A mathematical model of thermoplastic elastomers for analysing the topology of microstructures and mechanical properties during elongation

In this study, a mathematical model based on graph theory is developed to analyse the deformed structures and mechanical properties of thermoplastic elastomers (TPEs) using ABA-type triblock copolymers. TPEs exhibit a network structure formed by bridge chains; deformation of this network structure causes stress. During the deformation of TPEs, domain breakage and coalescence occur, accompanied by topological changes in the chains, such as conformational transitions between the bridge and loop chains. By employing the mathematical concepts of harmonic realization of graphs in the physical space and the tension tensor to quantify the stress in the bridge-chain network structure, an effective method for analysing topologicalchanges in microstructures caused by elongation is proposed. As an application of this method, optimal geometric structures of block copolymers with desired functionalities can be determined.

In this study, a mathematical model based on graph theory is developed to analyse the deformed structures and mechanical properties of thermoplastic elastomers (TPEs) using ABA-type triblock copolymers.TPEs exhibit a network structure formed by bridge chains; deformation of this network structure causes stress.During the deformation of TPEs, domain breakage and coalescence occur, accompanied by topological changes in the chains, such as conformational transitions between the bridge and loop chains.By employing the mathematical concepts of harmonic realization of graphs in the physical space and the tension tensor to quantify the stress in the bridge-chain network structure, an effective method for analysing topological

Introduction
Thermoplastic elastomers (TPEs) [1,2] are composite materials consisting of block copolymers.They possess the processing advantages of thermoplastics and the performance characteristics of elastomers.These properties make TPEs valuable in both academia and industries.In particular, researchers are interested in the hierarchical structures and physical properties of TPEs as these properties are influenced by the underlying structures.One type of block copolymer that has gained significant attention is the ABA-type block copolymer.This type of block copolymer exhibits microphase separation and two molecular conformations, i.e. bridge and loop chains [3][4][5][6][7][8].The physical properties of the microphase-separated structure are determined by the structures of both phases, which possess different physical properties.In the molecular chain structure, bridge chains are essential as they act as links between two distinct domains.Therefore, it is critical to accurately characterize the structural attributes of the phase-separated structure as well as the molecular conformations and their influence on the physical properties of TPEs.
Experimental analysis of individual molecular states in TPEs can be challenging owing to the limitations of observation systems.Consequently, usage of simulations is required [9][10][11][12][13][14].A previous study [9] used coarse-grained molecular dynamics (CGMD) [15,16] to study TPEs.To interpret the results of CGMD simulations, graph theory [17] was employed to analyse the network structure of the bridge chains connecting the hard spherical domains within the ABA-type chain structure.This analysis revealed that structural changes within the network affected stress alteration during stretching, demonstrating the effectiveness of graph theory in representing stress variation in TPEs.Other studies have also used graph theory to analyze the network structure of molecules, such as cross-linked rubber materials, and this research direction is a natural extension of our previous work.
Owing to the high computational cost of CGMD simulations, it is necessary to develop a mathematical model that captures the fundamental elements of graph theory to establish a direct relation between stress changes and the graph structure.The aim of this model is to advance the analysis of TPEs.A theory of quantifying stress within a network-based graph structure has been previously established using a tension tensor (TT) [18].By integrating the TT with modifications to the network structure as determined by graph theory, a comprehensive investigation of TPE deformation can be conducted using the mathematical model.
In this paper, we propose a mathematical model and methods for rational synthesis process of TPE-based predicting structures that exhibit expected mechanical properties in the elongation process, such as elasticity, stretchability and recovery from repeated deformations.Mathematical models are simpler than molecular dynamics simulations and involve lower computational costs.We developed a mathematical model that establishes relations between geometric data and the dynamic properties of TPEs.Furthermore, our model exhibits good qualitative correlation with CGMD simulations, making it an effective alternative to computationally expensive simulations.Our model reduces the involved computational costs as well as increasing the degree of freedom of the initial structure and iterated deformation, such as the elongation axis direction and number of elongation cycles, without any limitations.Moreover, this increase in the degree of freedom allows for a better understanding of the deformation limits of TPEs.Owing to its mathematical nature, this model illustrates how each material property is related to the geometric or topological structures of TPEs.As an application of our model, we identified two types of stress-strain (SS) curves based on the initial structures: a rapidly rising curve and slowly rising curve.The former is associated with soft and elongated properties, whereas the latter corresponds to high deformation In §2, we present the results of CGMD simulations, which are comparable with the results of our mathematical model.Specifically, we provide information on the stress, number of bridge chains (edges), number of domains (nodes) and length of bridge chains (edges).In §3, we introduce our mathematical model, which is based on graph theory.We model bridge-chain network structures using simple qualified ordered network structures, such as the body-centred cubic (BCC) structure.In addition, we employ harmonic realizations of topological graphs to achieve stable configurations that can withstand local deformations and represent the graph associated with the ABA-type triblock copolymer in the three-dimensional space.We ensure that there are sufficient similarities between the results of CGMD simulations and mathematical computations introduced in §2 for the initial BCC structure.Using our model, we calculate the SS curves for various initial structures of the graph model and investigate how these curves depend on geometric structures.In §4, we present the conclusions of this research.

