Cutting force is one of the research hotspots in direct sand mould milling because the cutting force directly affects the machining quality and tool wear. Unlike metals, sand mould is a heterogeneous discrete deposition material. There is still a lack of theoretical research on the cutting force. In order to realize the prediction and control of the cutting force in the sand mould milling process, an analytical model of cutting force is proposed based on the unequal division shear zone model of orthogonal cutting. The deformation velocity relations of the chip within the orthogonal cutting shear zone are analyzed first. According to the flow behavior of granular, the unequal division shear zone model of sand mould is presented, in which the governing equations of shear strain rate, strain and velocity are established. The constitutive relationship of quasi-solidliquid transition is introduced to build the 2D constitutive equation and deduce the cutting stress in the mould shear zone. According to the cutting geometric relations of up milling with straight cutting edge and the transformation relationship between cutting stress and cutting force, the dynamic cutting forces are predicted for different milling conditions. Compared with the experimental results, the predicted results show good agreement, indicating that the predictive model of cutting force in milling sand mould is validated. Therefore, the proposed model can provide the theoretical guidance for cutting force control in high efficiency milling sand mould.

Sand mould milling is the process of milling casting mould directly with CNC milling technology, in which the mould production does not use molds or patterns. Compared with the traditional casting process, a sand mould produced by milling can avoid wooden pattern making, eliminate the work steps of the casting process and reduce energy consumption [1]. So, the sand mould milling process is a typical example of green manufacturing, and is considered as a revolution in the field of traditional casting in single and mini-batch casting production. An accurate prediction of the dynamic cutting forces of milling sand mould is very important to study the ability of the process, surface quality, and monitoring the tool wear [2].

Milling is one of the most complex machining processes. At present, the models for predicting cutting forces mainly include empirical model, numerical model, artificial intelligence model and analytical model. The empirical model is expressed as an exponential function of cutting force and process parameters, based on the statistical and regression analysis of orthogonal experimental results [3]. FEM is the common numerical method for building the model, which can be used to analyze the chip deformation, tool wear, stress and temperature distribution, and optimize the process parameters. Through artificial neural network (ANN) [4] and particle swarm optimization (PSO) and other modern artificial intelligence algorithms, more accurate milling force model is established. The analytical model of milling force is established by the flow stress, constitutive equation of material and geometric relations of milling, which is the combination of theory such as thermo-elasticplastic, tribology, materials science and dynamics [5,6,7,8].

The analytical model is mainly used to predict the force of metal-cutting, for their deformation rules are analyzed clearly. In metal-cutting, the analytical model of cutting force is built based on the assumption of shear area. Merchant [9] proposed the single shear area model based on the flow rules of metal material in cutting. Oxley [5] modified the single shear area to a shear zone and established the parallel-sided shear zone model. Astakhov et al. [10], Bai et al. [7], Zhou et al. [11] divided the shear zone into two regions, i.e., wide zone and narrow zone, and predicted the cutting force by analyzing the flow rules and thermodynamics equations of material in the shear zone.

The sand mould for casting is a solidified mixture composed of original sand and binder, which can be considered as a discrete granular accumulating material with a skeletal system [2]. The deformation and flow rules of sand mould differ from those of continuum material. For granular materials, Masanobu et al. [12] and Gao et al. [13] studied the stress and strain of shear band based on both the simulation and the laboratory tests, Wang et al. [14] and, Campbell [15] studied the transition from quasi-static to inertial flow and proposed the constitutive equation of quasi-solidliquid transition. In addition, Discrete Element Method (DEM) as an effective numerical method was used to study the force and deformation of granular system [16,17,18]. Jerzy [19] set up the thermalmechanical coupling model of rock cutting based on DEM and simulated the process and dynamic cutting force. Nevertheless, there is a large number of computation, and the deformation and cutting mechanism of rock cutting were not studied, and the prediction model of cutting force was not presented.

