trivial distribution of residual stress.

##  4 Direct problem: examples

To get further insight into the findings of the Sec.3, we are now going to
consider some direct problems, where the final stress distributions is
obtained under different protocols are deduced and compared quantitatively and
qualitatively.

###  4.1 Fast glue

In the fast glue case, one can prescribe both the weight and the horizontal
prestress of each new block. By using the solutions (31) and (32) it is now
possible to compare the outcomes of different loading protocols.

To establish a meaningful comparison, we consider stacks with the same final
number $n$ of blocks and the same total weight. Specifically, we consider
blocks with increasing, constant and decreasing weight,

$b_{i}=iE,\qquad b_{i}=\frac{n+1}{2}E,\qquad
b_{i}=(n+1-i)E,\qquad(i=1,...,n).$ (37)

Note that in all cases above, the total weight of the column is the same but,
as we shall see, the stress distributions are significantly different.

We first compare stacks that are manufactured with
$\mathring{\sigma}_{x}^{j}=0$. The resulting stress distributions are reported
in Fig.6. All the stress distributions have, of course, zero resultant. By
changing the order of the weights one can change the qualitative behavior of
the $\sigma_{x}^{i}$ in the stack. In particular, if the $b_{i}$ are linearly
increasing the distribution of the $\sigma_{x}^{i}$ is also linearly
increasing, if the $b_{i}$ are uniform the $\sigma_{x}^{i}$ still increase,
but not linearly, and finally if the $b_{i}$ are decreasing, the
$\sigma_{x}^{i}$ are non-monotonic.

Figure 6:  Stress distributions resulting from the fast glue protocol in a
stack with $n=300$ layers, with $\mathring{\sigma}_{x}^{j}=0$ and different
distributions of $i$ sharing the same total weight (left); or with $b_{i}$
linear and different prescriptions for the deposition stress
$\mathring{\sigma}_{x}^{j}$ (right).

Another interesting effect results from the comparison of stacks that are
manufactured with the same linear distribution of $b_{i}=i\,E$, but that
differ in terms of deposition stress $\mathring{\sigma}_{x}^{j}$. In this case
one can obtain stacks where the resulting distribution of $\sigma_{x}^{i}$ is
linear if $\mathring{\sigma}_{x}^{j}=0$, but it becomes concave or convex if,
respectively, $\mathring{\sigma}_{x}^{j}>0$ or $\mathring{\sigma}_{x}^{j}<0$,
that is, if the blocks are pre-tensioned or pre-compressed prior to
deposition.

In passing we note that in both all distributions considered above, there are
special points where the various stress distributions intersect, two points in
the case where $\mathring{\sigma}_{x}^{j}=0$ and the $b_{i}$ are changed, and
one such point in the case where $b_{i}$ is linear and the
$\mathring{\sigma}_{x}^{j}$ are changed.

###  4.2 A remark on the (non) controllability of the stress in 3D printing

A key aspect of the fast glue protocol is represented by the fact that the
pre-deposition stress

$\mathring{\sigma}_{x}^{j}:=\sigma_{x}^{j,j-1}$ (38)

and the post-deposition stress $\sigma_{x}^{j,j}$ of the upper block will
differ in general. This happens because after the deposition of the $j$-th
block, the latter will deform together with the underlying stack to reach a
new equilibrium configuration, and therefore its level of stress will
generally change from the pre-deposition value.

Figure 7:  Differences between the pre-deposition stress
$\mathring{\sigma}_{x}^{j}$ (black) and the resulting post-deposition stress
$\sigma_{x}^{j,j}$ (red) in the newly deposited block, in two cases: $a)$:
$\mathring{\sigma}_{x}^{j}=0$ and $b)$: $\mathring{\sigma}_{x}^{j}=jE$. In
both cases the body force is uniform, $b_{i}=E$. The plots display the
dimensionless ratio $\sigma_{x}/E$.

This feature is illustrated in Fig.7 for two different loading scenarios, both
taking place with uniform weights ($b_{i}=E$). In both cases the pre-
deposition stress the post-deposition stress are different, although the
differences seem to be bounded if the number of blocks is very large.

The difference between the pre-deposition stress and the post-deposition
stress has relevant implications on 3D printing. Indeed, this result shows
that during additive manufacturing, the value of the surface stress right
after deposition can not be controlled directly, since it will depend in a
non-local fashion on the whole state of deformation of body. This fact was
already been highlighted in pioneering works on surface growth, specifically
we make reference to the seminal work of Trincher [2]. Further developments of
this aspect were discussed in the setting of continuous incompatible surface
growth in [41].