Review of coarse-grained simulation results and requirements for the mathematical model (a) TPE simulation results
Numerous studies have been conducted on the simulation of TPEs [9][10][11][12][13][14] using CGMD [15].Our aim is to construct a mathematical model that captures the characteristics observed in CGMD simulations.Although various features of TPEs have already been reported through CGMD simulations [9][10][11][12][13][14], we summarize the key features and construct a mathematical model that qualitatively reproduces the parameters representing these features.
To perform CGMD simulations of TPEs during elongation, we based our approach on previous studies [9,10,12] and calculated the parameters representing the features of deformed TPEs.An ABA-type block copolymer with a microphase-separated structure was selected as the target material, and the elongation of this structure was studied.To ensure a meaningful comparison between the results obtained from the mathematical model and CGMD simulations, matching the simulation conditions was crucial.The mathematical model operates under equilibrium conditions and can approximate equilibrium elongation.Therefore, the elongation rate in the CGMD simulation was set to 2.84436 × 10 −5 σ /τ , which represents the slowest possible rate for the CGMD simulation until reaching a maximum strain of 6.0.The other simulations conditions were the same as those in the previous study [9]; the simulations were performed using the COGNAC simulator [19] in OCTA [20].
Figure 1a illustrates the phase-separated structure of the TPE used in our CGMD simulations.In this structure, the A subchain forms a spherical domain that is arranged in a BCC lattice, while the B subchain is present within the matrix.The spherical A domain constitutes the hard domain, with A subchains anchored to it.ABA-type chains exhibit two types of chain conformations: loop and bridge chains, as illustrated in (figure 1b).A loop chain is a chain in which the two terminal A subchains are contained within the same A domain, while in contrast, a bridge chain is a chain in which the two terminal A subchains are contained within separate domains.These two types of chains are depicted in figure 1c; some bridge chains may experience stretching during the elongation process.
Initially, two separate A domains are connected via bridge chains.During uniaxial elongation, the applied strain causes elongation of the bridge chains and the domains undergo elongation owing to the elongated chains.Consequently, domain breaking occurs.However, the presence of numerous bridge chains leads to the formation of a network structure, which plays a crucial role in the propagation of stress within the TPE.Thus, to study stress during elongation, the bridge chain network must be analysed.To analyse the bridge chain network, we employed graph theory, which allowed us to study changes in the network through changes in the graph representation.In this representation, domains and bridge chains were represented as nodes and edges, respectively.Figure 2a illustrates the correspondence between the model network structure and graph representation.Similarly, in our CGMD simulations, the structure was also converted into a graph, as displayed in figure 2b.Each region surrounded by gray surface in the upper inset of figure 2b was designated as an A domain; the A domains were numbered and assigned as nodes.An exhaustive examination of all bridge chains within the system was conducted.Whenever a bridge chain linking two domains was identified, an edge connecting the respective nodes was established, resulting in the graph shown in the bottom inset of figure 2b.As demonstrated in a previous study [9], graph theory is a powerful tool to study the network structure of TPEs.By understanding the graph structure, we can gain a deep understanding of TPEs.