Sand mould milling is a new process for sand mould production, no in-depth study has yet been made for the cutting mechanism of sand mould. There is limited literature on the study of sand mould cutting force. The quasi-solidliquid transition feature of granular materials can be considered as an appropriate basis for further studying the deformation of shear area when cutting sand mould. In order to improve the integrity and accuracy of sand mould structure and reduce the tool wear by controlling the size of cutting force, in this paper, the quasi-solidliquid transition theory of granular system is introduced into the parallel-sided shear zone model of the cutting process to analyze the flow stress of shear zone. Through studying the shear strain rate of shear zone, the model of dynamic cutting forces in sand mould milling is established. Then, the cutting force of milling with straight cutting edges is predicted and discussed.

The sand mould is a piled mixture composed of silica sand and binder. The enlarged view of the surface of sand mould is shown in Figure1a. The strength and stiffness of sand are significantly higher than the curable binder, which means that the movement and separation of the chip are due to the deformation and fracture of the binding bridge among the sand particles [2]. The sand particles in the chip only move rigidly. The chip is presented by many disconnected sand particles each other as shown in Figure1b. On macro-scale, the shear feature in cutting sand mould is similar to that of cutting metal, so it must obey some rules of continuum mechanics such as the chips moving velocity. In the deformation analysis of metal-cutting, Oxley [5] proposed the parallel-sided shear zone model, which was divided into the wide zone and narrow zone by Astakhov et al. [10], as shown in Figure2. The cutting velocity changes slowly in the wide zone, while it changes quickly in the narrow zone. The shear strain rate in the shear zone follows the nonlinear relationship with the position as discussed in Ref. [10].

In Figure2, a is the thickness of cutting layer, v is the cutting velocity, 0 is the rake angle, is the shear angle, v0 and vn0 are the tangential and normal velocity of chip on the initial boundary of shear zone respectively, vh and vnh are the tangential and normal velocity of chip on the terminal edge of shear zone respectively. Assume that the volume remains constant during cutting by ignoring the dilatancy effect of granular system. According to the continuity conditions of chip flow, the velocity in the shear zone can be written as

In the cutting space, the shear zone can be seen as one-dimensional model discussed in Refs. [9, 10], so the variables of shear zone such as the velocity, stress, strain, and strain rate are only the functions of y coordinate. According to Eq.(1), vy within the shear zone can be considered constant.

In the analytical model of cutting force, the thickness of shear zone is the key to study the deformation of cutting. For metal-cutting, the thickness of shear zone affects the shear strain rate \(\dot{\gamma }\) [20]. By the analysis of quick stop micrograph from cutting experiments in Ref. [20], the thickness is regarded as half of the cutting layer thickness. In the sand mould milling process, the cutting layer thickness is changed to a sinusoid (Sect.2.4), and the cutting layer thickness varies from 0 to 1.4mm per cutting tooth. According to the thickness of shear zone in the metal cutting, the average thickness of shear zone can be approximated by 0.4mm, meaning the thickness of shear zone is just the thickness of 3 piled sand granules with the average mesh granularity of 100. For the granular system, Oda et al. [12] suggested that the shear zone width of the granular system without binding is about 78 granules, which was also reported by Vardoulakis [21]. Moreover, Oda et al. [12] pointed out that the thickness reduces if there is the rotational resistance at contacts according to the comparison between the simulation and laboratory tests. To the sand mould, there are some binding bridges among the sands, so its shear zone thickness must be smaller than that of the granular system without binding. Considering both the aspects of shear zone thickness in metal cutting and the bound granular system, the assumption that the shear zone thickness of sand mould cutting is 3D can be made, in which the thickness of wide zone is 2D, that of narrow zone is D, as shown in Figure2.

During the cutting process, the sands maintain rigid and move rigidly, therefore, the shear strain rate of the sand in the same layer (one sand granules thickness) can be considered as unchanged. So, the shear strain rate in shear zone can be defined as a piecewise constant function

where \(D = \sqrt 3 d/2\), d is the diameter of sand particle, \(\dot{\gamma }_{\text{m}}\) is the maximum shear strain rate within shear zone. Figure3 shows the distribution of shear strain rate in the shear zone.