###  4.3 Slow glue

While in the fast glue protocol the pre-deposition stress and post-deposition
stress can be both different than zero, and differ between each other, in the
slow glue protocol the post-deposition stress $\sigma_{x}^{j,j}$ will always
drop to zero throughout the process, despite any value given to the pre-
deposition stress $\mathring{\sigma}_{x}^{j}$. Indeed, as Fig.6a shows, in the
fast glue case even if the pre-deposition stress $\mathring{\sigma}_{x}^{j}$
is zero, the post-deposition stress $\sigma_{x}^{j,j}$ will generally differ
than zero, whereas it would always be zero in the slow glue case.

This marks a strong difference between fast and slow glue protocols, that may
be readily appreciated in by comparing the incompatibilities stored at the end
of the process. Consider the accretion of a stack with
$\mathring{\sigma}_{x}^{j}=0$ and, once again, three distributions of $b_{i}$
leading to the same total weight. As evidenced in Fig.8, in all cases the
distributions of incompatibilities are different in the fast and slow glue
protocols, above all for the first layers. Eventually however, for $b_{i}$
constant or decreasing, the incompatibilities would tend asymptotically to a
common value if the number of layers is very large, whereas the difference in
incompatibilities would remain constant in for $b_{i}$ linearly increasing.

Figure 8:  Comparison between the incompatibilities resulting from fast and
slow protocols, for walls that are manufactured with
$\mathring{\sigma}_{x}^{j}=0$. Here $\nu=0.3$.

Differences in terms of residual stress (that is, the stress distribution in
the stack upon removal of all external loading) may be appreciated in all
cases covered in Fig.8 in the plots of Fig.9. Upon calculating the
incompatibility, the stress distributions may be computed with $b_{i}=0$ from
(15). The results show that if the stacks that are manufactured with
$\mathring{\sigma}_{x}^{j}=0$, the effects of fast/slow glue are non-
negligible, but essentially of quantitative nature.

Figure 9:  Comparison between the residual stresses arising from fast and slow
protocols, for towers manufactured with $\mathring{\sigma}_{x}^{j}=0$. Here
$\nu=0.3$.

##  5 The inverse problem: programming residual stress

Thus far we have explored the problem of finding the residual stress arising
from a given deposition protocol. We now turn our attention to the following
inverse problem: find the deposition protocol

$\Pi=(\mathring{\sigma}^{2},\ldots,\mathring{\sigma}^{n};b^{1},\ldots,b^{n})$
(39)

which delivers a desired residual stress state. To avoid confusing the target
stress state with the current state of stress of the growing stack, we denote
the former by

$T_{x}^{n}=(\tau_{x}^{1},\ldots,\tau_{x}^{n}),\qquad\text{s.t.}\qquad\sum_{i=1}^{n}\tau_{x}^{i}=0.$
(40)

which is needed to achieve a desired residual stress state.

We remark that the target state of stress is solely sourced by the
incompatibilities, therefore if weights are applied on the stack during
manufacturing, the attainment of the target state is achieved by suppressing
all the weights. We also remark that by removal of the weights, the vertical
stresses $\sigma_{y}^{i,n}$ vanish, therefore the target state is defined
solely by a self-balanced distribution of horizontal stresses.

By using the definition (4) and the constitutive equations (2) (or by solving
(10)) we find that the incompatibilities needed to attain the target state
are:

$\delta_{i}=\frac{\tau_{x}^{i+1}-\tau_{x}^{i}}{E}\qquad i=1,...,n-1.$ (41)

Thus, our problem reduces to finding the deposition protocol that results in a
given distribution of incompatibilities $(\delta_{1},\ldots,\delta_{n-1})$.

The protocol depends on whether we are using a fast glue or a slow glue, and
on the loading applied to the stack during the deposition. We remark that
albeit the target state is unloaded, weights may still be used in the
manufacturing process to produce incompatibilities, but of course such weights
would be removed when the stack is complete. Weights are, of course, the only
possible type of control in the slow glue case, since here the horizontal pre-
deposition stress always drops to zero.