(b) Parameters describing the features of deformed TPEs
We performed one cycle of elongation-compression simulation and estimated the corresponding parameters obtained from the mathematical model based on graph theory.Figure 3 displays the calculated parameters, including the stress, number of domains (nodes), length of the edge (B subchain), and number of bridge chains (edges).As the strain increased, the stress also increased.However, at a strain of approximately 3.5, the stress exhibited plateaus.Similarly, as the strain decreased, the stress also decreased, becoming smaller than the initial stress in the undeformed structure.The number of domains decreased until a strain of approximately 1.5 and increased at strains above 1.5.In the reverse process, with decreasing strain, the number of domains monotonically decreased.The number of bridge chains exhibited a similar change as the number of domains.The lengths of the B subchains increased with increasing strain and reached a maximum value.The variations in the four parameters displayed in figure 3 can be attributed to the coalescence and breaking of domains.Figure 4 presents a plot of the lengths of all B subchains, where the curves for two typical chains illustrated in figure 4a,b are represented by green and blue lines, respectively.The chain in figure 4a shrinks at a strain of approximately 2, resulting in breaking of the right domain.The broken A domain subsequently joins a neighbouring A domain, and the chain continues to stretch.By contrast, the chain in figure 4b is stretched from a strain of 4 to 7, during which a conformational change from the loop chain to a bridge chain can be observed.The chain in figure 4b is also stretched from a strain of 0 to 1.5 as a bridge chain, exhibiting similar conformational changes as the chain in figure 4a at a strain of approximately 1.5.Owing to the coalescence of the A domain, the loop chain shown in figure 4b is formed at a strain of 4. These observations indicate that domain coalescence and breaking occur because of the stretching of bridge chains and conformational changes from bridge to loop chains or from loop to bridge chains also occur during TPE deformation.