The cutting velocity components along shear direction (x-direction) on both sides of the main shear plane change are in opposite direction as discussed in Ref. [22], so the velocity along shear direction on main shear plane may be regarded as equal to zero, that is

The physical property of sand mould can be considered as the same of continuous medium on a macro-scale. During the sand mould milling process, the rake face of milling cutter moving at high velocity squeezes the sand mould and the binding bridges among the sands in the shear zone are forced to bend and shear, causing the bonded sands to shear along the shear plane. When all the binding bridges break, the chip becomes a flowing granular system. So, the process of cutting sand mould is similar to the process of the granules transition from solid state to liquid state. In addition, the test results show that the temperature of cutter does not increase significantly during the process [2], so the effect of temperature on cutting force can be neglected.

Two types of generalized constitutive equationto describe the quasi-solidliquid transition of granular materials exist. Campbell [15] suggested that the volume fraction, shear rate, density, size, contact stiffness, friction coefficient, and restitution coefficient of granular affect the flow characteristics of granular materials and defined the generalized constitutive equation as

where Cv is the volume fraction, \(\dot{\gamma }\) is the shear rate, \(\rho_{\text{p}}\) is the density, d is the diameter, kn is the contact stiffness, is the friction coefficient among granules, e is the restitution coefficient. Equation(17) can be changed into two forms

Wang et al. [14] and Zhang et al. [23] proposed that the stress in different states is related to the shear rate \(\dot{\gamma }\), and the function fij in Eq.(17) can be expressed as the combination of the zero-order term, monomial term and quadratic term of shear rate \(\dot{\gamma }\). Wang et al. [14] proposed the following quasi-solidliquid transition constitutive equation of granular system

p is the grain density; d is the particle diameter; g is the gravitational acceleration; \(k_{\text{p1}} = \alpha_{ 1} C_{\text{vr}}^{{n_{ 1} }}\), \(k_{\text{p2}} = \alpha_{ 2} C_{\text{vr}}^{{n_{ 2} }}\), \(k_{\uptau 1} = \tan \theta_{1} k_{{{\text{p}}1}}\), and \(k_{\uptau 2} = \tan\theta_{2} k_{\text{p2}}\) are the stress coefficients related to relative concentration of grains, Cvr is critical volume fraction. According to Hanes and Inmans [24] experimental results through dry sand, the following coefficients have been given in Ref. [14]: n1=1, n2=1, \(\alpha_{1} = 0.7\), \(\alpha_{ 2} = 0. 5\), \({ \tan }\theta_{1} = 0. 5\), and \({ \tan }\theta_{ 2} = 0. 6\).

In Eq.(23), the zero-order term of shear rate is attributed to the static support among the sand particles, in which the deformation is the failure of binding bridges; the monomial term of shear rate is due to the relative sliding and extrusion among sands; the quadratic term is due to the collision and diffusion among sands.

The milling cutter with straight cutting edges is used to mill sand mould for avoiding the sand particles flying upwards. Figure4 shows the cutting layer sketch of up milling. The momentary cutting layer thickness of the ith cutter tooth can be approximately regarded as

where f is the feeding per tooth, vX is the spindle feeding speed, n is the spindle turning speed, Z is the teeth number, \(\varphi\)i is the rotation angle of the ith cutter tooth within the range \(\varphi_{i} \in \left[ {0,\;\uppi /2 + \arcsin \left( {a_{\text{e}} /R - 1} \right)} \right]\) for up milling, in which ae is the milling feeding and R is the radius of cutter.

For the end mill without helical angle, the shear stress and normal stress on the main shear plane can be considered as uniform. So, the instantaneous cutting force on the ith cutter tooth is proportional to the cutting area, and the expressions of the two components of the force are

To validate the model, the machining experiment of dynamic cutting force was carried out by up milling the coated sand with mesh granularity of 70/140. Table1 lists the sands physical parameters, while Table2 lists the tools parameters.

The machining experiments were performed in the sand mould machining center CAMTC-SMM1000, and the cutting force was measured using a three-axes piezoelectric dynamometer YDX-III9702 in which the sampling frequency was set to 60n (n is the spindle turning speed). The measurement system is shown in Figure7. Four representative experiments were designed and their process parameters are given in Table3.