###  5.1 Fast glue: inverse problem

Consider the deposition of a heavy stack with the fast-glue protocol. By
solving (31) where we now treat $\delta_{i}$ and $b_{i}$ as prescribed and
$\mathring{\sigma}_{x}^{i}$ as unknown, and bearing in mind that
$\mathring{\sigma}_{x}^{1}=0$, we obtain

$\mathring{\sigma}_{x}^{i}=\frac{1}{i-1}\sum_{k=1}^{i-1}\left(\nu\left(2kb_{k}-b_{k}-b_{k+1}\right)+Ek\delta_{k}\right),\qquad
i>1.$ (42)

The meaning of (42) is that to produce a desired distribution of
incompatibility $\delta_{i}$ in a heavy stack with given weights $b_{i}$, one
needs to provide the pre-deposition horizontal stress
$\mathring{\sigma}_{x}^{i}$ as dictated by (42). At the end of the process,
the weights would be removed and the only source of stress in the stack are
the incompatibilities. Note that if the stack is directly manufactured with
$b_{i}=0$, the final distribution of stress would coincide precisely with the
target residual stress state (40).

As a first application, we manufacture a stack with sinusoidal distribution of
residual stress,

$\tau_{x}^{i}=\sin\left(\frac{2\pi i}{n}\right)$ (43)

which is clearly self-balanced. The target residual stress is represented in
Fig.10a. Once the incompatibilities are calculated by (41), by making use of
(42) we can determine the required deposition protocol.

For illustration, we show that the same target may be achieved by two
different processes, involving both heavy and massless stacks. In particular
we manufacture the stack:

  * 1)

through the deposition of pre-stretched massless layers;

  * 2)

through the deposition of pre-stretched heavy layers with weight increasing as
$b_{i}=iE$.

Figure 10:  $a)$ target residual stress $\tau_{x}^{i}$ to be achieved through
the fast glue protocol. $b)$ Required deposition pre-stress to achieve the
target in the case of massless weights, and $c)$ required deposition pre-
stress in the case of linearly increasing weights. Here $\nu=0.3$.

The outcomes are illustrated in 10 for a stack of $n=20$ blocks. In Fig.10b we
illustrate the case $1)$, in Fig.10c we illustrate case $2)$. The required
deposition stress seems intuitively consistent with the final stress
distribution in the case $b_{i}=0$, but it is rather different from the target
stress in the case $b_{i}=iE$.

An interesting remark relative to (42) is that the distribution of deposition
stresses $\mathring{\sigma}_{x}^{i}$ required to produce a given
incompatibility is independent on the weights, if these satisfy

$b_{k}+b_{k+1}=2kb_{k},$ (44)

giving

$\tilde{b}_{i}=b_{1}\frac{2^{i-1}}{\Gamma(i)}{\prod_{k=0}^{i-2}(\tfrac{1}{2}+k)},$
(45)

where $\Gamma(i)$ is Euler’s Gamma function, with $b_{1}$ arbitrary. For
$b_{1}=0$ we obtain trivially the case of weightless blocks, but it is
interesting to note that even for $b_{1}\neq 0$ the deposition stress required
to produce a certain target residual stress would not depend on the weight
distribution.

###  5.2 Slow glue: inverse problem

In the slow glue paradigm the condition $\sigma_{x}(i,i)=0$ results into the
constraint $E\delta_{i}+\nu b_{i}=0$ between weights and incompatibilities,
therefore during deposition that weights may be used to obtain a target
residual stress $\tau_{x}^{i}$. Indeed, we simply need to prescribe weights
according to

$b_{i}=\frac{\tau_{x}^{i}-\tau_{x}^{i+1}}{\nu}.$ (46)

This makes the treatment of the inverse problem for the slow-glue case
definitely simpler than the fast-glue case. Moreover, at least conceptually,
(46) shows that any distribution of $\tau_{i}$ can be targeted through the
deposition of weights $b_{i}$.

There is, however, a subtlety in the practical implementation of this method.
Indeed, since $\tau_{x}^{i}$ must have zero average, it means that
$\tau_{x}^{i}$ cannot be nether constant, nor strictly increasing or
decreasing on the stack. This implies that there will be zones, where
$\tau_{x}^{i}$ is increasing, where one would need to deposit blocks with a
negative weight.