Results and discussion (a) Description of the mathematical model of TPEs
A mathematical model for TPEs with ABA-type chains was developed.As described in §2, a topological graph representation of the bridge chain network was used, with nodes representing domains and edges representing bridge chains.In the CGMD simulation, a loop chain sometimes changed to a bridge chain during domain decomposition; therefore, this conformational change must be incorporated into the model.Furthermore, to analyse the mechanical properties of the obtained abstract graph, it was necessary to map it onto physical space and assign lengths to the edges.The three-dimensional configurations of the nodes were also determined.A mathematical model satisfying the above conditions was proposed, as outlined below.
Let us assume that the initial structure was triply periodic in the x, y and z directions, representing the most stable structure under local deformations.The harmonic realization of a triply periodic graph [21] was employed to establish this initial configuration.Unlike in the CGMD simulation, the initial structure could be any crystal lattice; a triply periodic graph with an arbitrary finite graph as its initial fundamental structure was used.It should be noted that each node had loop chains that could transform into bridge chains during elongation.
The initial structures used in the mathematical model were based on different crystal lattice configurations, namely, BCC, cubic (Cube), face-centred cubic (FCC), diamond (Dia), K4 and double diamond (WDia).In the BCC structure obtained through the CGMD simulation, >97% of the structure consisted of bridge chains connecting to the first-or second-nearest domains, with 30% of the bridge chains connecting to the second-nearest domains.Therefore, bridge chains connecting to the first-and second-nearest domains were introduced into the BCC structure.By contrast, the remaining structures (Cube, FCC, Dia, K4 and WDia) only included bridge chains between the first-nearest domains.To ensure comparability between the structures during deformation process, we standardized the length of the bridge chains (edges) to ensure that the number of bridge chains or edges per unit volume was uniform across all structures.Consequently, the lengths of the bridge chains (edges) between the first-nearest domains in the BCC, Cube, FCC, Dia, K4, and WDia structures were set to 0.866, 0.598, 0.854, 0.453, 0.336 and 0.570, respectively.
In our model, the initial structure of the graph was elongated to analyse the deformed microphase separation, including the coalescence and breaking of domains.The elongation was mathematically described as an affine transformation applied to the nodes of the triply periodic graph in space while preserving the volume of the unit cell.The elongation direction was arbitrary.As the nodes shifted, the edges accordingly stretched or shrunk, representing the elongation of the bridge chains and enabling their length computation.
The energy of a graph is defined as the square sum of the lengths of its edges [21].It is equal to the elastic energy of a spring model, where the rest length of each edge is set to 0 and the spring constant k is set to 2. Energy is a scalar quantity and does not have a specific direction.However, in case of elongated TPEs, the elastic energy generates stresses in specific directions.To incorporate the concept of direction into energy, the concept of TT [18] is introduced.The TT is defined as the sum of the tensor product of each edge with itself; it is calculated by summing over all edges e in the graph: TT = e ⊗ e.
In the three-dimensional case, the tensor product is defined as while in the two-dimensional case, it is defined as The energy of an edge is given by 1/2 k||e|| 2 , and the trace of the TT is equal to the total energy ( e 2 ) of the graph.Hereafter, to represent the Cauchy stress tensor, we use the TT normalized by volume, T v , which is obtained by the volume corresponding to the periodic area occupied by the graph.
In [18], it is demonstrated that the Cauchy stress tensor Σ is given by As the TPE preserves its volume in our model, it is appropriate to use the traceless part of Σ: where , which is called the deviatoric stress tensor.The TT is symmetric by definition and can be diagonalised using rotation matrices to the following: Here, τ 1 , τ 2 and τ 3 represent the first, second and third eigenvalues, respectively, with ν 1 , υ 2 and υ 3 denoting the corresponding eigenvectors, respectively.
The TT represents directional stress as an ellipsoid.In the two-dimensional case, the normalized TT T e is obtained by dividing TT by the total number of edges.The equation for the corresponding ellipsoid (figure 5b) is given by This ellipsoid consists of a major axis (v 1 ) and minor axis (v 2 ), which correspond to the long and short radius, respectively.Domain coalescence or breakage in a TPE changes the topology of the underlying graph.In this model, when the edges shrink, two connected nodes are merged when the energy of an edge falls below the threshold E coale = 0.24, where the natural distance between the connected nodes is normalized to 1.0.The coalescence of the two domains induces a conformational change from bridge to loop chains.By contrast, breaking of a node (domain) occurs when the energy at the node, which is the sum of squares of the one-unit-subtracted lengths of all edges emerging at the domain and longer than length 1.0, exceeds the threshold E break = 10.0.If the breaking domain contains loop chains, the domain break induces a conformational change from loop to bridge chains.
At the breaking of the node (domain D), the local TT T local (D) is defined by the square sum of the TT of all edges emerging at D, that is, The breaking of domain D produces two domains-A and B. Domain A is located at a distance of 0.5 away from the original position of domain D in the direction of the first eigenvector v 1 of T local (D), while domain B is located at a distance of 0.5 away from the original position of domain D in the direction opposite to v 1 .Domains A and B remain disconnected if there are no loop chains at D. However, when there are loop chains at D, we assume that 60% of the loop chains transform into bridge chains between Domains A and B, 20% of the loop chains are located in Domain A, and the remaining 20% of the chains are located in Domain B (figure 5a).Assuming that the chain ends independently belong to either Domain A or B with equal probability, the probability of transitioning to a bridge chain is 50%.In CGMD simulations and real TPEs, often multiple loop chains exist in the initial configuration.With two loop chains, the probability of at least one of them transforming into a bridge chain reaches 75%.Meanwhile, in our model, only one loop chain is present in the initial configuration; therefore, we increased the probability of transitioning to a bridge chain to 60% to ensure correspondence with CGMD simulations and real TPEs.
The stress can be computed using the TT.In the case of uniaxial elongation, the direction of stress is almost equal to the direction of the first eigenvector of the TT; the stress (σ ) is calculated as follows: As described above, we constructed a mathematical model for representing the coalescence and breaking of domains during elongation process based on graphs.To validate our model, we conducted a feasibility study using a two-dimensional model.Snapshots of the model system are presented in figure 6.Each subfigure displays a network of bridge chains (represented as straight lines) and loop chains (represented as circles).In addition, a red ellipsoid defined by the first and second eigenvectors is displayed in each subfigure.The network structure during the stretching process is presented in figure 6a-h, while that during compression is presented in figure 6h-j.It should be noted that the maximum strain in this calculation was 4.46.As illustrated in figure 6e,f, domain coalescence occurred, resulting in a decrease in the number of domains.Figure 6g-i demonstrate the merging and splitting of domains, indicating that these processes were accurately represented in our model.During stretching, the ellipse representing the TT elongated.However, when splitting occurred, as illustrated in figure 6g-i, the major axis length of the ellipse decreased.This indicates that the stress decreased as the stretched edges diminished owing to domain splitting.It should be noted that similar calculations and processes can be performed in the three-dimensional case, although the resulting figures may appear more complex.