In the machining experiments, the self-excited vibration was carefully avoided, since it has a strong compact on the cutting forces. In order to avoid that, the spindle turning frequency was acquired far away from the region of self-excited vibrations. In addition, the measured signals of the cutting force were filtered by a software to remove noise influence.

Figure8 shows that the fluctuation cycles of the predicted and measured dynamic cutting force are in good agreement, and only slight deviations in dynamic curve exist. The main reasons are the following.

In the prediction model, the cutting edge is assumed absolutely sharp. Nevertheless, in the actual tool, the cutting edge is a circular arc transition with the radius of rn between the rake face and the flank that increases with the tool wear. The radii of cutting edge and sand particle are of the same order. When cutting the sand mould, some sand particles at the bottom of the cutting layer are forced into the sand workpiece by the cutting edge without becoming a chip, which is similarly to the pure ploughing process of micro-cutting [25]. The mechanism is shown in Figure9. The number of sand particles forced into the workpiece increases with the cutting edge radius. The interaction force F1 between the sand particles and the cutting edge increases linearly with its radius as discussed in Refs. [26,27,28] as ploughing force. From Figure9, F1 can be decomposed along the machine coordinate to

where \(\vartheta\) is the interaction angle between sand particle and cutting edge, \(\vartheta \in \left[ {0,\;\uppi /2} \right]\). According to Eq.(30), the measured cutting forces in X-direction are greater than the predicted results and the measured cutting forces in Y-direction offset the predicted results slightly. Besides, F1 also influences the amplitudes of cutting force. Figure10 shows the comparison between the predicted and measured amplitudes of cutting forces. In the experiments, the cutting edge radius was detected as rn=58.4m. The measured amplitudes of cutting forces in X-direction are larger than the predicted results by an average of 15%.

During milling, the thickness of cutting layer varies with the rotation of cutter (Eq.(24)), and the thickness of the shear zone is also changed. In the prediction model, the thickness of shear zone is regarded as the average of the 3D, which causes the curves of predicted cutting force separate the measured curve slightly.

According to the constitutive equation of sand mould shear zone (Eq.(23)), for the sand mould with uniform grain size, there are two aspects of the influence of particle size on cutting force. One is the static strength of sand mould, the other are the linear and quadric terms of shear rate which are the momentum and kinetic energy of the moving sand respectively. In general, the static strength of sand mould increases with the increase of the diameter of sand, and the momentum and kinetic energy of the single sand also increase with the diameter of sand particles. So the cutting force of the sand mould increases with the diameter increasing. However, the sand granularity in the sand mould is of 70/140, which means that the size of sand distributes randomly within a certain range. Therefore, the distributions of the sand particles and strength are non-uniform. In the sand mould cutting process, when cutting the large sand particle or the region with higher strength, the cutting forces will increase. Otherwise, cutting forces will decrease. In the constitutive equations of prediction model (Eq.(23)), the compression strength s is calculated as the macroscopical experimental results of the sand mould, and the sand particle diameter is calculated as an average diameter. That means the predictive model does not consider the effect of non-homogeneity, so the fluctuation of predicted cutting force is lower than the measured results.

The prediction model considers both sand mould and tool as rigid body, so the predicted results ignore the force fluctuation caused by their elastic deformation. On the other hand, collisions and impacts between the tool and sand particles will intensify the fluctuation of cutting force as discussed in Ref. [29].

The increasing of spindle feeding speed vX will lead to the higher thickness of cutting layer a derived from Eq.(24). So, the increase of cutting forces is proportional to that of spindle feeding speed as shown in Figure10(a).

According to Eq.(24), the feeding per tooth f is inversely proportional to the spindle turning speed n, thus the cutting forces decrease with the increase of spindle turning speed n. On the other hand, the cutting velocity v increases with spindle turning speed n, which can in turn increase the shear strain rate \(\dot{\gamma }\) of shear zone (Eq.(16)). According to the quasi-solidliquid transition constitutive equation of sand mould cutting (Eq.(23)), the stress in the shear zone will increase the cutting forces. Based on a comprehensive consideration of the above two factors, the cutting forces, therefore, decrease with the increase of spindle turning speed n but disproportionately, as Figure10(c) shows.