##  6 Conclusions

In this work we have formulated a discrete model that captures, in a minimal
setting where the material is treated as homogeneous, linearly elastic and
isotropic, and where plasticity and thermal effects are neglected, some
fundamental features of additively manufactured solids. By introducing a
simplified kinematics that allows for the onset of incompatible deformations
between the layers (schematized as elastic blocks) of an additively
manufactured stack, we have studied the role of deposition prestress applied
to the blocks, and of the weight of the blocks, on the resulting state of
incompatibility. From here, we have computed the final residual stress
patterns, and we could single out the factors behind qualitative and
quantitative differences emerging in their distributions.

To capture with minimal complexity the role played an “ideal glue” between the
blocks - which is required to maintain incompatible deformations, therefore
preventing the blocks to slide relative to each other - we have devised two
extreme behaviors: one, called “fast glue”, where the bond between the
prestressed block and the underlying stack occurs while the new block is still
maintained in equilibrium by some invisible “printing device”; the other,
called “slow glue”, where the block is first released and is left free to find
an equilibrium configuration together with the underlying stack, before the
glue perfects the bond.

The model proposed in this work is a very elementary approximation of the
complex aspects that occur in the process of additive manufacturing, for
example the role of the glue is overly simplified and the deformations are
constrained in a very special way, however its analytical simplicity allows to
obtain closed form results, that are very transparent in physical terms. In
particular, the model retains many features of richer continuous models, like
in particular the presence of a new type of boundary condition, that is the
control of “surface stress”, that is not contemplated in non-growing solids
[41, 42, 34].

The main findings of the model have been illustrated by comparing effects due
to different deposition protocols, showing that some controls (for example the
type of glue) will play a quantitative effect in the final stress distribution
of the stack, whereas other controls (for example the deposition surface pre-
stress) will play a qualitative effect. These results pave the way for new
studies on the fundamental role of the adhesive properties of the growth
surface.

The last part of the manuscript is devoted to the formulation of a
technologically important inverse problem, consisting in finding the
deposition protocol that is required to achieve a desired distribution of
residual stress, but also heterogeneous elastic properties, in an additively
manufactured stack [39, 8, 18].

The proposed model should be conceived as a template, that has the advantage
to highlight with physical and analytical transparency the non-trivial
mechanics involved in additive manufacturing of prestressed solids.

##  7 Appendix

###  7.1 Derivation of (15)

This section illustrates the steps to arrive at the determination of the
relation (15). Starting from the first relation of system (13), together with
the constitutive equations (2) and (4) and taking into account that the
problem along the vertical direction is solved by:
$\sigma_{y}^{i}=-\sum_{s=i}^{n}b_{s}$. The problem along the horizontal
direction can be expressed by the following system:

$\left\\{\begin{array}[]{lll}{\displaystyle\sum_{i=1}^{n}\sigma_{x}^{i}=0}\\\
\\\ {\epsilon_{x/y}^{i}=\frac{(\sigma_{x/y}^{i}-\nu\sigma_{y/x}^{i})}{E}\qquad
i=1,...,n}\\\ \\\
{\displaystyle\epsilon_{x}^{i+1}-\epsilon_{x}^{i}=\delta_{i}\qquad
i=1,...,n-1}\\\ \end{array}\right.$ (47)

Expressing the third relation of the system (47) through the constitutive
relation we can write:

$\sigma_{x}^{i+1}-\sigma_{x}^{i}=E\delta_{i}+\nu b_{i}$ (48)

From which we can write:

$\sigma_{x}^{i+1}=\sigma_{x}^{1}+\sum_{k=1}^{i}E\delta_{k}+\sum_{k=1}^{i}\nu
b_{k}$ (49)

Using now the first relation of the system (47) we determine the following
relation:

$\sigma_{x}^{1}=-\frac{1}{n}\sum_{i=2}^{n}\sum_{k=1}^{i-1}E\delta_{k}-\frac{1}{n}\sum_{i=2}^{n}\sum_{k=1}^{i-1}\nu
b_{k}$ (50)

Replacing the (50) in (49) we get:

$\sigma_{x}^{i+1}=\sum_{k=1}^{i}E\delta_{k}-\frac{1}{n}\sum_{a=2}^{n}\sum_{k=1}^{a-1}E\delta_{k}+\sum_{k=1}^{i}\nu
b_{k}-\frac{1}{n}\sum_{a=2}^{n}\sum_{k=1}^{a-1}\nu b_{k},$ (51)

from which we can write the following relation:

$\sigma_{x}^{i}=E\left(\sum_{k=1}^{i-1}\delta_{k}-\frac{1}{{n}}\sum_{a=2}^{{n}}\sum_{k=1}^{a-1}\delta_{k}\right)+\nu\left(\sum_{k=1}^{i-1}b_{k}-\frac{1}{{n}}\sum_{a=2}^{{n}}\sum_{k=1}^{a-1}b_{k}\right)$
(52)

rearranged it we can write:

$\sigma_{x}^{i}=\sum_{k=1}^{i-1}(E\delta_{k}+\nu
b_{k})-\frac{1}{{n}}\sum_{k=1}^{{n}-1}(({n}-k)(E\delta_{k}+\nu b_{k}))$ (53)

###  7.2 Derivation of (31)

This section illustrates the steps to arrive at the determination of the
relation (31). Starting from the problem $P_{j}$, the solution of the problem
along the vertical direction is that expressed by
$\sigma_{y}^{i,j+1}=-\sum_{k=i}^{j+1}b_{k}$. The problem along the horizontal
direction can be expressed by the following system:

$\left\\{\begin{array}[]{lll}{\displaystyle\sum_{i=1}^{j}\sigma_{x}^{i,j}=0}\\\
\\\
{\epsilon_{x/y}^{i,j}=\frac{(\sigma_{x/y}^{i,j}-\nu\sigma_{y/x}^{i,j})}{E}\qquad
i=1,...,j}\\\ \\\
{\displaystyle\epsilon_{x}^{i+1,j}-\epsilon_{x}^{i,j}=\delta_{i}\qquad
i=1,...,j-1}\\\ \end{array}\right.$ (54)

Expressing the third relation of the system (54) through the constitutive
relation we can write:

$\sigma_{x}^{i+1,j}-\sigma_{x}^{i,j}=E\delta_{i}+\nu b_{i}$ (55)

From which we can write:

$\sigma_{x}^{i+1,j}=\sigma_{x}^{1,j}+\sum_{k=1}^{i}E\delta_{k}+\sum_{k=1}^{i}\nu
b_{k}$ (56)

Using now the first relation of the system (54) we determine the following
relation:

$\sigma_{x}^{1,j}=-\frac{1}{j}\sum_{i=2}^{j}\sum_{k=1}^{i-1}E\delta_{k}-\frac{1}{j}\sum_{i=2}^{j}\sum_{k=1}^{i-1}\nu
b_{k}$ (57)

Replacing the (57) in (56) we get:

$\sigma_{x}^{i+1,j}=\sum_{k=1}^{i}E\delta_{k}-\frac{1}{j}\sum_{l=2}^{j}\sum_{k=1}^{l-1}E\delta_{k}+\sum_{k=1}^{i}\nu
b_{k}-\frac{1}{j}\sum_{l=2}^{j}\sum_{k=1}^{l-1}\nu b_{k}$ (58)

Using the constitutive relation we arrive at the relation

$\epsilon_{x}^{j,j}=\frac{1}{E}(\sigma_{x}^{j,j}-\nu\sigma_{y}^{j,j})=\sum_{k=1}^{j-1}\delta_{k}-\frac{1}{j}\sum_{l=2}^{j}\sum_{k=1}^{l-1}\delta_{k}+\frac{1}{E}\sum_{k=1}^{j-1}\nu
b_{k}-\frac{1}{jE}\sum_{l=2}^{j}\sum_{k=1}^{l-1}\nu b_{k}+\frac{\nu b_{j}}{E}$
(59)

which describes the horizontal deformation of the $j-th$ block when in the
tower there are $j$ blocks. From (59) we come to the incompatibility
$\delta_{j}$:

$\delta_{j}=\mathring{\epsilon}_{x}^{j+1}-\sum_{k=1}^{j-1}\delta_{k}+\frac{1}{j}\sum_{k=1}^{j-1}(j-k)\delta_{k}-\frac{1}{E}\sum_{k=1}^{j-1}\nu
b_{k}+\frac{\nu}{jE}\sum_{k=1}^{j-1}(j-k)b_{k}-\frac{\nu b_{j}}{E}$ (60)

The (60) can be rewritten as:

$\delta_{j}=\mathring{\epsilon}_{x}^{j+1}-\frac{1}{j}\sum_{k=1}^{j-1}k\delta_{k}-\frac{\nu}{jE}\sum_{k=1}^{j-1}kb_{k}-\frac{\nu
b_{j}}{E}$ (61)

We can further rewrite the formulation by writing:

$\begin{split}\delta_{j}&=\mathring{\epsilon}_{x}^{j+1}-\frac{1}{j}(j-1)\delta_{j-1}-\frac{1}{j}\sum_{k=1}^{j-2}k\delta_{k}-\frac{\nu}{jE}\sum_{k=1}^{j-1}kb_{k}-\frac{\nu
b_{j}}{E}\\\
&=\mathring{\epsilon}_{x}^{j+1}-\frac{j-1}{j}(\mathring{\epsilon}_{x}^{j}-\frac{1}{j-1}\sum_{k=1}^{j-2}k\delta_{k}-\frac{\nu}{(j-1)E}\sum_{k=1}^{j-2}kb_{k}-\frac{\nu
b_{j-1}}{E})-\frac{1}{j}\sum_{k=1}^{j-2}k\delta_{k}-\frac{\nu}{jE}\sum_{k=1}^{j-1}kb_{k}-\frac{\nu
b_{j}}{E}\\\
&=\mathring{\epsilon}_{x}^{j+1}-\frac{j-1}{j}\mathring{\epsilon}_{x}^{j}+\frac{1}{j}\sum_{k=1}^{j-2}k\delta_{k}+\frac{\nu}{jE}\sum_{k=1}^{j-2}kb_{k}+\frac{j-1}{j}\frac{\nu
b_{j-1}}{E}-\frac{1}{j}\sum_{k=1}^{j-2}k\delta_{k}-\frac{\nu}{jE}\sum_{k=1}^{j-1}kb_{k}-\frac{\nu
b_{j}}{E}\\\
&=\mathring{\epsilon}_{x}^{j+1}-\frac{j-1}{j}\mathring{\epsilon}_{x}^{j}+\frac{\nu}{jE}\sum_{k=1}^{j-2}kb_{k}+\frac{j-1}{j}\frac{\nu
b_{j-1}}{E}-\frac{\nu}{jE}\sum_{k=1}^{j-1}kb_{k}-\frac{\nu b_{j}}{E}\\\
&=\mathring{\epsilon}_{x}^{j+1}-\frac{j-1}{j}\mathring{\epsilon}_{x}^{j}+\frac{j-1}{j}\frac{\nu
b_{j-1}}{E}-\frac{j-1}{j}\frac{\nu b_{j-1}}{E}-\frac{\nu b_{j}}{E}\end{split}$
(62)

Simplifying the (62) we arrive at

$\delta_{j}=\mathring{\epsilon}_{x}^{j+1}-\frac{j-1}{j}\mathring{\epsilon}_{x}^{j}-\frac{\nu
b_{j}}{E},$ (63)

a relation which expresses the incompatibility in the case of fast glue.

###  7.3 Derivation of (36)

To prove that in the slow glue protocol the incompatibility arising in the
process is determined only by the weight of the underlying block, we recall
that in the slow glue case the horizontal stress of the last block has to be
zero at equilibrium. By imposing $\sigma_{x}^{i,i}=0$ for all $i\geq 2$, we
obtain the system

$\begin{array}[]{lll}\sigma_{x}^{1,1}=0\\\ \sigma_{x}^{2,2}=(E\delta_{1}+\nu
b_{1})=0\\\ \sigma_{x}^{3,3}=(E\delta_{1}+\nu b_{1})+2(E\delta_{2}+\nu
b_{2})=0\\\ \sigma_{x}^{4,4}=(E\delta_{1}+\nu b_{1})+2(E\delta_{2}+\nu
b_{2})+3(E\delta_{3}+\nu b_{3})=0\\\ \dots\end{array}$ (64)

Therefore, the result $E\delta_{i}+\nu b_{i}=0$ for all $i\geq 1$ follows at
once.

## Acknowledgements

SM thanks Regione Lazio for funding the project H-S3D (DTC 2021-2023 - TE1
Centro di Eccellenza, CUP F85F21001090003) and European Union for funding the
project Cultural Heritage Active Innovation for Next-Gen Sustainable Society -
CHANGES - PE00000020 PNRR – NextGenerationEU (CUP: F83C22001650006).

DR and SM thanks Regione Lazio for funding the projects: 3DH-solutions
(Progetti di Gruppi di Ricerca 2020, CUP F85F21001530009).

GZ gratefully acknowledges the support of GNFM (Gruppo Nazionale di Fisica
Matematica) of the INdAM F. Severi.

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