(b) Elongation of the TPE model for the BCC structure and comparison with CGMD simulation results
The BCC structure was used to demonstrate the deformation of TPEs using the mathematical model.Figure 7 presents the results for the stress, number of nodes (domains), length of edges (bridge chains), and number of edges (bridge chains), which are the same parameters analysed  in the CGMD simulation.Notably, to obtain the stress, number of domains, and number of bridges, 10 separate calculations were performed for one structure in various directions; the results represent the average values.The presented results correspond to one cycle of elongationcompression.However, for the edge lengths, the results are presented only for elongation.
As illustrated in the first panel of figure 7, the stress monotonically increased until a strain of approximately 2.3 was reached.Thereafter, the rate of stress increase decreased, and at a strain of 2.9, the stress decreased.As illustrated in the second panel of figure 7, the domains in the first-nearest neighbour relation initiate coalescence near a strain of 2.3, leading to a decrease in the number of domains.Subsequently, domain breaking occurred at strains of greater than 2.4, resulting in an increase in the number of nodes (domains).For larger strains, both coalescence and breaking occurred, but the number of domains continued to increase.Similar trends were also observed in the results for the number of edges (bridge chains).The number of bridges remained constant until a strain of 2.3 or greater was reached.However, at higher strains, the number of edges (bridge chains) decreased owing to domain coalescence and continued to decrease during compressive deformation.The edge length results indicate that stretching the graph led to both stretched and unstretched edges.The lengths of the stretched edges decreased because of domain splitting when they exceeded a certain value.Increasing the coalescence distance reduced the occurrence of domain splitting and edge shrinkage.These results indicate that as the domains broke, the stress on the edges decreased, resulting in an overall reduction in stress.This suggests that structural changes in the domains directly impact the stress.
Next, we compare the results obtained from the mathematical model with those obtained from the CGMD simulation.The results of the mathematical model are presented in figure 7 and are compared with the CGMD simulation results in figure 2. Although the numerical values of stress along a strain do not match, when they are scaled, they are qualitatively consistent with the simulation results.The overall features, such as bending of the stress plot during elongation and the presence of a hysteresis loop, can be reproduced using the mathematical model.Similarly, although the strain values corresponding to changes in the number of domains do not agree, when considered on a relative scale, the results are qualitatively consistent with the simulation results.The number of bridge chains decreased after a strain of 2.3 was reached and continued to monotonically decrease during compression process.These results qualitatively match the simulation results.This analysis indicates that the mathematical model qualitatively reproduced the phenomena observed from the structural data (numbers of domains and bridges) and physical property data (stress) obtained from the CGMD simulation.
However, although the results of the mathematical model and CGMD simulation were qualitatively the same, the involved computational costs were different.Each calculation in the mathematical model required less than one minute, whereas each calculation in the CGMD simulation required approximately two weeks owing to slow stretching.Consequently, the mathematical model was approximately 20 000 times faster than the CGMD simulation.Furthermore, the mathematical model offers a high degree of freedom in the directions of stretching, increasing the applicability of the model.Therefore, the mathematical model is a useful method for efficiently analysing qualitative results.

(c) Deformations of several types of graph structures
One feature of the proposed mathematical model is the wide applicability of various types of graphs.In this section, the results for various types of three-dimensional network structures are discussed.We prepared the initial structures for graphs with five types of three-dimensional network structures: FCC, Dia, K4, Cube, and WDia (figure 8a).Until now, the manufacturing technology for TPEs has only achieved the realization of BCC and FCC structures.However, we are conducting these computational experiments in anticipation of the possibility of manufacturing more-complex structures in the future.In the calculations, the initial structure was three times the size of a unit structure.Following the previous calculation for the BCC structure, we performed 10 elongation-compression calculations with varying elongation direction and obtained averaged stress-strain curves (figure 8b).
The stress increased as the structure was stretched and decreased during compression.As illustrated in figure 8b, two types of stress-strain curves were obtained: one with hysteresis and  three hysteresis-producing structures (BCC, FCC and Dia), length changes occurred at strains of approximately 3, which corresponded to the strain at which the increase in stress was suppressed.This result suggests that domain splitting is an irreversible structural change that results in hysteresis loss.
The results obtained from stretching various structures, as described in §3b, reveal the characteristics of the structures.Based on these results, we recommend structures for two specific requirements.For TPEs that exhibit high stress at low strains, the BCC structure is recommended because stress increases from low strain.For robust TPEs with small hysteresis loops and minimal energy dissipation, the Cube structure is recommended; this is because this structure exhibits an elastic stress-strain curve, indicating that even with changes in the network structure, the stress continues to increase with only a minimal reduction in stress.