Milling feeding depth ae mainly affects the interval of the cutter rotation angle \(\varphi\). At small milling feeding, the rotation angle \(\varphi\) of cutting the sand mould is so small that the milling forces do not reach the peaks and fall to zero. So, the milling forces are small as shown in Figure10d (ae=1.6mm). When the milling feeding ae is so large that the milling forces can reach the peak, the peak values will be the amplitudes. So, the amplitudes of milling forces become constant, as shown in Figure10(d) (ae>8mm).

In the process of sand mould milling, the optimal process should be chosen considering both the machining efficiency and the cutting force, which can help to deal with the related problems such as tool wear and machining precision. In this respect, the prediction model proposed in this paper can provide an effective way to calculate the cutting force.

The analytical model of cutting force is established successfully for sand mould milling. In this model, the static strength and dynamic effect are considered. The predicted results are verified by machining test.

Based on the unequal division shear zone model, the wide zone and narrow zone in the shear zone of sand mould cutting are defined. There are two layers of sands in the wide zone and a layer of sands in the narrow zone.

The shear strain rate in the shear zone is proposed as power relation to the position of the sand particle. The flow stress on the shear plane is calculated by the shear strain rate and the constitutive equation of quasi-solidliquid transition, thus the dynamic cutting forces of milling sand mould are predicted.

The amplitude of cutting force is proportional to the milling feeding depth and the spindle feeding speed, and reduce with the increasing of spindle turning speed, and is little influenced by milling feeding.

According to the prediction results of sand mould cutting force, the suitable processing parameters can be selected to achieve the effect of both efficiency and processing performance. The predicted cutting force can also be used to further study the tool wear and surface quality of the process in future work.

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Zhong-De Shan, born in 1970, is currently a research professorship and PhD candidate supervisor at State Key Laboratory of Advanced Forming Technology and Equipment, China Academy of Machinery Science and Technology, China. His main research interests include advanced forming process and intelligent equipment.

Fu-Xian Zhu, born in 1979, is currently a PhD candidate at State Key Laboratory of Advanced Forming Technology and Equipment, China Academy of Machinery Science and Technology, China; and is an associate professor at Jiangsu University of Technology. His main research interests include digital forming technology and equipment.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Shan, ZD., Zhu, FX. A Model for Predicting Dynamic Cutting Forces in Sand Mould Milling with Orthogonal Cutting. Chin. J. Mech. Eng. 31, 103 (2018). https://doi.org/10.1186/s10033-018-0306-6

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The technology used in modern parts manufacturing has come a long way during the past few decades. The introduction of computer numeric control (CNC) has delivered considerable improvements in machining precision, along with the advent of 3D drawings.

More recent innovations such as additive manufacturing and hybrid manufacturing have led to huge time-savings when creating foundry moulds as well as the components themselves. Maintenance engineers who are looking for new components suddenly have a glut of solutions that can deliver parts in record time.

Robotic solutions in all forms of life are no longer a fascination but a reality. Robots already play a big role in many manufacturing processes and soon enough, robots will use artificial intelligence (AI) to catapult industrial production into spheres that were previously unimaginable to us. The potential of an intelligent and sensitive robot is immeasurable.

Patternless castings Tinker Omega Sinto, an Omega Group Company, is taking advantage of the robotic and the new industrial revolution era that we are experiencing. The US company, in conjunction with the University of Northern Iowas (UNI) Metal Casting Center, have developed a robotic milling cell that is set to revolutionise the metal casting industrys approach towards the traditional manufacturing of patterns and ultimately the castings that they produce.

The Omega Group have developed a robotic milling cell that is set to revolutionise the metal casting industrys approach towards the traditional manufacturing of patterns and ultimately the castings that they produce

Jerry Thiel, the director of UNIs Metal Casting Center is very involved in additive manufacturing for the metal casting industry. They have an S-Max printer and offer printing services to foundries. Two years ago he approached us with a need he saw for producing larger, patternless castings, explained Wil Tinker, President of Tinker Omega Sinto.

Larger components are often created by pouring molten metal into a sand mould, which would traditionally have been made using a wooden template. Today, the 3D computer aided design (CAD) can be used with a 3D sand printer to rapidly build a mould that incorporates vents that are positioned to optimise the escape of gases from the mould, ensuring optimum quality of the base material, continued Tinker.

An alternative mould making process is to use a multi-axis CNC robot milling tool to create a precision mould from a solid block of sand. This process takes just a few hours, as opposed to the few weeks that would be required to create a traditional wooden pattern. Using the latest technology, lead times can be drastically reduced, especially when the various aspects of the process are well connected, or better, all on the same site.

The path generation/file preparation is via an off-the-shelf, third party company Robotmaster. Robotmaster develop CAD/CAM solutions for robots that seamlessly integrate off-line programming, simulation and code generation, delivering quick, error-free robot programmes.

The Machine Operating Software that then allows the user to synchronise the tool paths and programme in auto tool changing, auto tool inspection and other functions has been developed by Tinker Omega Sinto. This synchronisation of these two programmes Robotmaster and our Machine Operating Software allows the robot to run lights out, overnight completing the mould(s) automatically, Tinker described.

The Robot Sand Milling (RSM) platform comes standard with a Mastercam machining/milling software package, a company that Robotmaster has been working with for a number of years and their software is embedded in Mastercam.

We have developed the RSM platform with Kuka Robotics but based on the results of the GIFA 2019 exhibition we are now in discussions with other robot manufacturers. Very soon the customer will be able to select his preference of three brands.

RSM 25 to 50% of the cost of a printed mould The RSM is the start of technology for the future of the patternless foundry especially for foundries that have patterns made and dont use these patterns regularly. The biggest benefit of the RSM is that it provides the foundry with a cost-effective method of patternless casting. Automated robot milling systems can have flexible tooling designed to cater to specific material removal. Any object of any size or shape can be milled by simply adjusting the robot programming and end-of-arm-tooling.

Yes, the programming of the robots is very similar to the CNC programming of any other machining operation, but will depends on degree of freedom (or how many axes the robot-arm has) and also on the degree of complexity of the shape of the sand moulds.

Additive manufacturing and 3D printing took the foundry industry to another level bringing in big cost and time savings when developing a new product. The 3D printer enables us to create one-off prototypes without having to face the costs associated with traditional mould making.

However, during our research and development with UNI, when comparing the cost of a machined mould with a printed mould including equipment amortisation, material cost and run time, we found that the robotic milled moulds can be 25 to 50% of the cost of a printed mould.

This represents a huge saving for foundries and their clients. The potential going forward is exciting. We have a steel foundry customer where 80% of his large castings are cast from one-time polystyrene patterns. Our RSM platform will be very beneficial to his production processes as it will also be for art foundry customers who generally make nothing but one-off custom patterns.

The first RSM was commissioned at UNI 18 months ago and they then proceeded to fault it, crash it, push it and point out any process issues. We have just provided them with a third revision of the machine operating software.

Nowadays industrial robots are an appropriate technology for developing flexible and reconfigurable manufacturing systems which contribute to perform automatically operation such as milling, cutting, drilling, grinding, deburring and polishing. Machining robots symbolize a cost-saving and flexible alternative compared to conventional CNC machines which are the restricted working area and produced shape limitations. The improvement of individual elements and development of new devices has caused a perception change about the use of industrial robots to perform machining operations. The approach to this research was to analyze technical barriers of individual components that it was broken-down as well as full improving the system. This document is intended to provide technical constraints, current technology and future potential researches about robotic machining.

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The Feature Paper can be either an original research article, a substantial novel research study that often involves several techniques or approaches, or a comprehensive review paper with concise and precise updates on the latest progress in the field that systematically reviews the most exciting advances in scientific literature. This type of paper provides an outlook on future directions of research or possible applications.

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