Conclusion
In this study, we propose a mathematical model for TPEs based on graph theory and the concept of TT.We employ harmonic realizations of topological graphs to construct a graph associated with ABA-type triblock copolymers in the three-dimensional space.Our model provides stable configurations under local deformations.On the basis of this graph-based approach, the mathematical concept of strain is introduced and computed using the TT, which represents directional energy.The TT represents how energy is distributed in different directions at each vertex during stretching and visualizes changes in energy resulting from the separation and coalescence of hard domains.Our mathematical model produced qualitatively similar results to those obtained through CGMD simulations for the initial BCC structure.
The applicability of our mathematical model extends beyond the BCC structure.We applied it to other ordered initial structures, including FCC, Dia, K4, Cube and WDia, and calculated the stress-strain curves during uniaxial elongation.Results indicated that there were approximately two types of stress-strain curves depending on the initial structure.The first is a rapidly rising curve, which indicates that the topological structure is considerably deformed by elongation and does not return to the initial structure even after the external force is removed.In contrast, the second type of stress-strain curve is a slowly rising curve, which indicates that the deformed structure can restore its initial configuration when the external force is removed.The former structure exhibits soft and elongated properties, while the latter exhibits high resistance to deformation.
Analysis of these properties can be effectively performed using the proposed mathematical model, which allows for infinite iterations of the initial structural degrees of freedom and deformations while maintaining a low computational cost.The results of this study indicate that our analysis method can be effectively used for developing TPE materials.In the near future, we may apply this to the design of tough TPEs and contribute to the production of tough TPEs on demand.Some studies on topological invariants of graphs suggest a relation with physical properties (cf.reference [22][23][24][25]).Development of a mathematical theory incorporating these invariants should be explored in the future.

Figure 1 .
Figure 1.Thermoplastic elastomer structure.(a) Interfacial structure of the A domain.(b) Illustration of bridge and loop chains.(c) Snapshots at strains of 0.0 and 2.0 obtained by coarse-grained molecular dynamics simulations.

Figure 2 .Figure 3 .
Figure 2. Graphs of (a) a schematically represented structure and (b) a structure obtained through CGMD simulations.

Figure 4 .
Figure 4. Detailed analysis of deformation of ABA-type chains.Diagrams (a) and (b) display the structure of both domains and bridge chains.The upper plot displays the results of edge length, which are the same as those in the third panel of figure 3. The corresponding curves for the chains in (a) and (b) are depicted in green and blue, respectively.

Figure 5 .
Figure 5. (a) Modelling of domain breaking and coalescence during deformation.(b) Analysis using a tension tensor.In (b), a red ellipsoid defined by the first and second eigenvectors is depicted.

Figure 6 .
Figure 6.Snapshots of calculated results for a two-dimensional model during cyclic deformation.Strains are indicated in each snapshot.

Figure 7 .
Figure 7. Deformation results for the BCC domain structure.The four panels, from top to bottom, depict the stress, number of domains (nodes), length of edges (bridge chains) and number of edges (bridge chains).

Figure 8 .
Figure 8.(a) Initial structures of the unit cell for BCC, FCC, Dia, K4, Cube and WDia structures, and the (b) stress-strain curves obtained for these structures.

Figure 9 .
Figure 9. Results of model simulations for the BCC, FCC, Dia, K4, Cube and WDia structures during elongation-compression process.In each figure, the four panels, from top to bottom, depict the stress, number of domains (nodes), length of edges (bridge chains) and number of edges (bridge chains).
different stresses during elongation and compression and the other with elastic curves displaying no significant differences in stresses between elongation and compression.The former type of curve corresponds to the BCC, FCC and Dia structures, while the latter type corresponds to the K4, Cube and WDia structures.Plots of the results of other parameters are presented in figure9.For each structure, there are different results in terms of the number of domains, number of bridge chains and length of bridge chains (edges), indicating that the different structures altered the elongation properties.A substantial difference was observed in the bridge chain (edge) length.In K4, Cube and WDia, where hysteresis did not occur, many chains exhibited linear changes up to high strains; abrupt changes in length caused by domain splitting occurred only at high strains.In contrast, in the royalsocietypublishing.org/journal/rspa Proc.R. Soc.A 480: 20230389 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .